(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: lects1-5.dvi %%Pages: 31 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMBX10 CMR12 CMR10 CMMI10 CMSL10 CMSY10 CMTI10 CMR8 %%+ CMMI8 CMTT10 CMR6 CMR9 CMMI9 CMSY8 CMSY9 CMEX10 CMBX12 LINE10 %%+ LINEW10 CMMI6 CMTI9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips lects1-5.dvi -o lects1-5.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.08.29:2210 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: f7b6d320.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmb10 cmbx10 cmbx12 cmbx5 cmbx6 cmbx7 cmbx8 cmbx9 cmbxsl10 % cmdunh10 cmr10 cmr12 cmr17cmr6 cmr7 cmr8 cmr9 cmsl10 cmsl12 cmsl8 % cmsl9 cmss10cmss12 cmss17 cmss8 cmss9 cmssbx10 cmssdc10 cmssi10 % cmssi12 cmssi17 cmssi8cmssi9 cmssq8 cmssqi8 cmvtt10 % /TeXf7b6d320Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /exclamdown /equal /questiondown /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: aae443f0.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmmi10 cmmi12 cmmi5 cmmi6 cmmi7 cmmi8 cmmi9 cmmib10 % /TeXaae443f0Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /omega /epsilon /theta1 /pi1 /rho1 /sigma1 /phi1 /arrowlefttophalf /arrowleftbothalf /arrowrighttophalf /arrowrightbothalf /arrowhookleft /arrowhookright /triangleright /triangleleft /zerooldstyle /oneoldstyle /twooldstyle /threeoldstyle /fouroldstyle /fiveoldstyle /sixoldstyle /sevenoldstyle /eightoldstyle /nineoldstyle /period /comma /less /slash /greater /star /partialdiff /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /flat /natural /sharp /slurbelow /slurabove /lscript /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /dotlessi /dotlessj /weierstrass /vector /tie /psi /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /tie /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: bbad153f.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmsy10 cmsy5 cmsy6 cmsy7 cmsy8 cmsy9 % /TeXbbad153fEncoding [ /minus /periodcentered /multiply /asteriskmath /divide /diamondmath /plusminus /minusplus /circleplus /circleminus /circlemultiply /circledivide /circledot /circlecopyrt /openbullet /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal /precedesequal /followsequal /similar /approxequal /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows /arrowleft /arrowright /arrowup /arrowdown /arrowboth /arrownortheast /arrowsoutheast /similarequal /arrowdblleft /arrowdblright /arrowdblup /arrowdbldown /arrowdblboth /arrownorthwest /arrowsouthwest /proportional /prime /infinity /element /owner /triangle /triangleinv /negationslash /mapsto /universal /existential /logicalnot /emptyset /Rfractur /Ifractur /latticetop /perpendicular /aleph /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /union /intersection /unionmulti /logicaland /logicalor /turnstileleft /turnstileright /floorleft /floorright /ceilingleft /ceilingright /braceleft /braceright /angbracketleft /angbracketright /bar /bardbl /arrowbothv /arrowdblbothv /backslash /wreathproduct /radical /coproduct /nabla /integral /unionsq /intersectionsq /subsetsqequal /supersetsqequal /section /dagger /daggerdbl /paragraph /club /diamond /heart /spade /arrowleft /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /minus /periodcentered /multiply /asteriskmath /divide /diamondmath /plusminus /minusplus /circleplus /circleminus /.notdef /.notdef /circlemultiply /circledivide /circledot /circlecopyrt /openbullet /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal /precedesequal /followsequal /similar /approxequal /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows /arrowleft /spade /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: 74afc74c.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmbxti10 cmff10 cmfi10 cmfib8 cmti10 cmti12 cmti7 cmti8cmti9 cmu10 % /TeX74afc74cEncoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright /numbersign /sterling /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /exclamdown /equal /questiondown /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: 09fbbfac.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmsltt10 cmtt10 cmtt12 cmtt8 cmtt9 /TeX09fbbfacEncoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /arrowup /arrowdown /quotesingle /exclamdown /questiondown /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /visiblespace /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /dieresis /visiblespace /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /arrowup /arrowdown /quotesingle /exclamdown /questiondown /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /visiblespace /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: texps.pro %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-35 -250 1148 750}readonly def /UniqueID 5000827 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB3DF7BFF10C9BDA4EFE5F68A8CB1526990D1357AE6D2F7C2D2EF8496 4E47B39E6712EB8908A3265E5FAB40567E866C244814449F1E993AAB422C3F1D DFA8C7118584F2E5197FD4BFA3A8AE9E953C6CD4672C0FF51E41C3A919749C1A F06650DF4C5E17492164BDBCDF22609A74BFA7F69960A64B9F949FFC2A807458 8579366C4F41BDE1FDFBCC4845FA19BBB6963D65EE8532549274BAEBDFF24FA6 03235D1BE37C06B1938AF369DA75BF38DDBC87A1FF445EAA16E1895ABE9506B9 211955753E447865D33CEF007391D2666A046277A30A49804FFCED3FEA5EB2C3 E52EE14A9F75241EA10C91974CDA6236EB840FD44D6DDE4D9B3266C3B99BD38B D835BCA8CB819C073480FB972CC028D218F6A1D344CE1B63F4FBF2C826F412E1 6E0B05A26125865A14FD7B7030B478BB8BC6BC395335C3BA940E1C348267F4F9 0AF97BBEE253511940F1048E175D3569F7D05A28851B6F50765FEB6C9654FEDC 1BF52F535DB5BB90C1BD5D2EBF75E0AEBE82B20507F3C28A03746781018D4EB2 298E4F2C27ACF73FA73EBE43F014BB575AAD516C0407B29E1653375135ECB74D C91372F06FA8EF37C31AF3FA48AE65318EAA6C34830A5377ABB2DFA5DA53A574 433484BA1466709A4B186761655C8E482833B697673E847C691079E7F1DCB8D6 1AD91101D757B83E2090337D525AEECB028FB3C9F6A6E6AD2F322CFDC5A833E6 1CE4EDBF41FD34FD61630581D222F854A76C2EA9FD72796A7C9CC1F6C2FCCD16 E95CA05826A4ECFADA6A5FB83C41A7131E52BA6585DD6DD78515D8F7327DFC6F 9404F896CD850A7B3B3D71D485F5459E5A1D5C5BA2B33FE53AC0A660A0E747D6 B8121EFCEF5D53C913EE398525985ED5E4EE7AB530BEDFF1F42E6769001ECCFA E9D4AAE49D8C6793F56EEA2DD7878EF697A3E4BAAC949A10534EB73766AE6702 0784AE295A2A51EBE9BD0DFF1F7EB372938F202973C6840F4C38042B83C0AC1A FEECD2788D25D6CE7B5C9BF24151E1D46C302380CC4CCE751B3C3F8FAE5368E3 FBFF64235652D5883AED53E0FFF839EE76DF322401A6D121A7B36ABCECED2FC5 EE38AB90CF6DB566BC28991A2A769319F01DC8E0103B1A9F9F8272A3476A3AFE 870A52ADD066B66EEEE953F0CC3EE28214DE3B2C7C68458055B0AA2A0F81069E EFC3FE7C508AE43272A5039AFC79CEBEBAD76E5D76C387EBF1D34DBB5B34BAE8 762995379EF4EB60A9EE08D74D8C8EF06BF391397C55326EB89CBF633A940AA8 0385D1756AAC207040CE96A2A161743E55793A73C5045AF06BDDABDC5128BBE5 730A68A652CD91118925E4B5B0B26E49A289F59B3D311A6A7C1395371B46BC54 133DEFD5D09730CCE33579FD537F165E9BA38BCFE272B518BE5993D57E9C2E7F 81D7E21FD85F2E5D516E330DB9E0055C2D08137DE6F875AE9CA9555B0A410D62 23AFF5840BBC83B159301EB4EF5D1EC9F4485A5EAE555A91DD8D9C09C90D408F 6EE19D987B543DBC9107E21EAD50017A71D3F59426605E4D9233C05268B48D91 7070ED7899F6752F6699561B42C8FE72531E036F4693C11C54B683C69709BA94 C4EA1F2F9AABC5FF8AA5AF0FD1EC74AD82B94602D4ACCC2F75AD949339DE5571 88BABD333E35B110DC5B7E31998917C769995C98F48CC7FE5E55A97E8FA4E81E E4E02EFC3E6BE4D96B4756C1231FE24CEB92125F3665CA4A6E2DD94C79B07DE6 E6FFB9F0246DFA1591E0C67E932336F06442E1BD4E74372187084037E8E2FCFD C20B3D250F1379385D7284EC2BEDDE9E32EFD53E92B1419454626A6492102726 4397C0028EEABBA44C13A495A5CDEB2331BFCA340F19AA1CE3E234F32DB42578 930CAAAECDD0193915F5D7702BBFEA7A90AE9F59A29220D038949046DB921085 AAAC74D55C30E019ED09C9CC92D28C91E76A86F398D4F35235FFED0AA2788856 12EB66EF483DF7F3F77AC79F09E5878FECB1758F290B4AE2254B474C23DA6C6A 2373A5364CB21C141466AE389FF713EEF7EA7DCF022D12D3DD59C1B9A964EF26 F2EAE2C369F34BDB75EFB12937390EA8F431240F202A637B86311F025866D7EF A9C2E95301542720584D05087DDCA5A1BE7B4BC27A2DBB43A64DA4A6ED99DB45 23AD758930FEC2F501465023D4358FE5B2CC710FDD45E5B5DBF67204DE29F8A5 BFFBA80C047235CE85D9B2B333CE4C832D6A2D8A7719A3F0E4D3A9658CBDAD99 47B503B66FE97101FB0AE22A08BA17FB02EFDA2D393EB0458D5C72604DB9CC5C C5775E8124E2E9A9E259585D1DE423290FF739FE89E78807B690528AA2188B1C 87995E657052C6AF5805C64639E656F7C65BD08C9D21C93E954748BB2332B65A 00FBEB487CE889EFEEFA9F06C50D 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: LINEW10 %!PS-AdobeFont-1.1: LINEW10 1.001 %%CreationDate: 1992 Oct 23 20:22:06 %%RevisionDate: 2001 Jun 05 20:22:06 % Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LINEW10) readonly def /FamilyName (LaTeX) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /LINEW10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 54 /a54 put dup 63 /a63 put readonly def /FontBBox{-220 -220 1040 1040}readonly def /UniqueID 5011941 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: LINE10 %!PS-AdobeFont-1.1: LINE10 1.001 %%CreationDate: 1992 Oct 23 20:22:05 %%RevisionDate: 2001 Jun 05 20:22:05 % Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LINE10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /LINE10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /a0 put dup 8 /a8 put dup 64 /a64 put dup 65 /a65 put dup 80 /a80 put dup 97 /a97 put readonly def /FontBBox{-150 -150 1020 1020}readonly def /UniqueID 5011940 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 16 /parenleftBig put dup 17 /parenrightBig put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueID 5000819 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D6F5CA5FED4F9AC9ADE41E04F9FC52E758C9F45A92BED935 86F9CFDB57732045913A6422AD4206418610C81D882EE493DE9523CC1BFE1505 DD1390B19BC1947A01B93BC668BE9B2A0E69A968554239B88C00AF9FBDF09CCD 67D3B2094C11A04762FE8CC1E91D020A28B3C122D24BEAACF82313F4604F2FEF 6E176D730A879BE45DD0D4996EF0247AEB1CA0AB08FF374D99F06D47B36F9554 FAD9A2D3CE451B7791C3709D8A1DDDEFBD840C1B42AB824D5A0DFF0E0F15B0B7 22AEEB877FF489581DA6FA8DA64944555101EB16F7AB0B717E148B7B98D8DBFD 730C52937E226545CF8DC3E07C5BA30739BAFCD0F2B44275A6D503F582C0FB4F 449963D0AD2FAFDE33BA3D77BCA9D1DF878DDAFCA2E22CC4BACD542B282164C7 97C2BDE318AF9D501CA21F6E662E7AAB75A5F24D2C182E598D175D44E88AB19A E7CD59584F95B389183EE21B525BF52A3F23C0FE5383A5565A19361D716F508C AAB78411CA5A4D27552CC1C435760D5A89D535B71C593E755C616661363308DA A683F54ED0C23FB2C225A008392B0B719F66F11A946A090B7C00B662A3C69599 B4ECB0CC70C85C4BBBF207E0026F6C7A19F2ACFB7A60804FC98A4BFFD7BFFF2B 9529E6D9D4238002BBC255BC62959D6F3381FE06E0621B879D5FE5B541D45A1E 759A6E7DC32B1D1632368D09A97039DF255B6492B1B2B7E2C1434E8306ECA7D3 5A79B6D614B4979F10988BC76ED53A5F45315CD7DA216221F842FD0F3E050DD2 BAC23C984D506D8F7D614BCB6B244F5F41321549BB0BD041FBF3053307168680 3435E9C947D7F87B5D967FC8414668CB404B346CF11EA3DAFB945AA267ECA9FE 7E4C7C280ADEAAE83DB806F447DA9EA56DF3AFEEF6BA88EF315D9AFD9AC07AD3 1571E402D64E48E67961D955EE4304BC9D9CADAA9361D0EBD745F1B0CD192E89 0DF6AD4C99222CFF33CC7DE1134327A33AEE2864245397057CB9285219812665 637F3C229A7C4464D8F31982C39C747AB69A20A3A420E82153EC8DCE2A82681D A838108D2BC6CDA82C4BCBBBC518A60E9EAFC446509414547BA0B8ECC0E64A48 7859985FA5B6A69A0F822477526949F73B1A0039945032C46B455B30CF41D9E7 08AFFCB32F88AB7C7D812EA42C5CF9DF4A4BC52014E19CD63B93B95EBB22525F 3B4D19AD39EF1ECAB5D1F2F40C0F9447900C15BC2598311E781D90D04E8674A3 581006D6C0A16558AC8A63DE6325A9C05E53D1F673E9080A1D80963B25895A8D 2DAAEF0F665746AE2BB1075787D1BEC4586F0414D1BAB4487AB87AFDC3A40E41 FE0F193D2121F98484E27BAADE55D5111A33BD260261CE68B9B5BFAA6701BD09 168D9CDEDC8D1F207A2DB31D2C557C019649F9EFC85FBBF52DFF8F6539C0A912 FACE3A4CDA17575560B59A2FABCDB202ED797E14E5ECD7CDE80EB2EEAFE38E76 18613B606AD98AE0062A6E1BBD2413CC79B250BB4914D9B5F0D486D61D967631 B2468BA1054AB846B030721B305BAEC73DCDE50BBBA03461EDBB5618FF4ECF7A 356C8B08CDDD9AB920DF6E6E94B5D8B1A6D10939F397E49804E75E0115305CE1 E3DA5DD8794E7A83766E2726837ED318CAAA57361CB3BB5C81D7D8F7D7A5EC58 28A415EE8CC2BC8C632851F6AE0E93F4C8A95418E96A340388F5BBA0565ED189 164A76A59631CCA241029B2B9368676C74940CD636F43CCD85E048685E12A488 03766F8EC1E61D4BCD32F745DDAEAFA5D8AA2122F30B97FB4119B1344B718E66 21EE51202E6021B757F029141B80AB97343B707A4E7D6DAB87D2B73BA3046194 E98561BD0573449A3F80BB8EADD429D7A26A1F73B36288EDECE27D15AED13EBC E0A2A3AE37ADDA31BB814D557043977930F4CF9203F155A578A7A4F8CB9A88B2 DB3D841567A88D18BA8D19DF1228723C90AA60B624A408C04C5D78031586BD2F 6DB967C07DF32B22B3E0D9BEFBD6FE9A91DB3438CDFCE1F3C12C001EB5F2D6D5 E5699EBE42C5045F49154A05DA85C86A790EC28872DA9B95CD11546E831C5ECB 53C36B03BDA5F66BA99F3BDA78538B9C74E2794247D3C277C0F8C2C3FF56217F 14A79AD2451D2FE630312D90315394883B14518070F32CEEBDA11A9978F2B0DE 05E0537FC640032868E5134683E9A21BB6085A0DEF8696C28D684B0AC8679529 098D2437719679B29F193F07A5DF86A1622BC9DE004D68C7B36F5380E0383EBC A1BFA0957094CB567156284148E8FBBAA8582EDF437C146435234E730EA1F6F4 3D7A47F4B869194B5BFB8C0D24232F8DDC0171C56FE6801A4B4A662C3A69A70B 86E4A1BF9AD2F07185615E250B633A6A93FAC2A9F9D3DC2B43E6CBE6192CE32E ECDF83C5623AA2F446CCF197FC5D8B2B0EE5CD001D58BD3A3C696269391848EC A9849AB264464CE23A43727ABA5A085D2976166ED17E43269D4CED54C356C401 DC597A00CD6A065A8810D019882CC4F5C6D01233604F711F201EAF9E0A2E0748 BACC6609D64469E759B833893CFF8D73DACD2BB61F5BF3D1B7090A4799BAB16D 736DB3AB07AD73A42F4F83E6E12ADAE35CA462D1AAEB6F84EA7617D3A16174A2 DDD2E8E97C59297935B3691EC3F8736800BE6727143E62485BA785B05318A42E 3AA07B23FFE7AE4FDB7DA51CB0C384CB437B17FA7462DAF4144A95D344AC3E64 05835EA22F05786EB6E6620E495F51B457B16A1CF35B342129FC8539B1F8844E FCB012A09117E1BBDAA517F4C63821FCA7C5F42A78820AA3AF0E0AE1788AAF1E D36470DD9C450D90AE57191A56C83652A680074135654621B9BF627FC32F9517 638CECDAE485490EC98CA7C7BA44865D308D84CAB6135FA9DE7759024EAB0828 15DA8DD65DBB0AB3737C4E668E98F220FCA1BEF4FFDB7ADE16BAE87DAC8D1578 4600708BDE5B6B98C2FC46F07B7E50148EAB99670974A67F7E2A16D3106365B7 4FA86506A0B093D3819CB967A4DB0BB7F5A0A7393B57854128EA147DFEE8043B E9EA005C0C5791A0B1BD1946E9003D1E7B16FCEF87376527D8B3FF1DBA46BEF6 F05781F6A4472E5FA21F09F8AFE90D18753EBADBE3335C33990671A33A58B61B B9A75C87FA210DA4944C0D9F6579945A3ADCF54F15BBD4F6923176DE40072E02 AB62364A3BF5E816D18449AD623C20AEA59A8EA734934145876714B0803094A4 C612810137EC07D49B1355D4D79C7904A661691AC76F5D31DE1412D998536EE7 6A4DC726BE7522AC82FB41E2F886F118949A258E96FBC2030F797D8C0556422B 83C402D1FEE06D945DF1AF525B728DEE8AC3E49F91B5E45B60CE896A66D55A00 0D26BC1D12AD5A333B8BF82548ADD3ACC6D8C68CB924ACBA29D2D34B08B7E9D6 449475510EB5CEA25397C52BB6ADC647956B2970C93A7E6ED536D7B27D61C120 AC4D00CCEB8CB31021771C6AF0A2A10B5F89C6950379EF0105657498EE74326E 6476147C7E20A2678F90B50A94076466F2E0D84158947654863EE34ADA79BA42 2D287C136C2848981F07522089D4C6F9E7D337FA4DD6844B3F8D6BEC0E3494EB F5701387D77EFA0E7335DA281F3D11E18A67AEC831030EE19074E7B2022EC120 8E4331CFE96A395E5D87AFCA4B0E0368E52697E0C59270021F4B692C48032EA3 AC7D897CC18D81BB9512480870884850F1E99CE917DCAEC2E20CA701E72C0C6F E823DA51572AF11E4D3EDFE9B1463C3C3012F3848F02E0B4F0ED0728493DBD7B 26E91368F77D4AE0AD36DB6A110D131D6871B172FCCD506C2AEC007D34356C02 FBBA9EFFC5DBA0C1F953549067C5ADDC6AE6DAD196453486A13378034CF69E87 7B6B85CA1FC663D53B7573B2E785032E4391789B55633AAFC53E17F95905B6DA CBF15027E835D5B4EE3347D8A50CED79F76BDD8673E8EAD4145E90492910A325 2DF7BF4A1D4A805ECD1FF54ACF546E2CB5DFD91C395F17D01837FEEE46B8F159 BA96B830888CC4A5E879149DD6EA33143E9D5DF8CC94E98BA06F4E8CF32BB593 FC1C8D52D0715A7D2C3BE5E9FD7D7AB34CE42B7EA8D2C959281214A6D0A08848 E072DFB2D0A2A10F2A46DBF3C47A4DFA1E0D504894BEEDC2354263787F2C7A3B E74F5D51FC98B3D80CBCFA2C6A142B55619F94E0D31A83B4256F82E9EAD6BF5D 07F7B7DF9F10AEFA20EAD753269EB6554E921474E09F412A5D387521E07F08CE 6304FE77CB11F2A88D4F75BF2DB6713AB6D101D8ED87E0CBCC03AF8C084EF89C C2453CD93D96737EEE30961D549A28C94C050FAAE356B9B797B42B7F4D96AB7D 4AEBA8886EDE656A96BBFAE9E16DBE72E54CCD780A0FF9F0839CF10A04321BF1 DE1BEF163BDEF74A8AC13AF467C5817ECB2192C1B82DA11F9F50FFF6C292B951 6B2F994FC42763B4BA31715F8BC70EB7CF7F29A29E35B09BC7E69852CCA8C107 F4A12FC12216AF10AA38B1F23542921BA9DDE0AF18DF2D5FF27555680A8D5EBB 9CF0AA126B3F0ADD1883626EB75F0A79B762C2D7216A00E8860D721E0525A32A B3CE023584EDC893B1E010E33708A2729603EB49913E2B25D8ED8F94CF1C6CDE E2176D972E9DD4F189F22EDD00FD256B64A7A0ABE7C2564DB88946F7C89D1E0D DC50411A58FD1CAA9FC232FC224B986267100FC26910DCE3337480AE57B07E6D 11232588E67AF2A4FA102A7552C5F6FF37F1594F57A8AA29249BCB8E40456330 2C879312CE7A76CBD4614B034DE6F5B646A601B5F974087E0B0219C08002E6C9 A18AFC5B65020F8E05D76A29643EC144894DCA026BF4FAB66EE15E3FAA00547B F877CB5A912056E7E324B31ED2885273955EFC19A18D413702E98B286C57E0F0 82BA57248E15BB54C42031E6 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTT10 %!PS-AdobeFont-1.1: CMTT10 1.00B %%CreationDate: 1992 Apr 26 10:42:42 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -235 731 800}readonly def /UniqueID 5000832 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 3645B82392D5CAE11A7CB49D7E2E82DCD485CBA1772CE422BB1D7283AD675B65 48A7EA0069A883EC1DAA3E1F9ECE7586D6CF0A128CD557C7E5D7AA3EA97EBAD3 9619D1BFCF4A6D64768741EDEA0A5B0EFBBF347CDCBE2E03D756967A16B613DB 0FC45FA2A3312E0C46A5FD0466AB097C58FFEEC40601B8395E52775D0AFCD7DB 8AB317333110531E5C44A4CB4B5ACD571A1A60960B15E450948A5EEA14DD330F EA209265DB8E1A1FC80DCD3860323FD26C113B041A88C88A21655878680A4466 FA10403D24BB97152A49B842C180E4D258C9D48F21D057782D90623116830BA3 9902B3C5F2F2DD01433B0D7099C07DBDE268D0FFED5169BCD03D48B2F058AD62 D8678C626DC7A3F352152C99BA963EF95F8AD11DB8B0D351210A17E4C2C55AD8 9EB64172935D3C20A398F3EEEEC31551966A7438EF3FEE422C6D4E05337620D5 ACC7B52BED984BFAAD36EF9D20748B05D07BE4414A63975125D272FAD83F76E6 10FFF8363014BE526D580873C5A42B70FA911EC7B86905F13AFE55EB0273F582 83158793B8CC296B8DE1DCCF1250FD57CB0E035C7EDA3B0092ED940D37A05493 2EC54E09B984FCA4AB7D2EA182BCF1263AA244B07EC0EA901C077A059F709F30 4384CB5FA748F2054FAD9A7A43D4EA427918BD414F766531136B60C3477C6632 BEFE3897B58C19276A301926C2AEF2756B367319772C9B201C49B4D935A8267B 041D6F1783B6AEA4DAC4F5B3507D7032AA640AAB12E343A4E9BDCF419C04A721 3888B25AF4E293AACED9A6BDC78E61DA1C424C6503CC1885F762B87F4FA52F3F 4ADEAE2E622F8DC9FBA1B8EE1664BE56BE246A8A878AC9A76EE6D7BE7A137022 21AD1DD31BAAE575CBEDF1D0D3A3F5AF2540815699950B8F574F72810C1E4D9E 3C83E68BB70E6870C6E2782889D2BB02DECFC9B2FB8B3C8285DB719AC4D828F0 0DFAC8249E0540D83A68F2BFB4759E5E5EE3113A2090987B8FDE13D194FF749F E8E972DC81774638F051BA44A09A320DAD80F3B2C5DC2156A80E9DDCF12CECB7 E082A421DD3E728A52121444CD7712D5880A72397D13B4DD1D7EDF5D90E5091C 2C4BA3CDE94BF1BCCCF6E36B8ADFAD5DFA275ED37662D9953EA7242C59998ECB 1442469970DEDF5A4AC7BCE80C138E2BE4E993CBD99C8FD55A4EC2E40DDEB42C 89301B934CE30D5AF7F42A552EE0929E8241846E3BA58BD3AF49E0BA32D74AB8 4370BE224973ACDC8F5E8814475091FB5EB2FE095D6969219876329D289EBA35 3FAD769D9DB2B0B92A7FC041DA633F0B7AA2784611295C386354835ED5C15081 26CBE8B007093E38DC78E324A01EA831CBC8C600FC1A26108F5FAE567EE9A574 9745758B619C7C0CE89749B5240705C1F1A95DDD2F9CD1192E237820819BF7F9 47DFAE9810ECCBC0B6575BE940759A6E14618D72BC1D035BC95F46509B20669D A8E003AEFB27F4FE8CE47E0AD1FE50EDA486640A551462F04E586F190D56E3E3 A6615673540F042058BA963D48A6B1166C8614B80EC6DCF0E454ADB3BE6AFD5C B8311A0968EBBEFDF08065E189DBF2A590B58413CBE44E58A65140F42788EA3A FECBA3670E29717D54D37574EF06A400A4AC42D32638617080D6F22642E8B396 EF316CAAC75050DC9EF3B137A9023C6848D4CCD4F55AC1011AFA55E40508638E F3CE56DBA9026F8FDBF1FEBFF6D50CE701675A0CC9BCC51CFF90133D7D821C90 F416165A47EB555D60AF75DE5EFDE56F30FCB2E0B15D2B6AB492F3A0D736B860 70FC1C50369966C6C9A7973FBFC61E25AB872BF9DEE4794735BD5F51E54624B8 32C821E806DBC81CA4FE1369DE0EAEBE5EED922F16277D1B8C0113015BD24F30 EBA0BF4972DEBB891EE1D2391CB2A0129C82913C7A6E65D0B4FDD596A3BA234D F46A15C3CA34EF03A03F6742CD90E789E6100B92E820AA5FD304E84E3AC1F323 7B0AF13F39327C7D28412B7A6DED66F9FF6AEF3F2DCE8A81FABB83A7946E250E 9AF6BDDF56101EDCD318BA01FB0AEBBFE8DDB19F4ECCEC19C7C28A5D91819F62 4FA7711A57ACE5F5706D0018076F553E252B00A9F527649B05D4A5EEFF8AE3E4 258587AD063BC3D7D219A34FDF9CC092D4584294A88E5346FC61E2661B4F406E F78DA8FBDE5A40DC01F7951B01E6A447144E6282C01728313894467BAA6038D9 55BBFBEC7260999F46A5E22B162931578AC50F0C0F007114AC8A56005C50045C E62928B6C7DC3F245C8BB50ACEB56450AB8E64A3C24F6BD56CD3BEA26D9030BF C71C2EACF65CEA54612E3575BE98A1F9207DEFA0B93B36932A0B2F8387645E2B 5087E16B65E2A5C1A5A40F5175158ABC753FA33D91E2E82F4ADCF7ECD4EE0E5C 6946790CAA3DC7998DAF777CD0BC84CBB3CE46F94AF15AE230F42555B30BE539 7EC9596B6DBFF6B222E8DDF2578C69629C2BC104AB3D49C738C59AAA4827BDA4 04A5811557E09BCF45213324E8A90EBC84D9FE4600F6C336E955AC9393F3BF3E 027961A6D1C3676845562717E1CE2798529F5106A6560E005E71014493F383CD 6CC70CCB63112A131057A00DD9F02F2A5665569853B2BDDA153FF8E1833590A1 623E582E09C8E9996A9514D9174890EC0F7D76A61FF73D6631C8D81138C1C1BE 186CCC7A4562D1E06831DC105C307AF135D32D168BF1D08B1410CDFA8375F5E5 41412EC14BCDCC88E0415A4B15533705ED22786C0FF926D186E58F15974EC02A A586AA405DFCAFBD091DF533F1D921AE4C7846FD998CD74F9D2864D59FE2CA85 FC082EAADC9AFC744F4AF578663C283D304D0E2A3D63C4F33C0F47C95C57A06D 01AEA96C8C7703303AABF760E49383181E97E3CD110E70730300F093A06978FA 1C0FE5C8544152509CD618F2C36D4CADDA15E9984422066216B76C27094FE9A0 91F6B1F39F9EA5223D024E114E97B9CBE6FBFC56D985568673798B080CF24C59 5FCF4F8CDD11D4FAC24BCB2D626A5FBB7F96B9CE0FEA49F5DC5461AFCF70EC70 486146BA052EE319A0768FFA1E679B4DA74C605C812649079E9FDBE6F036E071 953894CC5FB811DCE31AD5A6855A972B29FF6413846A485ECE5A7CAC374590D9 D3E999999643CBCE04659742C280780EE1069A3E2658B5ECA0A6471C66528F0E BF206354B2CC85B85746DE1C538725DA69D84ED5A41B595B55D463D1EDDD9F9E B8A1B2BC7AA8DD2901289CF1B152DA89E459F5B791728A9A2D5DE75984B7C9F2 B122372EC673A450D82200586CA4D0A198A174FE9584F2A202520B33B4AADE4D 1A5DB209B5874563CDBA8772DBE7B6215535A9796E277FDF6F1E366EEA7D3872 0026FDA7C684CBC3B54A83C1EF316534C2BC856DA93030967D314BB23B324A1C 1F56D0D5774953767034F45FF70762615FEF903651C26A6D8882CCB2E38E9A6F 85B9CBE9341179041EF97A1D1766CB2C9F2AAFAD0224866BF040507AEFF426EC 1FCE4CBC08BB0DBF99773E4A0731F0F9224726D401E915E98BDDBEF696D98262 31F5C72DC396421AEF7097B938C30FE09348CD82A4D9A821F6383BA00F5F2A53 4B8E220137572EDE115A604B5135F8A8B49F53D3C6D1AB487884FE02D2DCF99B FDA38160E67CCE3840B037B34FEF988821A0C01EA9308729627DD73DCC8E4D11 136C55AEBF18053529172DDCD93F13410E579B85DBDAAB6DFDA610B819742BCA 379BC2AEB62F4141F6ACC1DF9879335D701043EB835CF9D0841D85B8750B587C 310D6B3E2D24688735BD196F5DE74B8BF88ECF559FA6CE0C51E000CDD742F7C6 1450A20CDBB58937DCAA389D0B4EFE3A3F2BF3D3221C239052ABAED920FB2DA2 3270C10F0A4EF1A140A512328911E6D8C680D07C79A232DD8154D00924D474E2 0293F02FA723C35B0D5F6A0789A2EA29065D8D088CA5598DCAC484E15FFACE39 6421CBB3DEAC2D691CD0B0C791E3AA44080959CC174FABABF5E151824D14DBE1 E5DEAAEBA6E943C094AAE9C4D4AE4F3C93C1BC993D7D7C0E567293DD500E2078 3B4C242C536AC5DA220703C4AB5BD95A73DE7907F0C74E0298CEF863DA211199 93ED1718B3E5BB3C7FE4BA1040AEF0AE45A7324E3D6209669D080F9AE859DD92 AC686D8B4CE4CE3F89A67C660A29089EA6C3DD418BD46CC7A5416F4F44B53484 BBF6D5E552E45FACEA614B00C75618E90F39219E7AA92DA261ABE77A49F1FB13 5F5710A22B666EC64A719E188FEFA9B7B9A6B5C05F4C5D42B687B7C1F4CC0FC0 4301232EDE7FA9170C614F3C14ACDFAE174BBED1705E2D450B60CB7E93C383D4 14FBF7BEF23FA34EB45F94AC3BB5E0F671BA23D2FEF1E4395C5C803AE494ED6D 8240556654981B50D0376151B98A5B48E8221E715B9A5CFFB69BB29C95FF2818 EA9DD6D0C5358275D04E565275B08B09D3E4194AFBCBC5ACE7B6BDA6002D0744 C8E9C294FE8976808EE28E1415DAE498224B2137F2E0EB6762DB352061DB60BF 8FCE00C992B973F20B1C688E20C51E4EC79EC9632287414B43010D4AF88B0452 C3204EF6E7CDB217B115D8823FFE2BA726C90925202B692D859BD2337FA122CD 194AE1C0162691794A60289B35F802A13787D4248D6750A15FED939BCC56AD3F 268A4D47766775047B5B983B81F389786006179195D2DA33B5569A0B02D0A52A 0C2507267E50612CAFD18F0096E11B1B9267D95CA4F64EA82D00651919BFE55F 00B67D58A7D1068DA36B7DFF6CC89B79098E25C27EE7837A53837373BD16F96E 19E0C850894F5E0A7330939E2E91DCA34534B0FDF570AB91AB2A20DD1091A1AF 0AE00D762E058EC3F9B282256DD5FB3EC9B86EE6770992CB80FECE871F3DBBEF 8B8DBFA728F02E360AADFEB457831C8575FBA47FDCC5CB3E4162F5FD6DEDA4E3 B01D7793AE7B32339F6EF093A913EA1C647DD25975049113E4ECED6B197B94EA 3043C37713B629D5441A5CE91FCA65B7167B5CA33BAC1F14A8E7C600E40D3829 6C86BC6135CE5335827680EFD47EB8513A4DD48A8C84AEA1A685B0BC9CADA4ED 65026861EA5C953CB77D1560E27E1A392FC3E3B9CE2661A8120CAE04DCB4B395 E5B0E2D9ED3756F34D3EB3D9F70FEEDB09E64CD6643ED14D4E9D1C1F8B160545 B09DDEADE163E988331C014E89AEADC9D8C9AC24066B742C67541B1D19DC2F75 119B6D3509253EB4F33E1B59849BBF211FF4ACD7F5E0922EF2B1E333B261AB5C 37228382E53A005230A2D6C73CC400F955048B8935253DCAF0E5E33477E80E66 7C7ACC9A6BA44297F7C60DD5BDE22F41014485DB239398EB8A0EAC64BA1BD1CB 2196A622761B2E63234E0AC1AAC512E9974A8A12280E2C41CF4F0CAB87D639E3 E0B2A5BA9656A33CC115ABCE7397C104376A51D55D7FAE898D1AF4591A278856 84746B0771648BA77B8626E595FBC5A3D9A940F1D07A78FC377957DFB6C7EB27 767AF46563CA503799528FFBC88B6CBB026BD1D71CAF13E9C71529B4895CBD0B 35984DF4C7CEE6A4368B5A45FAED067987FBE3D4795ED697932D8CAF9F961E5A 589D46E479AEC1395673D96C777DEACB6D39CB6C22E6D1B9A53A13741AEA8023 DBC617C985529724104EDDC5FE4E88D890CD306712120AA705A0229277700D55 791C13C3D7C2562A34CE619E11269674BDAB7BCAA1B1C92E20F4B862694A5D84 71E1E8255C0F1D1823E19341F1947E06C05947CC3320BD8340E937EA12CDB7D2 929735664C367C2139B6B86C317E9C9BF2A4DEDBB4F056CF7FDD7D900BF60F66 3F8A75A025DA54AF0CB1B2645913727AA06B70D1C2C057CE6480B6C18EA51CEC D5BF1BC0294580B6AB62E68FD45F77752B2DCC99C80AF1406ADEF13C4D5F8521 784E302627A5C6EEB90FE5FB99CF07DE94689151DC1DBAA8F8D40A898BAA328A 52546EF18B090E4E1A2B643D140DA2374CCE813812A7023800DE0AC3D42BB737 0F4E7706FACE6FAC8207468EDA092186446C47F3BCD3A23AE647C9A17D59278C E36AEE79454526BA0ADD82EC561BAE6620F983CDC45CDC1ED06EDFD9094B1FDA 00E7972B3046397694B00DA0BA171D014AD727AA3F615E4C8BAADC6A801BCFB9 67F255D9FF637C82A3A830916EBEEADC6F9DFCD2E72787A640E676762557678F 54B5405F38850DCAC5F2E704961B54498CE3FBB910DD7306D60EBFC23431A4D4 7B958C501B91EC5BD225B5F4849DE85EE2B8558AE287885A2B71B0000D0A1AD6 AB01A5A62C994FB1E6F84DDE4DE9B00C9622ABAA493BB4A62C27DFE4DDE39B6F 62695F459CC651D1CD3A68511D00440F7023A3C2C70ED1D66C0C217305800FC2 8558673B016AF00AC5207F698ECB7BFBA2ADBA9AFC1DE814D66A211B86B0FACB B528CC9CDB3D550FA57367BFC3AF051110A4935ED94B901E12DAAB2DD484C3BB B0DF8DF7D5EC5E0DC3A147CBA866F227E662EEBB769B94A0A532FE350599DDA5 7C12BCBCF66B52FE005ACA16294E697D20C5660EB98D99B8E00385B3EE0BF4BF D414D2172528C82904E3F1AE713E51968EFF2A2498052C8ACB2B0BED0138177F 36AE0AD22149471FBF5CF8D3C0E5F0B57E4B01AFC45B0AA34EE509E28CF88BEA EB7CC6E43DE08282B70ED359F1DC46F07EDE1BE4BED10750184C0C421E6FDBB2 E300F9C41E1167B6281E70EDE5F53467493537DF66C7BFFDAA539C4F028A0C90 4604FAF5D4ABE4E39D7EE92B1FF8C75B71BD150F137722B85718ED80FF251F74 EE7372A141479BA37C2EE484CC6E6B2B1185776D0B7CD062FBCE0B0D7960A4CF 8D7FBC8854DCB613048BD31EC2FE0EC9EF90A7AB4271177DB8B8237DE9108EE1 B4DAA3AA5B8068ED2E9CB2A1BC9524D5058BC26C531626D21801139950A7340B 564AA34162FEABD4D68609AD531EC699134ACE0A4B76872F51DF91B8AF86FA17 937737EEF89199BF3AF1F7CD77C567E144CDA617A11E7175DE1B29E8187A220E EECD9C65D355FE09F04376129BD0D15AE89E10B06762D2ECAA08A5F19899ADBD 3A37D42C77A2A5C0CE2CFD8360B669679A0203AED6CB0DB0D8562A37B670FB55 4715C46402ACBDD00A935D40FA0951846D7FE7A251D31AEAA408D9DCA81EB7C1 678B26F228A69B1139BF166BB1A1A9FF89315C726B28B36FC9D84B4B431E2CFE F16F592D52BB54B9992290A0B210FA5779665C8DCBE48955C5467054872E9E28 74C5A175427A35C9E8B4B4277D70177B8E47354023CF2ACEEC123EA6ABF62D77 2FDBFC9D3CB0F3F8DBABBDCB2A59EC9FA1EFCCA620F0D7823F605A112A0EE33F C3FAC4E6E34D4FB9D417D591DF776D46DE62D1E8636F55C3A03CC6F41A9F800C 97020EEA55B844B47056FC39117A23EFB6A6AD050197879AE7FBB5268A0E6160 BF6B8F602B83C8B1476DE0286D9F901586229224474C 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSL10 %!PS-AdobeFont-1.1: CMSL10 1.0 %%CreationDate: 1991 Aug 20 16:40:20 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueID 5000798 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9429B9D40924DC059325D9D4CC0344F3F997A99E6CC0676735EBCD685AAC9142 08DAFEC78BB41AFC2F1C219910BDF41D6279284EF600B69776CA15BC8A34347C 30783C52AFA60FBE3E353E2AE354CF87B558776A22C776C7A0B5AB5CE1F941EF C2D9CAC37294BF407A671F10E4743BF842143F4F7DFEE643BA3BBD8BB9E3F24A BCCF7F0ADF8BA500620C81033EAE8C4EF2C1DEF13AC575F1B3BBB66F093D3B78 5412B82B67FFA087AF57182B2230F9F2137180CA58A7D9B2C822FF04BE6CD01D 43B2CA7058C7B953F6D9B5D6E91ECBAA5CDE1159B0E59C83DBAD96D6C8C8BAB1 374EF652D10C0F3EE7104472C98DD3572AAF2D45A70BF7061447E21EE3C3BF23 DF39C2D1B35B42CD5297BEBE6BC94F7C9DC6E61EC67E4F677256FED9064BD3E4 B51A71B1D27CA4E5AA9E1D8080E6DAB5310711EEF87C40859FA935B19524AE83 63B163FA8397BDFF443227FEDF7DB27DC35D89FB1C5E435DA0619A5C88AFC73B 89A2DF5E767C5B536BC7167A840A0C32BD57A14DE69A7D0D819AC36FF32F908A 5070F32983BB007437E3500799DF5E0AD3710A4C0000F0098D5BE99F2EB9C1C2 C444FD9552D0DCA098A94B3BF176F511CEE13DB7EFFAED7C47B5ADCF8D4700F5 7A5FD1B49560969BF5C44F3749370663A04776F749DDD7B50674D93254426C4B EFE264BEE7810EC93784B7C01A7F29EFD92547E13A2C7851A2E709FBD5B87850 4A44F08F56A542DBE072D2FBC58D9E6468E1AB858DC35240E30D31C7AC13D6C5 7D2BB634BEE96FA0E10F842B11A789F72A333DD6DDCB1BC23227EBC406E50B40 30AF0C48E6359AB0C46898CDAF1118E46BFF8B00F54EACBC2AC262AB898C42B9 2E080C10DE923C1F753B824EFB048DCA92FD4766B67B26B16F5D2BE22D0B8919 C0F7C952E01BDBB713D2A0BAEC5489098147C6C715C1B7557B32EEE45245F89E 7C38EE33F995CF42B261D09A3A29227FE687BA825E64589E0C3D53D1B346813B C37A568FFBA248760923D2EFAE1AC87D099BCC1EC08448208BFF645110696ABB 69BDCE9AB882F24D1DA46D8C98A2BD65BA9BED75F402AE4E4001E74D371E33A1 FF22C967CF4D30956AB921468FE577B8D57BEADB18328C92AA5FCC8E7AF6A68E 4689265C78D80BF3A995EA677C0C8DE19BB885B4F3966492F4A795C9F26F6D41 2E9815B1C9101E5A2E1A10C34BCA9034DE20C50504D1BA6B614E626085D892FA 4494829CF19893FCDD21CE2F377C1264B3B25956D61798F2844974DE8F9312EC 6238B117EC4869ABD4DD8CC85FDF88CA04B32FFC247DCD68E5E3D8D664B455E7 D71B21A943C4A8B277459C932B1F4C1EDBB5DF6A2C19F48EF04AF69EEF10FD3D 218CAF031D54EECA208054235D9DE3BB0BDEEAA11F8D165883FC299B465006A7 641DD7BCF73931E88FB64DC1C50582EB1A32401F58AC34980B8E130F49F1E7FB CB062684BEB77BBC070F5586E1F6D25EFE3FE4A711C2F883C748E99B84844D91 08B1B8180EB2773894E43B6C1D87353D0F37B36C7024F8AB9E38CE33F4CF5F7E 4352FF857FA6BBA1B7673296A1EFF88B375A6D91F3EAA539C4A01943E1461695 6B9C1B0A8CED7061E9DDA1A7DD7285E56231220C0406B29EA4D42B1098F5843B CFA79EA8D4B752781E59BF132F0A90C9A4D2320A22BFB10177227E873AF10B14 BF2BA7EB6C32503B9EB0589C450028155EEE00C17F93C40193E34061FC735882 78E8F58DD04A1F3461B72E3C4DE34C14D7BC97729BDA7CE219CBCC0102392BD9 BD45547EFE9C11CE2FF9FB548D96ADF13699BE3C788710B77465D2483CCD31F8 B8B89554EB9F46B27B9A116FFE380B356E4A448CFDAC504221773F977DF592BB 8C1B5C69B9AE06A15225D5B513F3B4C11D74ECA0CA6D3E9EFFEE6E238256AF71 AAEEBBA4CF73C2D9FC5BCA9D56CEFE7FD45D70D738C72AAD578324F34AC328A5 1209A1257EF1B45DF1813BD12CF8B75EA4C1A29D230EB86B84A60609ED6AF0B7 E86F2769EB2F565D4F24041E42C6C879A6EEBA6C64F456B86AFBD271580CEBC9 76EC5F67871FDFCAA226A08582CDBB61BAB828200EE3BFC69C2C7175EB6ABFE7 478BF59D5CC40FAF2CB6F8D23775FB7915D93D1FA2DF6ADF611DE79DA50733BC CBB4F5CCF05990CB23124D0ED1335048918B5617B205E0128C48DA0FBBE5F769 698DA32F82D092858A6EC0FE593A50DF2F236BCF14DE715EC864DDB4A7EECF90 CA37C1329901C16A046D3E97F04C12BA30B4CF78D6BD222714EE113A679EF6BA 9B2CB452371D33E62D7A877FC15B7158564D314623B2A06690FE385973726D0A 4C48D53DC5F25B26F0C25EF552CBE6A57972542E9F4EE2BFBAFE74EB1038B8D6 C8197C9D93B81C717C2EB03C598557C691F2AF3136E16312205BB487EFF41398 29B9CD3610201231FCCCBAE0730E1A5CA967DCA79CADBACD1BDBD20D66D085DC F8AD75BFAEDFED61A9C70D54C5224A66FFEADBD0D70D1C434A06C5A494734040 54AED4CFEDB0F3C2A445E4D8D0807A4C3423BB4CDBFF6F6863C230A00CF2DE5D 74998A1DE68BA50DAB2143F806DBBBB4D266973918DC5D2EEDBB0E92134AF1AB AA3547F4F95BDCCCDF795ABB65B535575E75CADACD68BD52699CB05964175C40 4447A7F6D0F042D0E4FD73A3BB8733E817EED7AA93E3898F5424392B3D058592 5B8EEE7C6080827901C12B4533E727AA7C4BD59F27C6E2CB28C72541B53A2667 C4DF46BD9F589354790EF7B791D74B21C462C4B8720D2988B124FD64C3C2CDB7 60552E2E6387AF537886129299CB1C0CBFE0BE8687250D7B70F2F2CFC4504702 75F7EFE3A8154F609CD8349EFC430EAC1CFDEFED9C9B2B7C67186D3D5EA26FAF AEAE7E72C20117DD3BA5DD94767ED68E1A23A1274E6250895A6008529131CB5D A1892391537D4C4923DA067BCB4A21986FA1E8E4F1CE79C30973CE37E5BF4B45 FAFB6115A0190D864D00F55724727FEEF613DEC60187A12C8B16430F4E4A1856 1F9BB06B685BEAF5D5D71B7E0653A5BFA91BF2D7200615CF5AAB264AA64A02C0 18478C888E596510E0B311EF0373EB1CCB5B4CD053BA762B21053611B2C2BD20 27330F5052C0318884FC2282A0FA518663085E3BC8FB4DD8B5B5337E90DBC7CA BC6BC62FA779B891A8BD9CFAFDE8B934C17E5CA64018FEEEC1E2E924F2A3869A 281A921D1710D7C096B6F24AFBDA52C56D45CE922F797181F0E6091CB8AE338A 3A521B752E51982FB0C198B8E4AA9FCF800C3B7C2414A0D732DFC977A0CC153E 4D5D9A0B4E32996F04DDA55DBE319CA55682A12BB48EF9E92296C1F335F8314F 754387E35993ECCFC4A42940E727104D8ADA4D638EF1B4CB8235C5F651A611A4 10E3C3A48BA56CFF8F18196FFFC21C98FA2C3F4DAAE968AC8186C5B239E9D5AC 8F269003006F7776B69F0347E52E465841575EB95543004130D1B4EFE64F4A85 AF6084E41547E8487B9E364597F6BD13374F0786FA9A76EC1177D4098A80F0D0 6A16395606528CD096EFA6882AC0679BA85D5BF3076821D4279F0624B27AF46F 72CDE9F32A3F679C9D0776BF99872B6E5DD65D7258F2EC15F3BE7C2CEF4CC1B4 3C6DC7999FE93CFF7790DFBC30D40CFD38EC28DE8B145442768391E8F5AEDECB C33DB8FED6B8EF5D75DD3321F2E5386844140D2EDAA0349E7606F54DFDD34F28 0EA12BDED1E143C2970710111F2E44B40CB994750E32D663ACD2E3B3F10F26DA 296F4CA63A589B999B9DBAD38EE3AC524628E7630980AC0D87BB541EEB6D3F94 62F14DEB2AA0A7E3F688B81EB4C616753CA4F9B5C5D152FE8D982CF90F5899D6 B61279B7B013B11FBAD23CDA088449BE30DBECEAB4721A293B7F935496A80BA2 ED51754F3C8F5F5080C481573396C02DBA9B67AFCA7D8FC4CD97974A40ADE878 70477D15C3A85AA1CE7E29A290804A0A52D0EB56754ED5A3A13DD4D3E89CCC68 A6E121A4D1976A9422107E52C451B0155CCBBB88523422A1C498CCF50213CDEC 779A9867D97D5E2A7586A3A11D65127F95F333E29C7072BF2A11191674592181 81AFE5F74FB7E609774B40E4AD10225F16D3C474C5A5ABC1BF249EB93E97FEEA 3D706C01D2711C05B13E4A4C8D24115C20E6685E346E35CCD9139ACBD35BE652 019DB74655E87101976E886FDEFCCFD16582D155EED25158397644629486D4B0 A7ABDF0C7442D03451C00B0F478F82D1560C5688382723DCBB6D0BB277970EB7 E13EBA5AA73428AF4879E1A9E7B7BBD4573242E729010D0C0E7C70D1D4A89FB7 7A2EAAC01191A8934E80409C63F61FDCD21EAE8E65213951364609F6DA9ED41A F251421113D9C6183C158F68375A9AE6DA10FAC44795B1696ED52EF40DEF93C6 86AF211776DDCED9C0EA882730DF56DB7C3B1F0A12865D1C4FBAAA1739B30218 0BA95074066B77F8C9597D1504BDBC7D5D5AFAED1A6D526C71DA738C93285815 909FBF9EB55D9E6E4FF29C36E40956C87D821188908726B60093A1A5A7957BEA 5B3EE6296D4A596D9F8C59909A04CACD39F9C30F5D7053282095F79FB122CAEF 23E27AA9809E18EFCFB28E2C802CA86E0991F0386E676FD5A8C4933EC45A1B59 18EA2B6CE854370E5D7139B01FD485E776D769FB830F48582673DAB2B7646E74 220C8969522293ECA73AC6870339C91B8303A2FFB4CA66AD2AD63A37839B0664 01371C840A2E5777BF48CE6146267C15787AE56EC91F50C11DAD2AA3FFE051CF FDCF08AF1EA035AD0667745A12A3D27A9226F77FA691526C03E0B1B5F05E8727 B2ECBAC418ACC732FD61694AEA6D9480A3A3ABB66A0F8F69B6FBFB78A74AB767 9453F235D2459318D4BF2915EFF19148C5890675468FFC2BB23469ABDC35DAD8 3090DAA4E06E8B661B5A62570B1E57CF5EB935D091373770B99E6E4EADD8FCDA 18408EF5641B53B163BDE4EE18600F8CE92ED3E3DC707D29553CA573CF1F425E 74876892D0DB32A07EDF147DDA1BBD15EF9A44B6DE20AFFF6DE4A31DD752FA3E F7738BA61ABCF630FF96568A8C7130EE609851BB6B5E13B6CC4BD9F01A8D1668 BAD00615CD36AE5B46432BFF459E540AE75EBD009BCCB0633EDFF277A0620EE7 EF8C594F296632A61186314F0ECADFAE9D166DD982559ACF0F2ED5F777C7401D 27FEA3F2AE1E817FCB1109FC5AA99A7BDF799624FD8E50503E22A198795068AE A748633D2D8097C677D87B387641C9A4C35A17DC04C52BB4A6A17A3C8136B3AB 7AA410360D3AA63141A056D1B779CF0CDDD18612332BA839DE705E7A3B46301A 64B78403366861DC242B3E9066CCEEC8A13899D304C04EFEAFCA5C77319D5993 F9614301C33468200D56E9651C9C8DC1B271640108B7F759925883E6445F5C67 BE1189A8DAE235AF6EE4FFC32CA3F529B49CE06469DC2E8B019F9237783A147E E41EB069783891CEEBDD88C0972094A26294C51E714799469567B02D918FA197 C714F1D724A79F84588B696469EDABE6A6FBF1B8C4D033F5D82849EA4FD11427 DE5832C3983F774058DACAD29A0635C160E95D3A81C90F791197DA56A28BDD81 D6450316882322AE2264FD7E26A88F6EB7737A8B11917E8ADE5E5156A75E79F8 D76B20B5AA688406113A7C96E268CDC2956CC40AFF0FC3639607532311668D61 21CFB6A400E01A21631A6B787D9347EF841C7E2966CF7E16D2511A6E4A3BBABC 8E1308A87B898AEE995143D51B3AA21872A608A65D58483D9EE6D49847102480 B3244D2C81B3F682ADEA096BA7C199B6C768EC6EBBFF01A800C4B7B609B43731 918C25B309EE6F51D8ECB87B27CBDF497ACF0C81DFB15FCE0E153A905C4FB201 DD209DBFCFD91F67FE8C08CF1F0469BCF48F7A6B04BE92036901B44B54CA7FA9 1463243488571B4C441DB60954920DDBA7D10EFFBD8435DFB26A6BDE9711BA25 46AEDF125F077159401A5143DBAB94BEC77BAF888045527119476528430A634B A2D92738F23C85B933184ECAF88CD7C3E58A72B8D976CC6B1D60CCB6850383ED E8E64FAD5454F931BA3822833629F62ED0538AF6E71E7C3A413141436D3B6DF4 CB977F402227318B50B876F52F6539524A50EDEB7BF64EF6173D9BD8A4A1B7D5 C1F2DD10A4CF54EC91B93559E3612DD5DD904BF95CCFE9B41087244A1C81D2E3 D2A0B01B487E07878C604BD08303B22145BA8027EC8147AFF5BF1306FBA85360 A46F5CC4167B1780B5CD63E534E0E24A43E544BEBA42EE91E274882C598179A7 C4BBDD2888697E52E0584133C0054B72EE35E2879A58185ED4F0EA960AD24004 242149AAFE820F1AB3B6CB2084B302B4CE4BC90B9D3239DBE6FD3CF4C2A30FE6 0344A2CB55A82CE00E41AF7AA90F6A4B7DF941E072B03ADFDDAFB73DF303D7E4 40A22EE729D62D2FB9813446BE923C895EBFE00DC25B15593752508A0410BF2C ED577EEB7CB8E8D7FEE8B0C0E42F9E6F3AE03169E940D35C13EECEF83D2580B7 C3D970FA80D5F2BD4E70A4900813A6D52E4C635576ABFC4C74126CFBC95D55C5 54FA35D97E6B227A14A5D76DDBAE89717D45E795D903CBA1271B7C17B235D34C 539D28664FDF347D79780FCF99A7A14C0C5DFF62A57B49C130F9E8B25BCC22A3 69F764FEAF22B6A04D1A9189AB6BF5505176120FD782CD11EC633BC54D2CC1F2 4679BA21D46FF8AA9BCF1C346559ED4B7478EF386E5AB9C1CFEC3F044E0194BA CB7781CD0C377A3C9C536B6158C9751FFD81473677707D8E0D95DCFF50A56951 649E10518D22C22FD6CAEC03B49999DC8D5E6FB603032573E9E575D72BF875A1 54CC3B53BBCF91ECEA1F0F2D1352E221EB1FA5319EFE49E64102EB9EE72FBC4F BE5B0E5B43742B9CF0283FE39675EA709AB90B7BF4BEB79286B81E3FEE4A9745 98B950106BB0A1F45FEB8E2B400225F722B6229D8BB6E289F78B132A9C777DB1 34CE9B1060ED258ACD36B42D443777DC4DE239F6384836DBD7752F9EE68F6D6A 8F753ADBE4D92B600BA1B16B1F3AA4B6F7D21C45F84D4DA71476DDA8B4DADA5C 813380CD03F0EBF69FE1321B2FDDAFBD777459F6CCD599224732E353AA601D63 2483177436CE99FDA653D6EEF9317A5407380EE216A6AEFE62A51C7F6A9A4A93 0D36D6AED6A83E8B310430AC3FC21E5E780C52157F5ECA9E8270824CDD8D39B0 611586248455E48818CFC2F1C4B8FA7120C89E423E440011FE194312E3AACAC1 A23255AF1861A7D0BB4D4AE8689F574BD91AD7B5A4DFF45F6C3A75D01C301227 D8D76D904E0A35F84FB51E88FE61581C435FE06F84D50EE6EFEADC010CC3E4F1 694D782FBD2B5999F1BD8276A96ACE2E478319EC63AD56C4CF0CE02405EBDCF1 071E5DEB9DAE4492193686CD61318BF1F32A324F5601AA6F0FC0DFCA102A77E2 22006E3F38EB76542DCD5A87C4327822C52B017DD2C0E80D6E1BE7B60B4D18FB 6503FD08762186D52225CA014AD913667C7A62E798BC6C3120896D7E6C14826E 65278C28D549058A180577EA3F061E5A71EFDD728751464B2DE3D219E1DEF927 16B04D0549DABB93D45D0F868CA8636F00FDD7B351C1B6431C0E2DD0DF5274B7 1451068D7E94CB58696B7B3A55843C4934AAE34BD8057CEED08A24D0A13BE745 A1DBCAF86F39277E271EC22528A0DCDE774BDDDA8BAECFD1F74E691F5D40D92C 339173F5CD9F274598632EE8C7537401D85C1D94E69B6F5D9137A3BD0C060E3E 78C6262E9C3FCC4E9F6711225358D9091620B6F207D8A13C3D316091B2156EDE 7E1CCD2467639A701555341E096D2D37364BE6BA15EC2DEEADDC79F06B035059 2DB0FBD7DB87A7A93F5CF2B5DA0A1E67BED1D9AF533EE5F56699E90FEAC7C34D E2CD8C144DD19DB7A3A47F3CEF6D92F978985C404CD8C700F7838B5982238F02 60820F261EEDE520AE25308B98B99E0058B227AE8E00DD50C8E414D52C47ACC3 42C2B2D8E3582C6DF96112FEE9DB7F30587C33E443BF408D6F810B57E48C8567 87606C59EED375EC11337C73B8A398BEA3404E1379937B9DB8862B7D62B8E817 582F1117307C96B2E7B5A93F334DDC0A242B246468D1EFE70140EE42D7AC9193 F7C564ADE524381C55777F806D3E54D483459F4BEC1558FB477235E0F5527FE7 00531E3F1AF37B5289C9E8265A299D01D930E3640CD5B1326D0A0F8DBA9D859B 0901BBEDF17F5A4D0F6B3F572CF5EBD62E4C50AEFB296EDA2741E5AFEA28BDB9 4AEBB59D25AA3FE1AA7623D1835C861BDD902EBD54F270D0DDCDA49331581927 61911052730481394143B2C531D84068902FCD6132EE32C2D8510400376D5E38 B151CECE8624163A4EB16997EDB597032BC0625DDF469305B0BC82EBA022AB17 6D3FAB60D3C83E02B10DF8A83DD14B3257AEB6D4D9FCAC14CC4536A8386F12B2 F71EEEDFFB1CC55431680C9A5D5B77BB722496416291B239A1D94EE6A044CD34 FDBAE2DF3317E27D9DEC69689132078146D6C301C2706D7C603C36E26F001884 AF4E1ED79F3B9DF7C6B96C9B9AD9F87C2511F7A823690C204E14EE9143123D1A FA30FFE8AACD720B46FBB091CFF0D7C24893BEF1DBFD701EA5D2760602C42955 2095C3D15F2C0AEF5375E1C0E4962DBD615AA3A7496B844A9E9DB7F25589F252 EC13E75D987395E9DDE59F3387FFF65514E87DCB9694CCADFAC938D9330CACFF 594819E115BE517EB6A169A013DFF78B66D2D3FB5F458C9DFB6EDB4D508B01E1 89BD652EBE114EE8F8A8331E37B2612B616ABE5847F6BE68A35B0F3BB8CD1FF8 677B138C99293C7954722DAF0182616D1543F16225C3BCBE26E3AB712476948A 2E6E0D63D9C82347A191472F2060E4F609A94A46C3D9E2CCA53C8B4F29887339 C7E7714BFA82C7D75F7A6334CD8C278C44C80FC221584897F2E53C8016EF2720 2AB2CE2AE8682E4F927B808B73E782D31CAB7F3EC3F034D79C026D426F8125FC A2D5FDD00A46F484D4866D01269127C33FF1E0FFF43368A6C998FD241EAC7E53 F8CB16429A82F7B200FA16F8D7F065248E678C34160F55B07B8D6E16FBFD036B 3990329688FFD1575801DB8D128D5C794ED140C96FB8D2FE13E538E0999385EE DE3F13015260A86BBDD60B510A48099B40632CF68FF66CCA35F87D95402BC4B0 F7308449906CB265B802C34BCCD8FF94CDC181C4DE04C3340FB3775219A2B884 6BB949B66F997FD9B44A65949C04F30BA0904ABAA94B3C8B30A0B98D2B949EFC 13F3AF72C0EC92D491405DA1A53F60A2DFA572A548950AAEA5C2BD9CD3166929 7DCAB5FA0869E9D420F27533B9B0BA6D6FA950F5C9E82049409FDBE4EB9E3E5F 40AD987F72F78162B89AC831F177E9F88AB2D9761103B6878227F2679EFF2180 192351D27866A57CB27DAD62398C891F18837498A0E0D64D1CE27F6E91806327 24D8A4CD1882A57084B3100B8EB6229CC80F82EB72BEFB468AEE55496614B422 8AFE8B1DF28F4ADD6FAC4777D840BAE4BD3B845B559C03368D2F3AB600F42CF8 7336415D25289F4523980E746BB9913D75D220E833E0D8EAEDFCBEDB3302D56A 70C352BDA03D7E7FE17FEBED5E0926A4A8199F1DBBC5F556F4966A5A2EB67FD5 AA6A8DE8B4E36270A80617992A9BBB84AD21F04961ACC5231148FBDE3D83A88A F512B63ED6727AA4D316C2D44F820289BDD1443D980FD0F27DA9C8FFD084918D E304F8DD7A56AC5F90D6507E224273E272F22FEB2F8C0025927B771A41DD5A17 ED1F650B7D7F6665344A935A779A17B5BC75C41CF87E2AA17AB0DD6DC57F9257 BA40964B48068B892F1DE05C5B64857815DC402C2F4D729928887ACC3A5E9149 AAA5F6565BEF3BCFDF14996271CEDDEDD98C9104A376D659556EE4E4EDCE5046 5B9E354DB04DB953D8BE8FC177258439CDBDCD1B7FB85EC61F575F1F220E2080 875FD3E7BB0AD82046040A10D1A6AB4D5C880D9D48A02754678CBA1836A75F58 97BABFE5F9EFB2536EBABBBC09EEB9AB0B7937A0EF25FB2639FCA34F363806D0 49B664B09A096927101C4BCBBC8145D48CB76FD7B02B944395760021AC18D718 10C914E57EBBD2A7B1CAC6196CF8871830FC6C23871556530506865A27C6B142 7839589DC52F9C3D4F7F33F5300A4871D6BADF67CCC41B3435C5EBD98C53EFEF 92EF9E9C30D81F1DFC12FA4C9E6199C2187289CF5B8E9E37C19AB445AAC020DD 4438BF7CFAF41AD5F4AA02105596B8BEB4A9328FEBF8B7E3B55AF75DD823452D 8D43534E01696F94B90C51D6A42409590F3C71157349D288294C10DF372DE959 037739E12EE829CFD0C841E26EDAF259F379E2C9918485B8C2025CCA7DDA2455 6FE8A616D538329D899E1829640B7EEC10A362B4988794666456D2CA2E1F6119 BCE20E91315C8FAE98D027FD7715BC432DC28BCB58DF07EB1E80E78080297847 B9674CBDCA87E1B009DADE1A5B8D5308CB3E62CB1506 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 990CD249827D9648574955F61BAAA11263A91B6C3D47A5190165B0C25ABF6D3E 6EC187E4B05182126BB0D0323D943170B795255260F9FD25F2248D04F45DFBFB DEF7FF8B19BFEF637B210018AE02572B389B3F76282BEB29CC301905D388C721 59616893E774413F48DE0B408BC66DCE3FE17CB9F84D205839D58014D6A88823 D9320AE93AF96D97A02C4D5A2BB2B8C7925C4578003959C46E3CE1A2F0EAC4BF 8B9B325E46435BDE60BC54D72BC8ACB5C0A34413AC87045DC7B84646A324B808 6FD8E34217213E131C3B1510415CE45420688ED9C1D27890EC68BD7C1235FAF9 1DAB3A369DD2FC3BE5CF9655C7B7EDA7361D7E05E5831B6B8E2EEC542A7B38EE 03BE4BAC6079D038ACB3C7C916279764547C2D51976BABA94BA9866D79F13909 95AA39B0F03103A07CBDF441B8C5669F729020AF284B7FF52A29C6255FCAACF1 74109050FBA2602E72593FBCBFC26E726EE4AEF97B7632BC4F5F353B5C67FED2 3EA752A4A57B8F7FEFF1D7341D895F0A3A0BE1D8E3391970457A967EFF84F6D8 47750B1145B8CC5BD96EE7AA99DDC9E06939E383BDA41175233D58AD263EBF19 AFC0E2F840512D321166547B306C592B8A01E1FA2564B9A26DAC14256414E4C8 42616728D918C74D13C349F4186EC7B9708B86467425A6FDB3A396562F7EE4D8 40B43621744CF8A23A6E532649B66C2A0002DD04F8F39618E4F572819DD34837 B5A08E643FDCA1505AF6A1FA3DDFD1FA758013CAED8ACDDBBB334D664DFF5B53 956017667084E248B4E19A77B7C42DF6762413BB3B142E38BA5F7E0CE122DCA2 4D36660A76858D4BE350161BD0087399D584416AA00EEE3F9B5149A9022A5B7D 7C5274ADB4A5E9E33CF00A0B37E004AD169841C12EAF4E27526601C518AFC8E2 74CF1864A9486DD0CD3FE11B0CAFD7C2546211D154BBB9DDD351052B0F03D42B CD6E46FC7AB3B28E23FA251693985BDF7B79984F5BA2D862DF9A5F64FFC160CA 2F755C725A68D741787F4791E2731485F4E26454CC3581EE715B3D1834DA71B8 F151B6F13A23BB2599490C20D9A0A6675513CC1962A7363AFD4D98AA94E8E793 0FEDE343D361DF6AF762C50D88C4FD745544A7963721D7BD7F5506B805F87029 A3F082758F043FF419AA2664F6E1240B1638446A1B6BC62FFC301F6CE9C804AF CBE4CA67CC85851231A70AE8CF76D8C835A2A513FB21A8086C76CF0170A0AC3F 5905532FC17840900FAFEAF780089E729382CF21A6108912D93A6DF8132F812C B777B68149A4A7D8BD483EDEEF9D08DEBE4FECCCDD19557F107C94D5F462D12C 505322A6DAD0897D51AF17CD1630FBBD448B8DC3C5969DAE108E6C88FDA6AFE6 4908310D1C671D2E2FDCE8CC8E02FEDFB62BAE24DC82845A7F80ACDC8E76A281 C14AF63C4D08A556AF0C5F1CE2E963B1F739269159EC925D773352ABBD408002 55433485E9E0466E88D2B9620EC9F53F67A6CEFD1E884B73C83BE130CE5D9EA1 B277534DB39270BAC1415A05CA3F8FDC8223B848383EC00A3F2B6028CC373D75 24A4EEAB96F7F8B61721A37E411D3472B8A690E88FCC4D82EEF8A7C7DD121FE1 3882896A528479A98B5C848D03DEAE999F930EBF59701F3C143189512374C67F 8F6DBFDE7629625309EEA64A60AC7E8F8ECB4AB581764055963D8A65DA901E41 AB90D500DA2F7CC26058707C39A6E0BCB869B82B523C4D016898BDB6A4BB5931 EA36144207DC3DD30CD678DFEC3D0DBDC36764AA2AE15EB0CD79C163FA4570DE AE9F26E93708851B148E638E08844CF0E03E099E9996F5C1E5559C496146976B 5DFC6B2CCE2B305B124A58FF909E8CFCE6EF3A0195755F36B12A7BC95360CE43 73C2EDC10A17E9AC25FBD1887D5C711E433ACF35A1AE312E1DE82E399C693C83 C653F61F2A45A6DEC64359F55D8129E5BF02CE0F30E64E3218E9B81335DC9834 03139F28783C0D9CD77A1E87C02254DB2F99D2902B53D093A1BA1FA49A78D673 B329B2762D3A8E7DCCC38FF007F6AAD2BD7FF04E73ED5C2F3B6544EBAD3B4C31 E47BE33BFFCB4A16CEAE6C74923297A68709F61BC5841451CEB7CC528EB19D5F 274725D770DA0D15467279CBF36F8B01211CBC5EF4E3A154D7A280FBDDD9CE6A DF44F97D8C5382CAD7F1FB8634B534F212D2430B903447FCFC1EA296E1023CEF 541C08420849DCF1F86BE60D824CDC9BF1D60E67A4CDF9F5F4F5A64412144E3A 821B03A490EFDFF74063AE7046A4FFC2D06CEEC3ECEB2D2B2E95E1647A935761 5943D2C8E639E0614EBB491B58B7295A9883984ED24926E8FE7E86F6BBE5CC4D 50C44330D6A0684E4A37C4FDCD7B49C6EA2DB08FCB729436C97DDBFBCEA714A6 C7DEEF6AF2DAF876E526D07A073C455796A2D2670BB2E4294081F15D6B7B9442 EF915C5D4BA227D710EFB8820F8D066D2F27F7DCF49DA7E159315B132E4E5F38 95F28B7FB3D14ECFBF4ED9E5BB0F5710AE6EFCA4788770B5736064DF73F46886 CE89314131A79BAA80667A95CCEC77DB66C645707018CC9623E9E0759DF38481 8519FD8A00C185C925ECC2C384D1FE1F94288A1EA7DDDB14455021564B355662 0375F13D6ADB14D5BA51BE3AF99D2580382AD4CAE6049D1255D753BCCA8BBE8A DB4CE7A08FFA10315652E9F74C35E3C14019E1AFD84F676B1EB54F53F9A04F4A A27C142837A2E6DFB1725D54FCA0BDCF7E1631E6F33AD95B6E392C71725467AB 1872382F2DC496040FA599BEBA3864E0DC43DC9178DF8A6BFA5FFDC2DDAD8CC2 95ECAC475D2D456FB90066FD583C1B2ED778D4350DCA317D962CB0CCD770A12F 054A90CFEA6BDB2A23390DAD7D092AF1B7C0ABCE38EC14DCC44C8D3131B66B6C 7D96806FBA032521BA64A69C6B30DB46D07E46B927C2845639AAD9AF37046A23 6EFC18A09D8FB81B876B297A07F65AC06134FA48C741060C2A791BAA45A37AB4 67B23AA6DA1A64E1E2159C623A010FE52DAD87734A4CA1286BC52AEEB5851919 11406A7270973D021258134F3BB27BE8799C26C511D5CC8E9DB936E63A71B68C 6DDFC09EF7E47D13F3B0D4C8188B53E360DD3C20B3D1ABA80BAFBB678D120709 16B7FDC1E03A1E6BF983634C4ADB4C63D6A14ED54DD0940AA00480F0F5693406 C48971E85BCC086A9B6ADCE05197471CD3EBC31D6F9428042EA0060D0DB33875 3357663183DECAFE9FC1321B1BA8AE28370C6818E913E72641A072469E4BD755 15CB20BDEFD5658DA116EE1C54B565E13AABAA632E32A5C2CC0BB82792CF7BCE 3EB6D9ECFC5A97D63CF10171408E9D974B02A6ED6ADE7632E7106B6BFA975884 9497C623D0CAB93C80AD1267718D1C91CC9FFD7A3AC854C3692615E96289B65B B8FF537381A58BAC2E40685778874A6E474BF7F3D675F22B7A1AD7C014471FA0 79FF42D0A966E520DA1F52117C51E0169D3F775B1CD801276FE333FD75FEBCC3 7AA39C1DEA4BE81034D1CDB10856C47689E6FA2654D3D84E2B85806ED3FB5292 6EBE366BF3AF3071DFC61EF575643A381CF6C686FBA600A6ADF56341B89F3D87 E00A7886F28CD65681CC9E0A1C37564A646767060F9EEEC7095CBF21E8F72F54 C5CA08596952CBCA4851C0C910453F89BAD7B42927C61170FB2006446C8A8A12 015DD8622FE7B900B692ED1A197D0B4BCB2C3A49C3F6ED6AE2E4166C653601D4 30E7087191B50F89ACD78EAAD7BAC97794F957DE58A677D5A38F1DB9704A1725 95AAF8D6195D98C97AFC5FF3A090ABF711FF71181C737F57E1D2F181B130CAC7 3688B32B2E9D2C6C50DD2293D99108BEE0C71CFCAFDD1894DD44307663BDA92F EFD2BBD3EC3020FA4EF575D06A76C77E7B6A5424E234CA192A303ABF65224E50 CFCBD974E964D0CF2621E846B67742B51DFFFD062E332662B6D1F7044522BD7A 34AEACA170546E072ADE24A205DE7952F2A199A2C69B29953E43644E27166021 A12E132B0760C4F4E0763AE6BBCBD39012B1DF36F3EE9517336A471AECF6033B B6B2C3887E9B32B309E0EF912EF987ACF4D2CA4A49E60B8721D884637DDD0B46 F2B46DFF5CAE013365DC51224A7E48559615BF97A4EC89C0BB6BC6A9BC0C48DD 36A2E88880390C9BBB76807B5B8FA9852DD68ACE90F5DE207D6F59FA7D35E335 D9DCA70229B33C79E4347401BA7EC0295D38F72262411DFC4B6D77320BC5576F C55119D099401B2E0B7F7FCBCF4C9F4CD6A7110FB13A90B8D616F72F5A766A9B 8EC792EA7CD564D6D062D48821346608FC7ABD0A347FD8F82A2A696750123887 08929051558DB61D2E3CEBAAEF666C12E8C0DB1F65A3B0CFEC2C799A8B2C4F99 CB5AD76A7547F88606367E90D80F4C898A2A98FFB90C1904E62DDDFB6844658E B28AA07739C669A94C1DB88601F4A047C7517C05CE11B2B60E29069AD2D9BF7F 7E7CC5779E3B5B5D6AC394AFA7BB18B08CD138ED23BD9DF408E7962727DCBCB3 D5B6AC816C5BEDC6668731A698DFBA3444DD26AE94F688534C2B218F6F571E95 8AD6288FB054DCA5889B4E34B445EFF4BD1253DD7087833B4C892EFA78B20D70 2A0DA926DD479E8761632C4E572742196E09164B5A7D906065F0A71E3690D699 8877753ACE715C940C414B589F67B5FD40717C4699EC0C1E9BD4247E24F93CBA 4E4257B4C789CBE9C39E755F6DB3A7036632768534BFECEB0ADBF20DFAF53A0F 692DF9D90A6838D01DAABB9B7C78309399FDFDDD30D13CFA0A90F78EFA8660DF 7611ABB4F82A6A06642A4133060B272FD4E09134092B99615A36F458B109F53B A695112FF6C355A658AC9D805D21222540C456E35A943A8806A0D051C32677D3 33D67E504FD917C21283341614C49B9CBB6BD3A4F23BDAEE8E8227F36AFF6F27 A830DB59BAA57AB0331D1946E5C002770A3CDEB8064CE4404384CB324B8FE9F5 913A7D0082619524871678A9842C70AA548391AF4D63DA7E8F0BC5F7538955B1 3E40493BE713F53E87566EF7AF9A63EB573CE0FEFEF7062BE2AE6AD2AC4EDB73 1EC17F9B0F9469F19E6BFD0101F24ACC9601E8B4DE8A58128F465798DF1CB68F 6079F69F5B348E4FB726F01EE9A5868B699D6E0BFFA294AD76B427C7B4423843 EF661CFEAED2BA9EFF89A459CA2D9C19F2F6E8F95DB30F7323AD0657928CFE28 13F8C249D5286C055F362B46201B209415380ED9732A5A4ED6A5D632ED946E8D BABD2CA8E7FEA98B388428DE80F4F4869298E719AF9898DCC44E09065305D632 9BE013589669C4C5FAF8B2AD17D95330C040709DA31973828C111B1AD48A443C 6C095CC489E9BFBE2D2DCB696CD148B614C4B7DA62C6AB87D11001B6467403F2 F79E65F71CDF979B7A7C0E7A2655AE3B420BFDF520757B8AC178472C4FA45479 7E591662A3FD36BBA7D47C04167C82632B9DCFC22D540DFC15459137F7A620C1 6A6BA1F4872DEE5A5C649520BB69F2ACFACCA7CA1EB2059EDB742A66535C7667 4023EBE9C9E7477A589C5A985D6D430120F12B6EF1250CCD56956BD9DFF58463 A834C4FB06CF48F59A3F48FAAF65EC4795ED939C41CF023E4D2FC8C251A298EB 3BD084A2EA95BC345F1A03511F54BDF9C8AB48A3D149243F32F9A83C047D70BE 47E217F52A75F60F4375D680524DFA563E44520426AE8050B48EF605EB692871 FFAECCD11F269C1082E9772D4749FDB6C39F2D83C8E5E265D4F7CFFFB21E2D64 40B6330164F98CB0B13C4D4B7E7340A0B3ED0AA65682872FB48399B069570606 8E49628617403BEC38A2CF3C340BA45AD8C613F5375C42B4317DE2EE5C3216C1 A0E4E070FC4602C87B4E661F86B3D79FBE6F9137B7C207E1CF3F8D13BB638D8A E0388B8E26AC0570286207161AA2A1F756864C1307A6CA8CEE8F36DE8E2BB36C 6495CB9FCB6898BFD952E1695B38F09093B21D920B3EA0A0B696C21FB44AA506 2F27D31CAE3BEA1113D9C23EC0CC1CDF9BC4D445E2484D50178421998E8AAC1E D6C4583EF41356E8FD300AE3E6826F53320E664911C35A50F9E74E4326D22C7B 6F346A5E4C920140FD19B91C9A81FA38BCD0CCB57E0D6E597F276D80CFC61269 3241DF8CF6C8674B064D0B4B9FDA6C7C105224F13CC6F0D167088BF4B06FC236 2DF8C4F05BB2807606DC2CEC8942A8F0D8BAE9FE379E63FF91880A3D66A7F349 00B813DE5F75D749E7FEC7DA9135F3984A048A3C6925C1F4F3C050537E6473F6 A2CCDC06A7D999B1C44D301A126C965AC70506E6F6542FEEDFCB2BBDE4AE4667 BE501AB4718FE5B1B28F5192999BEDBCCC113752589E9E4DF6E3546BD3513440 C7D750B777EF8144D2BC37EECA86AD5005F6D85333FB261C6105CEE496651BE7 C4F1FD7B01DC6219B4386FA4C96370F6B390FAB7282A876FF3DF0D89530616F4 F1728171548DA461FD9C80EEB63A1C2BD9834A3B5878875552E0CC35F47B69B3 22FEA1617EBCE249A236D7D61A50938B22FD90C6D5A67DBDA1D5ECFAB57B5BEF 5927C5CE690AF8CC539B12462094E6A84284F581ECBD6B323F60824D6ACC7A4B FCCD78464A72FE1FE98929692D889049A6D0735F760D9EB406732F27B13E5F04 8690B5D8B287A5EBA86B1F47C31B8B96AE45BACFFC6B0353B00E08C3A990183C AD7CA8C443227F8188CA664329D79333FBA580B392FEF2967D9FA41EEA0C3A22 76F44D5C92B3392FC6175826C560B7E5F2FD89590D91A1DA72A935C78DADA689 C9536ED66362FA6818F4B853BE3F685343B897B5D8B9DE5B25F6C912AB2D9DF5 C5C265EB2DC06FCA045B21094796B591AA8EC83E26CF17BA0A630D61D09763AC 751567B3DBC500FA7AA5DD024656650B38630C80D1F3B83F1BC7FEF718BD5782 B7EB59E566F11A13318C305424DCF7998241C85A8E7B372A05CCA3A254B508AA 320156C0950A8AC27674B071954318D48C9466DB1564F5532CE2C266680E1890 27765FE870781D6E4A42C09A8E42EB4A8C3FFFF2D57B94B2AD7060D585A86011 A12909CDA1D0BE874DC247775F363CC6122BC6A8165AC8AB4608BE74D66A6F69 3F9F5FBEEDD388AC8E4EDAD4083C6AECF2ED326621FAA14E1069DB41A38C0CAC 30CBBE8B7A8B90168F41A1C562719458CCE5E12759CA99979A6E43EEACA3113D 8BFBFBE4D64DD05D74D477D12DC2E38A9ECB2BB0DE0D6340E028042351B52C56 D4E29F856FA5A957AA6A30B788B31A0FCBFADC62AE7BBBD43251ED45543F7525 53757720B5EC0017C6626337095742200F10E5DD0CCEA15F7F108289227E54FA 2605D85C7B3939D6860953B541AA2A90AB88B92F3E623BE3727FAC15F0677269 F1C5B4144821012B9FAB26663F8C400A65870946609ABA525C863D59C38D579B 59F1471497BB82CC2F37EB3DF042D05ADD4A9A13A6C2055D4AC0EBD80E811133 9A0956357DC2DAC8BC025372FADB43EBF7A39BBF64F7C190C4257EB944379760 BA538B879C95F6D67BA9EAA2A00F826A8E103ECD24CA9575CCA6235023B8F235 2F6B214ABE4418F1891E8DBF42FA9B6EFC910B1A875D8FB4D5F52118FEA8C78A 261BC3FD65E6AEF34EBAED1643593680FA54AB095CA9D3DDDD7F60ECB73CC9DC 563A2A75B3906CBE16EC40E11247660F6D1260B1F56ECEB273002E5FA8CB2B4E 30BA7EC93D6FB6392B61566FF9DF40E245BA14B98039FFCEA89234980CB82B34 A384500334A90362701928303027AC4450EC3D021185A13F43D67D0DF9EB4A9C 0EE166EC053D8D4438B4A50A3A5C59F059712EEAF976FA58113250C2460DB710 FD6C777F9359FDAE2855723E607518673DB4C73BCC9A4B2318A4363F28CA6413 4592110A17D5F7AFB6205411560475F68B6A456B663BA11DDECD9935490121E7 564B488F99282034FE53BF89695A2A14D9A81959D5B61DE8A2FDEF340C6F7E54 78CAD267071C2DBFCB49A84870567E5D3A076AD1920B31954914A533954A7B9D 2B643D3B0C75C5C57E58F84F45EB99FE2D7A4AE3EF800F21A747C167F8878821 2DD26F92E6FEF0C25065A4CEF404B862F4023C4EB5641ABCE974F72C6E95C42D 6C1989B45AD5C8D3A65370305ED5F5EB2E32C274A69BF939B579183501B0AB89 1C4C445BC5FA37B12D83119555E5F6FFCE943812505770A15A3543E046AEE95F 455C2538626DE3FB11DD0BB653F7E65834DCE3937BB3D7E4FBD1AF3BC21258C0 DB0E9A5F895C41AF519CEFB4807CF58679A817C11312706FACB348BA3BF60287 853BB00F6615C7E90EF74DE7C9E8098DB50FC6AAAC0F3C6604C4705A11006EB2 2257B24DCC339DD372B1B74DDC69E9B5B893E26D9268FE0C4B8C464AA2595A08 DEF00B6473E34DFA8D5D01C984C26FEC4604C9978457B4E228D6DAA2236E44E8 F824726CA22F131BDDDA694A5A1D1F7DA23F205FD93BB04C54F5F3FF9496E0E6 0994E8E6CED924EC616D975F9CF66F600598A28E37D091D4D85E74C7289BBFDD 5B19B8365240329C1C38F898BE95AF6DD214BA590CC165A91D36CE398839A791 BE636127BC1FEB4B39FC1DA3783766296B3FD37A7675428A024BE3DC7D06F34D C5B45AC9DB86DE352EFA10BF78E719E7B7C9D2D93D484BDF4803AE0B4008598E 5C120129EE50717610434651CDC0FE3766E4FA990545DA6EE1B933A5CF99AEEB E96DEA23E2B1FF456DF27982FE53FED1336D42DABCB108DCAA1B2B409CD3DAC0 C0AF7F0D75DC826927F6CF279236908612434215B617585F1F60510753F3A7FD B04C01DCD28F2311EE16C40E9E074214F49066515AFEE196A992EAE033C2BC70 C6F2E60F4E9B8F9D3DF2B6C69AB5DE8697F205D34E0B0AAF9112901D2A3FA69F 549E38A6378D69F94D529A4F73BA4F821063B20C18A1D78B77F8ABC3BE8EF8A1 3EB70867F28DB352EA8AD15E6FC22489CF2F2C8ACB45561D54CFA9C97F170CE3 0BC2D8C1A6DBB749018539CBA202A966DC1ACBEE6F51479F74C82F96782EED92 5165D14FD09FC060FA678DE68226C3FEE73ACBD58663E78C619266A5F2B75340 7153F7F61D85D53E732ED074F845DB9FC2020D3D2C79FD500071530D599F42A1 A2E1DB5718BB268D60D46B1124F1A93CE7E733C5993FA6E5760D3DF7340481E2 B15957DB7B20DDEAEC5AF747CE97B94526585073723B01BB75033F730EA854C3 A480A4F5934625D4A942173D0724534D8FF310C41DB7E8ABA83D144B1A81B548 5B6DEFAF9A390CAE2C0C6AF02CFEC1701463246E21082A0C363012A0F9815E22 5E07113DF7D4FCD38A7D7548B8FDA0C198675A05F091A2CA5449AFBEDA41A66D A3295D9FFA34ACAF54D2132634D5A22134BFA8730B20BF1430075F570F360AE0 FCEE5929AD24DB267BA75E5C3EB9D418E125A2AA626E642021D48C5B6132B686 C93AAF79D4B7B8931A140E7925B1D41CCA3D364D564530552959B4C2334EA5C1 BF85D178E8B25AAD4ED83FC322657B3885233220184C82A624293CB9DC181883 58C83117994A7E59B553E2C7E613D56C4CB2DF88A98E853B52F1C78D2F0A465D 7F512BF57D009ABCE966D60F6571DB680E96DBFEAAB49792C85A1277586B87D5 19AEE5A567B37B479D7B8135ED5C09090B8E615350785FA3DFF8EF738B1DC338 D65522425974823971D28FA459EEC9A21EC742097B32A9FBEB2AFA76DAD3CA33 EB92E2BFC5408016F88C415B22FD363D39C3180C058A038581295601085AD0C9 B205E6559BD4562EED00E412F5F6DFE8C2EA41FD3C33BD582215C438504D5F6B 84104360AFF591735B5D0A32ED125390D234498F4BDD99CB7B45AD1362AE5EA2 01E716509350BA31D214349B743C688FEC0E0863EEE557D3A061B0634CBFA93D 956C6F439004C27325CE3D15D5ABD49E156D2E87B2A57DA5B65C4E401E1D575B 56D664CE9A89445E84CB16DCC08FF76490B89CEF7F60F96B32716CB34FEF70C2 359B5351C2DBBB442A199106D0CCD0BF284ADA6A610B7791574981279048FE2A 957D2CC9C715236CC698B2CBA12215A2AD94D017C1B235E7834E13DA3B54778B E2B6119BDE7FEB971B9FB593C907E449AAC7331ED998D7415836E97710044096 3EA80F35CD2BA634ED80EAB754723D0A05928BB8D1D90BFB94C852C193B69F57 A549278E6B8C0BF3575D4D3C61F78F3F0A9E6E52C1DC1B22DD5D22E94DA7CCD3 5158657946DB58B4E8BF2C1A96A820B88EB9352BBDDDB7AE4CBF5C378922A212 6902E50D04F2BFDD471E89AA7D5997F972AB55353E55FCD0AF5367FF44758A46 AD4B5B757BC39829718BF62AB16FAB88C52651DE0416A703074779C6F7069280 4554FB43DCFF3BDDDF57054990C5B068633D8B6344E81EA463FADEE0D94FFB8C 2466F8A170B968E9A4EF1C52A18D0C03B18B8D0491F010C336D438CFE4FD2CC1 BB7A6F1548238E3820FEBC3B46B7D745AA7A22C3AE243AD44B24734C6A164939 67D729FB57DF94D9281C9DACFEC5901B020F14E72CF3186FB75184A5A5CBDD98 51396F650F7B6FE277E313285D886B5751D6B382DCBC5870B9A0B45FB40ABD4C CB8297640B2FA17BA6BE38B1385D20313B283D1ECD6F95ED959B53EA5D20D814 B4D0F7C1B585DE478DF676003223B22ED31B5D31BD3F4F7157D4698E5EC7CB94 3AD6CCE914EF4FEA6366E8C9C833F926E4CE72FD236E64413DEC9A9C75A1AA5F 322C602F224089F68E4A32CD89999A8874AC08AE11D8682DBABFB67D724203EB 8312815BB38E1A61E3F270B028128B778ECECFEFB9F829EC3DCD3FAF96C6492C B69FF3DE2A9330FD6057AB547667C3F7F8C3D27E60DF2FD4E89AB5A39549ABA0 9E2BB7EC1F381DF342CE5D0AE84F6A23B9AFE81AB22C904D2A5881D1B0924010 C58343518BF69D0C4C2E67891142C4AFA414483E6967C7949C97106C6A354E97 A66E72EE8A26A27CB885B487B357D30F8350A31E7AE3D5482C211751C441E901 9FA5E20B61BC10125F068DCA7F5B2ABC2E419B7D6EB8C4B8913D129C7D1FA57D D684740FA47BB39D0471700FD65A317CB6CD277FCA390B0ADB72344B67C15CAD 3D3FAC75ED60076825A6835412432E63FCBC76B0E22B2539F6E36A215318EBB4 DDCCEE36D72304582B5AEE4471DC05ED0CDF66602E13E823585CE6C96EDCF088 78D53095BABE07E7CBBE807B538AA1462EA3707BC2153A4CCCF0F4253F8985B0 BA1B939E82BF84683AA23BDA609F87A3780040FCFF9A0A52C79CF244791F7172 2A53460AE4373486684472B2B99B2945D3F34448085D98F48B30A4928BCADE19 F18DB65295AB546ED5E12E5AD47FD0D0095180B7FA557E085A54FE271985D477 094436B722D2A0632ABDBEFCE7FCFC4C70058B1EB287671834159F52548CC05D 844A904710A0C69C08E4CE1703B9500E897F8E300FA9784C1E96054F61908C92 77E1AAF62FA080B03B1BEF6386A39DF30F1475AE889E8A164B70A282C0C49C39 6ED28B03DB575BDF9770BCF9965E92231DAF6522B68E9DFE54B719E63B12BBD7 D9C6DA1EE70ABA0583642856AE1D6651890D29B540608C29E4936B012612FD2A 4A2897CA10BECFCF017695CEEEB70851B8F15DA8C35623161D18E233929B4D34 5634CA86ECB3616070B4517CC95A1977207E26D7AFE5E6AA2ACD4E74F2AF1320 8C5DDDE0CA007BDF5B7EC636C251318495FA7631A627A13A270C557080302DFD 69817D67A6A5BC2761B34AFD2A5E26BE8A208BDD4A6F52C9430D4A95AD33FEE4 73A353582DE8FC9E59E9A7873B5DEA6F458DDEC24492F56D3E1F95AE919C4BF1 A776D50E3A9466A52ED23C847901 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec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cleartomark %%EndFont TeXDict begin 40258431 52099146 1000 600 600 (lects1-5.dvi) @start /Fa 137[35 7[43 4[24 1[35 1[35 101[{ TeX74afc74cEncoding ReEncodeFont }5 74.7198 /CMTI9 rf /Fb 150[22 6[27 98[{ TeXaae443f0Encoding ReEncodeFont }2 49.8132 /CMMI6 rf /Fc 192[83 8[83 54[{}2 83.022 /LINEW10 rf /Fd 158[83 16[83 14[42 83 55[83 7[83{}6 83.022 /LINE10 rf /Fe 133[50 59 59 81 59 62 44 44 46 59 62 56 62 93 31 59 1[31 62 56 34 51 62 50 62 54 7[85 5[62 84 4[106 67 2[42 2[70 74 1[81 8[31 4[56 56 56 56 56 2[31 1[31 28[93 15[{ TeXf7b6d320Encoding ReEncodeFont }43 99.6264 /CMBX12 rf /Ff 143[83 6[39 39 13[46 1[120 5[39 1[88 62[50 50 16[{}9 83.022 /CMEX10 rf /Fg 255[60{ TeXbbad153fEncoding ReEncodeFont }1 74.7198 /CMSY9 rf /Fh 205[47 71 19 14[71 32[55{ TeXbbad153fEncoding ReEncodeFont }5 66.4176 /CMSY8 rf /Fi 142[34 2[46 12[41 24[64 60 49 57 63 55 58 58 65[{ TeXaae443f0Encoding ReEncodeFont }11 74.7198 /CMMI9 rf /Fj 131[77 2[41 41 55 41 43 30 30 30 41 43 38 43 64 21 41 23 21 43 38 23 34 43 34 43 38 9[79 2[55 43 2[52 60 1[70 48 2[28 2[50 52 2[54 58 3[60 10[38 3[21 26 21 60 1[30 30 6[21 20[43 12[{ TeXf7b6d320Encoding ReEncodeFont } 48 74.7198 /CMR9 rf /Fk 200[30 30 30 30 30 30 30 49[{ TeXf7b6d320Encoding ReEncodeFont }7 49.8132 /CMR6 rf /Fl 130[48 48 48 2[48 3[48 4[48 48 1[48 8[48 1[48 1[48 48 1[48 2[48 48 19[48 48 48 48 3[48 48 2[48 48 48 1[48 2[48 48 48 48 48 48 1[48 48 48 48 40[{ TeX09fbbfacEncoding ReEncodeFont }35 90.9091 /CMTT10 rf /Fm 139[25 33 32 3[43 2[37 29 24 3[33 36 1[30 37 13[43 2[45 6[31 7[53 3[35 2[20 58[{ TeXaae443f0Encoding ReEncodeFont }17 66.4176 /CMMI8 rf /Fn 136[51 37 39 27 28 28 37 39 35 39 1[20 2[20 39 35 1[31 1[31 1[35 35[55 4[35 35 35 35 1[35 35 35 35 4[55 1[27 27 40[{ TeXf7b6d320Encoding ReEncodeFont }29 66.4176 /CMR8 rf /Fo 133[37 44 42 60 42 49 30 37 38 42 46 46 51 74 23 42 1[28 46 42 28 42 46 42 42 46 12[65 51 66 70 10[59 1[69 65 1[68 6[28 11[28 33 28 2[37 37 27[51 12[{ TeX74afc74cEncoding ReEncodeFont }40 90.9091 /CMTI10 rf /Fp 143[76 5[25 2[45 45 40 40 40 40 32[73 14[61 24[71 3[71 71 4[45 12[71 25 71{ TeXbbad153fEncoding ReEncodeFont } 17 90.9091 /CMSY10 rf /Fq 133[61 73 73 99 73 76 53 54 57 1[76 69 76 115 38 73 1[38 76 69 42 63 76 61 76 67 9[142 2[96 76 103 103 94 103 1[131 83 108 1[52 2[87 1[105 1[98 104 6[38 4[69 69 69 69 69 2[38 46 38 31[76 12[{ TeXf7b6d320Encoding ReEncodeFont }49 119.552 /CMBX10 rf /Fr 138[110 77 1[82 12[91 1[88 22[119 22[99 99 99 99 99 49[{ TeXf7b6d320Encoding ReEncodeFont }11 172.188 /CMBX10 rf /Fs 165[56 68 68 93 68 68 66 51 67 71 62 71 68 83 57 71 1[33 68 71 59 62 69 66 64 68 6[25 4[45 45 45 45 45 2[25 30 3[35 35 1[71 38[{ TeXf7b6d320Encoding ReEncodeFont }36 90.9091 /CMSL10 rf /Ft 135[52 3[33 43 41 1[46 44 55 2[47 37 31 1[43 45 42 47 39 39 48 7[53 75 3[53 56 3[69 2[62 2[40 76 71 58 67 75 65 69 68 2[71 45 71 25 25 24[57 17[37 15[{ TeXaae443f0Encoding ReEncodeFont }39 90.9091 /CMMI10 rf /Fu 128[45 2[91 45 40 48 48 66 48 51 35 36 36 48 51 45 51 76 25 48 28 25 51 45 28 40 51 40 51 45 3[25 45 25 1[68 1[93 68 68 66 51 67 71 62 71 68 83 57 71 1[33 68 71 59 62 69 66 64 68 1[43 1[71 1[25 25 45 45 45 45 45 45 45 45 45 45 45 25 30 25 71 1[35 35 25 2[45 1[45 25 17[76 76 51 51 53 66 7[71 2[{ TeXf7b6d320Encoding ReEncodeFont }87 90.9091 /CMR10 rf /Fv 132[52 46 55 55 76 55 58 41 41 43 1[58 52 58 87 29 55 1[29 58 52 32 48 58 46 58 51 11[80 73 58 78 79 71 2[99 63 2[40 3[69 80 76 74 79 6[29 52 1[52 1[52 52 52 52 52 2[29 35 3[41 41 1[81 25[58 12[{ TeXf7b6d320Encoding ReEncodeFont }53 90.9091 /CMBX10 rf /Fw 133[106 125 1[172 1[132 92 94 98 1[132 119 132 198 66 125 1[66 132 119 73 109 132 106 132 115 13[132 178 178 162 3[143 2[90 5[172 169 180 6[66 12[79 3[92 92 1[185 25[132 12[{ TeXf7b6d320Encoding ReEncodeFont }37 206.559 /CMBX10 rf /Fx 137[51 54 2[38 4[81 27 2[27 38[69 73 65[{ TeXf7b6d320Encoding ReEncodeFont }8 99.6264 /CMR12 rf /Fy 138[92 64 65 68 2[82 1[137 46 2[46 92 82 1[76 1[73 1[80 20[99 5[104 4[125 11[82 82 1[82 82 82 2[55 46 44[{ TeXf7b6d320Encoding ReEncodeFont }23 143.462 /CMBX10 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin end %%EndSetup %%Page: 0 1 TeXDict begin 0 0 bop 911 2008 a Fy(15-451)58 b(Algorithms,)d(F)-14 b(all)55 b(2005)1506 2191 y(Lectures)g(1-5)1682 2619 y Fx(Avrim)34 b(Blum)p eop end %%Page: 1 2 TeXDict begin 1 1 bop 0 795 a Fw(Con)-7 b(ten)g(ts)0 1389 y Fv(1)84 b(In)m(tro)s(duction)36 b(to)e(Algorithms)2462 b(3)136 1544 y Fu(1.1)94 b(Ov)m(erview)84 b Ft(:)46 b(:)g(:)g(:)f(:)h (:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:) g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g (:)f(:)131 b Fu(3)136 1700 y(1.2)94 b(In)m(tro)s(duction)28 b Ft(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f (:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:) f(:)h(:)g(:)f(:)h(:)g(:)f(:)131 b Fu(3)136 1856 y(1.3)94 b(On)30 b(guaran)m(tees)i(and)e(sp)s(eci\014cations)70 b Ft(:)45 b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h (:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)131 b Fu(4)136 2012 y(1.4)94 b(An)30 b(example:)42 b(Karatsuba)30 b(Multiplication)40 b Ft(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g (:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)131 b Fu(5)136 2168 y(1.5)94 b(Matrix)32 b(m)m(ultiplication)25 b Ft(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h (:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:) g(:)f(:)131 b Fu(6)0 2429 y Fv(2)84 b(Asymptotic)36 b(Analysis)f(and)g (Recurrences)1970 b(9)136 2585 y Fu(2.1)94 b(Ov)m(erview)84 b Ft(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g (:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:) g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)131 b Fu(9)136 2741 y(2.2)94 b(Asymptotic)32 b(analysis)78 b Ft(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f (:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:) h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)131 b Fu(9)136 2897 y(2.3)94 b(Recurrences)49 b Ft(:)d(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g (:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:) f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(11)345 3053 y(2.3.1)106 b(Solving)31 b(b)m(y)f(unrolling)71 b Ft(:)45 b(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g (:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(11)345 3208 y(2.3.2)106 b(Solving)31 b(b)m(y)f(guess)h(and)e (inductiv)m(e)i(pro)s(of)44 b Ft(:)i(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:) h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(12)345 3364 y(2.3.3)106 b(Recursion)31 b(trees,)g(stac)m(king)h (bric)m(ks,)e(and)g(a)h(Master)g(F)-8 b(orm)m(ula)94 b Ft(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(13)0 3626 y Fv(3)f(Probabilistic)37 b(Analysis)e(and)g(Randomized) h(Quic)m(ksort)1383 b(15)136 3782 y Fu(3.1)94 b(Ov)m(erview)84 b Ft(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g (:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:) g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(15)136 3938 y(3.2)94 b(W)-8 b(orst-case,)34 b(a)m(v)m(erage-case,)g(and)c(randomized)g (algorithms)36 b Ft(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h (:)g(:)f(:)h(:)g(:)f(:)85 b Fu(15)136 4093 y(3.3)94 b(The)30 b(Basics)i(of)e(Probabilistic)i(Analysis)81 b Ft(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g (:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(16)345 4249 y(3.3.1)106 b(Linearit)m(y)32 b(of)e(Exp)s(ectation)39 b Ft(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f (:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(17)345 4405 y(3.3.2)106 b(Example)31 b(1:)41 b(Card)30 b(sh)m(u\017ing)35 b Ft(:)45 b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f (:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:) 85 b Fu(18)345 4561 y(3.3.3)106 b(Example)31 b(2:)41 b(In)m(v)m(ersions)31 b(in)f(a)g(random)g(p)s(erm)m(utation)83 b Ft(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:) 85 b Fu(18)136 4717 y(3.4)94 b(Analysis)31 b(of)g(Randomized)f(Quic)m (ksort)71 b Ft(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g (:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(18)345 4873 y(3.4.1)106 b(Metho)s(d)31 b(1)63 b Ft(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g (:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:) f(:)h(:)g(:)f(:)85 b Fu(18)345 5028 y(3.4.2)106 b(Metho)s(d)31 b(2)63 b Ft(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h (:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:) h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(19)136 5184 y(3.5)94 b(F)-8 b(urther)31 b(Discussion)64 b Ft(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h (:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:) g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(20)345 5340 y(3.5.1)106 b(More)31 b(linearit)m(y)h(of)f(exp)s (ectation:)42 b(a)31 b(random)e(w)m(alk)j(sto)s(c)m(k)f(mark)m(et)70 b Ft(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(20)1927 5568 y(1)p eop end %%Page: 2 3 TeXDict begin 2 2 bop 0 -165 a Fu(2)3339 b Fs(CONTENTS)345 91 y Fu(3.5.2)106 b(Y)-8 b(et)32 b(another)f(w)m(a)m(y)g(to)g(analyze)h (quic)m(ksort:)41 b(run)29 b(it)i(bac)m(kw)m(ards)67 b Ft(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(21)0 334 y Fv(4)f(Selection)36 b(\(det)e(&)h(rand\):)46 b(\014nding)36 b(the)e(median)h(in)g(linear)g(time)1034 b(23)136 486 y Fu(4.1)94 b(Ov)m(erview)84 b Ft(:)46 b(:)g(:)g(:)f(:)h (:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:) g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g (:)f(:)85 b Fu(23)136 638 y(4.2)94 b(The)30 b(problem)g(and)g(a)h (randomized)f(solution)60 b Ft(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g (:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(23)136 790 y(4.3)94 b(A)31 b(deterministic)g(linear-time)h (algorithm)c Ft(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f (:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(24)0 1033 y Fv(5)f(Comparison-based)36 b(Lo)m(w)m(er)e(Bounds)i (for)f(Sorting)1603 b(27)136 1185 y Fu(5.1)94 b(Ov)m(erview)84 b Ft(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g (:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:) g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(27)136 1338 y(5.2)94 b(Sorting)31 b(lo)m(w)m(er)h(b)s(ounds)j Ft(:)45 b(:)h(:)g(:)g(:)f(:)h (:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:) g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(27)136 1490 y(5.3)94 b(Av)m(erage-case)34 b(lo)m(w)m(er)e(b)s(ounds) 22 b Ft(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f (:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:) 85 b Fu(29)136 1642 y(5.4)94 b(Lo)m(w)m(er)32 b(b)s(ounds)c(for)i (randomized)g(algorithms)94 b Ft(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g (:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b Fu(29)p eop end %%Page: 3 4 TeXDict begin 3 3 bop 0 753 a Fr(Lecture)65 b(1)0 1207 y Fw(In)-7 b(tro)7 b(duction)79 b(to)g(Algorithms)0 1728 y Fq(1.1)137 b(Ov)l(erview)0 1976 y Fu(The)33 b(purp)s(ose)g(of)h(this) g(lecture)h(is)f(to)g(giv)m(e)i(a)e(brief)g(o)m(v)m(erview)h(of)f(the)g (topic)h(of)g(Algorithms)f(and)f(the)i(kind)e(of)0 2089 y(thinking)39 b(it)i(in)m(v)m(olv)m(es:)61 b(wh)m(y)39 b(w)m(e)h(fo)s(cus)f(on)h(the)g(sub)5 b(jects)39 b(that)h(w)m(e)g(do,)i (and)d(wh)m(y)h(w)m(e)g(emphasize)g(pro)m(ving)0 2202 y(guaran)m(tees.)h(W)-8 b(e)30 b(also)f(go)h(through)d(an)i(example)g (of)f(a)h(problem)f(that)h(is)f(easy)h(to)g(relate)h(to)f(\(m)m (ultiplying)h(t)m(w)m(o)0 2315 y(n)m(um)m(b)s(ers\))g(in)i(whic)m(h)f (the)h(straigh)m(tforw)m(ard)g(approac)m(h)g(is)g(surprisingly)e(not)i (the)f(fastest)i(one.)44 b(This)31 b(example)0 2428 y(leads)f (naturally)f(in)m(to)h(the)f(study)g(of)g(recurrences,)g(whic)m(h)g(is) g(the)h(topic)g(of)f(the)g(next)g(lecture,)i(and)d(pro)m(vides)i(a)0 2541 y(forw)m(ard)g(p)s(oin)m(ter)g(to)h(topics)g(suc)m(h)g(as)f(the)h (FFT)f(later)i(on)e(in)g(the)h(course.)0 2693 y(Material)h(in)e(this)h (lecture:)136 2939 y Fp(\017)46 b Fu(Administrivia)31 b(\(see)g(handouts\))136 3153 y Fp(\017)46 b Fu(What)31 b(is)g(the)f(study)g(of)g(Algorithms)h(all)h(ab)s(out?)136 3366 y Fp(\017)46 b Fu(Wh)m(y)31 b(do)f(w)m(e)h(care)g(ab)s(out)f(sp)s (eci\014cations)h(and)f(pro)m(ving)h(guaran)m(tees?)136 3580 y Fp(\017)46 b Fu(The)30 b(Karatsuba)g(m)m(ultiplication)j (algorithm.)136 3793 y Fp(\017)46 b Fu(Strassen's)30 b(matrix)h(m)m(ultiplication)h(algorithm.)0 4149 y Fq(1.2)137 b(In)l(tro)t(duction)0 4398 y Fu(This)41 b(course)85 b(is)42 b(ab)s(out)g(the)g(design)g(and)f(analysis)i(of)f(algorithms)h (|)f(ho)m(w)g(to)h(design)f(correct,)k(e\016cien)m(t)0 4510 y(algorithms,)32 b(and)d(ho)m(w)i(to)g(think)f(clearly)h(ab)s(out) f(analyzing)i(correctness)f(and)f(running)f(time.)0 4663 y(What)36 b(is)g(an)f(algorithm?)58 b(A)m(t)36 b(its)g(most)g(basic,)i (an)d(algorithm)i(is)f(a)g(metho)s(d)f(for)g(solving)h(a)g (computational)0 4775 y(problem.)j(Along)28 b(with)g(an)f(algorithm)h (comes)h(a)f(sp)s(eci\014cation)g(that)g(sa)m(ys)g(what)f(the)h (algorithm's)h(guaran)m(tees)0 4888 y(are.)43 b(F)-8 b(or)32 b(example,)h(w)m(e)e(migh)m(t)h(b)s(e)f(able)g(to)h(sa)m(y)g (that)g(our)e(algorithm)j(indeed)d(correctly)j(solv)m(es)f(the)g (problem)0 5001 y(in)g(question)g(and)g(runs)e(in)i(time)h(at)g(most)f Ft(f)10 b Fu(\()p Ft(n)p Fu(\))32 b(on)g(an)m(y)g(input)f(of)i(size)f Ft(n)p Fu(.)46 b(This)31 b(course)h(is)h(ab)s(out)e(the)i(whole)0 5114 y(pac)m(k)-5 b(age:)45 b(the)32 b(design)f(of)h(e\016cien)m(t)h (algorithms,)g Fo(and)f Fu(pro)m(ving)g(that)g(they)g(meet)g(desired)f (sp)s(eci\014cations.)45 b(F)-8 b(or)0 5227 y(eac)m(h)41 b(of)g(these)f(parts,)j(w)m(e)e(will)f(examine)h(imp)s(ortan)m(t)g(tec) m(hniques)g(that)f(ha)m(v)m(e)i(b)s(een)d(dev)m(elop)s(ed,)44 b(and)39 b(with)0 5340 y(practice)32 b(w)m(e)f(will)f(build)g(up)f(our) h(abilit)m(y)i(to)f(think)f(clearly)h(ab)s(out)f(the)h(k)m(ey)g(issues) f(that)h(arise.)1927 5568 y(3)p eop end %%Page: 4 5 TeXDict begin 4 4 bop 0 -165 a Fu(4)1662 b Fs(LECTURE)30 b(1.)71 b(INTR)m(ODUCTION)30 b(TO)g(ALGORITHMS)0 91 y Fu(The)23 b(main)g(goal)h(of)f(this)g(course)h(is)f(to)h(pro)m(vide)f (the)g(in)m(tellectual)j(to)s(ols)e(for)f(designing)g(and)g(analyzing)h (y)m(our)f(o)m(wn)0 204 y(algorithms)34 b(for)g(problems)e(y)m(ou)i (need)f(to)i(solv)m(e)f(in)f(the)h(future.)49 b(Some)34 b(to)s(ols)g(w)m(e)g(will)g(discuss)e(are)i(Dynamic)0 317 y(Programming,)48 b(Divide-and-Conquer,)f(Data)f(Structure)d (design)h(principles,)k(Randomization,)g(Net)m(w)m(ork)0 430 y(Flo)m(ws,)28 b(Linear)f(Programming,)g(and)f(the)g(F)-8 b(ast)28 b(F)-8 b(ourier)27 b(T)-8 b(ransform.)38 b(Some)27 b(analytical)h(to)s(ols)f(w)m(e)g(will)g(discuss)0 543 y(and)j(use)g(are)h(Recurrences,)f(Probabilistic)i(Analysis,)e (Amortized)i(Analysis,)f(and)e(P)m(oten)m(tial)k(F)-8 b(unctions.)0 695 y(There)38 b(is)h(also)h(a)f(dual)f(to)i(algorithm)f (design:)58 b(Complexit)m(y)39 b(Theory)-8 b(.)66 b(Complexit)m(y)40 b(Theory)e(lo)s(oks)h(at)h(the)0 808 y(in)m(trinsic)f(di\016cult)m(y)h (of)f(computational)i(problems)d(|)h(what)g(kinds)f(of)h(sp)s (eci\014cations)h(can)f(w)m(e)h(exp)s(ect)f Fo(not)0 921 y Fu(to)44 b(b)s(e)e(able)h(to)h(ac)m(hiev)m(e?)80 b(In)42 b(this)h(course,)j(w)m(e)e(will)f(delv)m(e)h(a)f(bit)g(in)m(to) h(complexit)m(y)h(theory)-8 b(,)46 b(fo)s(cusing)d(on)0 1034 y(the)h(somewhat)g(surprising)e(notion)i(of)g(NP-completeness.)82 b(W)-8 b(e)45 b(will)f(\(ma)m(y\))h(also)g(sp)s(end)d(some)i(time)g(on) 0 1147 y(cryptograph)m(y)-8 b(.)56 b(Cryptograph)m(y)35 b(is)h(in)m(teresting)h(from)d(the)i(p)s(oin)m(t)f(of)h(view)g(of)f (algorithm)i(design)e(b)s(ecause)g(it)0 1260 y(uses)c(a)g(problem)g (that's)h(assumed)e(to)i(b)s(e)f(in)m(trinsically)h(hard)e(to)i(solv)m (e)h(in)e(order)f(to)i(construct)g(an)f(algorithm)0 1372 y(\(e.g.,)h(an)f(encryption)f(metho)s(d\))g(whose)g(securit)m(y)h (rests)g(on)f(the)h(di\016cult)m(y)f(of)h(solving)g(that)g(hard)e (problem.)0 1729 y Fq(1.3)137 b(On)47 b(guaran)l(tees)f(and)h(sp)t (eci\014cations)0 1978 y Fu(One)23 b(fo)s(cus)h(of)g(this)g(course)g (is)g(on)g(pro)m(ving)g(correctness)h(and)e(running-time)g(guaran)m (tees)i(for)f(algorithms.)40 b(Wh)m(y)0 2091 y(is)35 b(ha)m(ving)g(suc)m(h)f(a)h(guaran)m(tee)h(useful?)51 b(Supp)s(ose)33 b(w)m(e)i(are)g(talking)h(ab)s(out)e(the)g(problem)g (of)h(sorting)g(a)g(list)g(of)0 2204 y Ft(n)g Fu(n)m(um)m(b)s(ers.)54 b(It)35 b(is)g(prett)m(y)h(clear)h(wh)m(y)d(w)m(e)i(at)g(least)h(w)m (an)m(t)f(to)g(kno)m(w)f(that)h(our)e(algorithm)j(is)e(correct,)j(so)d (w)m(e)0 2317 y(don't)29 b(ha)m(v)m(e)h(to)f(w)m(orry)g(ab)s(out)f (whether)g(it)h(has)g(giv)m(en)h(us)e(the)g(righ)m(t)i(answ)m(er)e(all) i(the)f(time.)41 b(But,)29 b(wh)m(y)g(analyze)0 2429 y(running)d(time?)40 b(Wh)m(y)28 b(not)g(just)f(co)s(de)h(up)e(our)i (algorithm)g(and)f(test)i(it)f(on)f(100)i(random)e(inputs)g(and)g(see)h (what)0 2542 y(happ)s(ens?)44 b(Here)32 b(are)h(a)f(few)g(reasons)g (that)h(motiv)-5 b(ate)34 b(our)d(concern)i(with)e(this)h(kind)f(of)i (analysis)f(|)g(y)m(ou)h(can)0 2655 y(probably)d(think)f(of)i(more)g (reasons)f(to)s(o:)0 2902 y Fv(Comp)s(osabilit)m(y)-9 b(.)46 b Fu(A)28 b(guaran)m(tee)h(on)e(running)f(time)i(giv)m(es)h(a)f (\\clean)h(in)m(terface".)41 b(It)28 b(means)f(that)i(w)m(e)e(can)h (use)227 3015 y(the)e(algorithm)h(as)f(a)g(subroutine)f(in)h(some)g (other)g(algorithm,)i(without)e(needing)f(to)i(w)m(orry)f(whether)f (the)227 3128 y(kinds)j(of)h(inputs)f(on)h(whic)m(h)g(it)g(is)h(b)s (eing)e(used)g(no)m(w)h(necessarily)h(matc)m(h)g(the)f(kinds)f(of)h (inputs)f(on)h(whic)m(h)227 3241 y(it)i(w)m(as)g(originally)h(tested.)0 3455 y Fv(Scaling.)46 b Fu(The)36 b(t)m(yp)s(es)h(of)f(guaran)m(tees)i (w)m(e)f(will)f(examine)h(will)g(tell)h(us)d(ho)m(w)i(the)f(running)f (time)i(scales)g(with)227 3568 y(the)27 b(size)h(of)f(the)g(problem)g (instance.)40 b(This)26 b(is)h(useful)f(to)i(kno)m(w)f(for)g(a)g(v)-5 b(ariet)m(y)29 b(of)e(reasons.)39 b(F)-8 b(or)28 b(instance,)227 3681 y(it)34 b(tells)h(us)d(roughly)h(ho)m(w)h(large)g(a)g(problem)f (size)h(w)m(e)g(can)f(reasonably)h(exp)s(ect)g(to)g(handle)f(giv)m(en)i (some)227 3794 y(amoun)m(t)c(of)g(resources.)0 4008 y Fv(Designing)36 b(b)s(etter)e(algorithms.)46 b Fu(Analyzing)31 b(the)e(asymptotic)i(running)d(time)i(of)g(algorithms)g(is)g(a)g (useful)227 4121 y(w)m(a)m(y)i(of)e(thinking)g(ab)s(out)g(algorithms)i (that)f(often)f(leads)h(to)g(nonob)m(vious)g(impro)m(v)m(emen)m(ts.)0 4335 y Fv(Understanding.)46 b Fu(An)30 b(analysis)g(can)g(tell)h(us)e (what)g(parts)h(of)f(an)h(algorithm)h(are)f(crucial)g(for)g(what)f (kinds)g(of)227 4448 y(inputs,)36 b(and)f(wh)m(y)-8 b(.)56 b(If)35 b(w)m(e)h(later)g(get)h(a)e(di\013eren)m(t)h(but)f(related)h (task,)i(w)m(e)e(can)f(often)h(use)f(our)g(analysis)227 4561 y(to)i(quic)m(kly)g(tell)h(us)d(if)i(a)f(small)h(mo)s (di\014cation)g(to)g(our)f(existing)h(algorithm)g(can)g(b)s(e)f(exp)s (ected)g(to)h(giv)m(e)227 4674 y(similar)31 b(p)s(erformance)f(to)h (the)f(new)g(problem.)0 4888 y Fv(Complexit)m(y-theoretic)35 b(motiv)-6 b(ation.)46 b Fu(In)e(Complexit)m(y)i(Theory)-8 b(,)49 b(w)m(e)c(w)m(an)m(t)h(to)g(kno)m(w:)70 b(\\ho)m(w)46 b(hard)e(is)227 5001 y(fundamen)m(tal)25 b(problem)g Ft(X)32 b Fu(really?")40 b(F)-8 b(or)26 b(instance,)h(w)m(e)e(migh)m(t) h(kno)m(w)f(that)h(no)e(algorithm)i(can)g(p)s(ossibly)227 5114 y(run)21 b(in)h(time)h Ft(o)p Fu(\()p Ft(n)15 b Fu(log)i Ft(n)p Fu(\))22 b(\(gro)m(wing)i(more)e(slo)m(wly)h(than)f Ft(n)15 b Fu(log)i Ft(n)22 b Fu(in)g(the)g(limit\))h(and)f(w)m(e)g(ha)m (v)m(e)i(an)e(algorithm)227 5227 y(that)33 b(runs)e(in)h(time)g Ft(O)s Fu(\()p Ft(n)1109 5194 y Fn(3)p Fm(=)p Fn(2)1219 5227 y Fu(\).)47 b(This)31 b(tells)i(us)f(ho)m(w)g(w)m(ell)h(w)m(e)g (understand)d(the)j(problem,)f(and)g(also)h(ho)m(w)227 5340 y(m)m(uc)m(h)e(ro)s(om)f(for)g(impro)m(v)m(emen)m(t)i(w)m(e)e(ha)m (v)m(e.)p eop end %%Page: 5 6 TeXDict begin 5 5 bop 0 -165 a Fs(1.4.)72 b(AN)31 b(EXAMPLE:)f(KARA)-8 b(TSUBA)31 b(MUL)-8 b(TIPLICA)g(TION)1530 b Fu(5)0 91 y(It)42 b(is)h(often)f(helpful)g(when)f(thinking)h(ab)s(out)g (algorithms)h(to)g(imagine)h(a)e(game)i(where)d(one)i(pla)m(y)m(er)g (is)g(the)0 204 y(algorithm)37 b(designer,)i(trying)d(to)h(come)h(up)d (with)h(a)h(go)s(o)s(d)g(algorithm)g(for)f(the)h(problem,)h(and)d(its)i (opp)s(onen)m(t)0 317 y(\(the)c(\\adv)m(ersary"\))h(is)f(trying)f(to)h (come)h(up)d(with)h(an)h(input)e(that)i(will)g(cause)g(the)g(algorithm) g(to)g(run)e(slo)m(wly)-8 b(.)0 430 y(An)28 b(algorithm)i(with)f(go)s (o)s(d)f(w)m(orst-case)j(guaran)m(tees)f(is)f(one)g(that)h(p)s(erforms) d(w)m(ell)j(no)e(matter)i(what)f(input)f(the)0 543 y(adv)m(ersary)j(c)m (ho)s(oses.)43 b(W)-8 b(e)32 b(will)f(return)f(to)i(this)e(view)h(in)g (a)g(more)g(formal)g(w)m(a)m(y)h(when)e(w)m(e)h(discuss)f(randomized)0 656 y(algorithms)h(and)f(lo)m(w)m(er)i(b)s(ounds.)0 979 y Fq(1.4)137 b(An)46 b(example:)61 b(Karatsuba)46 b(Multiplication)0 1221 y Fu(One)29 b(thing)h(that)g(mak)m(es)h(algorithm)g(design)e (\\Computer)h(Science")h(is)e(that)i(solving)f(a)g(problem)f(in)h(the)g (most)0 1334 y(ob)m(vious)k(w)m(a)m(y)g(from)f(its)g(de\014nitions)g (is)g(often)h(not)g(the)f(b)s(est)g(w)m(a)m(y)h(to)g(get)h(a)e (solution.)50 b(A)34 b(simple)f(example)h(of)0 1447 y(this)c(is)h(m)m (ultiplication.)0 1599 y(Sa)m(y)e(w)m(e)f(w)m(an)m(t)h(to)g(m)m (ultiply)g(t)m(w)m(o)g Ft(n)p Fu(-bit)f(n)m(um)m(b)s(ers:)39 b(for)28 b(example,)h(41)17 b Fp(\002)f Fu(42)29 b(\(or,)g(in)f(binary) -8 b(,)28 b(101001)19 b Fp(\002)d Fu(101010\).)0 1712 y(According)34 b(to)h(the)e(de\014nition)h(of)g(what)f(it)h(means)g(to) g(m)m(ultiply)-8 b(,)35 b(what)f(w)m(e)g(are)g(lo)s(oking)g(for)g(is)f (the)h(result)g(of)0 1825 y(adding)28 b(41)g(to)h(itself)g(42)g(times)f (\(or)g(vice)h(v)m(ersa\).)41 b(Y)-8 b(ou)29 b(could)f(imagine)h (actually)g(computing)f(the)g(answ)m(er)g(that)0 1938 y(w)m(a)m(y)39 b(\(i.e.,)j(p)s(erforming)37 b(41)i(additions\),)h(whic) m(h)e(w)m(ould)g(b)s(e)g(correct)h(but)e(not)i(particularly)f (e\016cien)m(t.)66 b(If)37 b(w)m(e)0 2051 y(used)c(this)g(approac)m(h)h (to)g(m)m(ultiply)f(t)m(w)m(o)i Ft(n)p Fu(-bit)f(n)m(um)m(b)s(ers,)f(w) m(e)g(w)m(ould)h(b)s(e)e(making)i(\002\(2)3073 2018 y Fm(n)3121 2051 y Fu(\))f(additions.)50 b(This)33 b(is)0 2164 y(exp)s(onen)m(tial)27 b(in)e Ft(n)h Fu(ev)m(en)g(without)g(coun)m (ting)h(the)f(n)m(um)m(b)s(er)e(of)i(steps)g(needed)f(to)i(p)s(erform)d (eac)m(h)j(addition.)40 b(And,)0 2277 y(in)28 b(general,)i(exp)s(onen)m (tial)g(is)e(bad.)1188 2244 y Fn(1)1267 2277 y Fu(A)h(b)s(etter)f(w)m (a)m(y)h(to)h(m)m(ultiply)e(is)h(to)g(do)f(what)h(w)m(e)f(learned)h(in) f(grade)h(sc)m(ho)s(ol:)811 2513 y Fl(101001)477 2626 y(x)286 b(101010)477 2739 y(-------------)764 2852 y(1010010)668 2965 y(101001)477 3078 y(+)48 b(101001)477 3190 y(-------------)573 3303 y(11010111010)c(=)k(1722)0 3540 y Fu(The)30 b(running)e(time)j (here)f(is)g Ft(O)s Fu(\()p Ft(n)1182 3507 y Fn(2)1221 3540 y Fu(\),)h(b)s(ecause)f(w)m(e)h(are)g(p)s(erforming)d Ft(n)i Fu(additions,)h(eac)m(h)g(of)f(whic)m(h)g(tak)m(es)i Ft(O)s Fu(\()p Ft(n)p Fu(\))0 3652 y(time.)63 b(So,)40 b(this)e(is)g(a)g(simple)f(example)i(where)e(ev)m(en)h(though)g(the)g (problem)f(is)h(de\014ned)e(\\algorithmically",)0 3765 y(using)30 b(the)g(de\014nition)g(is)h(not)g(the)f(b)s(est)g(w)m(a)m(y) h(of)g(solving)g(the)g(problem.)0 3917 y(Is)24 b(the)g(ab)s(o)m(v)m(e)h (metho)s(d)f(the)g(fastest)h(w)m(a)m(y)g(to)g(m)m(ultiply)f(t)m(w)m(o)i (n)m(um)m(b)s(ers?)36 b(It)25 b(turns)d(out)j(it)f(is)h(not.)38 b(Here)25 b(is)f(a)h(faster)0 4030 y(metho)s(d)32 b(called)h(Karatsuba) g(Multiplication,)i(disco)m(v)m(ered)e(b)m(y)g(Anatoli)g(Karatsuba,)g (in)f(Russia,)h(in)f(1962.)49 b(In)0 4143 y(this)33 b(approac)m(h,)h(w) m(e)g(tak)m(e)h(the)e(t)m(w)m(o)h(n)m(um)m(b)s(ers)e Ft(X)40 b Fu(and)33 b Ft(Y)53 b Fu(and)32 b(split)i(them)f(eac)m(h)h (in)m(to)g(their)f(most-signi\014can)m(t)0 4256 y(half)d(and)g(their)g (least-signi\014can)m(t)j(half.)1909 4472 y Fl(---------------)764 4585 y(X)47 b(=)h(A*2^{n/2})d(+)i(B)334 b(|)95 b(A)143 b(|)g(B)95 b(|)1909 4698 y(+-------------+)764 4811 y(Y)47 b(=)h(C*2^{n/2})d(+)i(D)334 b(|)95 b(C)143 b(|)g(D)95 b(|)1909 4923 y(---------------)p 0 4981 1560 4 v 104 5034 a Fk(1)138 5066 y Fj(This)31 b(is)f(reminiscen)n(t)g(of)h(an)f (exp)r(onen)n(tial-time)g(sorting)g(algorithm)i(I)d(once)h(sa)n(w)h(in) f(Prolog.)49 b(The)30 b(co)r(de)g(just)g(con)n(tains)g(the)0 5157 y(de\014nition)24 b(of)i(what)f(it)g(means)g(to)g(sort)g(the)f (input)g(|)g(namely)-6 b(,)25 b(to)g(pro)r(duce)f(a)h(p)r(erm)n (utation)g(of)g(the)g(input)f(in)g(whic)n(h)h(all)h(elemen)n(ts)0 5249 y(are)34 b(in)g(ascending)g(order.)58 b(When)33 b(handed)g(directly)g(to)h(the)f(in)n(terpreter,)j(it)d(results)i(in)e (an)h(algorithm)h(that)e(examines)h(all)g Fi(n)p Fj(!)0 5340 y(p)r(erm)n(utations)26 b(of)g(the)g(giv)n(en)f(input)g(list)h(un) n(til)g(it)g(\014nds)e(one)i(that)g(is)g(in)f(the)h(righ)n(t)f(order.)p eop end %%Page: 6 7 TeXDict begin 6 6 bop 0 -165 a Fu(6)1662 b Fs(LECTURE)30 b(1.)71 b(INTR)m(ODUCTION)30 b(TO)g(ALGORITHMS)0 91 y Fu(W)-8 b(e)32 b(can)e(no)m(w)h(write)f(the)h(pro)s(duct)e(of)h Ft(X)38 b Fu(and)30 b Ft(Y)50 b Fu(as)1074 307 y Ft(X)7 b(Y)104 b Fu(=)83 b(2)1512 269 y Fm(n)1559 307 y Ft(AC)27 b Fu(+)20 b(2)1855 269 y Fm(n=)p Fn(2)1973 307 y Ft(B)5 b(C)26 b Fu(+)20 b(2)2274 269 y Fm(n=)p Fn(2)2392 307 y Ft(AD)j Fu(+)d Ft(B)5 b(D)s(:)887 b Fu(\(1.1\))0 522 y(This)26 b(do)s(es)g(not)h(y)m(et)h(seem)f(so)g(useful:)38 b(if)26 b(w)m(e)i(use)e(\(1.1\))i(as)f(a)g(recursiv)m(e)g(m)m (ultiplication)i(algorithm,)g(w)m(e)e(need)f(to)0 635 y(p)s(erform)k(four)g Ft(n=)p Fu(2-bit)i(m)m(ultiplications,)i(three)d (shifts,)h(and)e(three)i Ft(O)s Fu(\()p Ft(n)p Fu(\)-bit)f(additions.) 44 b(If)30 b(w)m(e)i(use)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))31 b(to)0 748 y(denote)g(the)g(running)e(time)i(to)g(m)m(ultiply)g(t)m(w)m (o)h Ft(n)p Fu(-bit)f(n)m(um)m(b)s(ers)e(b)m(y)h(this)h(metho)s(d,)f (this)g(giv)m(es)i(us)e(a)h(recurrence)0 861 y(of)1457 1076 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b(=)g(4)p Ft(T)13 b Fu(\()p Ft(n=)p Fu(2\))22 b(+)e Ft(cn;)1270 b Fu(\(1.2\))0 1292 y(for)38 b(some)g(constan)m(t)i Ft(c)p Fu(.)64 b(\(The)37 b Ft(cn)h Fu(term)g(re\015ects)h(the)f(time)h(to)g(p)s(erform)d(the)j (additions)f(and)f(shifts.\))64 b(This)0 1405 y(recurrence)29 b(solv)m(es)h(to)g Ft(O)s Fu(\()p Ft(n)968 1372 y Fn(2)1007 1405 y Fu(\),)g(so)f(w)m(e)g(do)g(not)g(seem)h(to)f(ha)m(v)m(e)i(made)e (an)m(y)g(progress.)40 b(\(In)28 b(the)i(next)58 b(lecture)29 b(w)m(e)0 1518 y(will)i(go)g(in)m(to)g(the)g(details)g(of)g(ho)m(w)f (to)h(solv)m(e)h(recurrences)e(lik)m(e)i(this.\))0 1670 y(Ho)m(w)m(ev)m(er,)h(w)m(e)e(can)f(tak)m(e)i(the)f(form)m(ula)f(in)g (\(1.1\))j(and)c(rewrite)i(it)g(as)f(follo)m(ws:)858 1885 y(\(2)938 1848 y Fm(n)1006 1885 y Fp(\000)20 b Fu(2)1142 1848 y Fm(n=)p Fn(2)1260 1885 y Fu(\))p Ft(AC)27 b Fu(+)20 b(2)1591 1848 y Fm(n=)p Fn(2)1709 1885 y Fu(\()p Ft(A)h Fu(+)f Ft(B)5 b Fu(\)\()p Ft(C)27 b Fu(+)20 b Ft(D)s Fu(\))g(+)g(\(1)h Fp(\000)f Fu(2)2712 1848 y Fm(n=)p Fn(2)2830 1885 y Fu(\))p Ft(B)5 b(D)s(:)671 b Fu(\(1.3\))0 2101 y(It)32 b(is)h(not)f(hard)g(to)h(see)g(|)f(y)m(ou)h(just)e(need)i (to)g(m)m(ultiply)f(it)h(out)g(|)f(that)h(the)f(form)m(ula)h(in)f (\(1.3\))i(is)e(equiv)-5 b(alen)m(t)0 2213 y(to)35 b(the)f(expression)g (in)g(\(1.1\).)54 b(The)34 b(new)g(form)m(ula)g(lo)s(oks)h(more)f (complicated,)j(but,)e(it)f(results)g(in)g(only)h Fo(thr)-5 b(e)g(e)0 2326 y Fu(m)m(ultiplications)46 b(of)f(size)h Ft(n=)p Fu(2,)i(plus)c(a)h(constan)m(t)h(n)m(um)m(b)s(er)d(of)i(shifts) f(and)g(additions.)84 b(So,)48 b(the)d(resulting)0 2439 y(recurrence)30 b(is)1446 2655 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b(=)f(3)p Ft(T)13 b Fu(\()p Ft(n=)p Fu(2\))22 b(+)e Ft(c)2351 2617 y Fh(0)2375 2655 y Ft(n;)1258 b Fu(\(1.4\))0 2870 y(for)30 b(some)h(constan)m(t)h Ft(c)773 2837 y Fh(0)796 2870 y Fu(.)41 b(This)29 b(recurrence)i(solv)m(es)g(to) g Ft(O)s Fu(\()p Ft(n)2041 2837 y Fn(log)2132 2854 y Fk(2)2179 2837 y Fn(3)2218 2870 y Fu(\))26 b Fp(\031)f Ft(O)s Fu(\()p Ft(n)2537 2837 y Fn(1)p Fm(:)p Fn(585)2701 2870 y Fu(\).)0 3022 y(Is)g Fo(this)h Fu(metho)s(d)f(the)g(fastest)i(p) s(ossible?)38 b(Again)26 b(it)g(turns)e(out)i(that)g(one)g(can)f(do)h (b)s(etter.)39 b(In)25 b(fact,)i(Karp)e(disco)m(v-)0 3135 y(ered)33 b(a)h(w)m(a)m(y)g(to)g(use)f(the)h(F)-8 b(ast)34 b(F)-8 b(ourier)34 b(T)-8 b(ransform)33 b(to)h(m)m(ultiply)f (t)m(w)m(o)i Ft(n)p Fu(-bit)e(n)m(um)m(b)s(ers)f(in)h(time)h Ft(O)s Fu(\()p Ft(n)15 b Fu(log)3730 3098 y Fn(2)3785 3135 y Ft(n)p Fu(\).)0 3248 y(Sc)m(h\177)-45 b(onhage)30 b(and)f(Strassen)g(in)g(1971)j(impro)m(v)m(ed)e(this)f(to)i Ft(O)s Fu(\()p Ft(n)15 b Fu(log)i Ft(n)e Fu(log)i(log)h Ft(n)p Fu(\),)30 b(whic)m(h)f(is,)h(asymptotically)-8 b(,)33 b(the)0 3361 y(fastest)e(algorithm)h(kno)m(wn.)40 b(W)-8 b(e)32 b(will)e(discuss)g(the)g(FFT)h(later)g(on)g(in)f(this)g (course.)0 3513 y(Actually)-8 b(,)31 b(the)e(kind)e(of)i(analysis)g(w)m (e)g(ha)m(v)m(e)g(b)s(een)f(doing)h(really)g(is)g(meaningful)f(only)g (for)h(v)m(ery)g(large)g(n)m(um)m(b)s(ers.)0 3626 y(On)34 b(a)i(computer,)h(if)e(y)m(ou)h(are)g(m)m(ultiplying)g(n)m(um)m(b)s (ers)d(that)j(\014t)f(in)m(to)i(the)e(w)m(ord)g(size,)j(y)m(ou)e(w)m (ould)f(do)g(this)g(in)0 3739 y(hardw)m(are)e(that)i(has)e(gates)i(w)m (orking)f(in)g(parallel.)52 b(So)34 b(instead)g(of)f(lo)s(oking)i(at)g (sequen)m(tial)g(running)d(time,)j(in)0 3852 y(this)41 b(case)g(w)m(e)g(w)m(ould)g(w)m(an)m(t)g(to)g(examine)h(the)f(size)g (and)f(depth)g(of)h(the)f(circuit)i(used,)g(for)f(instance.)72 b(This)0 3965 y(p)s(oin)m(ts)26 b(out)h(that,)h(in)e(fact,)j(there)e (are)f(di\013eren)m(t)h(kinds)f(of)h(sp)s(eci\014cations)g(that)g(can)g (b)s(e)f(imp)s(ortan)m(t)g(in)g(di\013eren)m(t)0 4078 y(settings.)0 4421 y Fq(1.5)137 b(Matrix)45 b(m)l(ultiplication)0 4666 y Fu(It)31 b(turns)f(out)h(the)g(same)g(basic)g (divide-and-conquer)g(approac)m(h)g(of)g(Karatsuba's)g(algorithm)h(can) f(b)s(e)f(used)g(to)0 4779 y(sp)s(eed)22 b(up)g(matrix)h(m)m (ultiplication)i(as)e(w)m(ell.)39 b(T)-8 b(o)23 b(m)m(ultiply)g(t)m(w)m (o)h Ft(n)p Fu(-b)m(y-)p Ft(n)f Fu(matrices)g(in)g(the)g(usual)f(w)m(a) m(y)i(tak)m(es)g(time)0 4892 y Ft(O)s Fu(\()p Ft(n)162 4859 y Fn(3)201 4892 y Fu(\).)51 b(If)33 b(one)h(breaks)f(do)m(wn)g (eac)m(h)h Ft(n)f Fu(b)m(y)h Ft(n)f Fu(matrix)h(in)m(to)g(four)f Ft(n=)p Fu(2)h(b)m(y)f Ft(n=)p Fu(2)h(matrices,)i(then)d(the)h (standard)0 5005 y(metho)s(d)j(can)h(b)s(e)f(though)m(t)h(of)g(as)f(p)s (erforming)g(eigh)m(t)i Ft(n=)p Fu(2-b)m(y-)p Ft(n=)p Fu(2)g(m)m(ultiplications)g(and)e(four)g(additions)h(as)0 5118 y(follo)m(ws:)p 973 5141 316 4 v 971 5254 4 113 v 1016 5220 a Ft(A)p 1126 5254 V 87 w(B)p 1286 5254 V 973 5257 316 4 v 971 5370 4 113 v 1014 5336 a(C)p 1126 5370 V 90 w(D)p 1286 5370 V 973 5373 316 4 v 1333 5278 a Fp(\002)p 1450 5141 322 4 v 1448 5254 4 113 v 1491 5220 a Ft(E)p 1603 5254 V 94 w(F)p 1769 5254 V 1450 5257 322 4 v 1448 5370 4 113 v 1492 5336 a(G)p 1603 5370 V 83 w(H)p 1769 5370 V 1450 5373 322 4 v 1821 5278 a Fu(=)p 1943 5141 985 4 v 1941 5254 4 113 v 1988 5220 a Ft(AE)26 b Fu(+)20 b Ft(B)5 b(G)p 2428 5254 V 90 w(AF)34 b Fu(+)20 b Ft(B)5 b(H)p 2926 5254 V 1943 5257 985 4 v 1941 5370 4 113 v 1984 5336 a(C)i(E)25 b Fu(+)20 b Ft(D)s(G)p 2428 5370 V 83 w(C)7 b(F)33 b Fu(+)20 b Ft(D)s(H)p 2926 5370 V 1943 5373 985 4 v eop end %%Page: 7 8 TeXDict begin 7 7 bop 0 -165 a Fs(1.5.)72 b(MA)-8 b(TRIX)31 b(MUL)-8 b(TIPLICA)g(TION)2420 b Fu(7)0 91 y(Strassen)72 b(noticed)37 b(that,)i(as)d(in)g(Karatsuba's)h(algorithm,)i(one)d(can)h (clev)m(erly)h(rearrange)f(the)f(computation)0 204 y(to)41 b(in)m(v)m(olv)m(e)h(only)d Fo(seven)h Ft(n=)p Fu(2-b)m(y-)p Ft(n=)p Fu(2)h(m)m(ultiplications)h(\(and)d(14)i(additions\).)2790 171 y Fn(2)2899 204 y Fu(Since)f(adding)f(t)m(w)m(o)i Ft(n)p Fu(-b)m(y-)p Ft(n)0 317 y Fu(matrices)31 b(tak)m(es)h(time)f Ft(O)s Fu(\()p Ft(n)966 284 y Fn(2)1006 317 y Fu(\),)f(this)h(results)f (in)g(a)h(recurrence)f(of)1438 521 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b(=)f(7)p Ft(T)13 b Fu(\()p Ft(n=)p Fu(2\))22 b(+)e Ft(cn)2398 484 y Fn(2)2437 521 y Ft(:)1251 b Fu(\(1.5\))0 726 y(This)30 b(recurrence)g(solv)m(es)h(to)g(a)g(running)e(time)i(of)f (just)g Ft(O)s Fu(\()p Ft(n)2084 693 y Fn(log)2175 710 y Fk(2)2222 693 y Fn(7)2261 726 y Fu(\))c Fp(\031)f Ft(O)s Fu(\()p Ft(n)2580 693 y Fn(2)p Fm(:)p Fn(81)2709 726 y Fu(\))31 b(for)f(Strassen's)g(algorithm.)3743 693 y Fn(3)0 878 y Fu(Matrix)37 b(m)m(ultiplication)i(is)d(esp)s(ecially)i (imp)s(ortan)m(t)e(in)g(scien)m(ti\014c)i(computation.)60 b(Strassen's)36 b(algorithm)h(has)0 991 y(more)31 b(o)m(v)m(erhead)h (than)e(standard)g(metho)s(d,)h(but)f(it)h(is)g(the)g(preferred)e (metho)s(d)h(on)h(man)m(y)g(mo)s(dern)e(computers)0 1103 y(for)g(ev)m(en)i(mo)s(destly)e(large)i(matrices.)41 b(Asymptotically)-8 b(,)32 b(the)e(b)s(est)f(matrix)h(m)m(ultiply)g (algorithm)g(kno)m(wn)f(is)h(b)m(y)0 1216 y(Copp)s(ersmith)35 b(and)i(Winograd)g(and)f(has)h(time)h Ft(O)s Fu(\()p Ft(n)1890 1183 y Fn(2)p Fm(:)p Fn(376)2054 1216 y Fu(\),)h(but)e(is)g (not)g(practical.)62 b(Nob)s(o)s(dy)36 b(kno)m(ws)h(if)f(it)i(is)0 1329 y(p)s(ossible)30 b(to)h(do)f(b)s(etter)h(|)f(the)h(FFT)f(approac)m (h)h(do)s(esn't)f(seem)h(to)g(carry)f(o)m(v)m(er.)p 0 4615 1560 4 v 104 4669 a Fk(2)138 4701 y Fj(In)i(particular,)j(the)c (quan)n(tities)h(that)g(one)h(computes)f(recursiv)n(ely)g(are)h Fi(q)2351 4709 y Fk(1)2418 4701 y Fj(=)f(\()p Fi(A)21 b Fj(+)g Fi(D)r Fj(\)\()p Fi(E)k Fj(+)c Fi(H)6 b Fj(\),)34 b Fi(q)3177 4709 y Fk(2)3244 4701 y Fj(=)e Fi(D)r Fj(\()p Fi(G)21 b Fg(\000)h Fi(E)t Fj(\),)33 b Fi(q)3773 4709 y Fk(3)3840 4701 y Fj(=)0 4792 y(\()p Fi(B)25 b Fg(\000)c Fi(D)r Fj(\)\()p Fi(G)g Fj(+)g Fi(H)6 b Fj(\),)33 b Fi(q)669 4800 y Fk(4)736 4792 y Fj(=)e(\()p Fi(A)21 b Fj(+)g Fi(B)t Fj(\))p Fi(H)6 b Fj(,)33 b Fi(q)1267 4800 y Fk(5)1334 4792 y Fj(=)e(\()p Fi(C)26 b Fj(+)21 b Fi(D)r Fj(\))p Fi(E)t Fj(,)34 b Fi(q)1862 4800 y Fk(6)1929 4792 y Fj(=)d Fi(A)p Fj(\()p Fi(F)g Fg(\000)21 b Fi(H)6 b Fj(\),)33 b(and)f Fi(q)2613 4800 y Fk(7)2680 4792 y Fj(=)f(\()p Fi(C)26 b Fg(\000)21 b Fi(A)p Fj(\)\()p Fi(E)j Fj(+)d Fi(F)11 b Fj(\).)53 b(The)32 b(upp)r(er-left)0 4883 y(quadran)n(t)26 b(of)h(the)f(solution)h(is)g Fi(q)954 4891 y Fk(1)1007 4883 y Fj(+)17 b Fi(q)1118 4891 y Fk(2)1171 4883 y Fj(+)g Fi(q)1282 4891 y Fk(3)1335 4883 y Fg(\000)g Fi(q)1446 4891 y Fk(4)1481 4883 y Fj(,)27 b(the)f(upp)r(er-righ)n(t)f(is)i Fi(q)2180 4891 y Fk(4)2233 4883 y Fj(+)17 b Fi(q)2344 4891 y Fk(6)2379 4883 y Fj(,)27 b(the)f(lo)n(w)n(er-left)i(is)f Fi(q)3008 4891 y Fk(2)3061 4883 y Fj(+)17 b Fi(q)3172 4891 y Fk(5)3207 4883 y Fj(,)27 b(and)f(the)g(lo)n(w)n(er)i(righ)n(t)0 4975 y(is)e Fi(q)111 4983 y Fk(1)163 4975 y Fg(\000)17 b Fi(q)274 4983 y Fk(5)326 4975 y Fj(+)g Fi(q)437 4983 y Fk(6)489 4975 y Fj(+)g Fi(q)600 4983 y Fk(7)634 4975 y Fj(.)35 b(\(feel)26 b(free)h(to)f(c)n(hec)n(k!\))104 5034 y Fk(3)138 5066 y Fj(According)31 b(to)g(Man)n(uel)h(Blum,)h (Strassen)e(said)g(that)g(when)g(coming)g(up)g(with)g(his)g(algorithm,) j(he)c(\014rst)h(tried)g(to)g(solv)n(e)g(the)0 5157 y(problem)c(mo)r(d) g(2.)39 b(Solving)27 b(mo)r(d)g(2)g(mak)n(es)g(the)f(problem)h(easier)i (b)r(ecause)e(y)n(ou)f(only)h(need)g(to)g(k)n(eep)f(trac)n(k)g(of)i (the)f(parit)n(y)f(of)i(eac)n(h)0 5249 y(en)n(try)-6 b(,)30 b(and)g(in)f(particular,)k(addition)d(is)g(the)g(same)h(as)f (subtraction.)48 b(One)29 b(he)h(\014gured)f(out)h(the)g(solution)g(mo) r(d)g(2,)i(he)e(w)n(as)h(then)0 5340 y(able)26 b(to)g(mak)n(e)g(it)f(w) n(ork)h(in)g(general.)p eop end %%Page: 8 9 TeXDict begin 8 8 bop 0 -165 a Fu(8)1662 b Fs(LECTURE)30 b(1.)71 b(INTR)m(ODUCTION)30 b(TO)g(ALGORITHMS)p eop end %%Page: 9 10 TeXDict begin 9 9 bop 0 753 a Fr(Lecture)65 b(2)0 1207 y Fw(Asymptotic)79 b(Analysis)g(and)0 1456 y(Recurrences)0 1977 y Fq(2.1)137 b(Ov)l(erview)0 2222 y Fu(In)27 b(this)g(lecture)i(w) m(e)f(discuss)f(the)g(notion)h(of)g(asymptotic)h(analysis)f(and)f(in)m (tro)s(duce)h Ft(O)s Fu(,)g(\012,)g(\002,)g(and)f Ft(o)g Fu(notation.)0 2335 y(W)-8 b(e)28 b(then)f(turn)e(to)j(the)f(topic)h (of)e(recurrences,)i(discussing)e(sev)m(eral)i(metho)s(ds)e(for)h (solving)g(them.)40 b(Recurrences)0 2448 y(will)34 b(come)h(up)e(in)g (man)m(y)h(of)h(the)f(algorithms)g(w)m(e)h(study)-8 b(,)34 b(so)g(it)h(is)f(useful)f(to)i(get)g(a)f(go)s(o)s(d)g(in)m(tuition)g (for)g(them)0 2561 y(righ)m(t)h(at)g(the)g(start.)53 b(In)34 b(particular,)i(w)m(e)f(fo)s(cus)f(on)g(divide-and-conquer)h (st)m(yle)g(recurrences,)h(whic)m(h)e(are)h(the)0 2674 y(most)c(common)f(ones)h(w)m(e)g(will)f(see.)0 2826 y(Material)i(in)e (this)h(lecture:)136 3063 y Fp(\017)46 b Fu(Asymptotic)32 b(notation:)42 b Ft(O)s Fu(,)30 b(\012,)g(\002,)h(and)e Ft(o)p Fu(.)136 3264 y Fp(\017)46 b Fu(Recurrences)31 b(and)e(ho)m(w)i(to)g(solv)m(e)h(them.)330 3465 y Fv({)45 b Fu(Solving)31 b(b)m(y)f(unrolling.)330 3618 y Fv({)45 b Fu(Solving)31 b(with)f(a)h(guess)f(and)g(inductiv)m(e)h(pro)s(of.)330 3771 y Fv({)45 b Fu(Solving)31 b(using)f(a)h(recursion)f(tree.)330 3924 y Fv({)45 b Fu(A)31 b(master)f(form)m(ula.)0 4265 y Fq(2.2)137 b(Asymptotic)47 b(analysis)0 4510 y Fu(When)24 b(w)m(e)h(consider)g(an)f(algorithm)h(for)g(some)g(problem,)g(in)f (addition)h(to)g(kno)m(wing)g(that)g(it)g(pro)s(duces)e(a)i(correct)0 4623 y(solution,)35 b(w)m(e)g(will)f(b)s(e)f(esp)s(ecially)i(in)m (terested)f(in)g(analyzing)h(its)f(running)e(time.)51 b(There)33 b(are)i(sev)m(eral)g(asp)s(ects)0 4736 y(of)e(running)e (time)j(that)f(one)g(could)g(fo)s(cus)f(on.)48 b(Our)32 b(fo)s(cus)g(will)h(b)s(e)f(primarily)g(on)h(the)g(question:)46 b(\\ho)m(w)33 b(do)s(es)0 4849 y(the)28 b(running)f(time)i Fo(sc)-5 b(ale)29 b Fu(with)f(the)g(size)i(of)e(the)h(input?")39 b(This)27 b(is)i(called)g Fo(asymptotic)k(analysis)p Fu(,)d(and)e(the)g(idea)0 4962 y(is)35 b(that)h(w)m(e)f(will)h(ignore)g (lo)m(w-order)f(terms)g(and)g(constan)m(t)h(factors,)i(fo)s(cusing)c (instead)i(on)f(the)g(shap)s(e)f(of)i(the)0 5075 y(running)26 b(time)i(curv)m(e.)40 b(W)-8 b(e)28 b(will)g(t)m(ypically)h(use)e Ft(n)g Fu(to)h(denote)g(the)g(size)g(of)g(the)f(input,)h(and)e Ft(T)13 b Fu(\()p Ft(n)p Fu(\))28 b(to)g(denote)g(the)0 5188 y(running)h(time)i(of)f(our)g(algorithm)i(on)e(an)g(input)f(of)i (size)g Ft(n)p Fu(.)0 5340 y(W)-8 b(e)32 b(b)s(egin)e(b)m(y)g(presen)m (ting)g(some)h(con)m(v)m(enien)m(t)i(de\014nitions)d(for)g(p)s (erforming)f(this)h(kind)g(of)g(analysis.)1927 5568 y(9)p eop end %%Page: 10 11 TeXDict begin 10 10 bop 0 -165 a Fu(10)1102 b Fs(LECTURE)29 b(2.)71 b(ASYMPTOTIC)29 b(ANAL)-8 b(YSIS)30 b(AND)h(RECURRENCES)0 91 y Fv(De\014nition)k(2.1)46 b Ft(T)13 b Fu(\()p Ft(n)p Fu(\))32 b Fp(2)g Ft(O)s Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))36 b Fo(if)g(ther)-5 b(e)38 b(exist)e(c)-5 b(onstants)38 b Ft(c;)15 b(n)2416 105 y Fn(0)2488 91 y Ft(>)32 b Fu(0)37 b Fo(such)f(that)i Ft(T)13 b Fu(\()p Ft(n)p Fu(\))32 b Fp(\024)f Ft(cf)10 b Fu(\()p Ft(n)p Fu(\))36 b Fo(for)h(al)5 b(l)0 204 y Ft(n)25 b(>)g(n)231 218 y Fn(0)270 204 y Fo(.)0 429 y Fu(Informally)41 b(w)m(e)g(can)g(view)g(this)g(as)g(\\)p Ft(T)13 b Fu(\()p Ft(n)p Fu(\))41 b(is)g(prop)s(ortional)g(to)g Ft(f)10 b Fu(\()p Ft(n)p Fu(\),)44 b(or)c(b)s(etter,)k(as)d Ft(n)g Fu(gets)g(large.")74 b(F)-8 b(or)0 541 y(example,)33 b(3)p Ft(n)483 509 y Fn(2)544 541 y Fu(+)21 b(17)29 b Fp(2)e Ft(O)s Fu(\()p Ft(n)1005 509 y Fn(2)1044 541 y Fu(\))32 b(and)g(3)p Ft(n)1390 509 y Fn(2)1450 541 y Fu(+)21 b(17)29 b Fp(2)f Ft(O)s Fu(\()p Ft(n)1912 509 y Fn(3)1951 541 y Fu(\).)45 b(This)32 b(notation)h(is)f(esp)s(ecially)h (useful)e(in)h(discussing)0 654 y(upp)s(er)k(b)s(ounds)f(on)i (algorithms:)57 b(for)37 b(instance,)j(w)m(e)e(sa)m(w)g(last)h(time)f (that)g(Karatsuba)g(m)m(ultiplication)h(to)s(ok)0 767 y(time)31 b Ft(O)s Fu(\()p Ft(n)369 734 y Fn(log)460 751 y Fk(2)506 734 y Fn(3)546 767 y Fu(\).)0 919 y(Notice)42 b(that)e Ft(O)s Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))41 b(is)f(a)g(set)g(of)g(functions.)69 b(Nonetheless,)44 b(it)d(is)f(common)g(practice)h(to)g(write)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))41 b(=)0 1032 y Ft(O)s Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))31 b(to)h(mean)f(that)g Ft(T)13 b Fu(\()p Ft(n)p Fu(\))26 b Fp(2)g Ft(O)s Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\):)42 b(esp)s(ecially)32 b(in)f(con)m(v)m(ersation,)i(it)e(is)g(more)g(natural)g(to)h(sa)m(y)g (\\)p Ft(T)13 b Fu(\()p Ft(n)p Fu(\))0 1145 y(is)25 b Ft(O)s Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\)")25 b(than)f(to)h(sa)m (y)h(\\)p Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b(is)f(in)h Ft(O)s Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\)".)39 b(W)-8 b(e)26 b(will)f(t)m(ypically)h(use)e(this)h(common)g(practice,)i (rev)m(erting)0 1258 y(to)k(the)g(correct)g(set)g(notation)h(when)d (this)h(practice)i(w)m(ould)e(cause)h(confusion.)0 1506 y Fv(De\014nition)k(2.2)46 b Ft(T)13 b Fu(\()p Ft(n)p Fu(\))33 b Fp(2)f Fu(\012\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))36 b Fo(if)h(ther)-5 b(e)37 b(exist)g(c)-5 b(onstants)39 b Ft(c;)15 b(n)2413 1520 y Fn(0)2485 1506 y Ft(>)32 b Fu(0)37 b Fo(such)g(that)h Ft(T)13 b Fu(\()p Ft(n)p Fu(\))32 b Fp(\025)h Ft(cf)10 b Fu(\()p Ft(n)p Fu(\))36 b Fo(for)h(al)5 b(l)0 1619 y Ft(n)25 b(>)g(n)231 1633 y Fn(0)270 1619 y Fo(.)0 1868 y Fu(Informally)42 b(w)m(e)g(can)g(view)g(this)g(as)g(\\) p Ft(T)13 b Fu(\()p Ft(n)p Fu(\))42 b(is)g(prop)s(ortional)g(to)g Ft(f)10 b Fu(\()p Ft(n)p Fu(\),)45 b(or)d(w)m(orse,)j(as)d Ft(n)g Fu(gets)g(large.")77 b(F)-8 b(or)0 1980 y(example,)33 b(3)p Ft(n)483 1947 y Fn(2)543 1980 y Fp(\000)21 b Fu(2)p Ft(n)26 b Fp(2)h Fu(\012\()p Ft(n)1005 1947 y Fn(2)1044 1980 y Fu(\).)44 b(This)31 b(notation)h(is)g(esp)s(ecially)g(useful)f (for)g(lo)m(w)m(er)i(b)s(ounds.)41 b(In)31 b(Chapter)g(5,)h(for)0 2093 y(instance,)h(w)m(e)g(will)f(pro)m(v)m(e)h(that)f(an)m(y)h (comparison-based)f(sorting)g(algorithm)h(m)m(ust)f(tak)m(e)h(time)g (\012\()p Ft(n)15 b Fu(log)i Ft(n)p Fu(\))32 b(in)0 2206 y(the)f(w)m(orst)f(case)i(\(or)e(ev)m(en)h(on)g(a)m(v)m(erage\).)0 2455 y Fv(De\014nition)k(2.3)46 b Ft(T)13 b Fu(\()p Ft(n)p Fu(\))26 b Fp(2)e Fu(\002\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))33 b Fo(if)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b Fp(2)g Ft(O)s Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))33 b Fo(and)h Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b Fp(2)g Fu(\012\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))p Fo(.)0 2703 y Fu(Informally)30 b(w)m(e)h(can)g(view)f(this)h(as)f(\\)p Ft(T)13 b Fu(\()p Ft(n)p Fu(\))31 b(is)f(prop)s(ortional)h(to)g Ft(f)10 b Fu(\()p Ft(n)p Fu(\))30 b(as)g Ft(n)g Fu(gets)h(large.")0 2951 y Fv(De\014nition)k(2.4)46 b Ft(T)13 b Fu(\()p Ft(n)p Fu(\))35 b Fp(2)f Ft(o)p Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))39 b Fo(if)e(for)i(al)5 b(l)38 b(c)-5 b(onstants)40 b Ft(c)35 b(>)f Fu(0)p Fo(,)40 b(ther)-5 b(e)39 b(exists)f Ft(n)2938 2965 y Fn(0)3012 2951 y Ft(>)c Fu(0)k Fo(such)g(that)h Ft(T)13 b Fu(\()p Ft(n)p Fu(\))35 b Ft(<)0 3064 y(cf)10 b Fu(\()p Ft(n)p Fu(\))32 b Fo(for)i(al)5 b(l)33 b Ft(n)25 b(>)g(n)758 3078 y Fn(0)797 3064 y Fo(.)0 3312 y Fu(F)-8 b(or)33 b(example,)g(last)g(time)g(w)m(e)g(sa)m(w)f(that)h(w)m(e)f (could)h(indeed)e(m)m(ultiply)i(t)m(w)m(o)g Ft(n)p Fu(-bit)f(n)m(um)m (b)s(ers)f(in)g(time)i Ft(o)p Fu(\()p Ft(n)3697 3279 y Fn(2)3737 3312 y Fu(\))f(b)m(y)0 3425 y(the)k(Karatsuba)f(algorithm.) 57 b(V)-8 b(ery)36 b(informally)-8 b(,)38 b Ft(O)g Fu(is)e(lik)m(e)h Fp(\024)p Fu(,)g(\012)e(is)g(lik)m(e)i Fp(\025)p Fu(,)g(\002)e(is)h (lik)m(e)g(=,)h(and)e Ft(o)h Fu(is)f(lik)m(e)i Ft(<)p Fu(.)0 3538 y(There)30 b(is)g(also)i(a)e(similar)h(notation)h Ft(!)h Fu(that)e(corresp)s(onds)e(to)i Ft(>)p Fu(.)0 3690 y(In)36 b(terms)g(of)g(computing)h(whether)e(or)i(not)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))37 b(b)s(elongs)f(to)h(one)g(of)f(these) h(sets)g(with)f(resp)s(ect)g(to)h Ft(f)10 b Fu(\()p Ft(n)p Fu(\),)38 b(a)0 3803 y(con)m(v)m(enien)m(t)32 b(w)m(a)m(y)g(is)e(to)h (compute)g(the)g(limit:)1769 4047 y(lim)1740 4097 y Fm(n)p Fh(!1)1949 3986 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))p 1949 4026 192 4 v 1955 4110 a Ft(f)d Fu(\()p Ft(n)p Fu(\))2150 4047 y Ft(:)1538 b Fu(\(2.1\))0 4292 y(If)30 b(the)g(limit)i(exists,)f (then)f(w)m(e)h(can)f(mak)m(e)i(the)e(follo)m(wing)i(statemen)m(ts:)136 4517 y Fp(\017)46 b Fu(If)30 b(the)h(limit)g(is)f(0,)h(then)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))26 b(=)f Ft(o)p Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))30 b(and)g Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b(=)g Ft(O)s Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\).)136 4703 y Fp(\017)46 b Fu(If)31 b(the)g(limit)h(is)f(a)g(n)m(um)m(b)s(er)f(greater)i(than)f(0)g (\(e.g.,)i(17\))f(then)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))26 b(=)g(\002\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))31 b(\(and)g Ft(T)13 b Fu(\()p Ft(n)p Fu(\))26 b(=)g Ft(O)s Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))227 4816 y(and)30 b Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b(=)g(\012\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\)\))136 5003 y Fp(\017)46 b Fu(If)30 b(the)h(limit)g(is)f(in\014nit)m(y)-8 b(,)31 b(then)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b(=)g Ft(!)s Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))31 b(and)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b(=)g(\012\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\).)0 5227 y(F)-8 b(or)45 b(example,)k(supp)s(ose)44 b Ft(T)13 b Fu(\()p Ft(n)p Fu(\))49 b(=)f(2)p Ft(n)1392 5194 y Fn(3)1462 5227 y Fu(+)29 b(100)p Ft(n)1752 5194 y Fn(2)1807 5227 y Fu(log)1925 5249 y Fn(2)1979 5227 y Ft(n)h Fu(+)f(17)46 b(and)e Ft(f)10 b Fu(\()p Ft(n)p Fu(\))48 b(=)h Ft(n)2894 5194 y Fn(3)2933 5227 y Fu(.)83 b(The)44 b(ratio)i(of)f(these)g(is)0 5340 y(2)24 b(+)f(\(100)15 b(log)467 5362 y Fn(2)522 5340 y Ft(n)p Fu(\))p Ft(=n)23 b Fu(+)h(17)p Ft(=n)1020 5307 y Fn(3)1060 5340 y Fu(.)56 b(In)34 b(this)h(limit,)j(this)d(go)s(es)h(to)g(2.)56 b(Therefore,)37 b Ft(T)13 b Fu(\()p Ft(n)p Fu(\))33 b(=)g(\002\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\).)56 b(Of)35 b(course,)p eop end %%Page: 11 12 TeXDict begin 11 11 bop 0 -165 a Fs(2.3.)72 b(RECURRENCES)2891 b Fu(11)0 91 y(it)32 b(is)g(p)s(ossible)f(that)i(the)f(limit)g(do)s (esn't)g(exist)g(|)g(for)f(instance)i(if)e Ft(T)13 b Fu(\()p Ft(n)p Fu(\))28 b(=)f Ft(n)p Fu(\(2)21 b(+)g(sin)15 b Ft(n)p Fu(\))32 b(and)f Ft(f)10 b Fu(\()p Ft(n)p Fu(\))27 b(=)g Ft(n)k Fu(then)0 204 y(the)j(ratio)h(oscillates)h(b)s(et)m(w)m (een)e(1)g(and)f(3.)51 b(In)33 b(this)g(case)i(w)m(e)f(w)m(ould)f(go)i (bac)m(k)f(to)h(the)e(de\014nitions)h(to)g(sa)m(y)g(that)0 317 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b(=)g(\002\()p Ft(n)p Fu(\).)0 469 y(One)36 b(con)m(v)m(enien)m(t)j(fact)e(to)g(kno)m (w)f(\(whic)m(h)h(w)m(e)g(just)f(used)f(in)h(the)h(paragraph)f(ab)s(o)m (v)m(e)i(and)d(y)m(ou)i(can)g(pro)m(v)m(e)g(b)m(y)0 582 y(taking)22 b(deriv)-5 b(ativ)m(es\))22 b(is)f(that)g(for)f(an)m(y)h (constan)m(t)h Ft(k)s Fu(,)h(lim)1891 596 y Fm(n)p Fh(!1)2079 582 y Fu(\(log)18 b Ft(n)p Fu(\))2337 549 y Fm(k)2379 582 y Ft(=n)26 b Fu(=)f(0.)37 b(This)20 b(implies,)j(for)e(instance,)i (that)0 695 y Ft(n)15 b Fu(log)i Ft(n)25 b Fu(=)g Ft(o)p Fu(\()p Ft(n)512 662 y Fn(1)p Fm(:)p Fn(5)606 695 y Fu(\))31 b(b)s(ecause)f(lim)1135 709 y Fm(n)p Fh(!1)1323 695 y Fu(\()p Ft(n)15 b Fu(log)i Ft(n)p Fu(\))p Ft(=n)1750 662 y Fn(1)p Fm(:)p Fn(5)1870 695 y Fu(=)25 b(lim)2092 709 y Fm(n)p Fh(!1)2280 695 y Fu(\(log)18 b Ft(n)p Fu(\))p Ft(=)2583 630 y Fp(p)p 2659 630 55 4 v 65 x Ft(n)25 b Fu(=)g(lim)2961 709 y Fm(n)p Fh(!1)3165 622 y Ff(p)p 3248 622 398 4 v 73 x Fu(\(log)18 b Ft(n)p Fu(\))3506 669 y Fn(2)3545 695 y Ft(=n)25 b Fu(=)g(0.)0 847 y(So,)g(this)e (notation)h(giv)m(es)h(us)d(a)i(language)h(for)e(talking)h(ab)s(out)f (desired)g(or)g(ac)m(hiev)-5 b(able)25 b(sp)s(eci\014cations.)39 b(A)23 b(t)m(ypical)0 960 y(use)36 b(migh)m(t)g(b)s(e)f(\\w)m(e)i(can)f (pro)m(v)m(e)h(that)g Fo(any)f Fu(algorithm)h(for)f(problem)f Ft(X)43 b Fu(m)m(ust)36 b(tak)m(e)i(\012\()p Ft(n)15 b Fu(log)i Ft(n)p Fu(\))36 b(time)g(in)g(the)0 1073 y(w)m(orst)c(case.) 47 b(My)32 b(fancy)h(algorithm)g(tak)m(es)g(time)g Ft(O)s Fu(\()p Ft(n)15 b Fu(log)i Ft(n)p Fu(\).)46 b(Therefore,)32 b(m)m(y)h(algorithm)g(is)f(asymptotically)0 1186 y(optimal.")0 1504 y Fq(2.3)137 b(Recurrences)0 1746 y Fu(W)-8 b(e)30 b(often)e(are)h(in)m(terested)g(in)f(algorithms)i(expressed)d(in)h(a)h (recursiv)m(e)g(w)m(a)m(y)-8 b(.)41 b(When)29 b(w)m(e)f(analyze)i (them,)f(w)m(e)g(get)0 1859 y(a)36 b(recurrence:)52 b(a)36 b(description)g(of)f(the)h(running)e(time)j(on)f(an)f(input)g(of)h (size)h Ft(n)e Fu(as)h(a)g(function)g(of)g Ft(n)f Fo(and)i Fu(the)0 1972 y(running)29 b(time)i(on)f(inputs)f(of)i(smaller)g (sizes.)41 b(Here)31 b(are)g(some)g(examples:)0 2162 y Fv(Mergesort:)46 b Fu(T)-8 b(o)35 b(sort)f(an)g(arra)m(y)h(of)f(size) h Ft(n)p Fu(,)g(w)m(e)f(sort)g(the)h(left)f(half,)i(sort)e(righ)m(t)h (half,)g(and)e(then)h(merge)h(the)227 2275 y(t)m(w)m(o)c(results.)41 b(W)-8 b(e)31 b(can)f(do)g(the)g(merge)g(in)g(linear)g(time.)41 b(So,)31 b(if)e Ft(T)13 b Fu(\()p Ft(n)p Fu(\))30 b(denotes)g(the)h (running)d(time)i(on)g(an)227 2387 y(input)g(of)g(size)h Ft(n)p Fu(,)g(w)m(e)f(end)g(up)f(with)i(the)f(recurrence)g Ft(T)13 b Fu(\()p Ft(n)p Fu(\))26 b(=)e(2)p Ft(T)13 b Fu(\()p Ft(n=)p Fu(2\))22 b(+)e Ft(cn)p Fu(.)0 2560 y Fv(Selection)35 b(sort:)46 b Fu(In)30 b(selection)j(sort,)f(w)m(e)f (run)f(through)g(the)i(arra)m(y)f(to)h(\014nd)e(the)h(smallest)h (elemen)m(t.)44 b(W)-8 b(e)33 b(put)227 2673 y(this)c(in)f(the)i (leftmost)f(p)s(osition,)h(and)e(then)h(recursiv)m(ely)g(sort)g(the)g (remainder)f(of)h(the)g(arra)m(y)-8 b(.)42 b(This)28 b(giv)m(es)227 2786 y(us)i(a)h(recurrence)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b(=)g Ft(cn)20 b Fu(+)g Ft(T)13 b Fu(\()p Ft(n)20 b Fp(\000)g Fu(1\).)0 2959 y Fv(Multiplication:)46 b Fu(Here)41 b(w)m(e)h(split)f(eac)m(h)h(n)m(um)m(b)s(er)d(in)m(to)j (its)f(left)h(and)e(righ)m(t)h(halv)m(es.)73 b(W)-8 b(e)42 b(sa)m(w)g(in)e(the)h(last)227 3072 y(lecture)31 b(that)f(the)g (straigh)m(tforw)m(ard)g(w)m(a)m(y)h(to)f(solv)m(e)h(the)f(subproblems) e(ga)m(v)m(e)j(us)e Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b(=)g(4)p Ft(T)13 b Fu(\()p Ft(n=)p Fu(2\))20 b(+)f Ft(cn)p Fu(.)227 3185 y(Ho)m(w)m(ev)m(er,)33 b(rearranging)e(terms)f(in)g(a)h (clev)m(er)g(w)m(a)m(y)h(impro)m(v)m(ed)f(this)f(to)h Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b(=)g(3)p Ft(T)13 b Fu(\()p Ft(n=)p Fu(2\))22 b(+)e Ft(cn)p Fu(.)0 3374 y(What)34 b(ab)s(out)f(the)g(base)h(cases?)49 b(In)33 b(general,)i(once)f(the)f (problem)g(size)h(gets)g(do)m(wn)f(to)h(a)f(small)h(constan)m(t,)h(w)m (e)0 3487 y(can)i(just)f(use)g(a)h(brute)f(force)h(approac)m(h)g(that)g (tak)m(es)h(some)f(other)g(constan)m(t)h(amoun)m(t)f(of)g(time.)60 b(So,)38 b(almost)0 3600 y(alw)m(a)m(ys)c(w)m(e)f(can)f(sa)m(y)h(the)g (base)f(case)h(is)g(that)g Ft(T)13 b Fu(\()p Ft(n)p Fu(\))28 b Fp(\024)g Ft(c)33 b Fu(for)f(all)h Ft(n)28 b Fp(\024)g Ft(n)2502 3614 y Fn(0)2541 3600 y Fu(,)33 b(where)f Ft(n)2919 3614 y Fn(0)2990 3600 y Fu(is)h(a)f(constan)m(t)i(w)m(e)f(get)g(to)0 3713 y(c)m(ho)s(ose)e(\(lik)m(e)h(17\))g(and)e Ft(c)g Fu(is)h(some)g(other)f(constan)m(t)i(that)f(dep)s(ends)d(on)j Ft(n)2547 3727 y Fn(0)2586 3713 y Fu(.)0 3865 y(What)k(ab)s(out)f(the)g (\\in)m(tegralit)m(y")k(issue?)52 b(F)-8 b(or)35 b(instance,)i(what)d (if)g(w)m(e)h(w)m(an)m(t)g(to)g(use)f(mergesort)h(on)f(an)g(arra)m(y)0 3978 y(with)f(an)f(o)s(dd)g(n)m(um)m(b)s(er)g(of)h(elemen)m(ts)h(|)f (then)f(the)h(recurrence)g(ab)s(o)m(v)m(e)h(is)f(not)g(tec)m(hnically)i (correct.)50 b(Luc)m(kily)-8 b(,)0 4091 y(this)33 b(issue)h(turns)e (out)i(almost)g(nev)m(er)g(to)g(matter,)i(so)d(w)m(e)h(can)g(ignore)g (it.)51 b(In)33 b(the)g(case)i(of)e(mergesort)i(w)m(e)f(can)0 4203 y(argue)29 b(formally)h(b)m(y)f(using)f(the)h(fact)h(that)g Ft(T)13 b Fu(\()p Ft(n)p Fu(\))29 b(is)g(sandwic)m(hed)f(b)s(et)m(w)m (een)i Ft(T)13 b Fu(\()p Ft(n)2791 4170 y Fh(0)2814 4203 y Fu(\))29 b(and)f Ft(T)13 b Fu(\()p Ft(n)3209 4170 y Fh(00)3251 4203 y Fu(\))30 b(where)e Ft(n)3632 4170 y Fh(0)3684 4203 y Fu(is)h(the)0 4316 y(next)37 b(smaller)g(p)s(o)m(w)m (er)g(of)f(2)h(and)g Ft(n)1233 4283 y Fh(00)1311 4316 y Fu(is)g(the)g(next)f(larger)i(p)s(o)m(w)m(er)e(of)h(2,)i(b)s(oth)d (of)h(whic)m(h)f(di\013er)h(b)m(y)f(at)i(most)f(a)0 4429 y(constan)m(t)32 b(factor)f(from)f(eac)m(h)h(other.)0 4581 y(W)-8 b(e)32 b(no)m(w)e(describ)s(e)g(four)f(metho)s(ds)h(for)g (solving)h(recurrences)f(that)h(are)g(useful)f(to)h(kno)m(w.)0 4857 y Fe(2.3.1)112 b(Solving)39 b(b)m(y)f(unrolling)0 5068 y Fu(Man)m(y)30 b(times,)g(the)g(easiest)h(w)m(a)m(y)f(to)g(solv)m (e)h(a)f(recurrence)f(is)g(to)h(unroll)f(it)h(to)g(get)h(a)f (summation.)40 b(F)-8 b(or)30 b(example,)0 5181 y(unrolling)g(the)h (recurrence)f(for)g(selection)i(sort)f(giv)m(es)g(us:)1060 5340 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b(=)g Ft(cn)20 b Fu(+)g Ft(c)p Fu(\()p Ft(n)h Fp(\000)f Fu(1\))h(+)f Ft(c)p Fu(\()p Ft(n)g Fp(\000)g Fu(2\))h(+)f Ft(:)15 b(:)g(:)21 b Fu(+)f Ft(c:)874 b Fu(\(2.2\))p eop end %%Page: 12 13 TeXDict begin 12 12 bop 0 -165 a Fu(12)1102 b Fs(LECTURE)29 b(2.)71 b(ASYMPTOTIC)29 b(ANAL)-8 b(YSIS)30 b(AND)h(RECURRENCES)0 91 y Fu(Since)37 b(there)g(are)g Ft(n)f Fu(terms)h(and)f(eac)m(h)i(one) f(is)g(at)h(most)f Ft(cn)p Fu(,)i(w)m(e)e(can)g(see)g(that)h(this)f (summation)f(is)h(at)h(most)0 204 y Ft(cn)94 171 y Fn(2)133 204 y Fu(.)54 b(Since)35 b(the)g(\014rst)f Ft(n=)p Fu(2)h(terms)g(are)g (eac)m(h)h(at)f(least)h Ft(cn=)p Fu(2,)h(w)m(e)e(can)g(see)h(that)f (this)g(summation)f(is)h(at)h(least)0 317 y(\()p Ft(n=)p Fu(2\)\()p Ft(cn=)p Fu(2\))28 b(=)d Ft(cn)687 284 y Fn(2)726 317 y Ft(=)p Fu(4.)42 b(So,)30 b(it)h(is)g(\002\()p Ft(n)1378 284 y Fn(2)1417 317 y Fu(\).)41 b(Similarly)-8 b(,)31 b(a)g(recurrence)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b(=)g Ft(n)2805 284 y Fn(5)2864 317 y Fu(+)20 b Ft(T)13 b Fu(\()p Ft(n)20 b Fp(\000)g Fu(1\))31 b(unrolls)f(to:)1038 555 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b(=)f Ft(n)1520 518 y Fn(5)1579 555 y Fu(+)20 b(\()p Ft(n)h Fp(\000)e Fu(1\))1951 518 y Fn(5)2012 555 y Fu(+)h(\()p Ft(n)g Fp(\000)g Fu(2\))2384 518 y Fn(5)2444 555 y Fu(+)g Ft(:)15 b(:)g(:)21 b Fu(+)f(1)2797 518 y Fn(5)2837 555 y Ft(;)851 b Fu(\(2.3\))0 794 y(whic)m(h)26 b(solv)m(es)h(to)g(\002\()p Ft(n)778 761 y Fn(6)817 794 y Fu(\))f(using)g(the)g(same)h(st)m(yle)g (of)f(reasoning)h(as)f(b)s(efore.)78 b(Another)26 b(con)m(v)m(enien)m (t)i(w)m(a)m(y)f(to)g(lo)s(ok)0 907 y(at)33 b(man)m(y)g(summations)f (of)h(this)f(form)g(is)h(to)g(see)g(them)g(as)f(an)h(appro)m(ximation)g (to)h(an)e(in)m(tegral.)49 b(E.g.,)34 b(in)e(this)0 1019 y(last)f(case,)h(the)e(sum)g(is)g(at)h(most)g(the)g(in)m(tegral)h(of)e Ft(f)10 b Fu(\()p Ft(x)p Fu(\))26 b(=)e Ft(x)2105 986 y Fn(5)2175 1019 y Fu(ev)-5 b(aluated)32 b(at)f Ft(x)25 b Fu(=)g Ft(n)20 b Fu(+)g(1.)0 1355 y Fe(2.3.2)112 b(Solving)39 b(b)m(y)f(guess)g(and)g(inductiv)m(e)g(pro)s(of)0 1577 y Fu(Another)30 b(go)s(o)s(d)g(w)m(a)m(y)i(to)f(solv)m(e)g(recurrences) f(is)g(to)h(mak)m(e)h(a)e(guess)h(and)e(then)h(pro)m(v)m(e)h(the)g (guess)f(correct)i(induc-)0 1690 y(tiv)m(ely)-8 b(.)50 b(Or)32 b(if)h(w)m(e)g(get)h(in)m(to)g(trouble)f(pro)m(ving)g(our)f (guess)h(correct)h(\(e.g.,)h(b)s(ecause)e(it)g(w)m(as)g(wrong\),)h (often)f(this)0 1803 y(will)e(giv)m(e)h(us)d(clues)i(as)g(to)g(a)f(b)s (etter)h(guess.)41 b(F)-8 b(or)31 b(example,)g(sa)m(y)g(w)m(e)g(ha)m(v) m(e)h(the)e(recurrence)1469 2041 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b(=)g(7)p Ft(T)13 b Fu(\()p Ft(n=)p Fu(7\))22 b(+)e Ft(n;)1297 b Fu(\(2.4\))1478 2179 y Ft(T)13 b Fu(\(1\))84 b(=)f(1)p Ft(:)1746 b Fu(\(2.5\))0 2417 y(W)-8 b(e)34 b(migh)m(t)g(\014rst)f(try)g(a)g(solution)h(of)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))30 b Fp(\024)g Ft(cn)p Fu(,)k(where)f Ft(c)d Fp(\025)g Fu(1)j(to)h(handle)f(the)g(base)h(case.)50 b(W)-8 b(e)34 b(w)m(ould)f(then)0 2530 y(assume)d(it)h(holds)f(true)g (inductiv)m(ely)h(for)f Ft(n)1496 2497 y Fh(0)1545 2530 y Ft(<)24 b(n)30 b Fu(and)g(plug)g(in)g(to)h(our)f(recurrence)g (\(using)h Ft(n)3252 2497 y Fh(0)3300 2530 y Fu(=)25 b Ft(n=)p Fu(7\))31 b(to)g(get:)1503 2768 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(\024)g Fu(7\()p Ft(cn=)p Fu(7\))22 b(+)e Ft(n)1777 2906 y Fu(=)83 b Ft(cn)20 b Fu(+)g Ft(n)1777 3044 y Fu(=)83 b(\()p Ft(c)20 b Fu(+)g(1\))p Ft(n:)0 3282 y Fu(Unfortunately)-8 b(,)36 b(this)e(isn't)g(what)g(w)m (e)h(w)m(an)m(ted:)49 b(our)33 b(m)m(ultiplier)i(\\)p Ft(c)p Fu(")g(w)m(en)m(t)g(up)e(b)m(y)h(1)h(when)e Ft(n)h Fu(w)m(en)m(t)g(up)g(b)m(y)g(a)0 3395 y(factor)29 b(of)g(7.)40 b(In)28 b(other)g(w)m(ords,)h(our)f(m)m(ultiplier)h(is)f(acting)i(lik)m (e)f(log)2319 3417 y Fn(7)2358 3395 y Fu(\()p Ft(n)p Fu(\).)40 b(So,)29 b(let's)h(mak)m(e)f(a)g(new)e(guess)i(using)f(a)0 3508 y(m)m(ultiplier)j(of)g(this)g(form)f(|)g(or)h(log)1264 3530 y Fn(7)1304 3508 y Fu(\(7)p Ft(n)p Fu(\))g(to)g(get)h(the)f(base)g (case)g(of)g Ft(n)25 b Fu(=)h(1)31 b(righ)m(t.)42 b(So,)31 b(w)m(e)g(ha)m(v)m(e)h(a)f(new)f(guess)0 3621 y(of)1525 3859 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(\024)g Ft(n)15 b Fu(log)2140 3881 y Fn(7)2179 3859 y Fu(\(7)p Ft(n)p Fu(\))p Ft(:)1339 b Fu(\(2.6\))0 4097 y(If)30 b(w)m(e)h(assume)f(this)g(holds)g(true)g(inductiv)m(ely)h(for)g Ft(n)1808 4064 y Fh(0)1856 4097 y Ft(<)25 b(n)p Fu(,)30 b(then)g(w)m(e)h(get:)1349 4335 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(\024)f Fu(7[\()p Ft(n=)p Fu(7\))15 b(log)2196 4357 y Fn(7)2235 4335 y Fu(\()p Ft(n)p Fu(\)])21 b(+)f Ft(n)1623 4473 y Fu(=)82 b Ft(n)15 b Fu(log)1963 4495 y Fn(7)2003 4473 y Fu(\()p Ft(n)p Fu(\))20 b(+)g Ft(n)1623 4611 y Fu(=)82 b Ft(n)15 b Fu(log)1963 4633 y Fn(7)2003 4611 y Fu(\(7)p Ft(n)p Fu(\))p Ft(:)1515 b Fu(\(2.7\))0 4849 y(So,)31 b(w)m(e)f(ha)m(v)m(e)i(v)m(eri\014ed)e(our)g(guess.)0 5001 y(It)j(is)g(imp)s(ortan)m(t)g(in)f(this)h(t)m(yp)s(e)f(of)h(pro)s (of)f(to)i(b)s(e)e(careful.)48 b(F)-8 b(or)34 b(instance,)g(one)f (could)g(b)s(e)f(lulled)g(in)m(to)i(thinking)0 5114 y(that)40 b(our)g(initial)h(guess)f(of)g Ft(cn)g Fu(w)m(as)g(correct)h(b)m(y)f (reasoning)g(\\w)m(e)h(assumed)e Ft(T)13 b Fu(\()p Ft(n=)p Fu(7\))41 b(w)m(as)f(\002\()p Ft(n=)p Fu(7\))g(and)g(got)0 5227 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))38 b(=)g(\002\()p Ft(n)p Fu(\)".)65 b(The)37 b(problem)h(is)g(that)h(the)f(constan)m(ts)i (c)m(hanged)f(\()p Ft(c)f Fu(turned)f(in)m(to)i Ft(c)26 b Fu(+)f(1\))39 b(so)g(they)f(really)0 5340 y(w)m(eren't)31 b(constan)m(t)h(after)f(all!)p eop end %%Page: 13 14 TeXDict begin 13 13 bop 0 -165 a Fs(2.3.)72 b(RECURRENCES)2891 b Fu(13)0 91 y Fe(2.3.3)112 b(Recursion)38 b(trees,)f(stac)m(king)h (bric)m(ks,)g(and)g(a)g(Master)g(F)-9 b(orm)m(ula)0 302 y Fu(The)28 b(\014nal)h(metho)s(d)f(w)m(e)h(examine,)h(whic)m(h)f(is)f (esp)s(ecially)i(go)s(o)s(d)f(for)f(divide-and-conquer)h(st)m(yle)h (recurrences,)f(is)0 415 y(the)h(use)f(of)h(a)g(recursion)g(tree.)41 b(W)-8 b(e)31 b(will)f(use)f(this)h(to)g(metho)s(d)g(to)g(pro)s(duce)f (a)h(simple)f(\\master)i(form)m(ula")f(that)0 528 y(can)h(b)s(e)e (applied)i(to)g(man)m(y)f(recurrences)g(of)h(this)f(form.)0 680 y(Consider)g(the)g(follo)m(wing)i(t)m(yp)s(e)e(of)h(recurrence:) 1450 876 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))84 b(=)e Ft(aT)13 b Fu(\()p Ft(n=b)p Fu(\))21 b(+)f Ft(cn)2407 838 y Fm(k)3713 876 y Fu(\(2.8\))1460 1013 y Ft(T)13 b Fu(\(1\))84 b(=)e Ft(c;)0 1209 y Fu(for)26 b(p)s(ositiv)m(e)h(constan)m(ts)h Ft(a)p Fu(,)f Ft(b)p Fu(,)g Ft(c)p Fu(,)h(and)e Ft(k)s Fu(.)39 b(This)25 b(recurrence)i(corresp)s(onds)d(to)j(the)g(time)g(sp) s(en)m(t)f(b)m(y)g(an)g(algorithm)0 1322 y(that)34 b(do)s(es)f Ft(cn)502 1289 y Fm(k)577 1322 y Fu(w)m(ork)h(up)e(fron)m(t,)i(and)f (then)g(divides)g(the)g(problem)g(in)m(to)h Ft(a)f Fu(pieces)h(of)g (size)g Ft(n=b)p Fu(,)g(solving)g(eac)m(h)0 1435 y(one)29 b(recursiv)m(ely)-8 b(.)42 b(F)-8 b(or)29 b(instance,)i(mergesort,)f (Karatsuba)f(m)m(ultiplication,)i(and)e(Strassen's)f(algorithm)i(all)g (\014t)0 1548 y(this)36 b(mold.)58 b(A)37 b Fo(r)-5 b(e)g(cursion)39 b(tr)-5 b(e)g(e)37 b Fu(is)f(just)g(a)h(tree)f(that)h(represen)m(ts)f (this)h(pro)s(cess,)g(where)f(eac)m(h)h(no)s(de)f(con)m(tains)0 1661 y(inside)d(it)h(the)g(w)m(ork)g(done)f(up)f(fron)m(t)i(and)f(then) g(has)g(one)h(c)m(hild)g(for)f(eac)m(h)i(recursiv)m(e)f(call.)51 b(The)33 b(lea)m(v)m(es)j(of)e(the)0 1774 y(tree)d(are)g(the)f(base)h (cases)g(of)g(the)f(recursion.)41 b(A)30 b(tree)h(for)f(the)h (recurrence)f(\(2.9\))i(is)f(giv)m(en)g(b)s(elo)m(w.)3421 1741 y Fn(1)p 1087 1938 457 4 v 1087 2088 4 151 v 1540 2088 V 1087 2091 457 4 v 1240 2052 a Ft(cn)1334 2019 y Fm(k)1157 2131 y Fd(\010)1074 2172 y(\010)991 2214 y(\010)950 2234 y(\010)p 1314 2234 4 146 v 1390 2123 a(a)1474 2156 y(a)1557 2189 y(a)1640 2222 y(a)1672 2235 y(a)1645 2472 y(\000)1585 2532 y(\000)1840 2472 y(A)1871 2534 y(A)1878 2417 y(P)1961 2445 y(P)2044 2472 y(P)2127 2500 y(P)2210 2528 y(P)2235 2536 y(P)794 2472 y(\000)735 2532 y(\000)p 926 2489 4 101 v 977 2472 a(@)994 2489 y(@)1194 2472 y(\000)1177 2489 y(\000)p 1326 2489 V 1377 2472 a(@)1394 2489 y(@)p 750 2238 345 4 v 750 2388 4 151 v 1090 2388 V 750 2391 345 4 v 1162 2238 V 1162 2388 4 151 v 1503 2388 V 1162 2391 345 4 v 1612 2238 V 1612 2388 4 151 v 1953 2388 V 1612 2391 345 4 v 1404 2538 357 4 v 1404 2688 4 151 v 1757 2688 V 1404 2691 357 4 v 2195 2538 355 4 v 2195 2688 4 151 v 2546 2688 V 2195 2691 355 4 v 1820 2538 V 1820 2688 4 151 v 2171 2688 V 1820 2691 355 4 v 1444 2774 a(\000)1427 2791 y(\000)p 1576 2791 4 101 v 1627 2774 a(@)1644 2791 y(@)1844 2774 y(\000)1827 2791 y(\000)p 1976 2791 V 2028 2774 a(@)2044 2791 y(@)2244 2774 y(\000)2228 2791 y(\000)p 2376 2791 V 2428 2774 a(@)2444 2791 y(@)3078 2097 y Fc(6)p 3075 2952 7 938 v 3078 2952 a(?)790 2352 y Ft(c)p Fu(\()p Ft(n=b)p Fu(\))1038 2319 y Fm(k)1203 2352 y Ft(c)p Fu(\()p Ft(n=b)p Fu(\))1451 2319 y Fm(k)1653 2352 y Ft(c)p Fu(\()p Ft(n=b)p Fu(\))1901 2319 y Fm(k)1428 2652 y Ft(c)p Fu(\()p Ft(n=b)1641 2619 y Fn(2)1681 2652 y Fu(\))1716 2619 y Fm(k)1840 2652 y Ft(c)p Fu(\()p Ft(n=b)2053 2619 y Fn(2)2094 2652 y Fu(\))2129 2619 y Fm(k)2216 2652 y Ft(c)p Fu(\()p Ft(n=b)2429 2619 y Fn(2)2469 2652 y Fu(\))2504 2619 y Fm(k)3153 2539 y Fu(log)3270 2561 y Fm(b)3305 2539 y Fu(\()p Ft(n)p Fu(\))1127 2652 y Fp(\001)15 b(\001)g(\001)-695 b Ft(c)p Fu(\()p Ft(n=b)750 2619 y Fn(2)791 2652 y Fu(\))826 2619 y Fm(k)p 524 2538 357 4 v 524 2688 4 151 v 877 2688 V 524 2691 357 4 v 544 2774 a Fd(\000)527 2791 y(\000)p 676 2791 4 101 v 727 2774 a(@)744 2791 y(@)0 3205 y Fu(T)-8 b(o)33 b(compute)g(the)g(result)f(of)h(the)g(recurrence,)g(w)m(e)g (simply)f(need)g(to)h(add)f(up)g(all)h(the)g(v)-5 b(alues)33 b(in)f(the)h(tree.)48 b(W)-8 b(e)0 3318 y(can)33 b(do)f(this)g(b)m(y)g (adding)g(them)h(up)e(lev)m(el)j(b)m(y)e(lev)m(el.)49 b(The)32 b(top)g(lev)m(el)i(has)e(v)-5 b(alue)33 b Ft(cn)2914 3285 y Fm(k)2957 3318 y Fu(,)g(the)f(next)h(lev)m(el)h(sums)d(to)0 3430 y Ft(ca)p Fu(\()p Ft(n=b)p Fu(\))296 3397 y Fm(k)340 3430 y Fu(,)d(the)f(next)h(lev)m(el)h(sums)d(to)i Ft(ca)1372 3397 y Fn(2)1412 3430 y Fu(\()p Ft(n=b)1586 3397 y Fn(2)1625 3430 y Fu(\))1660 3397 y Fm(k)1703 3430 y Fu(,)h(and)d(so)i(on.)39 b(The)27 b(depth)f(of)i(the)f(tree)h(\(the)g(n)m(um)m(b)s(er)e(of)h (lev)m(els)0 3543 y(not)k(including)f(the)g(ro)s(ot\))h(is)g(log)1151 3565 y Fm(b)1185 3543 y Fu(\()p Ft(n)p Fu(\))p Ft(:)g Fu(Therefore,)g(w)m(e)f(get)i(a)f(summation)f(of:)902 3755 y Ft(cn)996 3717 y Fm(k)1054 3660 y Ff(h)1093 3755 y Fu(1)21 b(+)f Ft(a=b)1382 3717 y Fm(k)1445 3755 y Fu(+)g(\()p Ft(a=b)1703 3717 y Fm(k)1747 3755 y Fu(\))1782 3717 y Fn(2)1842 3755 y Fu(+)g(\()p Ft(a=b)2100 3717 y Fm(k)2143 3755 y Fu(\))2178 3717 y Fn(3)2238 3755 y Fu(+)g Ft(:::)h Fu(+)f(\()p Ft(a=b)2683 3717 y Fm(k)2727 3755 y Fu(\))2762 3717 y Fn(log)2853 3734 y Fb(b)2896 3717 y Fm(n)2943 3660 y Ff(i)3713 3755 y Fu(\(2.9\))0 3977 y(T)-8 b(o)31 b(help)f(us)g(understand)f(this,)i(let's)g(de\014ne)f Ft(r)e Fu(=)e Ft(a=b)1880 3944 y Fm(k)1923 3977 y Fu(.)41 b(Notice)33 b(that)e Ft(r)i Fu(is)d(a)h Fo(c)-5 b(onstant)p Fu(,)33 b(since)e Ft(a)p Fu(,)g Ft(b)p Fu(,)g(and)f Ft(k)k Fu(are)0 4090 y(constan)m(ts.)48 b(F)-8 b(or)34 b(instance,)f(for)g (Strassen's)f(algorithm)h Ft(r)f Fu(=)c(7)p Ft(=)p Fu(2)2279 4057 y Fn(2)2320 4090 y Fu(,)33 b(and)f(for)g(mergesort)i Ft(r)d Fu(=)e(2)p Ft(=)p Fu(2)h(=)e(1.)48 b(Using)0 4203 y(our)30 b(de\014nition)g(of)g Ft(r)s Fu(,)h(our)e(summation)i (simpli\014es)f(to:)1272 4409 y Ft(cn)1366 4371 y Fm(k)1424 4315 y Ff(h)1463 4409 y Fu(1)20 b(+)g Ft(r)j Fu(+)d Ft(r)1818 4371 y Fn(2)1877 4409 y Fu(+)g Ft(r)2012 4371 y Fn(3)2071 4409 y Fu(+)g Ft(:::)h Fu(+)f Ft(r)2393 4371 y Fn(log)2484 4388 y Fb(b)2527 4371 y Fm(n)2574 4315 y Ff(i)3668 4409 y Fu(\(2.10\))0 4615 y(W)-8 b(e)32 b(can)e(no)m(w)h(ev)-5 b(aluate)32 b(three)e(cases:)-107 4834 y(Case)h(1:)46 b Ft(r)37 b(<)d Fu(1.)57 b(In)35 b(this)g(case,)k(the)d(sum)e(is)i(a)g (con)m(v)m(ergen)m(t)i(series.)57 b(Ev)m(en)36 b(if)g(w)m(e)g(imagine)h (the)e(series)h(going)h(to)227 4947 y(in\014nit)m(y)-8 b(,)43 b(w)m(e)d(still)h(get)g(that)g(the)f(sum)f(1)27 b(+)f Ft(r)j Fu(+)d Ft(r)2016 4914 y Fn(2)2082 4947 y Fu(+)g Ft(:)15 b(:)g(:)42 b Fu(=)f(1)p Ft(=)p Fu(\(1)28 b Fp(\000)f Ft(r)s Fu(\).)69 b(So,)42 b(w)m(e)e(can)h(upp)s(er-b)s (ound)227 5060 y(form)m(ula)29 b(\(2.9\))h(b)m(y)e Ft(cn)992 5027 y Fm(k)1034 5060 y Ft(=)p Fu(\(1)17 b Fp(\000)f Ft(r)s Fu(\),)28 b(and)g(lo)m(w)m(er)h(b)s(ound)d(it)j(b)m(y)f(just)f (the)i(\014rst)e(term)h Ft(cn)3122 5027 y Fm(k)3165 5060 y Fu(.)40 b(Since)28 b Ft(r)j Fu(and)c Ft(c)i Fu(are)227 5173 y(constan)m(ts,)j(this)e(solv)m(es)i(to)f(\002\()p Ft(n)1363 5140 y Fm(k)1405 5173 y Fu(\).)p 0 5254 1560 4 v 104 5308 a Fk(1)138 5340 y Fj(This)c(tree)e(has)h(branc)n(hing)g (factor)h Fi(a)p Fj(.)p eop end %%Page: 14 15 TeXDict begin 14 14 bop 0 -165 a Fu(14)1102 b Fs(LECTURE)29 b(2.)71 b(ASYMPTOTIC)29 b(ANAL)-8 b(YSIS)30 b(AND)h(RECURRENCES)-107 91 y Fu(Case)g(2:)46 b Ft(r)28 b Fu(=)d(1.)39 b(In)22 b(this)h(case,)i(all)f(terms)f(in)f(the)h(summation)g(\(2.9\))i(are)e (equal)g(to)h(1,)h(so)e(the)g(result)g(is)g Ft(cn)3525 58 y Fm(k)3567 91 y Fu(\(log)3720 113 y Fm(b)3770 91 y Ft(n)5 b Fu(+)227 204 y(1\))26 b Fp(2)f Fu(\002\()p Ft(n)580 171 y Fm(k)638 204 y Fu(log)17 b Ft(n)p Fu(\).)-107 388 y(Case)31 b(3:)46 b Ft(r)28 b(>)e Fu(1.)41 b(In)30 b(this)g(case,)i(the)f(last)g(term)g(of)g(the)g(summation)f(dominates.) 42 b(W)-8 b(e)31 b(can)g(see)g(this)g(b)m(y)f(pulling)g(it)227 501 y(out,)h(giving)g(us:)1333 625 y Ft(cn)1427 587 y Fm(k)1470 625 y Ft(r)1514 587 y Fn(log)1605 604 y Fb(b)1648 587 y Fm(n)1710 531 y Ff(h)1749 625 y Fu(\(1)p Ft(=r)s Fu(\))1953 587 y Fn(log)2045 604 y Fb(b)2089 587 y Fm(n)2156 625 y Fu(+)20 b Ft(:)15 b(:)g(:)21 b Fu(+)f(1)p Ft(=r)j Fu(+)d(1)2754 531 y Ff(i)3668 625 y Fu(\(2.11\))227 802 y(Since)34 b(1)p Ft(=r)h(<)30 b Fu(1,)36 b(w)m(e)e(can)g(no)m(w)g(use)f (the)h(same)h(reasoning)f(as)g(in)f(Case)h(1:)48 b(the)35 b(summation)e(is)h(at)h(most)227 915 y(1)p Ft(=)p Fu(\(1)22 b Fp(\000)e Fu(1)p Ft(=r)s Fu(\))31 b(whic)m(h)f(is)h(a)f(constan)m(t.) 43 b(Therefore,)30 b(w)m(e)h(ha)m(v)m(e)1480 1125 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(2)g Fu(\002)1984 1031 y Ff(\020)2033 1125 y Ft(n)2088 1087 y Fm(k)2130 1125 y Fu(\()p Ft(a=b)2297 1087 y Fm(k)2341 1125 y Fu(\))2376 1087 y Fn(log)2467 1104 y Fb(b)2510 1087 y Fm(n)2557 1031 y Ff(\021)2622 1125 y Ft(:)227 1345 y Fu(W)-8 b(e)32 b(can)f(simplify)e(this)i(form)m(ula)f(b)m(y)h(noticing)g(that)g Ft(b)2116 1312 y Fm(k)13 b Fn(log)2257 1329 y Fb(b)2301 1312 y Fm(n)2373 1345 y Fu(=)25 b Ft(n)2524 1312 y Fm(k)2566 1345 y Fu(,)31 b(so)f(w)m(e)h(are)g(left)g(with)1627 1550 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(2)g Fu(\002)2131 1456 y Ff(\020)2181 1550 y Ft(a)2229 1512 y Fn(log)2320 1529 y Fb(b)2363 1512 y Fm(n)2410 1456 y Ff(\021)2475 1550 y Ft(:)1168 b Fu(\(2.12\))227 1754 y(W)-8 b(e)32 b(can)f(simplify)e(this)i(further)e(b)m(y)h(sw)m(apping)g(the)h(\\)p Ft(a)p Fu(")g(and)f(the)g(\\)p Ft(n)p Fu(")h(to)g(get:)1627 1959 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(2)f Fu(\002)2130 1865 y Ff(\020)2180 1959 y Ft(n)2235 1921 y Fn(log)2326 1938 y Fb(b)2369 1921 y Fm(a)2410 1865 y Ff(\021)2475 1959 y Ft(:)1168 b Fu(\(2.13\))227 2169 y(\(T)-8 b(ak)m(e)32 b(log)600 2190 y Fm(b)664 2169 y Fu(of)f(\(2.12\))i(and)c(\(2.13\))k (to)e(con)m(vince)h(y)m(ourself)e(this)h(is)f(legal!!\))227 2317 y(Note)40 b(that)e(Case)h(3)f(is)g(what)g(w)m(e)h(used)e(for)h (Karatsuba)g(m)m(ultiplication)i(\()p Ft(a)e Fu(=)g(3)p Ft(;)15 b(b)39 b Fu(=)f(2)p Ft(;)15 b(k)42 b Fu(=)c(1\))h(and)227 2430 y(Strassen's)30 b(algorithm)i(\()p Ft(a)25 b Fu(=)g(7)p Ft(;)15 b(b)26 b Fu(=)f(2)p Ft(;)15 b(k)30 b Fu(=)25 b(2\).)0 2648 y(Com)m(bining)30 b(the)h(three)f(cases)i(ab)s(o)m(v)m(e) f(giv)m(es)h(us)e(the)g(follo)m(wing)i(\\master)f(theorem".)0 2867 y Fv(Theorem)k(2.1)46 b Fo(The)33 b(r)-5 b(e)g(curr)g(enc)g(e)1450 3061 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))84 b(=)e Ft(aT)13 b Fu(\()p Ft(n=b)p Fu(\))21 b(+)f Ft(cn)2407 3023 y Fm(k)1460 3198 y Ft(T)13 b Fu(\(1\))84 b(=)e Ft(c;)0 3393 y Fo(wher)-5 b(e)34 b Ft(a)p Fo(,)e Ft(b)p Fo(,)h Ft(c)p Fo(,)f(and)i Ft(k)i Fo(ar)-5 b(e)33 b(al)5 b(l)33 b(c)-5 b(onstants,)35 b(solves)e(to:)1335 3587 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(2)f Fu(\002\()p Ft(n)1913 3549 y Fm(k)1956 3587 y Fu(\))33 b Fo(if)f Ft(a)25 b(<)g(b)2320 3549 y Fm(k)1335 3724 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(2)f Fu(\002\()p Ft(n)1913 3687 y Fm(k)1971 3724 y Fu(log)18 b Ft(n)p Fu(\))32 b Fo(if)g Ft(a)26 b Fu(=)f Ft(b)2523 3687 y Fm(k)1335 3862 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(2)f Fu(\002\()p Ft(n)1913 3825 y Fn(log)2004 3842 y Fb(b)2047 3825 y Fm(a)2089 3862 y Fu(\))33 b Fo(if)f Ft(a)26 b(>)f(b)2454 3825 y Fm(k)0 4095 y Fu(A)35 b(nice)g(in)m(tuitiv) m(e)h(w)m(a)m(y)g(to)g(think)e(of)h(the)f(computation)i(ab)s(o)m(v)m(e) g(is)f(to)g(think)f(of)h(eac)m(h)h(no)s(de)e(in)h(the)f(recursion)0 4208 y(tree)e(as)g(a)f(bric)m(k)h(of)f(heigh)m(t)h(1)g(and)f(width)f (equal)i(to)g(the)g(v)-5 b(alue)31 b(inside)g(it.)44 b(Our)31 b(goal)h(is)g(no)m(w)f(to)h(compute)g(the)0 4321 y(area)g(of)f(the)g(stac)m(k.)44 b(Dep)s(ending)30 b(on)h(whether)f(w)m(e)h(are)h(in)e(Case)i(1,)f(2,)h(or)f(3,)h(the)f (picture)f(then)h(lo)s(oks)g(lik)m(e)i(one)0 4434 y(of)e(the)f(follo)m (wing:)239 4652 y Fl(+-----------------+)138 b(+---------------+)329 b(+------------+)286 4765 y(\\_______________/)187 b(|_______________|) 281 b(/______________\\)334 4878 y(\\______|______/)235 b(|_______|_______|)f(/_______|________\\)382 4991 y(\\__|__|__|__/)283 b(|___|___|___|___|)186 b(/____|___|___|_____\\)0 5209 y Fu(In)33 b(the)h(\014rst)f(case,)j(the)e(area)h(is)f(dominated)g(b)m (y)g(the)g(top)g(bric)m(k;)i(in)e(the)g(second)g(case,)i(all)f(lev)m (els)g(pro)m(vide)f(an)0 5322 y(equal)d(con)m(tribution,)g(and)f(in)g (the)g(last)i(case,)f(the)g(area)g(is)g(dominated)f(b)m(y)g(the)h(b)s (ottom)g(lev)m(el.)p eop end %%Page: 15 16 TeXDict begin 15 15 bop 0 753 a Fr(Lecture)65 b(3)0 1207 y Fw(Probabilistic)79 b(Analysis)g(and)0 1456 y(Randomized)g(Quic)-7 b(ksort)0 1977 y Fq(3.1)137 b(Ov)l(erview)0 2219 y Fu(In)27 b(this)h(lecture)g(w)m(e)g(b)s(egin)g(b)m(y)f(discussing)h(the)g (di\013erence)g(b)s(et)m(w)m(een)g(w)m(orst-case)i(and)d(a)m(v)m (erage-case)32 b(b)s(eha)m(vior,)0 2332 y(and)k(in)m(tro)s(duce)h (randomized)g(\(probabilistic\))i(algorithms)f(and)e(the)h(notion)h(of) f(w)m(orst-case)i(exp)s(ected)e(time)0 2445 y(b)s(ounds.)h(W)-8 b(e)31 b(mak)m(e)f(this)f(concrete)h(with)f(a)h(discussion)e(of)i(a)f (randomized)g(v)m(ersion)h(of)f(the)h(Quic)m(ksort)g(sorting)0 2558 y(algorithm,)41 b(whic)m(h)d(w)m(e)g(pro)m(v)m(e)g(has)g(w)m (orst-case)i(exp)s(ected)e(running)e(time)i Ft(O)s Fu(\()p Ft(n)15 b Fu(log)j Ft(n)p Fu(\).)63 b(In)37 b(the)h(pro)s(cess,)h(w)m (e)0 2671 y(discuss)30 b(basic)g(probabilistic)h(concepts)g(suc)m(h)g (as)f(ev)m(en)m(ts,)i(random)e(v)-5 b(ariables,)31 b(and)f(linearit)m (y)i(of)e(exp)s(ectation.)0 2992 y Fq(3.2)137 b(W)-11 b(orst-case,)46 b(a)l(v)l(erage-case,)f(and)h(randomized)g(algorithms)0 3234 y Fu(The)36 b(last)i(lecture)g(discussed)e(the)h(notions)g(of)g Ft(O)s Fu(,)h(\012,)h(and)d(\002)g(b)s(ounds,)h(and)f(ho)m(w)h(to)g (compute)h(them)e(using)0 3347 y(recurrences.)54 b(W)-8 b(e)37 b(b)s(egin)d(this)h(lecture)h(with)f(a)g(di\013eren)m(t)h (issue:)50 b(w)m(orst-case)36 b(v)m(ersus)f(a)m(v)m(erage)j(case)e(b)s (ounds.)0 3460 y(Note)30 b(that)g(for)e(comparison-based)i(algorithms)f (lik)m(e)i(Quic)m(ksort)e(and)f(Mergesort,)j(w)m(e)f(express)e(running) f(time)0 3573 y(in)j(terms)g(of)h(the)f(n)m(um)m(b)s(er)g(of)g (comparisons)g(made.)0 3725 y(Sa)m(y)h Ft(I)37 b Fu(is)31 b(some)f(input)g(and)g Ft(T)13 b Fu(\()p Ft(I)7 b Fu(\))30 b(is)h(running)d(time)j(of)g(our)f(algorithm)h(on)g(input)e Ft(I)7 b Fu(.)41 b(W)-8 b(e)31 b(can)g(then)f(de\014ne:)1194 3902 y Ft(T)1247 3916 y Fn(w)n(orstcase)1544 3902 y Fu(\()p Ft(n)p Fu(\))84 b(=)329 b(max)1906 3979 y(inputs)30 b Fm(I)35 b Fu(of)30 b(size)i Fm(n)2585 3902 y Ft(T)13 b Fu(\()p Ft(I)7 b Fu(\))1132 4107 y Ft(T)1185 4121 y Fn(a)n(v)n(eragecase)1544 4107 y Fu(\()p Ft(n)p Fu(\))84 b(=)345 b(a)m(vg)1906 4201 y(inputs)30 b Fm(I)35 b Fu(of)30 b(size)i Fm(n)2585 4107 y Ft(T)13 b Fu(\()p Ft(I)7 b Fu(\))0 4368 y(F)-8 b(or)39 b(instance,)i(Mergesort)f(has)e(b)s(oth)g (w)m(orst-case)i(and)e(a)m(v)m(erage-case)k(time)d(\002\()p Ft(n)15 b Fu(log)j Ft(n)p Fu(\).)64 b(It)39 b(do)s(esn't)f(really)0 4481 y(dep)s(end)d(on)i(the)g(input)f(at)i(all.)61 b(On)37 b(the)g(other)g(hand,)h(for)f(some)g(algorithms,)j(the)d(running)e (time)j(dep)s(ends)0 4594 y(critically)33 b(on)d(the)g(input.)40 b(One)30 b(example)h(is)g(Quic)m(ksort.)0 4798 y Fv(Quic)m(ksort:)46 b Fu(Giv)m(en)31 b(arra)m(y)g(of)g(some)g(length)f Ft(n)p Fu(,)311 4977 y(1.)46 b(Pic)m(k)32 b(an)e(elemen)m(t)i Ft(p)e Fu(of)g(the)h(arra)m(y)g(as)f(the)h(piv)m(ot)g(\(or)g(halt)g(if) f(the)h(arra)m(y)g(has)f(size)h(0)g(or)f(1\).)311 5114 y(2.)46 b(Split)31 b(the)g(arra)m(y)g(in)m(to)h(sub-arra)m(ys)e(LESS,)g (EQUAL,)h(and)f(GREA)-8 b(TER)31 b(b)m(y)g(comparing)g(eac)m(h)h(ele-) 427 5227 y(men)m(t)i(to)h(the)e(piv)m(ot.)52 b(\(LESS)32 b(has)i(all)g(elemen)m(ts)h(less)f(than)f Ft(p)p Fu(,)h(EQUAL)g(has)f (all)h(elemen)m(ts)h(equal)427 5340 y(to)c Ft(p)p Fu(,)g(and)e(GREA)-8 b(TER)31 b(has)f(all)h(elemen)m(ts)h(greater)g(than)e Ft(p)p Fu(\).)1905 5568 y(15)p eop end %%Page: 16 17 TeXDict begin 16 16 bop 0 -165 a Fu(16)477 b Fs(LECTURE)29 b(3.)71 b(PR)m(OBABILISTIC)30 b(ANAL)-8 b(YSIS)30 b(AND)h(RANDOMIZED)g (QUICKSOR)-8 b(T)311 91 y Fu(3.)46 b(recursiv)m(ely)31 b(sort)g(LESS)e(and)h(GREA)-8 b(TER.)0 326 y(The)30 b(Quic)m(ksort)h (algorithm)h(giv)m(en)f(ab)s(o)m(v)m(e)h(is)f(not)f(y)m(et)i(fully)e (sp)s(eci\014ed)g(b)s(ecause)h(w)m(e)g(ha)m(v)m(e)h(not)e(stated)i(ho)m (w)e(w)m(e)0 439 y(will)d(pic)m(k)g(the)f(piv)m(ot)h(elemen)m(t)h Ft(p)p Fu(.)39 b(F)-8 b(or)27 b(the)g(\014rst)e(v)m(ersion)i(of)g(the)f (algorithm,)j(let's)e(alw)m(a)m(ys)h(c)m(ho)s(ose)f(the)g(leftmost)0 552 y(elemen)m(t.)0 787 y Fv(Basic-Quic)m(ksort:)47 b Fu(Run)d(the)i(Quic)m(ksort)f(algorithm)i(as)e(giv)m(en)i(ab)s(o)m(v)m (e,)j(alw)m(a)m(ys)d(c)m(ho)s(osing)f(the)g(leftmost)227 899 y(elemen)m(t)32 b(in)e(the)h(arra)m(y)g(as)f(the)h(piv)m(ot.)0 1134 y(What)c(is)e(w)m(orst-case)j(running)c(time)j(of)f(Basic-Quic)m (ksort?)41 b(W)-8 b(e)27 b(can)f(see)h(that)f(if)g(the)g(arra)m(y)g(is) g(already)h(sorted,)0 1247 y(then)j(in)g(Step)h(2,)g(all)g(the)g (elemen)m(ts)h(\(except)f Ft(p)p Fu(\))g(will)g(go)g(in)m(to)h(the)e (GREA)-8 b(TER)31 b(buc)m(k)m(et.)42 b(F)-8 b(urthermore,)31 b(since)0 1360 y(the)37 b(GREA)-8 b(TER)36 b(arra)m(y)h(is)g(in)f (sorted)h(order,)1621 1327 y Fn(1)1698 1360 y Fu(this)f(pro)s(cess)g (will)h(con)m(tin)m(ue)h(recursiv)m(ely)-8 b(,)39 b(resulting)e(in)f (time)0 1473 y(\012\()p Ft(n)156 1440 y Fn(2)195 1473 y Fu(\).)42 b(W)-8 b(e)32 b(can)f(also)h(see)g(that)f(the)g(running)e (time)j(is)e Ft(O)s Fu(\()p Ft(n)2106 1440 y Fn(2)2146 1473 y Fu(\))h(on)f(an)m(y)i(arra)m(y)f(of)g Ft(n)f Fu(elemen)m(ts)i(b) s(ecause)f(Step)g(1)0 1586 y(can)25 b(b)s(e)f(executed)h(at)h(most)f Ft(n)f Fu(times,)i(and)e(Step)h(2)g(tak)m(es)h(at)f(most)g Ft(n)f Fu(steps)h(to)g(p)s(erform.)37 b(Th)m(us,)25 b(the)g(w)m (orst-case)0 1699 y(running)k(time)i(is)f(\002\()p Ft(n)798 1666 y Fn(2)837 1699 y Fu(\).)0 1851 y(On)g(the)i(other)f(hand,)g(it)h (turns)d(out)j(\(and)f(w)m(e)h(will)f(pro)m(v)m(e\))i(that)e(the)h(a)m (v)m(erage-case)j(running)29 b(time)j(for)f(Basic-)0 1964 y(Quic)m(ksort)c(\(a)m(v)m(eraging)i(o)m(v)m(er)e(all)g (di\013eren)m(t)g(initial)g(orderings)f(of)g(the)h Ft(n)e Fu(elemen)m(ts)j(in)e(the)g(arra)m(y\))h(is)f Ft(O)s Fu(\()p Ft(n)15 b Fu(log)j Ft(n)p Fu(\).)0 2077 y(So,)31 b(Basic-Quic)m(ksort)h(has)e(go)s(o)s(d)g(a)m(v)m(erage)j(case)f(p)s (erformance)e(but)f(not)i(go)s(o)s(d)f(w)m(orst-case)i(p)s(erformance.) 0 2229 y(The)f(fact)i(that)f(this)g(algorithm)h(w)m(orks)f(w)m(ell)g (on)g Fo(most)h Fu(inputs)e(ma)m(y)h(b)s(e)f(small)h(consolation)i(if)e (the)g(inputs)e(w)m(e)0 2342 y(are)f(faced)f(with)g(are)h(the)f(bad)g (ones)g(\(e.g.,)j(if)d(our)g(lists)h(are)f(nearly)h(sorted)f (already\).)41 b(One)28 b(w)m(a)m(y)h(w)m(e)g(can)f(try)g(to)0 2454 y(get)k(around)d(this)h(problem)g(is)g(to)h(add)f(randomization)h (in)m(to)h(the)e(algorithm)i(itself:)0 2717 y Fv(Randomized-Quic)m (ksort:)47 b Fu(Run)30 b(the)g(Quic)m(ksort)i(algorithm)f(as)g(giv)m (en)h(ab)s(o)m(v)m(e,)g(eac)m(h)g(time)f(pic)m(king)g(a)g Fo(r)-5 b(an-)227 2830 y(dom)32 b Fu(elemen)m(t)g(in)e(the)g(arra)m(y)h (as)g(the)f(piv)m(ot.)0 3092 y(W)-8 b(e)43 b(will)f(pro)m(v)m(e)h(that) f(for)f Fo(any)i Fu(giv)m(en)f(arra)m(y)h(input)d(arra)m(y)j Ft(I)48 b Fu(of)42 b Ft(n)g Fu(elemen)m(ts,)k(the)c(exp)s(ected)g(time) g(of)g(this)0 3205 y(algorithm)37 b Fv(E)p Fu([)p Ft(T)13 b Fu(\()p Ft(I)7 b Fu(\)])38 b(is)e Ft(O)s Fu(\()p Ft(n)15 b Fu(log)j Ft(n)p Fu(\).)59 b(This)35 b(is)i(called)g(a)g(W)-8 b(orst-case)39 b(Exp)s(ected-Time)e(b)s(ound.)57 b(Notice)38 b(that)0 3318 y(this)26 b(is)h(b)s(etter)g(than)f(an)g(a)m(v)m (erage-case)31 b(b)s(ound)24 b(b)s(ecause)j(w)m(e)g(are)g(no)f(longer)h (assuming)g(an)m(y)f(sp)s(ecial)i(prop)s(erties)0 3431 y(of)33 b(the)f(input.)46 b(E.g.,)34 b(it)f(could)g(b)s(e)f(that)h(in)f (our)g(desired)g(application,)i(the)f(input)e(arra)m(ys)i(tend)f(to)i (b)s(e)d(mostly)0 3543 y(sorted)c(or)f(in)g(some)h(sp)s(ecial)h(order,) f(and)f(this)g(do)s(es)g(not)h(a\013ect)h(our)e(b)s(ound)f(b)s(ecause)h (it)h(is)g(a)g Fo(worst-c)-5 b(ase)28 b Fu(b)s(ound)0 3656 y(with)j(resp)s(ect)h(to)g(the)g(input.)44 b(It)31 b(is)h(a)g(little)h(p)s(eculiar:)43 b(making)32 b(the)g(algorithm)h (probabilistic)f(giv)m(es)h(us)e Fo(mor)-5 b(e)0 3769 y Fu(con)m(trol)32 b(o)m(v)m(er)g(the)e(running)f(time.)0 3921 y(T)-8 b(o)31 b(pro)m(v)m(e)g(these)g(b)s(ounds,)d(w)m(e)j (\014rst)f(detour)g(in)m(to)h(the)g(basics)f(of)h(probabilistic)g (analysis.)0 4259 y Fq(3.3)137 b(The)46 b(Basics)g(of)g(Probabilistic)h (Analysis)0 4504 y Fu(Consider)34 b(rolling)h(t)m(w)m(o)h(dice)f(and)f (observing)g(the)h(results.)53 b(There)34 b(are)h(36)g(p)s(ossible)f (outcomes:)50 b(it)35 b(could)g(b)s(e)0 4617 y(that)k(the)g(\014rst)f (die)h(comes)h(up)e(1)h(and)f(the)h(second)g(comes)h(up)d(2,)42 b(or)d(that)g(the)g(\014rst)f(comes)i(up)d(2)j(and)e(the)0 4730 y(second)c(comes)h(up)f(1,)h(and)f(so)g(on.)53 b(Eac)m(h)35 b(of)f(these)h(outcomes)g(has)f(probabilit)m(y)h(1)p Ft(=)p Fu(36)h(\(assuming)e(these)h(are)0 4843 y(fair)f(dice\).)51 b(Supp)s(ose)32 b(w)m(e)i(care)h(ab)s(out)e(some)h(quan)m(tit)m(y)h (suc)m(h)f(as)f(\\what)h(is)g(the)g(probabilit)m(y)g(the)g(sum)f(of)h (the)0 4956 y(dice)c(equals)f(7?")41 b(W)-8 b(e)30 b(can)g(compute)f (that)h(b)m(y)f(adding)f(up)g(the)i(probabilities)f(of)h(all)g(the)f (outcomes)h(satisfying)0 5068 y(this)g(condition)h(\(there)g(are)g(six) f(of)h(them,)f(for)h(a)f(total)j(probabilit)m(y)d(of)h(1)p Ft(=)p Fu(6\).)p 0 5163 1560 4 v 104 5217 a Fk(1)138 5249 y Fj(T)-6 b(ec)n(hnically)g(,)31 b(this)e(dep)r(ends)g(on)g(ho)n (w)g(the)g(partitioning)h(step)f(is)h(implemen)n(ted,)g(but)e(will)j(b) r(e)e(the)g(case)h(for)g(an)n(y)e(reasonable)0 5340 y(implemen)n (tation.)p eop end %%Page: 17 18 TeXDict begin 17 17 bop 0 -165 a Fs(3.3.)72 b(THE)30 b(BASICS)g(OF)g(PR)m(OBABILISTIC)f(ANAL)-8 b(YSIS)1631 b Fu(17)0 91 y(In)30 b(the)h(language)h(of)f(probabilit)m(y)h(theory)-8 b(,)31 b(an)m(y)h(probabilistic)f(setting)h(is)f(de\014ned)e(b)m(y)i(a) g Fo(sample)k(sp)-5 b(ac)g(e)32 b Ft(S)k Fu(and)0 204 y(a)k Fo(pr)-5 b(ob)g(ability)44 b(me)-5 b(asur)g(e)42 b Ft(p)p Fu(.)69 b(The)40 b(p)s(oin)m(ts)g(of)g(the)g(sample)g(space)h (are)f(called)h Fo(elementary)i(events)p Fu(.)70 b(E.g.,)43 b(in)0 317 y(our)38 b(case,)j(the)e(elemen)m(tary)g(ev)m(en)m(ts)h(are) e(the)h(36)g(p)s(ossible)f(outcomes)h(for)f(the)g(pair)g(of)h(dice.)64 b(In)38 b(a)g(discrete)0 430 y(probabilit)m(y)h(distribution)e(\(as)i (opp)s(osed)e(to)i(a)g(con)m(tin)m(uous)f(one\),)j(the)e(probabilit)m (y)f(measure)h(is)f(a)g(function)0 543 y Ft(p)p Fu(\()p Ft(e)p Fu(\))29 b(o)m(v)m(er)h(elemen)m(tary)g(ev)m(en)m(ts)g Ft(e)f Fu(suc)m(h)f(that)h Ft(p)p Fu(\()p Ft(e)p Fu(\))d Fp(\025)f Fu(0)k(for)f(all)h Ft(e)d Fp(2)f Ft(S)5 b Fu(,)29 b(and)2645 479 y Ff(P)2732 566 y Fm(e)p Fh(2)p Fm(S)2878 543 y Ft(p)p Fu(\()p Ft(e)p Fu(\))d(=)f(1.)40 b(W)-8 b(e)30 b(will)f(also)g(use)0 656 y(Pr)o(\()p Ft(e)p Fu(\))j(in)m(terc)m (hangeably)g(with)e Ft(p)p Fu(\()p Ft(e)p Fu(\).)0 808 y(An)f Fo(event)f Fu(is)h(a)g(subset)g(of)g(the)g(sample)g(space.)41 b(F)-8 b(or)29 b(instance,)h(one)g(ev)m(en)m(t)g(w)m(e)f(migh)m(t)h (care)g(ab)s(out)e(is)h(the)g(ev)m(en)m(t)0 921 y(that)36 b(the)f(\014rst)f(die)i(comes)g(up)e(1.)55 b(Another)35 b(is)g(the)h(ev)m(en)m(t)g(that)g(the)f(t)m(w)m(o)i(dice)e(sum)g(to)g (7.)56 b(The)34 b(probabilit)m(y)0 1034 y(of)c(an)g(ev)m(en)m(t)i(is)e (just)f(the)i(sum)e(of)h(the)g(probabilities)h(of)f(the)g(elemen)m (tary)i(ev)m(en)m(ts)g(con)m(tained)f(inside)f(it)g(\(again,)0 1147 y(this)g(is)h(just)e(for)i(discrete)g(distributions)1424 1114 y Fn(2)1462 1147 y Fu(\).)0 1299 y(A)g Fo(r)-5 b(andom)35 b(variable)c Fu(is)f(a)h(function)f(from)g(elemen)m(tary)i(ev)m(en)m (ts)g(to)f(in)m(tegers)h(or)e(reals.)41 b(F)-8 b(or)31 b(instance,)h(another)0 1412 y(w)m(a)m(y)38 b(w)m(e)g(can)g(talk)g (formally)g(ab)s(out)f(these)g(dice)h(is)g(to)g(de\014ne)e(the)i (random)e(v)-5 b(ariable)38 b Ft(X)3176 1426 y Fn(1)3253 1412 y Fu(represen)m(ting)g(the)0 1525 y(result)30 b(of)h(the)f (\014rst)g(die,)g Ft(X)947 1539 y Fn(2)1017 1525 y Fu(represen)m(ting)h (the)f(result)h(of)f(the)h(second)f(die,)h(and)e Ft(X)k Fu(=)25 b Ft(X)3121 1539 y Fn(1)3181 1525 y Fu(+)20 b Ft(X)3347 1539 y Fn(2)3417 1525 y Fu(represen)m(ting)0 1637 y(the)31 b(sum)e(of)i(the)f(t)m(w)m(o.)42 b(W)-8 b(e)32 b(could)e(then)g(ask:)41 b(what)30 b(is)h(the)f(probabilit)m(y)h (that)g Ft(X)i Fu(=)25 b(7?)0 1789 y(One)34 b(prop)s(ert)m(y)f(of)i(a)f (random)g(v)-5 b(ariable)35 b(w)m(e)f(often)h(care)g(ab)s(out)f(is)g (its)h Fo(exp)-5 b(e)g(ctation)p Fu(.)54 b(F)-8 b(or)35 b(a)g(discrete)g(random)0 1902 y(v)-5 b(ariable)31 b Ft(X)38 b Fu(o)m(v)m(er)32 b(sample)e(space)h Ft(S)5 b Fu(,)31 b(the)f(exp)s(ected)h(v)-5 b(alue)31 b(of)f Ft(X)38 b Fu(is:)1444 2116 y Fv(E)p Fu([)p Ft(X)7 b Fu(])84 b(=)1886 2035 y Ff(X)1883 2219 y Fm(e)p Fh(2)p Fm(S)2024 2116 y Fu(Pr\()p Ft(e)p Fu(\))p Ft(X)7 b Fu(\()p Ft(e)p Fu(\))p Ft(:)1260 b Fu(\(3.1\))0 2404 y(In)26 b(other)i(w)m(ords,)f (the)g(exp)s(ectation)i(of)e(a)h(random)e(v)-5 b(ariable)28 b Ft(X)34 b Fu(is)28 b(just)e(its)i(a)m(v)m(erage)h(v)-5 b(alue)28 b(o)m(v)m(er)g Ft(S)5 b Fu(,)28 b(where)f(eac)m(h)0 2517 y(elemen)m(tary)40 b(ev)m(en)m(t)g Ft(e)e Fu(is)h(w)m(eigh)m(ted)h (according)f(to)g(its)g(probabilit)m(y)-8 b(.)66 b(F)-8 b(or)39 b(instance,)i(if)d(w)m(e)h(roll)g(a)g(single)g(die)0 2630 y(and)34 b(lo)s(ok)i(at)f(the)g(outcome,)j(the)d(exp)s(ected)g(v) -5 b(alue)36 b(is)f(3)p Ft(:)p Fu(5,)i(b)s(ecause)e(all)h(six)f(elemen) m(tary)h(ev)m(en)m(ts)g(ha)m(v)m(e)g(equal)0 2743 y(probabilit)m(y)-8 b(.)53 b(Often)34 b(one)g(groups)g(together)h(the)f(elemen)m(tary)i(ev) m(en)m(ts)g(according)f(to)f(the)h(di\013eren)m(t)f(v)-5 b(alues)35 b(of)0 2856 y(the)c(random)e(v)-5 b(ariable)31 b(and)f(rewrites)h(the)f(de\014nition)g(lik)m(e)i(this:)1417 3069 y Fv(E)p Fu([)p Ft(X)7 b Fu(])83 b(=)1855 2988 y Ff(X)1896 3163 y Fm(a)1990 3069 y Fu(Pr\()p Ft(X)33 b Fu(=)25 b Ft(a)p Fu(\))p Ft(a:)1230 b Fu(\(3.2\))0 3343 y(More)37 b(generally)-8 b(,)41 b(for)36 b(an)m(y)h(partition)g(of)g (the)g(probabilit)m(y)g(space)g(in)m(to)h(disjoin)m(t)f(ev)m(en)m(ts)h Ft(A)3226 3357 y Fn(1)3266 3343 y Ft(;)15 b(A)3374 3357 y Fn(2)3414 3343 y Ft(;)g(:)g(:)g(:)q Fu(,)39 b(w)m(e)e(can)0 3456 y(rewrite)31 b(the)f(exp)s(ectation)i(of)f(random)e(v)-5 b(ariable)31 b Ft(X)38 b Fu(as:)989 3670 y Fv(E)p Fu([)p Ft(X)7 b Fu(])26 b(=)1312 3589 y Ff(X)1360 3772 y Fm(i)1466 3589 y Ff(X)1447 3773 y Fm(e)p Fh(2)p Fm(A)1580 3783 y Fb(i)1621 3670 y Fu(Pr)o(\()p Ft(e)p Fu(\))p Ft(X)7 b Fu(\()p Ft(e)p Fu(\))28 b(=)2148 3589 y Ff(X)2196 3772 y Fm(i)2283 3670 y Fu(Pr\()p Ft(A)2484 3684 y Fm(i)2512 3670 y Fu(\))p Fv(E)p Fu([)p Ft(X)7 b Fp(j)p Ft(A)2816 3684 y Fm(i)2846 3670 y Fu(])p Ft(;)817 b Fu(\(3.3\))0 3979 y(where)37 b Fv(E)p Fu([)p Ft(X)7 b Fp(j)p Ft(A)539 3993 y Fm(i)568 3979 y Fu(])38 b(is)f(the)h(exp)s(ected)g(v)-5 b(alue)38 b(of)g Ft(X)45 b Fu(giv)m(en)38 b Ft(A)2064 3993 y Fm(i)2093 3979 y Fu(,)h(de\014ned)e(to)h(b)s(e)2830 3943 y Fn(1)p 2737 3958 223 4 v 2737 4011 a Fm(P)10 b(r)r Fn(\()p Fm(A)2906 4021 y Fb(i)2932 4011 y Fn(\))2984 3915 y Ff(P)3072 4002 y Fm(e)p Fh(2)p Fm(A)3205 4012 y Fb(i)3250 3979 y Fu(Pr\()p Ft(e)p Fu(\))p Ft(X)d Fu(\()p Ft(e)p Fu(\).)64 b(The)0 4104 y(form)m(ula)32 b(\(3.3\))h(will)e(b)s(e) g(useful)g(when)f(w)m(e)i(analyze)g(Quic)m(ksort.)45 b(In)30 b(particular,)i(note)g(that)g(the)g(running)d(time)0 4217 y(of)i(Randomized)f(Quic)m(ksort)h(is)f(a)h(random)f(v)-5 b(ariable,)31 b(and)f(our)g(goal)i(is)e(to)h(analyze)h(its)f(exp)s (ectation.)0 4515 y Fe(3.3.1)112 b(Linearit)m(y)39 b(of)e(Exp)s (ectation)0 4729 y Fu(An)25 b(imp)s(ortan)m(t)h(fact)h(ab)s(out)e(exp)s (ected)i(v)-5 b(alues)26 b(is)f(Linearit)m(y)i(of)f(Exp)s(ectation:)40 b(for)25 b(an)m(y)h(t)m(w)m(o)h(random)e(v)-5 b(ariables)0 4842 y Ft(X)41 b Fu(and)33 b Ft(Y)20 b Fu(,)35 b Fv(E)p Fu([)p Ft(X)30 b Fu(+)22 b Ft(Y)e Fu(])32 b(=)e Fv(E)p Fu([)p Ft(X)7 b Fu(])24 b(+)e Fv(E)p Fu([)p Ft(Y)e Fu(].)51 b(This)33 b(fact)i(is)f(incredibly)f(imp)s(ortan)m(t)h(for)g(analysis)g (of)g(algorithms)0 4954 y(b)s(ecause)k(it)g(allo)m(ws)h(us)e(to)i (analyze)g(a)f(complicated)i(random)d(v)-5 b(ariable)38 b(b)m(y)g(writing)g(it)g(as)g(a)g(sum)f(of)h(simple)0 5067 y(random)c(v)-5 b(ariables)35 b(and)e(then)h(separately)i (analyzing)g(these)e(simple)h(R)-10 b(Vs.)52 b(Let's)35 b(\014rst)f(pro)m(v)m(e)h(this)f(fact)i(and)0 5180 y(then)30 b(see)h(ho)m(w)f(it)h(can)g(b)s(e)f(used.)p 0 5254 1560 4 v 104 5308 a Fk(2)138 5340 y Fj(F)-6 b(or)26 b(a)g(con)n(tin)n(uous)f (distribution,)h(the)g(probabilit)n(y)f(w)n(ould)h(b)r(e)g(an)g(in)n (tegral.)p eop end %%Page: 18 19 TeXDict begin 18 18 bop 0 -165 a Fu(18)477 b Fs(LECTURE)29 b(3.)71 b(PR)m(OBABILISTIC)30 b(ANAL)-8 b(YSIS)30 b(AND)h(RANDOMIZED)g (QUICKSOR)-8 b(T)0 91 y Fv(Theorem)35 b(3.1)g(\(Linearit)m(y)g(of)g (Exp)s(ectation\))45 b Fo(F)-7 b(or)26 b(any)g(two)h(r)-5 b(andom)28 b(variables)e Ft(X)33 b Fo(and)26 b Ft(Y)20 b Fo(,)27 b Fv(E)p Fu([)p Ft(X)12 b Fu(+)5 b Ft(Y)20 b Fu(])25 b(=)0 204 y Fv(E)p Fu([)p Ft(X)7 b Fu(])21 b(+)f Fv(E)p Fu([)p Ft(Y)g Fu(])p Fo(.)0 434 y Fv(Pro)s(of)31 b Fu(\(for)g(discrete)g(R)-10 b(Vs\):)41 b(This)29 b(follo)m(ws)j (directly)f(from)f(the)g(de\014nition)g(as)h(giv)m(en)g(in)f(\(3.1\).) 184 640 y Fv(E)p Fu([)p Ft(X)d Fu(+)20 b Ft(Y)g Fu(])26 b(=)694 560 y Ff(X)691 743 y Fm(e)p Fh(2)p Fm(S)832 640 y Fu(Pr\()p Ft(e)p Fu(\)\()p Ft(X)7 b Fu(\()p Ft(e)p Fu(\))23 b(+)d Ft(Y)g Fu(\()p Ft(e)p Fu(\)\))26 b(=)1731 560 y Ff(X)1727 743 y Fm(e)p Fh(2)p Fm(S)1869 640 y Fu(Pr)o(\()p Ft(e)p Fu(\))p Ft(X)7 b Fu(\()p Ft(e)p Fu(\))23 b(+)2389 560 y Ff(X)2386 743 y Fm(e)p Fh(2)p Fm(S)2528 640 y Fu(Pr)o(\()p Ft(e)p Fu(\))p Ft(Y)e Fu(\()p Ft(e)p Fu(\))26 b(=)f Fv(E)p Fu([)p Ft(X)7 b Fu(])21 b(+)f Fv(E)p Fu([)p Ft(Y)g Fu(])p Ft(:)p 3666 640 50 59 v 0 1107 a Fe(3.3.2)112 b(Example)39 b(1:)50 b(Card)38 b(sh)m(u\017ing)0 1318 y Fu(Supp)s(ose)g(w)m(e)h(un)m (wrap)f(a)i(fresh)e(dec)m(k)i(of)g(cards)f(and)g(sh)m(u\017e)g(it)g(un) m(til)h(the)g(cards)f(are)g(completely)i(random.)0 1431 y(Ho)m(w)31 b(man)m(y)f(cards)g(do)g(w)m(e)g(exp)s(ect)h(to)g(b)s(e)e (in)h(the)g(same)h(p)s(osition)f(as)g(they)g(w)m(ere)h(at)g(the)f (start?)41 b(T)-8 b(o)30 b(solv)m(e)i(this,)0 1544 y(let's)f(think)f (formally)h(ab)s(out)g(what)f(w)m(e)h(are)g(asking.)41 b(W)-8 b(e)32 b(are)e(lo)s(oking)i(for)e(the)h(exp)s(ected)f(v)-5 b(alue)31 b(of)g(a)g(random)0 1657 y(v)-5 b(ariable)36 b Ft(X)42 b Fu(denoting)35 b(the)g(n)m(um)m(b)s(er)e(of)i(cards)g(that) g(end)g(in)f(the)h(same)g(p)s(osition)g(as)g(they)g(started.)55 b(W)-8 b(e)36 b(can)0 1770 y(write)c Ft(X)39 b Fu(as)31 b(a)h(sum)f(of)h(random)e(v)-5 b(ariables)32 b Ft(X)1623 1784 y Fm(i)1652 1770 y Fu(,)g(one)g(for)f(eac)m(h)i(card,)f(where)f Ft(X)2791 1784 y Fm(i)2846 1770 y Fu(=)c(1)32 b(if)g(the)f Ft(i)p Fu(th)h(card)g(ends)e(in)0 1883 y(p)s(osition)i Ft(i)h Fu(and)f Ft(X)666 1897 y Fm(i)722 1883 y Fu(=)d(0)j(otherwise.) 47 b(These)32 b Ft(X)1685 1897 y Fm(i)1746 1883 y Fu(are)g(easy)h(to)g (analyze:)46 b(Pr\()p Ft(X)2787 1897 y Fm(i)2844 1883 y Fu(=)28 b(1\))h(=)f(1)p Ft(=n)33 b Fu(where)f Ft(n)f Fu(is)i(the)0 1996 y(n)m(um)m(b)s(er)c(of)i(cards.)40 b(Pr\()p Ft(x)893 2010 y Fm(i)946 1996 y Fu(=)25 b(1\))31 b(is)g(also)g Fv(E)p Fu([)p Ft(X)1596 2010 y Fm(i)1625 1996 y Fu(].)40 b(No)m(w)32 b(w)m(e)e(use)g(linearit)m(y)i(of)f(exp)s (ectation:)888 2202 y Fv(E)p Fu([)p Ft(X)7 b Fu(])26 b(=)f Fv(E)p Fu([)p Ft(X)1380 2216 y Fn(1)1440 2202 y Fu(+)20 b Ft(:)15 b(:)g(:)21 b Fu(+)f Ft(X)1823 2216 y Fm(n)1870 2202 y Fu(])26 b(=)e Fv(E)p Fu([)p Ft(X)2185 2216 y Fn(1)2225 2202 y Fu(])d(+)f Ft(:)15 b(:)g(:)21 b Fu(+)f Fv(E)p Fu([)p Ft(X)2748 2216 y Fm(n)2795 2202 y Fu(])25 b(=)g(1)p Ft(:)0 2409 y Fu(So,)h(this)f(is)g(in)m(teresting:) 40 b(no)24 b(matter)i(ho)m(w)f(large)h(a)g(dec)m(k)f(w)m(e)h(are)f (considering,)i(the)e(exp)s(ected)g(n)m(um)m(b)s(er)f(of)h(cards)0 2522 y(that)31 b(end)f(in)g(the)g(same)h(p)s(osition)g(as)f(they)h (started)g(is)f(1.)0 2809 y Fe(3.3.3)112 b(Example)39 b(2:)50 b(In)m(v)m(ersions)38 b(in)g(a)g(random)g(p)s(erm)m(utation)0 3020 y Fu([hmm,)30 b(lets)h(lea)m(v)m(e)i(this)d(for)g(homew)m(ork])0 3350 y Fq(3.4)137 b(Analysis)47 b(of)e(Randomized)i(Quic)l(ksort)0 3593 y Fu(W)-8 b(e)33 b(no)m(w)e(giv)m(e)i(t)m(w)m(o)g(metho)s(ds)d (for)i(analyzing)g(randomized)g(quic)m(ksort.)44 b(The)31 b(\014rst)g(is)g(more)h(in)m(tuitiv)m(e)h(but)e(the)0 3706 y(details)37 b(are)f(messier.)58 b(The)35 b(second)h(is)g(a)h (neat)f(tric)m(ky)h(w)m(a)m(y)g(using)e(the)i(p)s(o)m(w)m(er)f(of)g (linearit)m(y)h(of)f(exp)s(ectation:)0 3819 y(this)30 b(will)h(b)s(e)f(less)g(in)m(tuitiv)m(e)j(but)c(the)i(details)g(come)g (out)g(nicer.)0 4105 y Fe(3.4.1)112 b(Metho)s(d)39 b(1)0 4317 y Fu(F)-8 b(or)39 b(simplicit)m(y)-8 b(,)42 b(let)d(us)f(assume)g (no)g(t)m(w)m(o)h(elemen)m(ts)h(in)e(the)g(arra)m(y)h(are)f(equal)h(|)f (when)f(w)m(e)i(are)f(done)g(with)0 4430 y(the)k(analysis,)j(it)d(will) g(b)s(e)g(easy)g(to)g(lo)s(ok)h(bac)m(k)f(and)f(see)h(that)h(allo)m (wing)g(equal)f(k)m(eys)h(could)e(only)h(impro)m(v)m(e)0 4543 y(p)s(erformance.)e(W)-8 b(e)32 b(no)m(w)e(pro)m(v)m(e)h(the)g (follo)m(wing)h(theorem.)0 4772 y Fv(Theorem)j(3.2)46 b Fo(The)32 b(exp)-5 b(e)g(cte)g(d)33 b(numb)-5 b(er)32 b(of)g(c)-5 b(omp)g(arisons)35 b(made)e(by)f(r)-5 b(andomize)g(d)35 b(quicksort)d(on)g(an)g(arr)-5 b(ay)34 b(of)0 4885 y(size)f Ft(n)f Fo(is)g(at)h(most)h Fu(2)p Ft(n)15 b Fu(ln)g Ft(n)p Fo(.)0 5114 y Fv(Pro)s(of:)50 b Fu(First)34 b(of)h(all,)i(when)c(w)m(e) i(pic)m(k)g(the)g(piv)m(ot,)i(w)m(e)e(p)s(erform)e Ft(n)22 b Fp(\000)h Fu(1)35 b(comparisons)f(\(comparing)h(all)h(other)0 5227 y(elemen)m(ts)e(to)f(it\))g(in)f(order)g(to)h(split)g(the)f(arra)m (y)-8 b(.)48 b(No)m(w,)34 b(dep)s(ending)d(on)h(the)g(piv)m(ot,)i(w)m (e)f(migh)m(t)g(split)g(the)f(arra)m(y)0 5340 y(in)m(to)e(a)g(LESS)e (of)i(size)g(0)g(and)e(a)i(GREA)-8 b(TER)29 b(of)h(size)g Ft(n)18 b Fp(\000)g Fu(1,)30 b(or)f(in)m(to)i(a)e(LESS)f(of)i(size)g(1) g(and)f(a)g(GREA)-8 b(TER)30 b(of)p eop end %%Page: 19 20 TeXDict begin 19 19 bop 0 -165 a Fs(3.4.)72 b(ANAL)-8 b(YSIS)30 b(OF)h(RANDOMIZED)f(QUICKSOR)-8 b(T)1718 b Fu(19)0 91 y(size)27 b Ft(n)13 b Fp(\000)g Fu(2,)27 b(and)f(so)h(on,)g (up)f(to)h(a)g(LESS)f(of)g(size)i Ft(n)13 b Fp(\000)g Fu(1)25 b(and)h(a)h(GREA)-8 b(TER)27 b(of)g(size)g(0.)40 b(All)27 b(of)g(these)g(are)g(equally)0 204 y(lik)m(ely)32 b(with)f(probabilit)m(y)g(1)p Ft(=n)g Fu(eac)m(h.)44 b(Therefore,)30 b(w)m(e)i(can)f(write)g(a)g(recurrence)g(for)f(the)h (exp)s(ected)g(n)m(um)m(b)s(er)f(of)0 317 y(comparisons)g Ft(T)13 b Fu(\()p Ft(n)p Fu(\))31 b(as)f(follo)m(ws:)996 601 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b(=)g(\()p Ft(n)20 b Fp(\000)g Fu(1\))h(+)1831 540 y(1)p 1827 580 55 4 v 1827 663 a Ft(n)1906 495 y Fm(n)p Fh(\000)p Fn(1)1913 520 y Ff(X)1916 703 y Fm(i)p Fn(=0)2039 601 y Fu(\()p Ft(T)13 b Fu(\()p Ft(i)p Fu(\))22 b(+)d Ft(T)13 b Fu(\()p Ft(n)20 b Fp(\000)g Ft(i)h Fp(\000)f Fu(1\)\))p Ft(:)810 b Fu(\(3.4\))0 883 y(F)-8 b(ormally)g(,)31 b(w)m(e)e(are)f(using)g(the) h(expression)f(for)g(Exp)s(ectation)h(giv)m(en)h(in)e(\(3.3\),)i(where) e(the)h Ft(n)f Fu(di\013eren)m(t)g(p)s(ossible)0 996 y(splits)i(are)h(the)g(ev)m(en)m(ts)g Ft(A)890 1010 y Fm(i)919 996 y Fu(.)944 963 y Fn(3)1024 996 y Fu(W)-8 b(e)31 b(can)g(rewrite)g(equation)g(\(3.4\))h(b)m(y)e(regrouping)g(and) g(getting)i(rid)e(of)g Ft(T)13 b Fu(\(0\):)1337 1280 y Ft(T)g Fu(\()p Ft(n)p Fu(\))83 b(=)g(\()p Ft(n)20 b Fp(\000)g Fu(1\))h(+)2172 1219 y(2)p 2168 1259 V 2168 1343 a Ft(n)2247 1174 y Fm(n)p Fh(\000)p Fn(1)2254 1199 y Ff(X)2257 1382 y Fm(i)p Fn(=1)2395 1280 y Ft(T)13 b Fu(\()p Ft(i)p Fu(\))1151 b(\(3.5\))0 1557 y(No)m(w,)32 b(w)m(e)g(can)g(solv)m(e)g(this)f(b)m(y)g(the)h(\\guess)f(and)g(pro)m (v)m(e)h(inductiv)m(ely")h(metho)s(d.)42 b(In)31 b(order)f(to)i(do)f (this,)h(w)m(e)g(\014rst)0 1670 y(need)d(a)g(go)s(o)s(d)g(guess.)40 b(In)m(tuitiv)m(ely)-8 b(,)32 b(most)d(piv)m(ots)h(should)e(split)i (their)f(arra)m(y)g(\\roughly")h(in)f(the)g(middle,)g(whic)m(h)0 1783 y(suggests)i(a)g(guess)f(of)h(the)f(form)g Ft(cn)15 b Fu(ln)g Ft(n)30 b Fu(for)g(some)h(constan)m(t)h Ft(c)p Fu(.)40 b(Once)31 b(w)m(e'v)m(e)h(made)e(our)g(guess,)h(w)m(e)f(will)h (need)0 1896 y(to)i(ev)-5 b(aluate)34 b(the)f(resulting)g(summation.)46 b(One)32 b(of)h(the)g(easiest)h(w)m(a)m(ys)f(of)f(doing)h(this)f(is)h (to)g(upp)s(er-b)s(ound)c(the)0 2009 y(sum)g(b)m(y)i(an)f(in)m(tegral.) 42 b(In)30 b(particular)h(if)f Ft(f)10 b Fu(\()p Ft(x)p Fu(\))30 b(is)h(an)f(increasing)h(function,)f(then)1456 2187 y Fm(n)p Fh(\000)p Fn(1)1463 2212 y Ff(X)1465 2394 y Fm(i)p Fn(=1)1604 2293 y Ft(f)10 b Fu(\()p Ft(i)p Fu(\))84 b Fp(\024)1997 2178 y Ff(Z)2080 2204 y Fm(n)2043 2367 y Fn(1)2142 2293 y Ft(f)10 b Fu(\()p Ft(x)p Fu(\))p Ft(dx;)0 2570 y Fu(whic)m(h)27 b(w)m(e)h(can)g(see)g(b)m(y)f(dra)m(wing)h(a)g (graph)f(of)g Ft(f)37 b Fu(and)27 b(recalling)i(that)f(an)f(in)m (tegral)j(represen)m(ts)d(the)h(\\area)g(under)0 2683 y(the)j(curv)m(e".)41 b(In)30 b(our)g(case,)h(w)m(e)g(will)g(b)s(e)f (using)f(the)i(fact)g(that)2173 2614 y Ff(R)2229 2683 y Fu(\()p Ft(cx)15 b Fu(ln)g Ft(x)p Fu(\))p Ft(dx)26 b Fu(=)f(\()p Ft(c=)p Fu(2\))p Ft(x)3020 2650 y Fn(2)3077 2683 y Fu(ln)15 b Ft(x)20 b Fp(\000)g Ft(cx)3422 2650 y Fn(2)3461 2683 y Ft(=)p Fu(4.)0 2835 y(So,)34 b(let's)g(no)m(w)f(do)g (the)g(analysis.)49 b(W)-8 b(e)34 b(are)g(guessing)f(that)h Ft(T)13 b Fu(\()p Ft(i)p Fu(\))30 b Fp(\024)f Ft(ci)15 b Fu(ln)h Ft(i)33 b Fu(for)g Ft(i)d Fp(\024)g Ft(n)21 b Fp(\000)h Fu(1.)49 b(This)32 b(guess)h(w)m(orks)0 2948 y(for)f(the)g(base)g(case)h Ft(T)13 b Fu(\(1\))29 b(=)f(0)k(\(if)g (there)g(is)g(only)h(one)f(elemen)m(t,)i(then)e(there)g(are)g(no)g (comparisons\).)46 b(Arguing)0 3061 y(b)m(y)30 b(induction)g(w)m(e)h (ha)m(v)m(e:)922 3340 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(\024)f Fu(\()p Ft(n)20 b Fp(\000)g Fu(1\))h(+)1757 3278 y(2)p 1752 3319 V 1752 3402 a Ft(n)1832 3234 y Fm(n)p Fh(\000)p Fn(1)1838 3259 y Ff(X)1841 3441 y Fm(i)p Fn(=1)1965 3340 y Fu(\()p Ft(ci)15 b Fu(ln)h Ft(i)p Fu(\))1196 3603 y Fp(\024)82 b Fu(\()p Ft(n)20 b Fp(\000)g Fu(1\))h(+)1757 3541 y(2)p 1752 3582 V 1752 3665 a Ft(n)1832 3488 y Ff(Z)1915 3514 y Fm(n)1878 3677 y Fn(1)1962 3603 y Fu(\()p Ft(cx)15 b Fu(ln)h Ft(x)p Fu(\))p Ft(dx)1196 3830 y Fp(\024)82 b Fu(\()p Ft(n)20 b Fp(\000)g Fu(1\))h(+)1757 3768 y(2)p 1752 3809 V 1752 3892 a Ft(n)1832 3736 y Ff(\020)1881 3830 y Fu(\()p Ft(c=)p Fu(2\))p Ft(n)2135 3792 y Fn(2)2192 3830 y Fu(ln)14 b Ft(n)20 b Fp(\000)g Ft(cn)2542 3792 y Fn(2)2581 3830 y Ft(=)p Fu(4)i(+)e Ft(c=)p Fu(4)2913 3736 y Ff(\021)1196 3989 y Fp(\024)82 b Ft(cn)15 b Fu(ln)g Ft(n;)96 b Fu(for)30 b Ft(c)c Fu(=)f(2)p Ft(:)p 2186 3989 50 59 v 0 4237 a Fu(In)k(terms)g(of)g(the)h(n)m(um)m(b)s(er)e(of)h (comparisons)h(it)f(mak)m(es,)i(Randomized)e(Quic)m(ksort)h(is)g(equiv) -5 b(alen)m(t)30 b(to)g(randomly)0 4350 y(sh)m(u\017ing)g(the)g(input)g (and)g(then)g(handing)g(it)h(o\013)g(to)g(Basic)h(Quic)m(ksort.)42 b(So,)30 b(w)m(e)h(ha)m(v)m(e)h(also)g(pro)m(v)m(en)f(that)g(Basic)0 4463 y(Quic)m(ksort)g(has)f Ft(O)s Fu(\()p Ft(n)15 b Fu(log)j Ft(n)p Fu(\))30 b Fo(aver)-5 b(age-c)g(ase)31 b Fu(running)e(time.)0 4753 y Fe(3.4.2)112 b(Metho)s(d)39 b(2)0 4965 y Fu(Here)i(is)g(a)g(neat)h(alternativ)m(e)h(w)m(a)m(y)f(to) f(analyze)h(randomized)f(quic)m(ksort)h(that)f(is)g(v)m(ery)g(similar)g (to)h(ho)m(w)e(w)m(e)0 5078 y(analyzed)31 b(the)g(card-sh)m(u\017ing)f (example.)p 0 5163 1560 4 v 104 5217 a Fk(3)138 5249 y Fj(In)e(addition,)i(w)n(e)f(are)h(using)f(Linearit)n(y)f(of)i(Exp)r (ectation)f(to)g(sa)n(y)g(that)f(the)g(exp)r(ected)g(time)h Fa(given)h Fj(one)e(of)i(these)f(ev)n(en)n(ts)f(can)0 5340 y(b)r(e)e(written)g(as)g(the)f(sum)h(of)g(t)n(w)n(o)g(exp)r (ectations.)p eop end %%Page: 20 21 TeXDict begin 20 20 bop 0 -165 a Fu(20)477 b Fs(LECTURE)29 b(3.)71 b(PR)m(OBABILISTIC)30 b(ANAL)-8 b(YSIS)30 b(AND)h(RANDOMIZED)g (QUICKSOR)-8 b(T)0 91 y Fv(Theorem)35 b(3.3)46 b Fo(F)-7 b(or)33 b(R)-5 b(andomize)g(d)35 b(Quicksort,)d(the)g(exp)-5 b(e)g(cte)g(d)33 b(numb)-5 b(er)32 b(of)g(c)-5 b(omp)g(arisons)36 b(is)31 b(at)i(most)f Fu(2)p Ft(n)15 b Fu(ln)h Ft(n)p Fo(.)0 321 y Fv(Pro)s(of:)39 b Fu(As)27 b(b)s(efore,)g(let's)h(assume)e (no)g(t)m(w)m(o)i(elemen)m(ts)g(in)e(the)h(arra)m(y)g(are)g(equal)g (since)g(it)g(is)f(the)h(w)m(orst)g(case)h(and)0 434 y(will)35 b(mak)m(e)h(our)f(notation)h(simpler.)54 b(The)35 b(tric)m(k)h(will)f(b)s(e)f(to)i(write)f(the)g(quan)m(tit)m(y)i(w)m(e)e (care)h(ab)s(out)f(\(the)g(total)0 547 y(n)m(um)m(b)s(er)d(of)h (comparisons\))g(as)h(a)f(sum)f(of)h(simpler)g(random)f(v)-5 b(ariables,)35 b(and)d(then)h(just)g(analyze)h(the)f(simpler)0 659 y(ones.)0 812 y(De\014ne)27 b(random)f(v)-5 b(ariable)28 b Ft(X)1021 826 y Fm(ij)1109 812 y Fu(to)g(b)s(e)e(1)i(if)f(the)g (algorithm)h Fo(do)-5 b(es)28 b Fu(compare)g(the)f Ft(i)p Fu(th)g(smallest)h(and)e Ft(j)5 b Fu(th)28 b(smallest)0 924 y(elemen)m(ts)c(in)f(the)g(course)g(of)g(sorting,)i(and)d(0)h(if)g (it)g(do)s(es)g(not.)38 b(Let)23 b Ft(X)30 b Fu(denote)24 b(the)f(total)h(n)m(um)m(b)s(er)e(of)h(comparisons)0 1037 y(made)30 b(b)m(y)h(the)f(algorithm.)42 b(Since)30 b(w)m(e)h(nev)m(er)g(compare)g(the)f(same)h(pair)f(of)h(elemen)m(ts)g (t)m(wice,)i(w)m(e)d(ha)m(v)m(e)1534 1278 y Ft(X)90 b Fu(=)1892 1172 y Fm(n)1853 1197 y Ff(X)1856 1380 y Fm(i)p Fn(=1)2068 1172 y Fm(n)2029 1197 y Ff(X)1988 1380 y Fm(j)t Fn(=)p Fm(i)p Fn(+1)2205 1278 y Ft(X)2280 1292 y Fm(ij)2341 1278 y Ft(;)0 1545 y Fu(and)30 b(therefore,)1415 1785 y Fv(E)p Fu([)p Ft(X)7 b Fu(])83 b(=)1892 1680 y Fm(n)1853 1705 y Ff(X)1856 1887 y Fm(i)p Fn(=1)2068 1680 y Fm(n)2029 1705 y Ff(X)1988 1887 y Fm(j)t Fn(=)p Fm(i)p Fn(+1)2205 1785 y Fv(E)p Fu([)p Ft(X)2374 1799 y Fm(ij)2435 1785 y Fu(])p Ft(:)0 2052 y Fu(Let)29 b(us)e(consider)h(one)h(of)f(these)h Ft(X)1199 2066 y Fm(ij)1260 2052 y Fu('s)f(for)g Ft(i)e(<)f(j)5 b Fu(.)40 b(Denote)30 b(the)e Ft(i)p Fu(th)h(smallest)g(elemen)m(t)h (in)e(the)g(arra)m(y)h(b)m(y)f Ft(e)3697 2066 y Fm(i)3754 2052 y Fu(and)0 2165 y(the)35 b Ft(j)5 b Fu(th)36 b(smallest)g(elemen)m (t)g(b)m(y)f Ft(e)1192 2179 y Fm(j)1229 2165 y Fu(,)i(and)d (conceptually)i(imagine)g(lining)g(up)e(the)h(elemen)m(ts)h(in)f (sorted)g(order.)0 2278 y(If)30 b(the)i(piv)m(ot)f(w)m(e)h(c)m(ho)s (ose)g(is)f(b)s(et)m(w)m(een)g Ft(e)1390 2292 y Fm(i)1450 2278 y Fu(and)f Ft(e)1669 2292 y Fm(j)1737 2278 y Fu(then)g(these)i(t)m (w)m(o)g(end)e(up)g(in)h(di\013eren)m(t)g(buc)m(k)m(ets)h(and)e(w)m(e)i (will)0 2391 y(nev)m(er)j(compare)h(them)f(to)h(eac)m(h)g(other.)55 b(If)35 b(the)g(piv)m(ot)h(w)m(e)f(c)m(ho)s(ose)h Fo(is)f Fu(either)h Ft(e)2795 2405 y Fm(i)2858 2391 y Fu(or)f Ft(e)3016 2405 y Fm(j)3088 2391 y Fu(then)g(w)m(e)h Fo(do)f Fu(compare)0 2503 y(them.)40 b(If)29 b(the)h(piv)m(ot)g(is)f(less)h (than)f Ft(e)1254 2517 y Fm(i)1312 2503 y Fu(or)g(greater)i(than)e Ft(e)1983 2517 y Fm(j)2049 2503 y Fu(then)g(b)s(oth)f Ft(e)2510 2517 y Fm(i)2568 2503 y Fu(and)h Ft(e)2786 2517 y Fm(j)2852 2503 y Fu(end)f(up)h(in)g(the)g(same)h(buc)m(k)m(et)0 2616 y(and)36 b(w)m(e)i(ha)m(v)m(e)g(to)g(pic)m(k)f(another)g(piv)m (ot.)62 b(So,)39 b(w)m(e)e(can)g(think)g(of)g(this)g(lik)m(e)h(a)g (dart)e(game:)55 b(w)m(e)38 b(thro)m(w)f(a)g(dart)0 2729 y(at)c(random)f(in)m(to)i(the)f(arra)m(y:)45 b(if)33 b(w)m(e)g(hit)g Ft(e)1480 2743 y Fm(i)1541 2729 y Fu(or)f Ft(e)1696 2743 y Fm(j)1766 2729 y Fu(then)g Ft(X)2050 2743 y Fm(ij)2143 2729 y Fu(b)s(ecomes)h(1,)h(if)f(w)m(e)g(hit)f(b)s (et)m(w)m(een)i Ft(e)3372 2743 y Fm(i)3433 2729 y Fu(and)e Ft(e)3654 2743 y Fm(j)3723 2729 y Fu(then)0 2842 y Ft(X)75 2856 y Fm(ij)170 2842 y Fu(b)s(ecomes)j(0,)h(and)e(otherwise)g(w)m(e)h (thro)m(w)g(another)f(dart.)53 b(Therefore,)35 b(the)g(probabilit)m(y)g (that)g Ft(X)3564 2856 y Fm(ij)3656 2842 y Fu(=)d(1)j(is)0 2955 y(exactly)d(2)p Ft(=)p Fu(\()p Ft(j)27 b Fp(\000)20 b Ft(i)h Fu(+)f(1\).)41 b(So,)31 b(w)m(e)f(ha)m(v)m(e:)1303 3201 y Fv(E)p Fu([)p Ft(X)7 b Fu(])84 b(=)1780 3095 y Fm(n)1742 3120 y Ff(X)1745 3302 y Fm(i)p Fn(=1)1956 3095 y Fm(n)1918 3120 y Ff(X)1877 3302 y Fm(j)t Fn(=)p Fm(i)p Fn(+1)2093 3201 y Fu(2)p Ft(=)p Fu(\()p Ft(j)28 b Fp(\000)19 b Ft(i)i Fu(+)f(1\))1588 3515 y(=)83 b(2)1841 3409 y Fm(n)1802 3434 y Ff(X)1805 3616 y Fm(i)p Fn(=1)1937 3409 y Fm(n)p Fh(\000)p Fm(i)p Fn(+1)1983 3434 y Ff(X)1980 3619 y Fm(d)p Fn(=2)2164 3515 y Fu(1)p Ft(=d)1588 3803 y(<)g Fu(2)p Ft(n)1899 3697 y Fm(n)1860 3722 y Ff(X)1857 3906 y Fm(d)p Fn(=2)1999 3803 y Fu(1)p Ft(=d:)0 4063 y Fu(The)31 b(quan)m(tit)m(y)i(1)22 b(+)e(1)p Ft(=)p Fu(2)j(+)e(1)p Ft(=)p Fu(3)h(+)f Ft(:)15 b(:)g(:)22 b Fu(+)f(1)p Ft(=n)p Fu(,)32 b(denoted)g Ft(H)2052 4077 y Fm(n)2098 4063 y Fu(,)h(is)e(called)i(the)f(\\)p Ft(n)p Fu(th)g(harmonic)f(n)m(um)m(b)s(er")g(and)g(is)0 4176 y(in)e(the)g(range)h([ln)15 b Ft(n;)g Fu(1)j(+)f(ln)e Ft(n)p Fu(])29 b(\(this)g(can)h(b)s(e)e(seen)i(b)m(y)f(considering)g (the)g(in)m(tegral)i(of)e Ft(f)10 b Fu(\()p Ft(x)p Fu(\))26 b(=)e(1)p Ft(=x)p Fu(\).)42 b(Therefore,)1239 4361 y Fv(E)p Fu([)p Ft(X)7 b Fu(])51 b Ft(<)g Fu(2)p Ft(n)p Fu(\()p Ft(H)1824 4375 y Fm(n)1891 4361 y Fp(\000)20 b Fu(1\))51 b Fp(\024)f Fu(2)p Ft(n)15 b Fu(ln)g Ft(n:)p 2611 4361 50 59 v 0 4771 a Fq(3.5)137 b(F)-11 b(urther)46 b(Discussion)0 5016 y Fe(3.5.1)112 b(More)38 b(linearit)m(y)g(of)g(exp) s(ectation:)50 b(a)38 b(random)h(w)m(alk)f(sto)s(c)m(k)f(mark)m(et)0 5227 y Fu(Supp)s(ose)29 b(there)j(is)f(a)h(sto)s(c)m(k)g(with)f(the)g (prop)s(ert)m(y)g(that)g(eac)m(h)i(da)m(y)-8 b(,)32 b(it)g(has)f(a)g (50:50)i(c)m(hance)g(of)e(going)h(either)g(up)0 5340 y(or)f(do)m(wn)g(b)m(y)h($1,)g(unless)f(the)g(sto)s(c)m(k)i(is)e(at)h (0)g(in)f(whic)m(h)g(case)i(it)f(sta)m(ys)g(there.)44 b(Y)-8 b(ou)32 b(start)g(with)f($m.)44 b(Eac)m(h)32 b(da)m(y)p eop end %%Page: 21 22 TeXDict begin 21 21 bop 0 -165 a Fs(3.5.)72 b(FUR)-8 b(THER)31 b(DISCUSSION)2556 b Fu(21)0 91 y(y)m(ou)32 b(can)f(buy)g(or)g(sell)h(as)g(m)m(uc)m(h)f(as)h(y)m(ou)f(w)m(an)m(t,)i (un)m(til)f(at)g(the)f(end)g(of)g(the)h(y)m(ear)g(all)g(y)m(our)g (money)f(is)g(con)m(v)m(erted)0 204 y(bac)m(k)g(in)m(to)g(cash.)41 b(What)31 b(is)g(the)f(b)s(est)g(strategy)i(for)e(maximizing)i(y)m(our) e(exp)s(ected)h(gain?)0 356 y(The)d(answ)m(er)h(is)f(that)i(no)e (matter)i(what)e(strategy)j(y)m(ou)e(c)m(ho)s(ose,)h(y)m(our)e(exp)s (ected)i(gain)f(b)m(y)f(the)h(end)f(of)h(the)g(y)m(ear)0 469 y(is)h(0)g(\(i.e.,)i(y)m(ou)e(exp)s(ect)g(to)h(end)e(with)h(the)g (same)g(amoun)m(t)h(of)f(money)g(as)g(y)m(ou)g(started\).)41 b(Let's)31 b(pro)m(v)m(e)g(that)f(this)0 582 y(is)g(the)h(case.)0 734 y(De\014ne)g(random)g(v)-5 b(ariable)32 b Ft(X)1034 748 y Fm(t)1095 734 y Fu(to)g(b)s(e)f(the)g(gain)h(of)f(our)g (algorithm)h(on)g(da)m(y)f Ft(t)p Fu(.)43 b(Let)32 b Ft(X)39 b Fu(b)s(e)30 b(the)i(o)m(v)m(erall)h(gain)f(at)0 847 y(the)f(end)e(of)i(the)f(y)m(ear.)42 b(Then,)1464 1051 y Ft(X)90 b Fu(=)83 b Ft(X)1858 1065 y Fn(1)1918 1051 y Fu(+)20 b Ft(:)15 b(:)g(:)21 b Fu(+)f Ft(X)2301 1065 y Fn(365)2411 1051 y Ft(:)0 1256 y Fu(Notice)33 b(that)e(the)g Ft(X)715 1270 y Fm(t)745 1256 y Fu('s)f(can)h(b)s(e)f (highly)h(dep)s(enden)m(t,)f(based)g(on)g(our)h(strategy)-8 b(.)43 b(F)-8 b(or)31 b(instance,)h(if)e(our)h(strategy)0 1368 y(is)f(to)g(pull)g(all)g(our)g(money)g(out)g(of)g(the)g(sto)s(c)m (k)h(mark)m(et)f(the)g(momen)m(t)h(that)f(our)g(w)m(ealth)h(exceeds)f ($m,)h(then)e Ft(X)3860 1382 y Fn(2)0 1481 y Fu(dep)s(ends)g(strongly)h (on)h(the)f(outcome)i(of)e Ft(X)1528 1495 y Fn(1)1568 1481 y Fu(.)41 b(Nonetheless,)31 b(b)m(y)g(linearit)m(y)h(of)e(exp)s (ectation,)1285 1686 y Fv(E)p Fu([)p Ft(X)7 b Fu(])83 b(=)g Fv(E)p Fu([)p Ft(X)1892 1700 y Fn(1)1932 1686 y Fu(])20 b(+)g Ft(:)15 b(:)g(:)22 b Fu(+)d Fv(E)p Fu([)p Ft(X)2454 1700 y Fn(365)2565 1686 y Fu(])p Ft(:)0 1890 y Fu(Finally)-8 b(,)41 b(no)c(matter)h(ho)m(w)f(man)m(y)g(shares)g Ft(s)g Fu(of)g(sto)s(c)m(k)h(w)m(e)g(hold)f(at)h(time)g Ft(t)p Fu(,)h Fv(E)p Fu([)p Ft(X)2885 1904 y Fm(t)2914 1890 y Fp(j)p Ft(s)p Fu(])e(=)g(0.)61 b(So,)39 b(using)e(\(3.3\),)0 2003 y(whatev)m(er)29 b(probabilit)m(y)h(distribution)e(o)m(v)m(er)i Ft(s)e Fu(is)g(induced)g(b)m(y)h(our)f(strategy)-8 b(,)31 b Fv(E)p Fu([)p Ft(X)2857 2017 y Fm(t)2887 2003 y Fu(])25 b(=)g(0.)41 b(Since)28 b(this)h(holds)f(for)0 2116 y(ev)m(ery)j Ft(t)p Fu(,)g(w)m(e)f(ha)m(v)m(e)i Fv(E)p Fu([)p Ft(X)7 b Fu(])26 b(=)f(0.)0 2268 y(This)31 b(analysis)i(can)f(b)s(e)g (generalized)h(to)g(the)g(case)g(of)f(gam)m(bling)h(in)f(a)g(\\fair)h (casino".)47 b(In)31 b(a)i(fair)f(casino,)i(there)0 2381 y(are)k(a)g(n)m(um)m(b)s(er)e(of)i(games)g(with)f(di\013eren)m(t)h (kinds)f(of)h(pa)m(y)m(o\013s,)j(but)36 b(eac)m(h)j(one)f(has)f(the)h (prop)s(ert)m(y)f(that)h(y)m(our)0 2494 y(exp)s(ected)27 b(gain)g(for)f(pla)m(ying)h(it)g(is)f(zero.)40 b(E.g.,)29 b(there)d(migh)m(t)h(b)s(e)f(a)h(game)g(where)f(with)g(probabilit)m(y)h (99)p Ft(=)p Fu(100)i(y)m(ou)0 2606 y(lose)j(but)f(with)f(probabilit)m (y)i(1)p Ft(=)p Fu(100)i(y)m(ou)d(win)g(99)h(times)f(y)m(our)h(b)s(et.) 42 b(In)31 b(that)h(case,)g(no)f(matter)h(what)g(strategy)0 2719 y(y)m(ou)c(use)e(for)h(whic)m(h)g(game)h(to)g(pla)m(y)g(and)f(ho)m (w)g(m)m(uc)m(h)g(to)h(b)s(et,)g(the)f(exp)s(ected)h(amoun)m(t)f(of)g (money)h(y)m(ou)f(will)h(ha)m(v)m(e)0 2832 y(at)j(the)g(end)e(of)i(the) f(da)m(y)h(is)g(the)f(same)h(as)g(the)f(amoun)m(t)h(y)m(ou)g(had)e (going)j(in.)0 3115 y Fe(3.5.2)112 b(Y)-9 b(et)37 b(another)h(w)m(a)m (y)f(to)g(analyze)i(quic)m(ksort:)50 b(run)38 b(it)f(bac)m(kw)m(ards)0 3326 y Fu(Here's)k(another)g(w)m(a)m(y)h(to)g(analyze)g(quic)m(ksort)f (|)g(run)e(the)i(algorithm)h(bac)m(kw)m(ards.)72 b(Actually)-8 b(,)46 b(to)41 b(do)g(this)0 3438 y(analysis,)30 b(it)g(is)f(b)s(etter) h(to)g(think)f(of)g(a)h(v)m(ersion)f(of)h(Quic)m(ksort)g(that)f (instead)h(of)f(b)s(eing)g(recursiv)m(e,)h(at)g(eac)m(h)h(step)0 3551 y(it)g(pic)m(ks)h(a)f(random)f(buc)m(k)m(et)h(in)g(prop)s(ortion)f (to)h(its)h(size)f(to)h(w)m(ork)f(on)f(next.)42 b(The)31 b(reason)g(this)f(v)m(ersion)i(is)e(nice)0 3664 y(is)i(that)h(if)f(y)m (ou)h(imagine)g(w)m(atc)m(hing)h(the)e(piv)m(ots)h(get)h(c)m(hosen)e (and)g(where)g(they)g(w)m(ould)g(b)s(e)g(on)g(a)g(sorted)h(arra)m(y)-8 b(,)0 3777 y(they)34 b(are)g(coming)g(in)f(completely)j(at)e(random.)50 b(Lo)s(oking)34 b(at)g(the)g(algorithm)h(run)d(bac)m(kw)m(ards,)j(at)f (a)g(generic)0 3890 y(p)s(oin)m(t)c(in)h(time,)g(w)m(e)g(ha)m(v)m(e)h Ft(k)h Fu(piv)m(ots)f(\(pro)s(ducing)d Ft(k)24 b Fu(+)c(1)31 b(buc)m(k)m(ets\))g(and)f(w)m(e)h(\\undo")f(one)h(of)g(our)f(piv)m(ot)h (c)m(hoices)0 4003 y(at)g(random,)g(merging)f(the)h(t)m(w)m(o)h (adjoining)f(buc)m(k)m(ets.)42 b([The)31 b(tric)m(ky)g(part)g(here)f (is)h(sho)m(wing)g(that)g(this)f(is)h(really)0 4116 y(a)37 b(legitimate)i(w)m(a)m(y)e(of)g(lo)s(oking)g(at)g(Quic)m(ksort)g(in)f (rev)m(erse.])59 b(The)36 b(cost)i(for)e(an)g(undo)f(op)s(eration)i(is) f(the)h(sum)0 4229 y(of)f(the)h(sizes)g(of)f(the)h(t)m(w)m(o)g(buc)m(k) m(ets)g(joined)f(\(since)h(this)f(w)m(as)h(the)f(n)m(um)m(b)s(er)f(of)i (comparisons)f(needed)g(to)h(split)0 4342 y(them\).)53 b(Notice)36 b(that)f(for)f(eac)m(h)h(undo)e(op)s(eration,)j(if)f(y)m (ou)f(sum)g(the)g(costs)h(o)m(v)m(er)h(all)f(of)f(the)h Ft(k)i Fu(p)s(ossible)d(piv)m(ot)0 4455 y(c)m(hoices,)h(y)m(ou)d(coun)m (t)h(eac)m(h)h(buc)m(k)m(et)f(t)m(wice)h(\(or)e(once)h(if)f(it)h(is)f (the)h(leftmost)g(or)f(righ)m(tmost\))i(and)e(get)h(a)g(total)h(of)0 4568 y Ft(<)25 b Fu(2)p Ft(n)p Fu(.)40 b(Since)28 b(w)m(e)g(are)g(pic)m (king)h(one)f(of)g(these)g Ft(k)j Fu(p)s(ossibilities)e(at)f(random,)g (the)g Fo(exp)-5 b(e)g(cte)g(d)30 b Fu(cost)f(is)e(2)p Ft(n=k)s Fu(.)41 b(So,)28 b(w)m(e)0 4681 y(get)152 4616 y Ff(P)239 4703 y Fm(k)297 4681 y Fu(2)p Ft(n=k)h Fu(=)c(2)p Ft(nH)790 4695 y Fm(n)836 4681 y Fu(.)p eop end %%Page: 22 23 TeXDict begin 22 22 bop 0 -165 a Fu(22)477 b Fs(LECTURE)29 b(3.)71 b(PR)m(OBABILISTIC)30 b(ANAL)-8 b(YSIS)30 b(AND)h(RANDOMIZED)g (QUICKSOR)-8 b(T)p eop end %%Page: 23 24 TeXDict begin 23 23 bop 0 753 a Fr(Lecture)65 b(4)0 1207 y Fw(Selection)79 b(\(det)g(&)g(rand\):)106 b(\014nding)79 b(the)0 1456 y(median)g(in)h(linear)f(time)0 1977 y Fq(4.1)137 b(Ov)l(erview)0 2221 y Fu(Giv)m(en)26 b(an)f(unsorted)f(arra)m(y)-8 b(,)27 b(ho)m(w)e(quic)m(kly)h(can)f(one)g(\014nd)f(the)h(median)g (elemen)m(t?)40 b(Can)24 b(one)i(do)f(it)g(more)g(quic)m(kly)0 2334 y(than)f(b)m(y)g(sorting?)39 b(This)23 b(w)m(as)i(an)f(op)s(en)f (question)i(for)f(some)g(time,)j(solv)m(ed)e(a\016rmativ)m(ely)g(in)f (1972)i(b)m(y)e(\(Man)m(uel\))0 2447 y(Blum,)31 b(Flo)m(yd,)g(Pratt,)h (Riv)m(est,)g(and)e(T)-8 b(arjan.)41 b(In)29 b(this)i(lecture)g(w)m(e)g (describ)s(e)f(t)m(w)m(o)i(linear-time)g(algorithms)f(for)0 2559 y(this)41 b(problem:)61 b(one)41 b(randomized)g(and)g(one)g (deterministic.)73 b(More)42 b(generally)-8 b(,)45 b(w)m(e)d(solv)m(e)g (the)f(problem)g(of)0 2672 y(\014nding)29 b(the)i Ft(k)s Fu(th)f(smallest)h(out)g(of)g(an)f(unsorted)f(arra)m(y)i(of)g Ft(n)f Fu(elemen)m(ts.)0 3005 y Fq(4.2)137 b(The)46 b(problem)g(and)g (a)g(randomized)g(solution)0 3249 y Fu(A)23 b(related)i(problem)e(to)h (sorting)f(is)h(the)f(problem)g(of)h(\014nding)e(the)h Ft(k)s Fu(th)h(smallest)g(elemen)m(t)h(in)e(an)g(unsorted)g(arra)m(y)-8 b(.)0 3362 y(\(Let's)27 b(sa)m(y)g(all)g(elemen)m(ts)g(are)g(distinct)f (to)h(a)m(v)m(oid)h(the)e(question)g(of)h(what)f(w)m(e)g(mean)g(b)m(y)g (the)h Ft(k)s Fu(th)f(smallest)h(when)0 3475 y(w)m(e)k(ha)m(v)m(e)i (equalities\).)44 b(One)31 b(w)m(a)m(y)h(to)f(solv)m(e)i(this)d (problem)h(is)g(to)g(sort)h(and)e(then)h(output)f(the)h Ft(k)s Fu(th)g(elemen)m(t.)44 b(Is)0 3588 y(there)27 b(something)g(faster)h({)f(a)g(linear-time)h(algorithm?)41 b(The)26 b(answ)m(er)h(is)g(y)m(es.)40 b(W)-8 b(e)28 b(will)f(explore)h(b)s(oth)e(a)h(simple)0 3701 y(randomized)j(solution) h(and)f(a)h(more)f(complicated)i(deterministic)f(one.)0 3853 y(The)j(idea)g(for)g(the)g(randomized)g(algorithm)h(is)f(to)h (notice)h(that)e(in)g(Randomized-Quic)m(ksort,)i(after)f(the)f(par-)0 3966 y(titioning)f(step)f(w)m(e)h(can)f(tell)i(whic)m(h)d(subarra)m(y)h (has)g(the)g(item)h(w)m(e)f(are)g(lo)s(oking)h(for,)g(just)e(b)m(y)h (lo)s(oking)h(at)g(their)0 4079 y(sizes.)40 b(So,)27 b(w)m(e)g(only)f(need)g(to)h(recursiv)m(ely)g(examine)g(one)g(subarra)m (y)-8 b(,)27 b(not)f(t)m(w)m(o.)41 b(This)25 b(algorithm)j(is)e(often)h (called)0 4191 y(Randomized-Select,)32 b(or)f(Quic)m(kSelect.)0 4450 y Fv(Quic)m(kSelect:)92 b Fu(Giv)m(en)32 b(arra)m(y)f Ft(A)f Fu(of)h(size)g Ft(n)f Fu(and)f(in)m(teger)j Ft(k)d Fp(\024)c Ft(n)p Fu(,)311 4644 y(1.)46 b(Pic)m(k)32 b(a)e(piv)m(ot)i (elemen)m(t)f Ft(p)g Fu(at)g(random)e(from)h Ft(A)p Fu(.)311 4793 y(2.)46 b(Split)c Ft(A)g Fu(in)m(to)h(subarra)m(ys)f(LESS)e(and)i (GREA)-8 b(TER)42 b(b)m(y)g(comparing)g(eac)m(h)i(elemen)m(t)f(to)g Ft(p)f Fu(as)g(in)427 4906 y(Quic)m(ksort.)g(While)31 b(w)m(e)g(are)f(at)h(it,)h(coun)m(t)f(the)f(n)m(um)m(b)s(er)f Ft(L)h Fu(of)h(elemen)m(ts)h(going)f(in)f(to)h(LESS.)311 5055 y(3.)55 b(\(a\))46 b(If)30 b Ft(L)25 b Fu(=)g Ft(k)f Fp(\000)c Fu(1,)31 b(then)f(output)g Ft(p)p Fu(.)431 5186 y(\(b\))45 b(If)30 b Ft(L)25 b(>)g(k)f Fp(\000)c Fu(1,)31 b(output)f(Quic)m(kSelect\(LESS,)h Ft(k)s Fu(\).)441 5317 y(\(c\))46 b(If)30 b Ft(L)25 b(<)g(k)f Fp(\000)c Fu(1,)31 b(output)f(Quic)m(kSelect\(GREA)-8 b(TER,)32 b Ft(k)24 b Fp(\000)c Ft(L)g Fp(\000)g Fu(1\))1905 5568 y(23)p eop end %%Page: 24 25 TeXDict begin 24 24 bop 0 -165 a Fu(24)67 b Fs(LECTURE)29 b(4.)72 b(SELECTION)28 b(\(DET)j(&)f(RAND\):)h(FINDING)h(THE)e(MEDIAN)h (IN)f(LINEAR)g(TIME)0 91 y Fv(Theorem)35 b(4.1)46 b Fo(The)33 b(exp)-5 b(e)g(cte)g(d)34 b(numb)-5 b(er)33 b(of)g(c)-5 b(omp)g(arisons)36 b(for)d(QuickSele)-5 b(ct)32 b(is)h Ft(O)s Fu(\()p Ft(n)p Fu(\))p Fo(.)0 354 y Fv(Pro)s(of:)98 b Fu(Let)34 b Ft(T)13 b Fu(\()p Ft(n;)i(k)s Fu(\))33 b(denote)g(the)g(exp)s(ected)g(time)h(to)f(\014nd)f(the)h Ft(k)s Fu(th)f(smallest)i(in)f(an)f(arra)m(y)i(of)f(size)g Ft(n)p Fu(,)h(and)0 467 y(let)d Ft(T)13 b Fu(\()p Ft(n)p Fu(\))26 b(=)f(max)613 482 y Fm(k)671 467 y Ft(T)13 b Fu(\()p Ft(n;)i(k)s Fu(\).)41 b(W)-8 b(e)31 b(will)g(sho)m(w)f(that)h Ft(T)13 b Fu(\()p Ft(n)p Fu(\))26 b Fp(\024)f Fu(4)p Ft(n)p Fu(.)0 619 y(First)35 b(of)g(all,)h(it)f(tak)m(es)i Ft(n)22 b Fp(\000)h Fu(1)35 b(comparisons)f(to)i(split)e(in)m(to)i(the) f(arra)m(y)g(in)m(to)g(t)m(w)m(o)h(pieces)f(in)g(Step)f(2.)54 b(As)34 b(with)0 732 y(Quic)m(ksort,)e(these)g(pieces)g(migh)m(t)h(ha)m (v)m(e)f(size)h(0)e(and)g Ft(n)21 b Fp(\000)f Fu(1,)32 b(or)g(1)g(and)e Ft(n)21 b Fp(\000)g Fu(2,)32 b(or)f(2)h(and)f Ft(n)20 b Fp(\000)h Fu(3,)32 b(and)f(so)h(on)f(up)0 845 y(to)f Ft(n)19 b Fp(\000)f Fu(1)30 b(and)f(0.)41 b(Eac)m(h)30 b(of)g(these)g(is)g(equally)g(lik)m(ely)-8 b(.)43 b(No)m(w,)31 b(the)e(piece)i(w)m(e)f(recurse)f(on)h(will)g(dep)s(end)e(on)h Ft(k)s Fu(,)i(but)0 958 y(since)c(w)m(e)f(are)h(only)f(giving)i(an)e (upp)s(er)e(b)s(ound,)h(w)m(e)i(can)g(b)s(e)e(p)s(essimistic)i(and)e (imagine)j(that)f(w)m(e)f(alw)m(a)m(ys)i(recurse)0 1070 y(on)i(the)h(larger)g(piece.)82 b(Therefore)30 b(w)m(e)h(ha)m(v)m(e:) 1002 1353 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(\024)g Fu(\()p Ft(n)20 b Fp(\000)g Fu(1\))h(+)1837 1291 y(2)p 1833 1332 55 4 v 1833 1415 a Ft(n)1942 1247 y Fm(n)p Fh(\000)p Fn(1)1949 1272 y Ff(X)1912 1460 y Fm(i)p Fn(=)p Fm(n=)p Fn(2)2120 1353 y Ft(T)13 b Fu(\()p Ft(i)p Fu(\))1276 1586 y(=)83 b(\()p Ft(n)20 b Fp(\000)g Fu(1\))h(+)f(a)m(vg)e([)p Ft(T)13 b Fu(\()p Ft(n=)p Fu(2\))p Ft(;)i(:)g(:)g(:)i(;)e(T)e Fu(\()p Ft(n)21 b Fp(\000)f Fu(1\)])c Ft(:)0 1799 y Fu(W)-8 b(e)40 b(can)f(solv)m(e)h(this)e(using)g(the)h(\\guess)g(and)f(c)m(hec) m(k")j(metho)s(d.)65 b(Assume)38 b(inductiv)m(ely)h(that)g Ft(T)13 b Fu(\()p Ft(i)p Fu(\))40 b Fp(\024)f Fu(4)p Ft(i)g Fu(for)0 1912 y Ft(i)26 b(<)f(n)p Fu(.)40 b(Then,)806 2124 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))83 b Fp(\024)f Fu(\()p Ft(n)21 b Fp(\000)e Fu(1\))i(+)f(a)m(vg)e([4\()p Ft(n=)p Fu(2\))p Ft(;)d Fu(4\()p Ft(n=)p Fu(2)24 b(+)c(1\))p Ft(;)15 b(:)g(:)g(:)i(;)e Fu(4\()p Ft(n)21 b Fp(\000)f Fu(1\)])1080 2262 y Fp(\024)82 b Fu(\()p Ft(n)21 b Fp(\000)e Fu(1\))i(+)f(4\(3)p Ft(n=)p Fu(4\))1080 2400 y Ft(<)82 b Fu(4)p Ft(n;)0 2612 y Fu(and)30 b(w)m(e)h(ha)m(v)m(e)g(v)m(eri\014ed) g(our)f(guess.)p 1365 2612 50 59 v 0 2875 a(One)42 b(w)m(a)m(y)h(to)g (think)e(in)m(tuitiv)m(ely)j(ab)s(out)e(this)g(b)s(ound)e(is)i(that)h (if)f(w)m(e)h(split)f(a)g(candy)g(bar)g(at)h(random)e(in)m(to)0 2987 y(t)m(w)m(o)e(pieces,)h(then)d(the)h(exp)s(ected)g(size)g(of)f (the)h(larger)g(piece)g(is)g(3)p Ft(=)p Fu(4)h(of)e(the)h(bar.)61 b(If)37 b(the)h(size)g(of)g(the)f(larger)0 3100 y(subarra)m(y)30 b(after)h(our)f(partition)h(w)m(as)g(alw)m(a)m(ys)h(3)p Ft(=)p Fu(4)g(of)f(the)f(arra)m(y)-8 b(,)32 b(then)e(w)m(e)h(w)m(ould)g (ha)m(v)m(e)h(a)e(recurrence)h Ft(T)13 b Fu(\()p Ft(n)p Fu(\))25 b Fp(\024)0 3213 y Fu(\()p Ft(n)f Fp(\000)f Fu(1\))h(+)g Ft(T)13 b Fu(\(3)p Ft(n=)p Fu(4\))37 b(whic)m(h)e(solv)m (es)i(to)f Ft(T)13 b Fu(\()p Ft(n)p Fu(\))34 b Ft(<)f Fu(4)p Ft(n)p Fu(.)57 b(No)m(w,)37 b(this)f(is)f(not)h(quite)g(the)g (case)h(for)e(our)g(algorithm)0 3326 y(b)s(ecause)h(3)p Ft(=)p Fu(4)i(is)e(only)g(an)g(exp)s(ected)h(v)-5 b(alue,)38 b(but)e(b)s(ecause)g(the)g(answ)m(er)g(is)g(linear)h(in)f Ft(n)p Fu(,)h(the)g(a)m(v)m(erage)h(of)f(the)0 3439 y Ft(T)13 b Fu(\()p Ft(i)p Fu(\)'s)31 b(turns)e(out)i(to)g(b)s(e)f(the)g (same)h(as)g Ft(T)13 b Fu(\(a)m(v)m(erage)33 b(of)d(the)h Ft(i)p Fu('s\).)0 3778 y Fq(4.3)137 b(A)46 b(deterministic)i (linear-time)e(algorithm)0 4022 y Fu(What)37 b(ab)s(out)f(a)h (deterministic)g(linear-time)h(algorithm?)60 b(F)-8 b(or)37 b(a)g(long)g(time)g(it)g(w)m(as)g(though)m(t)g(this)f(w)m(as)h(im-)0 4135 y(p)s(ossible)f({)h(that)g(there)f(w)m(as)h(no)f(metho)s(d)g (faster)h(than)f(\014rst)f(sorting)i(the)g(arra)m(y)-8 b(.)59 b(In)36 b(the)g(pro)s(cess)g(of)h(trying)0 4248 y(to)32 b(pro)m(v)m(e)h(this)e(claim)i(it)f(w)m(as)g(disco)m(v)m(ered)h (that)f(this)f(thinking)h(w)m(as)g(incorrect,)h(and)e(in)g(1972)i(a)f (deterministic)0 4361 y(linear)f(time)g(algorithm)g(w)m(as)g(dev)m (elop)s(ed.)0 4513 y(The)37 b(idea)h(of)f(the)h(algorithm)g(is)g(that)g (one)f(w)m(ould)g(lik)m(e)i(to)f(pic)m(k)g(a)g(piv)m(ot)g (deterministically)h(in)e(a)h(w)m(a)m(y)g(that)0 4626 y(pro)s(duces)32 b(a)i(go)s(o)s(d)f(split.)50 b(Ideally)-8 b(,)35 b(w)m(e)f(w)m(ould)f(lik)m(e)i(the)e(piv)m(ot)h(to)g(b)s(e)f (the)h(median)f(elemen)m(t)i(so)e(that)h(the)g(t)m(w)m(o)0 4739 y(sides)c(are)h(the)f(same)h(size.)41 b(But,)31 b(this)f(is)g(the)g(same)h(problem)f(w)m(e)g(are)h(trying)f(to)h(solv)m (e)g(in)f(the)h(\014rst)e(place!)42 b(So,)0 4852 y(instead,)31 b(w)m(e)f(will)g(giv)m(e)i(ourselv)m(es)f(leew)m(a)m(y)h(b)m(y)d(allo)m (wing)j(the)e(piv)m(ot)h(to)g(b)s(e)e(an)m(y)i(elemen)m(t)g(that)g(is)f (\\roughly")g(in)0 4965 y(the)k(middle:)46 b(at)35 b(least)f(3)p Ft(=)p Fu(10)i(of)d(the)h(arra)m(y)g(b)s(elo)m(w)g(the)g(piv)m(ot)g (and)f(at)h(least)h(3)p Ft(=)p Fu(10)h(of)d(the)h(arra)m(y)g(ab)s(o)m (v)m(e.)51 b(The)0 5077 y(algorithm)31 b(is)g(as)f(follo)m(ws:)0 5340 y Fv(DeterministicSelect:)46 b Fu(Giv)m(en)31 b(arra)m(y)g Ft(A)g Fu(of)f(size)h Ft(n)f Fu(and)g(in)m(teger)i Ft(k)c Fp(\024)d Ft(n)p Fu(,)p eop end %%Page: 25 26 TeXDict begin 25 25 bop 0 -165 a Fs(4.3.)72 b(A)31 b(DETERMINISTIC)d (LINEAR-TIME)j(ALGORITHM)1467 b Fu(25)311 91 y(1.)46 b(Group)40 b(the)h(arra)m(y)h(in)m(to)f Ft(n=)p Fu(5)g(groups)f(of)h (size)h(5)f(and)f(\014nd)f(the)i(median)g(of)g(eac)m(h)h(group.)71 b(\(F)-8 b(or)427 204 y(simplicit)m(y)g(,)33 b(w)m(e)d(will)h(ignore)g (in)m(tegralit)m(y)i(issues.\))311 346 y(2.)46 b(Recursiv)m(ely)-8 b(,)32 b(\014nd)d(the)i(true)f(median)g(of)h(the)f(medians.)41 b(Call)30 b(this)h Ft(p)p Fu(.)311 489 y(3.)46 b(Use)31 b Ft(p)f Fu(as)h(a)f(piv)m(ot)i(to)f(split)f(the)h(arra)m(y)g(in)m(to)g (subarra)m(ys)f(LESS)f(and)h(GREA)-8 b(TER.)311 631 y(4.)46 b(Recurse)31 b(on)f(the)h(appropriate)f(piece.)0 869 y Fv(Theorem)35 b(4.2)46 b Fo(DeterministicSele)-5 b(ct)38 b(makes)g Ft(O)s Fu(\()p Ft(n)p Fu(\))g Fo(c)-5 b(omp)g(arisons)41 b(to)e(\014nd)f(the)g Ft(k)s Fo(th)h(smal)5 b(lest)39 b(in)f(an)g(arr)-5 b(ay)0 982 y(of)33 b(size)f Ft(n)p Fo(.)0 1220 y Fv(Pro)s(of:)120 b Fu(Let)38 b Ft(T)13 b Fu(\()p Ft(n;)i(k)s Fu(\))38 b(denote)g(the)g(w)m(orst-case)i(time)e (to)h(\014nd)d(the)i Ft(k)s Fu(th)g(smallest)g(out)g(of)g Ft(n)p Fu(,)i(and)d Ft(T)13 b Fu(\()p Ft(n)p Fu(\))37 b(=)0 1333 y(max)169 1348 y Fm(k)227 1333 y Ft(T)13 b Fu(\()p Ft(n;)i(k)s Fu(\))31 b(as)g(b)s(efore.)0 1485 y(Step)g(1)h(tak)m(es)g(time)g Ft(O)s Fu(\()p Ft(n)p Fu(\),)g(since)f(it)h(tak)m(es)h(just)e(constan)m(t)h(time)g(to)g (\014nd)e(the)h(median)g(of)g(5)h(elemen)m(ts.)45 b(Step)31 b(2)0 1598 y(tak)m(es)g(time)f(at)f(most)h Ft(T)13 b Fu(\()p Ft(n=)p Fu(5\).)41 b(Step)29 b(3)h(again)g(tak)m(es)h(time)f Ft(O)s Fu(\()p Ft(n)p Fu(\).)40 b(No)m(w,)30 b(w)m(e)g(claim)g(that)g (at)g(least)g(3)p Ft(=)p Fu(10)h(of)f(the)0 1711 y(arra)m(y)36 b(is)f Fp(\024)e Ft(p)p Fu(,)k(and)e(at)h(least)g(3)p Ft(=)p Fu(10)h(of)f(the)f(arra)m(y)h(is)g Fp(\025)d Ft(p)p Fu(.)55 b(Assuming)35 b(for)g(the)g(momen)m(t)h(that)g(this)g(claim)g (is)0 1823 y(true,)30 b(Step)g(4)h(tak)m(es)h(time)f(at)g(most)g Ft(T)13 b Fu(\(7)p Ft(n=)p Fu(10\),)32 b(and)e(w)m(e)h(ha)m(v)m(e)h (the)e(recurrence:)1230 2016 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))84 b Fp(\024)e Ft(cn)20 b Fu(+)g Ft(T)13 b Fu(\()p Ft(n=)p Fu(5\))21 b(+)f Ft(T)13 b Fu(\(7)p Ft(n=)p Fu(10\))p Ft(;)1061 b Fu(\(4.1\))0 2208 y(for)25 b(some)g(constan)m(t)h Ft(c)p Fu(.)40 b(Before)26 b(solving)f(this)g(recurrence,)h(lets)g(pro) m(v)m(e)g(the)f(claim)h(w)m(e)g(made)f(that)g(the)h(piv)m(ot)g(will)0 2321 y(b)s(e)31 b(roughly)h(near)f(the)i(middle)e(of)h(the)g(arra)m(y) -8 b(.)46 b(So,)32 b(the)h(question)f(is:)43 b(ho)m(w)32 b(bad)f(can)i(the)f(median)f(of)h(medians)0 2434 y(b)s(e?)0 2586 y(Let's)i(\014rst)f(do)g(an)g(example.)51 b(Supp)s(ose)32 b(the)h(arra)m(y)h(has)f(15)i(elemen)m(ts)g(and)d(breaks)i(do)m(wn)f (in)m(to)h(three)g(groups)0 2699 y(of)d(5)f(lik)m(e)i(this:)990 2812 y Fp(f)p Fu(1)p Ft(;)15 b Fu(2)p Ft(;)g Fu(3)p Ft(;)g Fu(10)p Ft(;)g Fu(11)p Fp(g)p Ft(;)72 b Fp(f)p Fu(4)p Ft(;)15 b Fu(5)p Ft(;)g Fu(6)p Ft(;)g Fu(12)p Ft(;)g Fu(13)p Fp(g)p Ft(;)72 b Fp(f)p Fu(7)p Ft(;)15 b Fu(8)p Ft(;)g Fu(9)p Ft(;)g Fu(14)p Ft(;)g Fu(15)p Fp(g)p Ft(:)0 2973 y Fu(In)36 b(this)h(case,)i(the)e(medians)g(are)g(3,)i(6,)g(and)d (9,)j(and)e(the)g(median)f(of)h(the)g(medians)g Ft(p)f Fu(is)h(6.)60 b(There)36 b(are)i(\014v)m(e)0 3086 y(elemen)m(ts)32 b(less)e(than)h Ft(p)f Fu(and)f(nine)h(elemen)m(ts)i(greater.)0 3238 y(In)25 b(general,)j(what)e(is)g(the)g(w)m(orst)g(case?)40 b(If)26 b(there)g(are)g Ft(g)j Fu(=)c Ft(n=)p Fu(5)h(groups,)h(then)e (w)m(e)i(kno)m(w)f(that)g(in)g(at)g(least)i Fp(d)p Ft(g)s(=)p Fu(2)p Fp(e)0 3351 y Fu(of)i(them)g(\(those)g(groups)g(whose)f(median)h (is)g Fp(\024)25 b Ft(p)p Fu(\))30 b(at)g(least)h(three)f(of)g(the)g (\014v)m(e)g(elemen)m(ts)i(are)e Fp(\024)25 b Ft(p)p Fu(.)40 b(Therefore,)0 3464 y(the)23 b(total)h(n)m(um)m(b)s(er)e(of)h (elemen)m(ts)h Fp(\024)h Ft(p)d Fu(is)h(at)g(least)h(3)p Fp(d)p Ft(g)s(=)p Fu(2)p Fp(e)k(\025)d Fu(3)p Ft(n=)p Fu(10.)40 b(Similarly)-8 b(,)25 b(the)e(total)h(n)m(um)m(b)s(er)e(of)h (elemen)m(ts)0 3577 y Fp(\025)i Ft(p)30 b Fu(is)g(also)i(at)f(least)g (3)p Fp(d)p Ft(g)s(=)p Fu(2)p Fp(e)e(\025)24 b Fu(3)p Ft(n=)p Fu(10.)0 3729 y(No)m(w,)39 b(\014nally)-8 b(,)38 b(let's)g(solv)m(e)g(the)e(recurrence.)59 b(W)-8 b(e)38 b(ha)m(v)m(e)g(b)s(een)d(solving)i(a)g(lot)h(of)e(recurrences)g(b)m(y)h (the)f(\\guess)0 3842 y(and)31 b(c)m(hec)m(k")j(metho)s(d,)d(whic)m(h)h (w)m(orks)f(here)h(to)s(o,)h(but)e(ho)m(w)g(could)h(w)m(e)g(just)f (stare)h(at)h(this)e(and)g Fo(know)i Fu(that)f(the)0 3955 y(answ)m(er)d(is)f(linear)h(in)g Ft(n)p Fu(?)39 b(One)28 b(w)m(a)m(y)i(to)f(do)g(that)g(is)g(to)g(consider)g(the)f (\\stac)m(k)j(of)e(bric)m(ks")g(view)g(of)g(the)f(recursion)0 4068 y(tree)j(discussed)f(in)g(Lecture)g(2.)0 4220 y(In)j(particular,)j (let's)f(build)e(the)h(recursion)f(tree)i(for)f(the)g(recurrence)g (\(4.1\),)j(making)d(eac)m(h)h(no)s(de)f(as)g(wide)g(as)0 4333 y(the)d(quan)m(tit)m(y)g(inside)f(it:)477 4550 y Fl(+------------------------)o(----)o(----)o(---)o(---+)477 4663 y(|)811 b(cn)907 b(|)477 4775 y(+---------+--------------)o(----)o (----)o(-+-)o(---+)477 4888 y(|)143 b(cn/5)95 b(|)286 b(7cn/10)523 b(|)143 b(<-)47 b(total)f(9cn/10)477 5001 y(+--+----+-+-----+--------)o(----)o(-+--)o(-+)477 5114 y(|)95 b(|)191 b(|)334 b(|)620 b(|)334 b(<-)47 b(total)f(\(81/100\)cn) 477 5227 y(+--+----+-------+--------)o(----)o(-+)477 5340 y(...)1813 b(...)p eop end %%Page: 26 27 TeXDict begin 26 26 bop 0 -165 a Fu(26)67 b Fs(LECTURE)29 b(4.)72 b(SELECTION)28 b(\(DET)j(&)f(RAND\):)h(FINDING)h(THE)e(MEDIAN)h (IN)f(LINEAR)g(TIME)0 91 y Fu(Notice)i(that)f(ev)m(en)g(if)g(this)f (stac)m(k-of-bric)m(ks)j(con)m(tin)m(ues)e(do)m(wn)m(w)m(ard)f(forev)m (er,)i(the)e(total)i(sum)e(is)g(at)h(most)1139 296 y Ft(cn)p Fu(\(1)21 b(+)f(\(9)p Ft(=)p Fu(10\))i(+)e(\(9)p Ft(=)p Fu(10\))2038 258 y Fn(2)2100 296 y Fu(+)g(\(9)p Ft(=)p Fu(10\))2441 258 y Fn(3)2504 296 y Fu(+)f Ft(:)c(:)g(:)r Fu(\))p Ft(;)0 500 y Fu(whic)m(h)30 b(is)h(at)g(most)f(10)p Ft(cn)p Fu(.)42 b(This)29 b(pro)m(v)m(es)i(the)g(theorem.)p 2061 500 50 59 v 0 751 a(Notice)c(that)f(in)f(our)f(analysis)i(of)f (the)h(recurrence)f(\(4.1\))i(the)e(k)m(ey)h(prop)s(ert)m(y)f(w)m(e)g (used)g(w)m(as)g(that)h Ft(n=)p Fu(5)10 b(+)g(7)p Ft(n=)p Fu(10)26 b Ft(<)0 864 y(n)p Fu(.)40 b(More)29 b(generally)-8 b(,)31 b(w)m(e)e(see)g(here)g(that)g(if)g(w)m(e)g(ha)m(v)m(e)h(a)f (problem)f(of)h(size)g Ft(n)f Fu(that)i(w)m(e)f(can)g(solv)m(e)h(b)m(y) e(p)s(erforming)0 977 y(recursiv)m(e)36 b(calls)g(on)g(pieces)g(whose)f (total)i(size)f(is)g(at)g(most)g(\(1)24 b Fp(\000)f Ft(\017)p Fu(\))p Ft(n)35 b Fu(for)h(some)f(constan)m(t)i Ft(\017)d(>)f Fu(0)j(\(plus)f(some)0 1090 y(additional)41 b Ft(O)s Fu(\()p Ft(n)p Fu(\))f(w)m(ork\),)j(then)d(the)g(total)i(time)f(sp)s (en)m(t)e(will)i(b)s(e)e(just)h(linear)g(in)g Ft(n)p Fu(.)69 b(This)39 b(giv)m(es)j(us)d(a)h(nice)0 1203 y(extension)31 b(to)g(our)f(\\Master)i(theorem")f(from)f(Lecture)h(2.)0 1455 y Fv(Theorem)k(4.3)46 b Fo(F)-7 b(or)34 b(c)-5 b(onstants)34 b Ft(c)f Fo(and)h Ft(a)1489 1469 y Fn(1)1528 1455 y Ft(;)15 b(:)g(:)g(:)i(;)e(a)1778 1470 y Fm(k)1853 1455 y Fo(such)33 b(that)h Ft(a)2294 1469 y Fn(1)2354 1455 y Fu(+)20 b Ft(:)15 b(:)g(:)h(a)2614 1470 y Fm(k)2682 1455 y Ft(<)25 b Fu(1)p Fo(,)33 b(the)g(r)-5 b(e)g(curr)g(enc)g(e)1042 1659 y Ft(T)13 b Fu(\()p Ft(n)p Fu(\))84 b Fp(\024)e Ft(T)13 b Fu(\()p Ft(a)1619 1673 y Fn(1)1659 1659 y Ft(n)p Fu(\))20 b(+)g Ft(T)13 b Fu(\()p Ft(a)2009 1673 y Fn(2)2048 1659 y Ft(n)p Fu(\))21 b(+)f Ft(:)15 b(:)g(:)h(T)d Fu(\()p Ft(a)2520 1674 y Fm(k)2563 1659 y Ft(n)p Fu(\))20 b(+)g Ft(cn)0 1863 y Fo(solves)33 b(to)g Ft(T)13 b Fu(\()p Ft(n)p Fu(\))26 b(=)f(\002\()p Ft(n)p Fu(\))p Fo(.)p eop end %%Page: 27 28 TeXDict begin 27 27 bop 0 753 a Fr(Lecture)65 b(5)0 1207 y Fw(Comparison-based)78 b(Lo)-7 b(w)g(er)79 b(Bounds)g(for)0 1456 y(Sorting)0 1977 y Fq(5.1)137 b(Ov)l(erview)0 2225 y Fu(In)31 b(this)g(lecture)h(w)m(e)g(discuss)f(the)g(notion)h(of)g Fo(lower)i(b)-5 b(ounds)p Fu(,)33 b(in)f(particular)f(for)g(the)h (problem)f(of)g(sorting.)45 b(W)-8 b(e)0 2338 y(sho)m(w)30 b(that)g(an)m(y)g(deterministic)g(comparison-based)g(sorting)g (algorithm)h(m)m(ust)f(tak)m(e)h(\012\()p Ft(n)15 b Fu(log)i Ft(n)p Fu(\))30 b(time)g(to)h(sort)0 2451 y(an)g(arra)m(y)g(of)f Ft(n)h Fu(elemen)m(ts)h(in)e(the)h(w)m(orst)g(case.)42 b(W)-8 b(e)32 b(then)e(extend)h(this)g(result)f(to)h(a)m(v)m(erage)j (case)d(p)s(erformance,)0 2564 y(and)i(to)h(randomized)f(algorithms.)51 b(In)32 b(the)i(pro)s(cess,)g(w)m(e)f(in)m(tro)s(duce)h(the)f(2-pla)m (y)m(er)i(game)g(view)e(of)h(algorithm)0 2677 y(design)c(and)g (analysis.)0 3032 y Fq(5.2)137 b(Sorting)47 b(lo)l(w)l(er)f(b)t(ounds)0 3280 y Fu(So)e(far)h(w)m(e)g(ha)m(v)m(e)g(b)s(een)f(fo)s(cusing)g(on)h (the)g(question:)69 b(\\giv)m(en)46 b(some)f(problem)f Ft(X)7 b Fu(,)48 b(can)d(w)m(e)g(construct)g(an)0 3393 y(algorithm)38 b(that)g(runs)e(in)h Ft(O)s Fu(\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))38 b(time)g(on)f(inputs)g(of)g(size)h Ft(n)p Fu(?")62 b(This)36 b(is)i(often)g(called)g(an)g(upp)s(er)d(b)s (ound)0 3506 y(problem)d(b)s(ecause)h(w)m(e)h(are)f(determining)g(an)g (upp)s(er)e(b)s(ound)g(on)i(the)g(inheren)m(t)g(di\016cult)m(y)h(of)f (problem)g Ft(X)7 b Fu(.)49 b(In)0 3618 y(this)30 b(lecture)g(w)m(e)g (examine)h(the)f(\\lo)m(w)m(er)h(b)s(ound)d(problem.")40 b(Here,)31 b(the)f(goal)h(is)f(to)h(pro)m(v)m(e)f(that)h Fo(any)f Fu(algorithm)0 3731 y(m)m(ust)g(tak)m(e)i(time)f(\012\()p Ft(f)10 b Fu(\()p Ft(n)p Fu(\)\))31 b(time)g(to)g(solv)m(e)g(the)g (problem.)0 3883 y(T)-8 b(o)s(da)m(y)31 b(w)m(e)f(consider)h(the)f (class)h(of)g(comparison-based)f(sorting)h(algorithms.)42 b(These)30 b(are)g(sorting)h(algorithms)0 3996 y(that)e(only)g(op)s (erate)g(on)g(the)g(input)f(arra)m(y)h(b)m(y)f(comparing)i(pairs)e(of)h (elemen)m(ts,)h(and)e(mo)m(ving)i(elemen)m(ts)g(around)0 4109 y(based)35 b(on)g(the)g(results)g(of)g(these)h(comparisons.)55 b(F)-8 b(or)36 b(instance,)h(Quic)m(ksort,)g(Mergesort,)h(and)d (Insertionsort)0 4222 y(are)c(all)g(comparison-based)g(sorting)f (algorithms.)42 b(What)31 b(w)m(e)g(will)g(sho)m(w)f(is)g(the)h(follo)m (wing)h(theorem.)0 4499 y Fv(Theorem)j(5.1)46 b Fo(A)n(ny)23 b(deterministic)h(c)-5 b(omp)g(arison-b)g(ase)g(d)27 b(sorting)d(algorithm)i(must)e(p)-5 b(erform)25 b Fu(\012\()p Ft(n)15 b Fu(log)i Ft(n)p Fu(\))23 b Fo(c)-5 b(om-)0 4612 y(p)g(arisons)35 b(to)f(sort)f Ft(n)g Fo(elements)g(in)g(the)g (worst)h(c)-5 b(ase.)43 b(Sp)-5 b(e)g(ci\014c)g(al)5 b(ly,)34 b(for)g(any)f(deterministic)h(c)-5 b(omp)g(arison-b)g(ase)g(d) 0 4725 y(sorting)41 b(algorithm)i Fp(A)p Fo(,)e(for)g(al)5 b(l)41 b Ft(n)e Fp(\025)g Fu(2)i Fo(ther)-5 b(e)41 b(exists)f(an)h (input)g Ft(I)47 b Fo(of)41 b(size)f Ft(n)g Fo(such)h(that)g Fp(A)f Fo(makes)h(at)g(le)-5 b(ast)0 4838 y Fu(log)117 4859 y Fn(2)157 4838 y Fu(\()p Ft(n)p Fu(!\))26 b(=)f(\012\()p Ft(n)15 b Fu(log)i Ft(n)p Fu(\))32 b Fo(c)-5 b(omp)g(arisons)36 b(to)d(sort)h Ft(I)7 b Fo(.)0 5114 y Fu(T)-8 b(o)38 b(pro)m(v)m(e)h (this)e(theorem,)k(w)m(e)d(cannot)g(assume)f(the)h(sorting)h(algorithm) f(is)g(going)h(to)f(necessarily)h(c)m(ho)s(ose)g(a)0 5227 y(piv)m(ot)25 b(as)f(in)g(Quic)m(ksort,)i(or)e(split)h(the)f (input)f(as)i(in)e(Mergesort)j(|)e(w)m(e)g(need)g(to)h(someho)m(w)g (analyze)g Fo(any)j(p)-5 b(ossible)0 5340 y Fu(\(comparison-based\))26 b(algorithm)g(that)g(migh)m(t)g(exist.)39 b(The)25 b(w)m(a)m(y)h(w)m(e) f(will)h(do)f(this)g(is)g(b)m(y)g(sho)m(wing)g(that)h(in)e(order)1905 5568 y(27)p eop end %%Page: 28 29 TeXDict begin 28 28 bop 0 -165 a Fu(28)742 b Fs(LECTURE)30 b(5.)71 b(COMP)-8 b(ARISON-BASED)30 b(LO)m(WER)h(BOUNDS)g(F)m(OR)f(SOR) -8 b(TING)0 91 y Fu(to)30 b(sort)g(its)g(input,)f(the)g(sorting)h (algorithm)h(is)e(implicitly)i(pla)m(ying)f(a)g(game)g(of)g(20)g (questions)g(with)f(the)h(input,)0 204 y(trying)g(to)i(\014gure)d(out)i (in)f(what)g(the)h(order)f(its)h(elemen)m(ts)g(are)g(b)s(eing)f(giv)m (en.)0 356 y Fv(Pro)s(of:)129 b Fu(Let's)40 b(sa)m(y)h(the)f(input)e (arra)m(y)j(con)m(tains)g(the)f(elemen)m(ts)h Ft(a)2437 370 y Fn(1)2477 356 y Ft(;)15 b(a)2565 370 y Fn(2)2604 356 y Ft(;)g(:)g(:)g(:)i(;)e(a)2854 370 y Fm(n)2941 356 y Fu(in)40 b(some)g(unkno)m(wn)e(order)0 469 y(\(assume)f(the)h Ft(a)568 483 y Fm(i)633 469 y Fu(are)g(all)g(distinct\).)62 b(There)37 b(are)h Ft(n)p Fu(!)f(di\013eren)m(t)h(orders)e(these)i (elemen)m(ts)h(could)e(b)s(e)g(in.)61 b(No)m(w,)0 582 y(supp)s(ose)40 b(that)h(t)m(w)m(o)i(di\013eren)m(t)e(initial)i (orderings)d(of)h(these)h(n)m(um)m(b)s(ers)e Ft(I)2567 596 y Fn(1)2606 582 y Fu(,)k Ft(I)2715 596 y Fn(2)2754 582 y Fu(,)g(are)e(consisten)m(t)g(with)f(all)h(the)0 695 y(comparisons)23 b(the)h(sorting)f(algorithm)i(has)e(made)g(so)g (far.)38 b(In)23 b(this)g(case,)j(w)m(e)e(claim)g(that)g(the)f(sorting) h(algorithm)0 808 y(cannot)36 b(y)m(et)h(b)s(e)e(done.)55 b(The)35 b(reason)h(it)g(cannot)g(y)m(et)h(b)s(e)e(done)g(is)g(that)h (the)g(output)f(of)h(a)g(comparison-based)0 921 y(sorting)i(algorithm)h (has)e(to)h(b)s(e)f(a)h(p)s(erm)m(utation)g(of)g(the)g(input)e(\(the)j (sorter)f(can't)g(mak)m(e)h(up)d(new)h(n)m(um)m(b)s(ers)0 1034 y(to)f(output)f(since)g(all)h(it)g(kno)m(ws)f(ab)s(out)g(the)h (inputs)e Ft(a)1909 1048 y Fm(i)1972 1034 y Fu(is)h(ho)m(w)h(they)f (relate)i(to)f(eac)m(h)g(other\).)56 b(F)-8 b(urthermore,)0 1147 y(whatev)m(er)31 b(p)s(erm)m(utation)g(of)f(the)h(input)e(it)i (pro)s(duces)e(as)i(output)f(cannot)h(b)s(e)e(correct)j(for)e(b)s(oth)g Ft(I)3414 1161 y Fn(1)3483 1147 y Fu(and)g Ft(I)3700 1161 y Fn(2)3740 1147 y Fu(.)0 1299 y(So,)22 b(the)f(sorting)g (algorithm)g(needs)f(at)h(least)g(implicitly)h(to)f(ha)m(v)m(e)g (pinned)e(do)m(wn)h(whic)m(h)g(ordering)g(of)g Fp(f)p Ft(a)3529 1313 y Fn(1)3569 1299 y Ft(;)15 b(:)g(:)g(:)i(;)e(a)3819 1313 y Fm(n)3866 1299 y Fp(g)0 1412 y Fu(w)m(as)34 b(giv)m(en)h(in)e (the)h(input.)50 b(There)33 b(are)h Ft(n)p Fu(!)f(di\013eren)m(t)i(p)s (ossible)e(input)g(orderings,)h(and)f(eac)m(h)i(comparison)f(is)g(a)0 1525 y(binary)d(y)m(es/no)i(question)e(that)i(remo)m(v)m(es)g(some)f (of)g(these)g(orderings)f(from)g(consideration.)46 b(If)31 b(the)h(answ)m(er)f(to)0 1637 y(this)36 b(question)h(is)f(alw)m(a)m(ys) i(the)e(one)h(consisten)m(t)g(with)f(the)h(ma)5 b(jorit)m(y)37 b(of)f(the)h(orderings)f(still)h(remaining,)h(the)0 1750 y(algorithm)27 b(will)g(need)f(to)g(mak)m(e)i(log)1243 1772 y Fn(2)1282 1750 y Fu(\()p Ft(n)p Fu(!\))f(questions)f(to)h(narro) m(w)f(do)m(wn)g(the)g(input)f(to)i(a)g(single)g(p)s(ossibilit)m(y)-8 b(.)40 b(W)-8 b(e)0 1863 y(can)31 b(then)f(solv)m(e:)961 2059 y(log)1079 2081 y Fn(2)1118 2059 y Fu(\()p Ft(n)p Fu(!\))84 b(=)e(log)1623 2081 y Fn(2)1662 2059 y Fu(\()p Ft(n)p Fu(\))21 b(+)f(log)2016 2081 y Fn(2)2056 2059 y Fu(\()p Ft(n)g Fp(\000)g Fu(1\))h(+)f Ft(:)15 b(:)g(:)21 b Fu(+)f(log)2783 2081 y Fn(2)2823 2059 y Fu(\(2\))1352 2197 y(=)82 b(\012\()p Ft(n)15 b Fu(log)j Ft(n)p Fu(\))p Ft(:)p 2015 2197 50 59 v 0 2393 a Fu(Let's)31 b(do)f(an)h(example)g (with)f Ft(n)24 b Fu(=)h(3.)42 b(In)29 b(this)h(case,)i(there)f(are)f (six)h(p)s(ossible)f(input)f(orderings:)1155 2589 y Fp(f)p Fu(123)p Fp(g)p Ft(;)15 b Fp(f)p Fu(132)p Fp(g)p Ft(;)g Fp(f)p Fu(213)q Fp(g)p Ft(;)h Fp(f)p Fu(23)q(1)p Fp(g)q Ft(;)f Fp(f)q Fu(31)q(2)p Fp(g)q Ft(;)g Fp(f)p Fu(3)q(21)q Fp(g)p Ft(:)0 2785 y Fu(Supp)s(ose)26 b(the)h(sorting)h(algorithm)g (\014rst)f(compares)g Ft(A)p Fu([0])i(with)e Ft(A)p Fu([1].)41 b(If)26 b(the)i(answ)m(er)f(is)h(that)f Ft(A)p Fu([1])g Ft(>)e(A)p Fu([0])j(then)0 2897 y(w)m(e)j(ha)m(v)m(e)g(narro)m(w)m(ed)g (do)m(wn)f(the)g(input)g(to)h(the)f(three)h(p)s(ossibilities:)1556 3093 y Fp(f)p Fu(123)p Fp(g)p Ft(;)15 b Fp(f)p Fu(132)p Fp(g)p Ft(;)g Fp(f)p Fu(23)q(1)p Fp(g)q Ft(:)0 3289 y Fu(Supp)s(ose)33 b(the)h(next)h(comparison)f(is)g(b)s(et)m(w)m(een)h Ft(A)p Fu([1])h(and)d Ft(A)p Fu([2].)54 b(In)34 b(this)g(case,)i(the)f (most)g(p)s(opular)e(answ)m(er)h(is)0 3402 y(that)d Ft(A)p Fu([1])26 b Ft(>)f(A)p Fu([2],)32 b(whic)m(h)e(remo)m(v)m(es)i(just)e (one)g(ordering,)h(lea)m(ving)h(us)d(with:)1690 3598 y Fp(f)p Fu(132)p Fp(g)p Ft(;)15 b Fp(f)p Fu(231)p Fp(g)p Ft(:)0 3794 y Fu(It)30 b(no)m(w)h(tak)m(es)h(one)e(more)h(comparison)f (to)i(\014nally)e(isolate)i(the)f(input)e(ordering.)0 3946 y(Notice)i(that)e(our)g(pro)s(of)f(is)h(lik)m(e)h(a)f(game)h(of)f (20)h(questions)f(in)f(whic)m(h)h(the)g(resp)s(onder)e(do)s(esn't)i (actually)i(decide)0 4059 y(what)i(he)g(is)h(thinking)f(of)g(un)m(til)h (there)f(is)h(only)f(one)h(option)f(left.)50 b(This)33 b(is)g(legitimate)j(b)s(ecause)e(w)m(e)f(just)g(need)0 4172 y(to)39 b(sho)m(w)e(that)i(there)f(is)g Fo(some)g Fu(input)f(that)i(w)m(ould)e(cause)i(the)f(algorithm)h(to)f(tak)m(e)i (a)e(long)h(time.)64 b(In)37 b(other)0 4285 y(w)m(ords,)24 b(since)g(the)f(sorting)h(algorithm)g(is)g(deterministic,)i(w)m(e)d (can)h(tak)m(e)h(that)f(\014nal)e(remaining)i(option)g(and)e(then)0 4398 y(re-run)31 b(the)h(algorithm)g(on)g(that)g(sp)s(eci\014c)g (input,)f(and)h(the)f(algorithm)i(will)f(mak)m(e)h(the)f(same)g(exact)i (sequence)0 4510 y(of)d(op)s(erations.)0 4663 y(Y)-8 b(ou)38 b(can)f(also)h(think)f(of)g(the)g(ab)s(o)m(v)m(e)h(pro)s(of)f (in)g(terms)g(of)g(the)g(n)m(um)m(b)s(er)f(of)h(p)s(ossible)g Fo(outputs)h Fu(of)f(the)h(sorting)0 4775 y(algorithm.)j(An)m(y)27 b(comparison-based)g(sorting)h(algorithm)g(can)g(b)s(e)f(though)m(t)g (of)h(as)f(pro)s(ducing)f(a)i(p)s(erm)m(utation)0 4888 y(as)34 b(its)h(output)f(\(the)g(p)s(erm)m(utation)g(that,)i(when)e (applied)f(to)i(the)g(input,)f(pro)s(duces)f(a)h(sorted)h(arra)m(y\).) 52 b(There)0 5001 y(are)37 b Ft(n)p Fu(!)f(p)s(erm)m(utations)g(and)g (only)g(one)h(of)f(them)g(can)h(b)s(e)f(correct)h(for)f(an)m(y)h(giv)m (en)g(input.)58 b(Eac)m(h)37 b(comparison)0 5114 y(breaks)g(the)h(set)f (of)h(p)s(ossible)f(outputs)f(in)m(to)j(t)m(w)m(o)f(classes,)j(and)36 b(the)i(resp)s(onse)e(to)i(the)g(question)f(sa)m(ys)h(whic)m(h)0 5227 y(class)e(the)g(correct)g(output)f(is)g(in.)55 b(By)36 b(alw)m(a)m(ys)h(giving)f(the)g(answ)m(er)f(corresp)s(onding)f(to)i (the)f(larger)h(class,)i(an)0 5340 y(adv)m(ersary)31 b(can)f(force)h(the)g(algorithm)g(to)g(mak)m(e)h(log)1836 5362 y Fn(2)1876 5340 y Fu(\()p Ft(n)p Fu(!\))e(comparisons.)p eop end %%Page: 29 30 TeXDict begin 29 29 bop 0 -165 a Fs(5.3.)72 b(A)-10 b(VERA)m(GE-CASE)31 b(LO)m(WER)f(BOUNDS)2068 b Fu(29)0 91 y Fq(5.3)137 b(Av)l(erage-case)45 b(lo)l(w)l(er)h(b)t(ounds)0 339 y Fu(In)27 b(fact,)j(w)m(e)e(can)h (generalize)h(the)e(ab)s(o)m(v)m(e)h(theorem)g(to)f(sho)m(w)g(that)h (an)m(y)f(comparison-based)g(sorting)h(algorithm)0 452 y(m)m(ust)h(tak)m(e)i(\012\()p Ft(n)15 b Fu(log)j Ft(n)p Fu(\))30 b(time)h Fo(on)i(aver)-5 b(age)p Fu(,)31 b(not)g(just)f(in)g (the)g(w)m(orst)h(case.)0 730 y Fv(Theorem)k(5.2)46 b Fo(F)-7 b(or)34 b(any)f(deterministic)g(c)-5 b(omp)g(arison-b)g(ase)g (d)37 b(sorting)c(algorithm,)i(the)e(A)n(ver)-5 b(age-Case)32 b(num-)0 842 y(b)-5 b(er)24 b(of)g(c)-5 b(omp)g(arisons)27 b(\(the)e(numb)-5 b(er)24 b(of)g(c)-5 b(omp)g(arisons)27 b(on)e(aver)-5 b(age)24 b(on)h(a)f(r)-5 b(andomly)27 b(chosen)d(input)g(p)-5 b(ermutation\))0 955 y(is)33 b(at)g(le)-5 b(ast)34 b Fp(b)p Fu(log)571 977 y Fn(2)611 955 y Fu(\()p Ft(n)p Fu(!\))p Fp(c)p Fo(.)0 1233 y Fv(Pro)s(of:)43 b Fu(Let's)32 b(build)f(out)g(the)h(en)m(tire)g(decision)g(tree:)44 b(the)32 b(tree)g(w)m(e)g(get)g(b)m(y)f(lo)s(oking)i(at)f(all)g(p)s (ossible)f(series)h(of)0 1346 y(answ)m(ers)f(that)g(one)h(migh)m(t)f (get)h(from)f(some)g(ordering)g(of)g(the)g(input.)42 b(By)31 b(the)h(previous)e(argumen)m(t,)i(eac)m(h)g(leaf)0 1458 y(of)27 b(this)f(tree)h(corresp)s(onds)e(to)i(a)g(single)g(input)e (p)s(erm)m(utation)i(\(w)m(e)g(can't)g(ha)m(v)m(e)h(t)m(w)m(o)g(p)s (erm)m(utations)e(at)h(the)g(same)0 1571 y(leaf,)j(else)f(the)f (algorithm)h(w)m(ould)f(not)h(b)s(e)e(\014nished\).)39 b(The)28 b(depth)f(of)i(the)f(leaf)h(is)f(the)h(n)m(um)m(b)s(er)e(of)h (comparisons)0 1684 y(p)s(erformed)h(b)m(y)h(the)h(sorting)f(algorithm) i(on)e(that)h(input.)0 1836 y(If)36 b(the)g(tree)g(is)g(completely)i (balanced,)g(then)e(eac)m(h)h(leaf)g(is)f(at)g(depth)g Fp(d)p Fu(log)2649 1858 y Fn(2)2689 1836 y Fu(\()p Ft(n)p Fu(!\))p Fp(e)h Fu(or)f Fp(b)p Fu(log)3190 1858 y Fn(2)3230 1836 y Fu(\()p Ft(n)p Fu(!\))p Fp(c)h Fu(and)e(w)m(e)i(are)0 1949 y(done.)212 1916 y Fn(1)292 1949 y Fu(T)-8 b(o)31 b(pro)m(v)m(e)g(the)f(theorem,)h(w)m(e)g(just)f(need)g(to)h(sho)m(w)f (that)h(out)g(of)f(all)i(binary)d(trees)i(on)f(a)h(giv)m(en)g(n)m(um)m (b)s(er)0 2062 y(of)37 b(lea)m(v)m(es,)j(the)d(one)g(that)g(minimizes)g (their)g(a)m(v)m(erage)i(depth)d(is)h(a)g(completely)h(balanced)f (tree.)60 b(This)36 b(is)h(not)0 2175 y(to)s(o)30 b(hard)f(to)i(see:)41 b(giv)m(en)31 b(some)f(un)m(balanced)f(tree,)i(w)m(e)f(tak)m(e)i(t)m(w) m(o)f(sibling)f(lea)m(v)m(es)i(at)e(largest)h(depth)e(and)g(mo)m(v)m(e) 0 2288 y(them)35 b(to)h(b)s(e)e(c)m(hildren)h(of)g(the)g(leaf)h(of)f (smallest)h(depth.)53 b(Since)35 b(the)g(di\013erence)h(b)s(et)m(w)m (een)f(the)g(largest)i(depth)0 2401 y(and)29 b(the)g(smallest)i(depth)d (is)i(at)g(least)g(2)g(\(otherwise)g(the)g(tree)g(w)m(ould)f(b)s(e)g (balanced\),)h(this)g(op)s(eration)f(reduces)0 2514 y(the)f(a)m(v)m (erage)i(depth)d(of)h(the)g(lea)m(v)m(es.)42 b(Sp)s(eci\014cally)-8 b(,)30 b(if)e(the)g(smaller)g(depth)f(is)h Ft(d)g Fu(and)f(the)h (larger)g(depth)f(is)h Ft(D)s Fu(,)g(w)m(e)0 2627 y(ha)m(v)m(e)35 b(remo)m(v)m(ed)g(t)m(w)m(o)g(lea)m(v)m(es)i(of)d(depth)f Ft(D)k Fu(and)c(one)h(of)g(depth)g Ft(d)p Fu(,)h(and)e(w)m(e)i(ha)m(v)m (e)g(added)e(t)m(w)m(o)i(lea)m(v)m(es)i(of)d(depth)0 2740 y Ft(d)21 b Fu(+)e(1)31 b(and)f(one)h(of)f(depth)g Ft(D)23 b Fp(\000)d Fu(1.)p 1330 2740 50 59 v 0 3017 a(In)30 b(fact,)h(if)g(w)m(e)f(are)h(a)g(bit)f(more)h(clev)m(er)h(in)e (the)g(pro)s(of,)g(w)m(e)h(can)g(get)g(rid)f(of)g(the)h(\015o)s(or)f (in)g(the)g(b)s(ound.)0 3373 y Fq(5.4)137 b(Lo)l(w)l(er)46 b(b)t(ounds)h(for)e(randomized)h(algorithms)0 3621 y Fv(Theorem)35 b(5.3)46 b Fo(The)33 b(ab)-5 b(ove)33 b(b)-5 b(ound)34 b(holds)g(for)f(r)-5 b(andomize)g(d)36 b(algorithms)f(to)-5 b(o.)0 3898 y Fv(Pro)s(of:)94 b Fu(W)-8 b(e)33 b(can)g(think)e(of)h(a)g (randomized)g(algorithm)h(as)f(a)g(probabilit)m(y)h(distribution)e(o)m (v)m(er)i(deterministic)0 4011 y(algorithms.)51 b(E.g.,)36 b(with)d(some)h(probabilit)m(y)g(it)g(mak)m(es)g(sequence)g(of)g(c)m (hoices)h Ft(s)2831 4025 y Fn(1)2870 4011 y Fu(,)g(with)e(some)h (probabilit)m(y)g(it)0 4124 y(mak)m(es)e(sequence)f(of)g(c)m(hoices)h Ft(s)1104 4138 y Fn(2)1143 4124 y Fu(,)f(and)g(so)g(on.)41 b(The)31 b(exp)s(ected)g(running)e(time)i(of)g(the)g(randomized)g (algorithm)0 4237 y(on)f(some)h(input)e Ft(I)38 b Fu(is)30 b(just)1199 4379 y Ff(X)981 4564 y Fn(c)n(hoice)21 b(sequences)h Fm(s)1551 4460 y Fu(Pr\()p Ft(s)p Fu(\)\(Running)29 b(time)i(on)g Ft(I)37 b Fu(giv)m(en)31 b Ft(s)p Fu(\))p Ft(:)0 4767 y Fu(If)22 b(y)m(ou)h(recall)h(the)f(de\014nition)f(of)h(exp)s (ectation,)j(the)d(running)e(time)i(of)g(the)g(randomized)f(algorithm)i (is)f(a)g(random)0 4880 y(v)-5 b(ariable)31 b(and)f(the)g(sequences)h Ft(s)f Fu(corresp)s(ond)f(to)i(the)g(elemen)m(tary)h(ev)m(en)m(ts.)0 5032 y(So,)37 b(the)e(exp)s(ected)h(running)d(time)j(of)f(the)h (randomized)f(algorithm)h(is)f(just)g(an)g(a)m(v)m(erage)j(o)m(v)m(er)e (deterministic)0 5145 y(algorithms.)53 b(Since)34 b(eac)m(h)h (deterministic)g(algorithm)g(has)f(a)m(v)m(erage-case)k(running)33 b(time)h(at)h(least)h Fp(b)p Fu(log)3644 5167 y Fn(2)3684 5145 y Fu(\()p Ft(n)p Fu(!\))p Fp(c)p Fu(,)p 0 5254 1560 4 v 104 5308 a Fk(1)138 5340 y Fj(Ev)n(erything)25 b(w)n(ould)h(b)r(e)g (easier)h(if)f(w)n(e)g(could)g(someho)n(w)h(assume)f Fi(n)p Fj(!)g(w)n(as)h(a)f(p)r(o)n(w)n(er)g(of)h(2....)p eop end %%Page: 30 31 TeXDict begin 30 30 bop 0 -165 a Fu(30)742 b Fs(LECTURE)30 b(5.)71 b(COMP)-8 b(ARISON-BASED)30 b(LO)m(WER)h(BOUNDS)g(F)m(OR)f(SOR) -8 b(TING)0 91 y Fu(their)30 b(a)m(v)m(erage)j(do)s(es)d(to)s(o.)42 b(F)-8 b(ormally)g(,)32 b(the)f(a)m(v)m(erage-case)j(running)29 b(time)i(of)g(the)f(randomized)g(algorithm)i(is)70 296 y(a)m(vg)15 372 y Fn(inputs)20 b Fm(I)351 215 y Ff(X)278 399 y Fn(c)n(hoices)h Fm(s)560 296 y Fu([Pr)o(\()p Ft(s)p Fu(\)\(Running)30 b(time)h(on)f Ft(I)38 b Fu(giv)m(en)31 b Ft(s)p Fu(\)])83 b(=)2189 215 y Ff(X)2233 389 y Fm(s)2324 296 y Fu(a)m(vg)2375 372 y Fm(I)2477 296 y Fu([Pr\()p Ft(s)p Fu(\)\(Running)29 b(time)i(on)g Ft(I)37 b Fu(giv)m(en)31 b Ft(s)p Fu(\)])2035 530 y(=)2189 449 y Ff(X)2233 623 y Fm(s)2324 530 y Fu(Pr\()p Ft(s)p Fu(\))15 b(a)m(vg)2601 607 y Fm(I)2688 530 y Fu(\(Running)29 b(time)i(on)f Ft(I)38 b Fu(giv)m(en)31 b Ft(s)p Fu(\))2035 754 y Fp(\025)2189 673 y Ff(X)2233 848 y Fm(s)2324 754 y Fu(Pr\()p Ft(s)p Fu(\))p Fp(b)p Fu(log)2693 776 y Fn(2)2732 754 y Fu(\()p Ft(n)p Fu(!\))p Fp(c)2035 957 y Fu(=)83 b Fp(b)p Fu(log)2347 979 y Fn(2)2386 957 y Fu(\()p Ft(n)p Fu(!\))p Fp(c)p Ft(:)p 2693 957 50 59 v 0 1161 a Fu(One)35 b(w)m(a)m(y)i(to)f(think)f (of)h(the)g(kinds)f(of)g(b)s(ounds)f(w)m(e)i(ha)m(v)m(e)h(b)s(een)e (pro)m(ving)g(is)h(to)g(think)g(of)f(a)h(matrix)g(with)g(one)0 1274 y(ro)m(w)h(for)g(ev)m(ery)h(p)s(ossible)e(deterministic)i (comparison-based)g(sorting)f(algorithm)h(\(there)g(could)f(b)s(e)f(a)i (lot)g(of)0 1387 y(ro)m(ws!\))46 b(and)32 b(one)g(column)g(for)g(ev)m (ery)h(p)s(ossible)f(p)s(erm)m(utation)g(of)g(the)g Ft(n)g Fu(inputs)f(\(there)i(are)f(a)h(lot)g(of)f(columns)0 1500 y(to)s(o\).)52 b(En)m(try)33 b(\()p Ft(i;)15 b(j)5 b Fu(\))36 b(in)d(this)h(matrix)f(con)m(tains)i(the)f(running)e(time)i (of)g(algorithm)h Ft(i)f Fu(on)f(input)g Ft(j)5 b Fu(.)51 b(The)33 b(w)m(orst-)0 1613 y(case)i(deterministic)f(lo)m(w)m(er)h(b)s (ound)c(tells)k(us)d(that)i(for)g(eac)m(h)g(ro)m(w)g Ft(i)g Fu(there)f(exists)h(a)g(column)f Ft(j)3303 1627 y Fm(i)3366 1613 y Fu(suc)m(h)g(that)h(the)0 1726 y(en)m(try)g(\()p Ft(i;)15 b(j)384 1740 y Fm(i)414 1726 y Fu(\))34 b(is)h(large.)53 b(The)33 b(a)m(v)m(erage-case)38 b(deterministic)d(lo)m(w)m(er)h(b)s (ound)c(tells)j(us)e(that)i(for)f(eac)m(h)h(ro)m(w)f Ft(i)p Fu(,)i(the)0 1839 y(a)m(v)m(erage)42 b(of)e(the)f(elemen)m(ts)i (in)e(the)g(ro)m(w)h(is)f(large.)69 b(The)39 b(randomized)g(lo)m(w)m (er)i(b)s(ound)c(sa)m(ys)j(\\w)m(ell,)j(since)d(the)0 1952 y(ab)s(o)m(v)m(e)34 b(statemen)m(t)h(holds)e(for)g(ev)m(ery)h(ro)m (w,)g(it)f(m)m(ust)g(also)h(hold)f(for)g(an)m(y)h(w)m(eigh)m(ted)g(a)m (v)m(erage)i(of)d(the)g(ro)m(ws.")50 b(In)0 2065 y(the)32 b(language)i(of)e(game-theory)-8 b(,)34 b(one)f(could)f(think)f(of)h (this)g(as)g(a)h(t)m(w)m(o-pla)m(y)m(er)h(game)f(\(m)m(uc)m(h)g(lik)m (e)g(ro)s(c)m(k-pap)s(er-)0 2178 y(scissors\))f(b)s(et)m(w)m(een)h(an)e (\\algorithm)j(pla)m(y)m(er")f(who)e(gets)i(to)g(pic)m(k)f(a)g(ro)m(w)g (and)f(an)h(adv)m(ersarial)h(\\input)e(pla)m(y)m(er")0 2291 y(who)39 b(gets)i(to)f(pic)m(k)g(a)g(column.)68 b(Eac)m(h)40 b(pla)m(y)m(er)h(mak)m(es)g(their)e(c)m(hoice)j(and)d(the) g(en)m(try)h(in)f(the)h(matrix)g(is)g(the)0 2403 y(cost)35 b(to)f(the)g(algorithm-pla)m(y)m(er)i(whic)m(h)d(w)m(e)h(can)g(think)g (of)f(as)h(ho)m(w)g(m)m(uc)m(h)g(money)f(the)h(algorithm-pla)m(y)m(er)i (has)0 2516 y(to)e(pa)m(y)f(the)h(input)e(pla)m(y)m(er.)50 b(W)-8 b(e)35 b(ha)m(v)m(e)f(sho)m(wn)f(that)h(there)f(is)g(a)h (randomized)f(strategy)h(for)f(the)h(input)e(pla)m(y)m(er)0 2629 y(\(namely)-8 b(,)40 b(pic)m(k)e(a)f(column)g(at)h(random\))e (that)i(guaran)m(tees)g(it)g(an)f(exp)s(ected)g(gain)h(of)f(\012\()p Ft(n)15 b Fu(log)i Ft(n)p Fu(\))37 b(no)g(matter)0 2742 y(what)30 b(strategy)i(the)f(algorithm-pla)m(y)m(er)h(c)m(ho)s(oses.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF