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(ogram)i(using)f(locks)0 748 y Fx(L)p Fw(MV)e Fx(is)g(a)h(parallel)f (program)g(based)e(on)i(a)g(shared)f(memory)h(model,)g(where)g(a)g (program)g(consists)e(of)j(a)f(collection)g(of)g Fr(p)0 861 y Fx(threads)c(that)g(share)g(a)g(single)g(address)f(space.)30 b(The)19 b(number)g(of)h(threads)e(is)i(arbitrary)g(and)f(can)g(be)g (speci\002ed)e(at)j(runtime.)0 974 y(Each)30 b(thread)g Fr(i)h Fx(is)g(assigned)d(a)j(disjoint)g(and)f(contiguous)f(set)h(of)h (ro)n(ws)f(of)h(the)g(sparse)e(matrix)i(so)g(that)g(each)e(set)h(has)0 1087 y(roughly)25 b(the)g(same)g(number)g(of)g(nonzeros.)40 b(Since)25 b(there)g(are)g(on)g(the)h(order)f(of)h(thousands)d(of)j(ro) n(ws)f(and)g(only)g(tens)g(of)0 1200 y(nonzeros)e(per)j(ro)n(w)-6 b(,)26 b(it)g(is)g(tri)n(vial)h(to)e(generate)f(good)h(partitions.)41 b(Thread)25 b Fr(i)h Fx(performs)f(the)g(SMVP)h(on)f(its)h(set)f(of)h (ro)n(ws,)0 1313 y(using)c(the)g(follo)n(wing)h(call)g(sequence:)120 1541 y Fe(barrier\(\);)120 1620 y(zero)p 284 1620 20 4 v 23 w(vector\(w,)38 b(firstrow[i],)g(numrows[i]\);)120 1699 y(barrier\(\);)120 1778 y(local)p 324 1778 V 23 w(smvp\(n,)h(A,)g(Acol,)g(Aindex,)g(v,)h(w,)f(firstrow[i],)f (numrows[i]\);)141 2119 y Fn(F)l(ir)o(str)l(ow[i])26 b Fx(and)h Fn(numr)l(ows[i])e Fx(describe)h(the)h(set)g(of)h(ro)n(ws)f (assigned)e(to)j(thread)f Fr(i)p Fx(,)i Fn(barrier\(\))e Fx(performs)g(a)h(barrier)0 2232 y(synchronization)22 b(among)h(all)i(of)g(the)f(threads.)37 b(Since)24 b(dif)n(ferent)g (threads)g(might)h(update)e(the)h(same)f(sub)o(v)o(ector)f(of)j Fr(w)r Fx(,)g(a)0 2345 y(set)d(of)h Fr(L)i(>)g Fx(0)e(locks)e (guarantees)g(consistent)g(updates)g(of)i Fr(w)r Fx(:)261 2478 y Fe(setlock\(col)38 b(\045)i(L\);)141 2557 y(22)g(w[col][0])e(+=) i(A[Anext][0][0]*v[i][0])c(+)k(...)141 2636 y(23)g(w[col][1])e(+=)i (A[Anext][0][1]*v[i][0])c(+)k(...)141 2714 y(24)g(w[col][2])e(+=)i (A[Anext][0][2]*v[i][0])c(+)k(...)261 2793 y(unsetlock\(col)e(\045)h (L\);)2276 2478 y(setlock\(i)g(\045)g(L\);)2156 2557 y(27)h(w[i][0])f(+=)g(sum0;)2156 2636 y(28)h(w[i][1])f(+=)g(sum1;)2156 2714 y(29)h(w[i][2])f(+=)g(sum2;)2276 2793 y(unsetlock\(i)f(\045)i (L\);)141 3017 y Fx(If)20 b(lock)f Fr(k)k Fx(is)c(unset,)f(then)h Fn(setloc)n(k\(k\))g Fx(atomically)g(sets)f(lock)h Fr(k)j Fx(and)c(returns.)32 b(Otherwise,)19 b Fn(setloc)n(k\(k\))f Fx(w)o(aits)i(until)f(the)0 3130 y(lock)g(becomes)f(unset,)h(then)g (atomically)g(sets)g(it.)33 b Fn(Unsetloc)n(k\(k\))18 b Fx(unsets)h(lock)g Fr(k)s Fx(.)32 b(In)20 b(general,)f(more)h(locks)e (means)g(fe)n(wer)0 3243 y(synchronizations)d(between)i(threads,)h (with)h(the)f(best)g(case)f(being)g(a)h(unique)g(lock)f(for)i(each)e (sub)o(v)o(ector)-5 b(.)29 b(Ho)n(we)n(v)o(er)l(,)17 b(most)0 3356 y(systems)k(limit)j(the)f(number)f(of)h(locks.)31 b(Thus)22 b(the)g(number)g(of)h(locks)e Fr(L)i Fx(can)f(be)g (speci\002ed)e(at)j(runtime.)141 3519 y(It)18 b(is)e(interesting)g(to)h (observ)o(e)d(that)j(the)f Fn(only)g Fx(reason)f(we)h(need)f(locks)h 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g(e)o(xtra)h(reduction)f(operation.)51 b(The)29 b(idea)f(is)h(for)h (each)e(thread)g Fr(i)i Fx(to)g(update)d(its)j(o)n(wn)0 4720 y(pri)n(v)n(ate)22 b Fr(w)j Fx(v)o(ector)-5 b(.)32 b(These)21 b Fr(p)h Fx(pri)n(v)n(ate)g(v)o(ectors)f(are)i(then)f (summed)f(\(reduced\))h(to)h(form)g(the)g(\002nal)f(result)h(v)o(ector) e Fr(w)r Fx(.)120 4948 y Fe(barrier\(\);)120 5027 y(zero)p 284 5027 V 23 w(vector\(privatew[i],)37 b(0,)i(n\);)120 5106 y(local)p 324 5106 V 23 w(smvp\(n,)g(A,)g(Acol,)g(Aindex,)g(v,)h (privatew[i],)e(firstrow[i],)g(numrows[i]\);)120 5185 y(barrier\(\);)120 5264 y(add)p 244 5264 V 23 w(vectors\(privatew,)f (w,)j(firstrow[i],)e(numrows[i]\);)p eop %%Page: 8 12 8 11 bop 0 -228 a Fx(8)1521 b Fl(Da)n(vid)22 b(R.)g(O'Hallaron)p 0 -191 3900 4 v 0 175 a Fe(1)80 b Fd(v)o(oid)41 b Fe(add)p 405 175 20 4 v 23 w(vectors\()p Fd(double)f Fe(\(**tmpw\)[3],)e Fd(double)i Fe(\(*w\)[3],)f Fd(int)h Fe(firstrow,)f Fd(int)h Fe(numrows\))f Ff(f)0 254 y Fe(2)199 b Fd(int)40 b Fe(i,)g(j;)0 333 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array def /_pcmyk 4 array def /_bkrgb [1 1 1] def /_bkcmyk [0 0 0 0] def /fg{/_fg exch def /filltype 0 def/fills{_fg setgray}def}def /frgb{_frgb astore pop /filltype 0 def/fills{_frgb aload pop setrgbcolor}def}def /fcmyk{_fcmyk astore pop /filltype 0 def/fills{_fcmyk aload pop setcmykcolor}def}def /pg{/_pg exch def /stroketype 0 def/pens{_pg setgray}def}def /prgb{_prgb astore pop /stroketype 0 def/pens{_prgb aload pop setrgbcolor}def}def /pcmyk{_pcmyk astore pop /stroketype 0 def/pens{_pcmyk aload pop setcmykcolor}def}def /fpat{/fstr edf/filltype 1 def/fills{/patstr fstr def}bdf}bdf /ppat{/sstr edf/stroketype 1 def/pens{/patstr sstr def}bdf}bdf /bkg{ /_bkg exch def /_bp{gsave _bkg setgray fill grestore}def}def /bkrgb{_bkrgb astore pop/_bp{gsave _bkrgb aload pop setrgbcolor fill grestore}def}def /bkcmyk{_bkcmyk astore pop/_bp{gsave _bkcmyk aload pop setcmykcolor fill grestore}def}def /frg{ /_frg exch def /_fp{_frg setgray}def}def /frrgb{_frrgb astore pop/_fp{_frrgb aload pop setrgbcolor}def}def /frcmyk{_frcmyk astore pop/_fp{_frcmyk aload pop setcmykcolor}def}def /icomp{/ncolors edf ncolors 1 gt{/proc0 edf dup dup 0 get ncolors div cvi exch 0 3 -1 roll put 4 -1 roll ncolors div cvi 4 1 roll{proc0 dup/st0 edf 0 exch ncolors exch length dup ncolors sub exch ncolors div cvi string/st1 edf {dup 0 exch dup 1 exch 2 add{st0 exch get add}bind for 3 div ncolors 4 eq{exch dup 3 1 roll 3 add st0 exch get add 255 exch sub dup 0 lt{pop 0}if}if cvi dup 255 gt{pop 255}if exch ncolors div cvi exch st1 3 1 roll put}bind for st1}}if i}bdf /ci {/colorimage where {pop false exch colorimage} {icomp} ifelse}bdf /impat {/cnt 0 def /MySave save def currot 0 ne{currot neg rotate}if clip flattenpath pathbbox 3 -1 roll 8 div floor 8 mul dup/starty edf sub abs 8 div ceiling 8 mul cvi/height edf exch 8 div floor 8 mul dup/startx edf sub abs 8 div ceiling 8 mul cvi/width edf startx starty translate width height scale /height height 8 mul def /st0 width string def width height T [width 0 0 height neg 0 height] {patstr cnt 8 mod get/st1 edf 0 1 st0 length 1 sub dup 0 le{pop 1}if {st0 exch st1 put}bind for/cnt cnt 1 add def st0}bind imagemask MySave restore newpath}bdf /cm{/ncolors edf translate scale/height edf/colorimage where {pop} {ncolors mul}ifelse/width edf /tbitstr width string def width height 8 [width 0 0 height neg 0 height] {currentfile tbitstr readhexstring pop}bind ncolors dup 3 eq {ci}{icomp}ifelse}bdf /im{translate scale /height edf /width edf /tbitstr width 7 add 8 div cvi string def width height 1 [width 0 0 height neg 0 height] {currentfile tbitstr readhexstring pop}bind i}bdf /imk{/invFlag edf translate scale /height edf /width edf /tbitstr width 7 add 8 div cvi string def width height invFlag [width 0 0 height neg 0 height] {currentfile tbitstr readhexstring pop}bind imagemask}bdf /BeginEPSF {/MySave save def /dict_count countdictstack def /op_count count 1 sub def userdict begin /showpage {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath /languagelevel where {pop languagelevel 1 ne{false setstrokeadjust false setoverprint}if}if }bdf /EndEPSF {count op_count sub {pop}repeat countdictstack dict_count sub {end}repeat MySave restore}bdf /rectpath {/cv_r edf/cv_b edf/cv_l edf/cv_t edf cv_l cv_t moveto cv_r cv_t lineto cv_r cv_b lineto cv_l cv_b lineto cv_l cv_t lineto closepath}bdf /setpen{/penh edf/penv edf/penv2 penv 2 div def/penh2 penh 2 div def}bdf /dostroke{not pens 1.0 currentgray ne or {s}{newpath}ifelse}bdf /dodashfill{not fills 1.0 currentgray ne or {gsave f grestore gsave [] 0 setdash stroketype/stroketype filltype def s/stroketype edf grestore}{newpath}ifelse}bdf /dofill{not fills 1.0 currentgray ne or {f}{newpath}ifelse}bdf /dofillsave{not fills 1.0 currentgray ne or {gsave f grestore}if}bdf /doline{not pens 1.0 currentgray ne or {filltype/filltype stroketype def f/filltype edf}{newpath}ifelse}bdf /spx{SpaceExtra 0 32 4 -1 roll widthshow}bdf /lsx{SpaceExtra 0 32 LetterSpace 0 6 -1 roll awidthshow}bdf /Rjust{stringwidth pop JustOffset exch sub /JustOffset edf}bdf /Cjust{stringwidth pop 2 div JustOffset exch sub /JustOffset edf}bdf /adjfit{stringwidth pop LetterSpace StringLength 1 sub mul add SpaceExtra NumSpaces mul add dup /pw edf JustOffset exch sub dup /wdif edf StringLength div LetterSpace add /LetterSpace edf}bdf /ulb{currentpoint pop /underlinpt edf}bdf /ule{gsave currentpoint newpath moveto currentfont dup /ft1 known{dup /ft1 get begin /FontMatrix get FontMatrix tpmx concatmatrix pop} {begin FontMatrix tpmx copy pop}ifelse currentfont /FontInfo known {FontInfo begin UnderlinePosition UnderlineThickness end}{100 1}ifelse end dup tpmx dtransform pop setlinewidth dup tpmx dtransform pop 0 exch rmoveto underlinpt currentpoint pop sub 0 rlineto stroke grestore}bdf /fittext{ /SpaceExtra edf /LetterSpace edf /StringLength edf /NumSpaces edf /JustOffset edf not 1 currentgray ne or {dup {ulb}if exch dup adjfit lsx {ule}if}{pop pop}ifelse}bdf /cvRecFont{/encod edf FontDirectory 2 index known{cleartomark}{findfont dup length 1 add dict begin {1 index/FID ne{def}{pop pop}ifelse}forall encod{/Encoding CVvec def}if currentdict end definefont cleartomark}ifelse}bdf /wrk1 ( ) def/wdict 16 dict def /Work75 75 string def /Nmk{Work75 cvs dup}bdf /Npt{put cvn}bdf /dhOdh{Nmk 2 79 Npt}bdf /dhodh{Nmk 2 111 Npt}bdf /dhSdh{Nmk 2 83 Npt}bdf /sfWidth{gsave 0 0 moveto 0 0 lineto 0 0 lineto 0 0 lineto closepath clip stringwidth grestore}bdf /MakOF{dup dhodh FontDirectory 1 index known{exch pop}{exch findfont dup length 1 add dict begin {1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall /PaintType 2 def /StrokeWidth .24 1000 mul ftSize div dup 12 lt{pop 12}if def dup currentdict end definefont pop}ifelse}bdf /fts{dup/ftSize edf}def /mkFT{/tempFT 11 dict def tempFT begin /FontMatrix [1 0 0 1 0 0] def/FontType 3 def FontDirectory 3 index get /Encoding get/Encoding exch def /proc2 edf/ft2 exch findfont def/ft1 exch findfont def/FontBBox [0 0 1 1] def /BuildChar{wdict begin/chr edf/ftdt edf/chrst wrk1 dup 0 chr put def ftdt/proc2 get exec end}def end tempFT definefont pop}bdf /OLFt{dup dhOdh FontDirectory 1 index known{exch pop} {dup 3 -1 roll dup MakOF {outproc} mkFT}ifelse}bdf /mshw{moveto show}bdf /outproc{ftdt/ft1 get setfont gsave chrst sfWidth grestore setcharwidth dblsh}bdf /dblsh{currentgray 1 setgray chrst 0 0 mshw setgray ftdt/ft2 get setfont chrst 0 0 mshw}bdf /ShadChar{ftdt/ft1 get setfont gsave chrst sfWidth 1 index 0 ne{exch .05 add exch}if grestore setcharwidth chrst .06 0 mshw 0 .05 translate dblsh}bdf /ShFt{dup dhSdh FontDirectory 1 index known{exch pop} {dup 3 -1 roll dup MakOF {ShadChar} mkFT}ifelse}bdf /LswUnits{72 75 div dup scale}bdf /erasefill{_bp}def /CVvec 256 array def /NUL/SOH/STX/ETX/EOT/ENQ/ACK/BEL/BS/HT/LF/VT/FF/CR/SO/SI/DLE/DC1/DC2/DC3/DC4/NAK/SYN/ETB/CAN/EM/SUB/ESC/FS/GS/RS/US CVvec 0 32 getinterval astore pop CVvec 32/Times-Roman findfont/Encoding get 32 96 getinterval 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b(ar)o(allel)21 b(Pr)l(ocessing)e(Symposium)i Fx(\(Santa)g(Barbara,)h(April)h(1995\),)e (IEEE,)198 3964 y(pp.)h(418\226427.)0 4152 y([11])46 b(G)p Fw(R)m(OPP)-8 b Fx(,)25 b(W)-8 b(.,)27 b Fw(AND)f Fx(S)p Fw(MITH)p Fx(,)h(B.)42 b(The)26 b(design)f(of)h (data\255structure\255neutral)g(libraries)h(for)g(the)f(iterati)n(v)o (e)g(solution)g(of)198 4265 y(sparse)21 b(linear)h(systems.)31 b Fn(Scienti\002c)22 b(Pr)l(o)o(gr)o(amming)e(5)i Fx(\(1996\),)g (329\226336.)0 4452 y([12])46 b(H)p Fw(ENDRICKSON)p Fx(,)22 b(B.,)j Fw(AND)f Fx(L)p Fw(ELAND)p Fx(,)f(R.)37 b(An)25 b(impro)o(v)o(ed)d(spectral)i(graph)g(partitioning)g(algorithm)h(for)g (mapping)198 4565 y(parallel)d(computations.)31 b Fn(SIAM)23 b(J)n(.)f(Sci.)g(Comput.)g(16)p Fx(,)g(2)g(\(1995\),)h(452\226469.)0 4753 y([13])46 b(H)p Fw(INRICHS)p Fx(,)21 b(S.,)g(K)n Fw(OSAK)p Fx(,)g(C.,)g(O'H)p Fw(ALLAR)m(ON)p Fx(,)g(D.,)g(S)p Fw(TRICKER)p Fx(,)f(T)-7 b(.,)22 b Fw(AND)f Fx(T)-7 b Fw(AKE)p Fx(,)21 b(R.)31 b(An)22 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