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%!PS-Adobe-2.0 %%Creator: dvips 5.490 Copyright 1986, 1992 Radical Eye Software %%Title: mdynhash.dvi %%Pages: 33 1 %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips mdynhash %%BeginProcSet: tex.pro%!PS-Adobe /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N} B /TR{translate}N /isls false N /vsize 11 72 mul N /@rigin{isls{[0 -1 1 0 0 0] concat}if 72 Resolution div 72 VResolution div neg scale isls{Resolution hsize -72 div mul 0 TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 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(useful)f(for)h(v)m(ariations)h(of)f(the)g(sc)o(heme)d(presen)o(ted)46 2430 y(in)i(this)h(pap)q(er,)g(whic)o(h)f(yield)f(constructions)i(of)g (dynamic)e(dictionaries)h(for)g(parallel)g(and)h(dis-)46 2490 y(tributed)i(mac)o(hine)f(mo)q(dels)h(as)h(w)o(ell)f(as)h(further)g(impro)o (v)o(em)o(en)n(ts)e(of)i(the)f(sequen)o(tial)g(sc)o(heme)46 2551 y([DM89,)16 b(DM90a)q(,)g(DM90b)q(].)46 2776 y Fk(2)83 b(Dynamic)24 b(p)r(erfect)j(hashing)46 2945 y Fo(W)l(e)20 b(b)q(egin)g(b)o(y) f(reviewing)g(the)h(FKS)g(sc)o(heme)e(for)i(statically)f(storing)i(a)f(set)g Fj(S)j Fo(of)e(size)e Fj(n)p Fo(.)32 b(Let)46 3005 y Fi(H)88 3012 y Fl(s)121 3005 y Fo(=)13 b Fi(f)p Fj(h)h Fo(:)g Fj(U)19 b Fi(!)13 b(f)p Fo(1)p Fj(;)8 b(:)g(:)g(:)g(;)g(s)p Fi(g)14 b(j)f Fj(h)p Fo(\()p Fj(x)p Fo(\))h(=)g(\()p Fj(k)r(x)f Fo(mo)q(d)h Fj(p)p Fo(\))g(mo)q(d)f Fj(s;)20 b Fo(1)15 b Fi(\024)e Fj(k)j Fi(\024)e Fj(p)s Fi(\000)s Fo(1)p Fi(g)p Fo(,)f(where)f Fj(p)h Fo(is)f(prime)46 3065 y(and)i Fj(p)h Fi(\025)e Fj(N)5 b Fo(.)21 b(The)14 b(sc)o(heme)d(has)k(t)o(w)o(o)f(lev)o(els.)k(A)o(t)13 b(the)h(top)g(lev)o(el,)e(a)i(hash)h(function)f(partitions)g(the)p eop %%Page: 4 4 3 bop 46 188 a Fx(2)49 b(D)o(YNAMIC)14 b(PERFECT)j(HASHING)989 b Fo(4)46 342 y(elemen)o(ts)14 b(b)q(eing)i(stored)h(in)o(to)f Fj(s)h Fo(sets.)22 b(The)17 b(second)f(lev)o(el)f(consists)i(of)g(a)g(p)q (erfect)e(hash)j(function)46 402 y(for)c(eac)o(h)g(of)g(these)g(sets.)21 b(Sp)q(eci\014cally)l(,)12 b(a)j(function)e Fj(h)i Fo(c)o(hosen)f(uniformly)e (at)i(random)g(from)f Fi(H)1811 409 y Fl(s)1844 402 y Fo(is)46 462 y(used)h(to)f(partition)h(the)f(set)h Fj(S)i Fo(in)o(to)d Fj(s)h Fo(blo)q(c)o(ks.)20 b(Let)13 b Fj(W)1049 444 y Fl(h)1042 474 y(j)1085 462 y Fo(=)h Fi(f)p Fj(x)g Fi(2)g Fj(S)i Fi(j)e Fj(h)p Fo(\()p Fj(x)p Fo(\))g(=)f Fj(j)s Fi(g)p Fo(;)h(the)g(sup)q(erscript) 46 522 y Fj(h)k Fo(is)g(omitted)f(when)h Fj(h)g Fo(is)g(understo)q(o)q(d.)29 b(F)l(redman,)17 b(Koml\023)-24 b(os,)18 b(and)h(Szemer)o(\023)-23 b(edi)15 b(sho)o(w)k(that)f(if)g(a)46 582 y(function)e Fj(h)g Fo(is)g(c)o(hosen)g(from)g Fi(H)644 589 y Fl(s)679 582 y Fo(uniformly)e(at)j (random)e(then)626 717 y Fj(E)665 644 y Fg( )729 675 y(X)707 767 y Fm(0)p Ff(\024)p Fl(j)r(<s)820 2="" 25="" 182="" 206="" 644="" 683="" 690="" 705="" 717="" 751="" 865="" 925="" 931="" 962="" 970="" 977="" 990="" 992="" 1004="" 1038="" 1054="" 1081="" 1110="" 1131="" 1138="" 1189="" 1192="" 1197="" 1229="" 1237="" 1244="" 1270="" 1305="" 1320="" 1333="" 1395="" 1402="" 1455="" 1462="" 1515="" 1522="" 1533="" 1592="" 1623="" 1631="" 1638="" 1665="" 1699="" 1715="" 1726="" 1824="" 1831="" 1884="" 1891="" 1902="" 1944="" 1951="" 2010="" 2042="" 2049="" 2056="" 2083="" 2090="" 2117="" 2133="" 2228="" 2347="" 2408="" 2527="" 2587="" 2647="" y="" fg(="" )853="" fi(j)p="" fj(w)913="" fl(j)931="" fi(j)887="" fo(2)945="" fg(!!)1038="" fi(\024)1110="" fj(n)p="" fo(\()p="" fj(n)c="" fi(\000)g="" fo(1\))p="" v="" a="" fj(s)1814="" fo(\(2\))46="" y(\(where)20="" b="" fj(e)s="" fj(x)t="" fo(\))h(denotes)g(the)f(exp)q="" (ectation)g(of)h(the)f(random)g(v)m(ariable)g="" fo(\),)i(and)f="" (consequen)o(tly)46="" y(that)555="" y(pr)607="" )671="" y(x)648="" fm(0)p="" ff(\024)p="" fl(j)r(<s)762="" )795="" fj(w)855="" fl(j)873="" fi(j)828="" fo(2)887="" fg(!)933="" fj(<)990="" fo(2)p="" fj(n)12="" fi(\000)e="" fj(s)1201="" fg(!)1261="" fi(\025)1333="" fo(1)p="" a(2)1362="" fj(:)438="" fo(\(1\))46="" y(cho)q(osing)15="" fj(s)f="" fo(=")f(2\()p" fj(n)5="" fo(1\),)14="" b(relation)f(\()p="" fi(\003)p="" fo(\))g(implies)e(that)i(for)h(at)g="" (least)f(half)g(of)g(the)g(functions)h="" fj(h)g="" fi(2)g(h)1858="" fl(s)46="" fo(one)i(has)794="" fg(x)772="" fl(j)r(<s)885="" )918="" fj(w)978="" fl(j)996="" fi(j)952="" fo(2)1010="" fg(!)1056="" fj(<)e(n:)46="" fo(suc)o(h)j(a)i(function)e(is)h(used)="" g(to)g(partition)g="" fj(s)j="" fo(in)o(to)d(blo)q(c)o(ks)f="" fj(w)1177="" fl(j)1196="" fo(,)h(0)f="" fi(\024)f="" fj(j)k(<)c(s)p="" fo(.)26="" b(f)l(or)18="" b(eac)o(h)g(blo)q(c)o(k)f="" fj(w)1858="" fl(j)46="" fo(one)j(uses)f(relation)g(\()p="" fo(\))h(with)f="" fj(s)647="" fl(j)685="" fi(f)p="" fj(;)8="" fj(w)988="" fl(j)1006="" fj(w)1099="" fl(j)1117="" fi(j)13="" b(\000)h="" fi(g)p="" fo(.)31="" b(it)19="" b(follo)o(ws)g(that)h(for)="" g(at)g(least)46="" y(half)c(of)h(the)f(functions)g="" fj(h)e="" fi(2)g(h)623="" fl(s)639="" y(j)676="" fo(one)j(has)787="" fg(x)758="" fl(l<s)857="" y(j)884="" )917="" fj(w)977="" fl(j;l)1014="" fi(j)960="" fo(2)1028="" fg(!)1075="" fj(<)c="" fj(;)46="" fo(where)h="" fj(w)231="" fl(j;l)282="" fj(x)f="" fi(2)h="" fj(w)493="" fl(j)526="" fi(j)f="" fj(h)p="" fj(x)p="" fo(\))h(=")g" fj(l)q="" fo(,)g(i.)8="" b(e.,)13="" fj(w)941="" fl(j;l)977="" fi(j)h(\024)g="" fo(1)h(for)g(all)f="" fo(.)20="" b(f)l(or)15="" b(eac)o(h)f="" fj(j)k="" fo(therefore)c(at)h(least)46="" y(half)i(of)h(the)g="" (functions)f(in)h="" fi(h)598="" fl(s)614="" y(j)653="" fo(are)f(injectiv)o(e)e(on)k="" fj(w)1046="" fl(j)1064="" fo(.)25="" b(one)18="" b(uses)g(one)f(suc)o(h)h(function)f(for)h(eac)o(h)46="" fj(w)92="" fl(j)110="" fo(.)j(the)c(total)f(space)h="" (requiremen)n(t)c(is)k(linear)e(since)580="" fg(x)558="" fl(j)r(<s)671="" fj(s)694="" fl(j)726="" fj(s)d="" fo(+)g(4)g="" fi(\001)945="" fg(x)922="" fl(j)r(<s)1036="" )1068="" fj(w)1128="" fl(j)1147="" fi(j)1102="" fo(2)1160="" fg(!)1207="" fj(o)q="" fo(\))46="" y(b)o(y)i(the)g(c)o(hoice)f(of)h(the)g(hash)h(functions.)46="" y(f)l(or)e(the)g(dynamic)e(case,)i(w)o(e)f(use)h(the)g(standard)h="" (doubling)f(metho)q(d)g(to)g(deal)g(with)f(the)h(fact)g(that)46="" y(w)o(e)f(do)h(not)g(kno)o(w)g(in)f(adv)m(ance)h(ho)o(w)g(big)f(the)h="" (top-lev)o(el)e(table)h(or)h(an)o(y)f(of)h(the)g(subtables)g(will)e(get.)46="" y(supp)q(ose)h(that)g="" fj(n)g="" fo(is)f(the)h(curren)o(t)e(n)o(um)o(b)q(er)="" g(of)i(elemen)o(ts)d(stored)j(in)f(the)g(table.)20="" b(the)14="" b(fks)f(sc)o(heme)46="" y(in)19="" b(use)h(will)f(accommo)q(date)f(up)i(to)g="" fj(m)25="" fo(elemen)o(ts.)k(the)20="" b(v)m(alue)g(of)g="" fo(will)19="" b(initially)f(b)q(e)i(set)f(to)46="" y(\(1)6="" b(+)g="" fj(c)p="" fo(\))g="" fi(\001)g="" fj(n)14="" fo(for)g(some)f="" fj(c)h(="">)f Fo(0)i(and)f(as)g Fj(n)g Fo(c)o(hanges)g(will)f(nev)o(er)g(b)q(e)h(more)e (than)1452 2628 y Fm(1+)p Fl(c)p 1452 2636 61 2 v 1452 2664 a Fm(1)p Ff(\000)p Fl(c)1524 2647 y Fi(\001)6 b Fj(n)p Fo(.)20 b(Let)14 b Fj(s)p Fo(\()p Fj(M)5 b Fo(\),)14 b(to)46 2707 y(b)q(e)f(sp)q (eci\014ed,)g(b)q(e)h(the)f(n)o(um)o(b)q(er)e(of)j(sets)g(in)o(to)f(whic)o(h) f(the)h(top)h(lev)o(el)d(hash)k(function)e(is)g(to)h(partition)46 2768 y(the)h(elemen)o(ts)d(of)k Fj(S)s Fo(.)k(The)15 b(function)g Fj(h)h Fo(will)e(b)q(e)h(a)h(random)e(elemen)o(t)e(of)k Fi(H)1431 2775 y Fl(s)p Fm(\()p Fl(M)t Fm(\))1515 2768 y Fo(.)k(Th)o(us,)c(the)f(set)g Fj(S)46 2828 y Fo(is)h(partitioned)g(b)o(y)g Fj(h)g Fo(in)o(to)g(the)g (subsets)h Fj(W)857 2835 y Fl(j)889 2828 y Fo(=)d Fi(f)p Fj(x)f Fi(2)h Fj(S)j Fi(j)d Fj(h)p Fo(\()p Fj(x)p Fo(\))f(=)h Fj(i)p Fi(g)p Fo(,)h(0)f Fi(\024)g Fj(j)j(<)d(s)p Fo(\()p Fj(M)5 b Fo(\).)46 2947 y(Let)19 b Fj(T)165 2954 y Fl(j)203 2947 y Fo(b)q(e)g(the)h (blo)q(c)o(k)e(of)i(memory)d(used)i(for)h(storing)g Fj(W)1146 2954 y Fl(j)1164 2947 y Fo(.)31 b(The)20 b(amoun)o(t)e(of)i(space)g(allo)q (cated)46 3007 y(to)g Fj(T)138 3014 y Fl(j)176 3007 y Fo(is)f Fj(s)251 3014 y Fl(j)269 3007 y Fo(,)i(where)e Fj(s)471 3014 y Fl(j)509 3007 y Fo(=)h(2)p Fj(m)634 3014 y Fl(j)652 3007 y Fo(\()p Fj(m)714 3014 y Fl(j)745 3007 y Fi(\000)14 b Fo(1\),)20 b(and)h Fj(m)1017 3014 y Fl(j)1054 3007 y Fo(is)f(the)f(maximal)e(size)i(of)h Fj(W)1598 3014 y Fl(j)1636 3007 y Fo(the)f(curren)o(t)46 3067 y(table)g Fj(T)200 3074 y Fl(j)236 3067 y Fo(is)g(mean)o(t)f(to)h(manage.)29 b(The)19 b(quan)o(tit)o(y)f Fj(m)1051 3074 y Fl(j)1088 3067 y Fo(is)g(alw)o(a)o(ys)h(as)h(least)f(as)g(big)g(as)h Fi(j)p Fj(W)1747 3074 y Fl(j)1765 3067 y Fi(j)f Fo(and)p eop %%Page: 5 5 4 bop 46 188 a Fx(2)49 b(D)o(YNAMIC)14 b(PERFECT)j(HASHING)989 b Fo(5)46 342 y(is)16 b(at)h(most)e(t)o(wice)h(the)g(n)o(um)o(b)q(er)e(of)j (all)f(elemen)o(ts)d(ev)o(er)j(mapp)q(ed)f(to)i Fj(j)i Fo(b)o(y)d(the)g (curren)o(t)g(top)h(lev)o(el)46 402 y(function)i Fj(h)p Fo(.)29 b(The)19 b(subset)g Fj(W)612 409 y Fl(j)649 402 y Fo(is)g(resolv)o(ed)f (within)g Fj(T)1074 409 y Fl(j)1111 402 y Fo(b)o(y)g(using)h(a)h(p)q(erfect)e (hash)i(function)e Fj(h)1858 409 y Fl(j)46 462 y Fo(from)e Fi(H)204 469 y Fl(s)220 480 y(j)241 462 y Fo(.)24 b(If)16 b(the)h(v)m(alue)g Fj(k)565 469 y Fl(j)601 462 y Fo(sp)q(eci\014es)f(whic)o(h)h(hash)h(function) e Fj(h)1265 469 y Fl(j)1301 462 y Fo(is)h(b)q(eing)g(used,)g(then)g Fj(x)e Fi(2)g Fj(W)1858 469 y Fl(j)46 528 y Fo(is)j(stored)i(in)e(lo)q (cation)h(\()p Fj(k)538 535 y Fl(j)557 528 y Fj(x)13 b Fo(mo)q(d)h Fj(p)p Fo(\))g(mo)q(d)f Fj(s)893 535 y Fl(j)930 528 y Fo(of)19 b(subtable)g Fj(T)1215 535 y Fl(j)1233 528 y Fo(.)29 b(It)18 b(will)g(b)q(e)h(arranged)h(that)f(the)46 588 y(follo)o(wing)d(condition)g (is)g(alw)o(a)o(ys)g(satis\014ed:)720 684 y Fg(X)665 778 y Fm(0)p Ff(\024)p Fl(j)r(<s)p 2="" 5="" 6="" 18="" 46="" 121="" 188="" 312="" 467="" 527="" 587="" 673="" 691="" 713="" 725="" 732="" 759="" 813="" 881="" 931="" 942="" 975="" 979="" 1002="" 1121="" 1181="" 1241="" 1302="" 1362="" 1422="" 1482="" 1542="" 1603="" 1663="" 1670="" 1723="" 1730="" 1831="" 1842="" 1902="" 1963="" 2023="" 2083="" 2143="" 2203="" 2264="" 2424="" 2485="" 2645="" 2705="" 2766="" 2826="" 2833="" 2886="" 2893="" 2927="" 2935="" 2946="" 2953="" 2963="" 3040="" 3061="" 3068="" fm(\()p="" fl(m)t="" fm(\))835="" y="" fj(s)858="" fl(j)904="" fi(\024)975="" fo(32)p="" fj(m)1075="" fm(2)p="" v="" a="" fj(s)p="" fo(\()p="" fj(m)5="" b="" fo(\))1112="" y(+)11="" b(4)p="" fj(m)20="" b(:)548="" fo(\(7\))46="" y(the)21="" b(parameter)g="" fo(\))22="" b(will)e(b)q(e)i(c)o(hosen)f(to)h(b)q(e)g(\002\()p="" fj(n)p="" fo(\))f(so)h(that)g="" (the)f(righ)o(t)h(hand)g(side)f(of)h(this)46="" y(equation)c(is)f="" fj(o)q="" fo(\).)27="" b(w)l(e)17="" b(will)g(see)h(that)g(this)g="" (guaran)o(tees)g(that)g(the)g(total)g(space)g(used)g(is)f(linear)46="" y(in)f(the)g(n)o(um)o(b)q(er)e(of)j(elemen)o(ts)c(curren)o(tly)i(stored)="" h(in)g(the)g(table.)46="" y(the)k(algorithm)g(can)g(b)q(e)h(sp)q(eci\014ed)="" f(more)f(precisely)g(as)i(describ)q(ed)f(in)g(the)g(program)g(giv)o(en)g(in)="" y(figures)c(1)i(and)f(2)g(b)q(elo)o(w.)23="" b(the)17="" b(v)m(ariable)f="" fw(c)n(ount)23="" fo(k)o(eeps)15="" b(trac)o(k)i(of)g(the)f(n)o="" (um)o(b)q(er)f(of)i(up)q(dates)h(p)q(er-)46="" y(formed)11="" b(in)i(the)g(hash)g(table)g(of)g(the)g(presen)o(t)f(size)g="" fo(.)20="" b(f)l(rom)12="" b(time)f(to)i(time)e(it)h(b)q(ecomes)g="" (necessary)46="" y(to)19="" b(restructure)e(the)h(whole)g(table.)27="" b(this)19="" b(is)f(the)g(case)g(when)h="" fw(c)n(ount)k="" fo(reac)o(hes)18="" fj(m)24="" fo(or)18="" b(when)h(\()p="" fi(\003\003)p="" fo(\))46="" y(b)q(ecomes)f(wrong.)32="" b(in)19="" b(b)q(oth)i(cases,)f(w)o(e)f(start)h(a)g="" fw(new)i(phase)t="" fo(,)d(resetting)g="" fj(m)25="" fo(to)20="" b(the)f(new)h(v)m="" (alue)46="" y(\(1)13="" b(+)f="" fj(c)p="" fo(\))g="" fi(\001)g="" fo(,)19="" b(where)f="" fj(n)g="" fo(is)g(the)g(n)o(um)o(b)q(er)e(of)j(elemen)o(ts)c="" (curren)o(tly)h(stored)j(in)f(the)g(dictionary;)g(the)46="" y(v)m(ariable)h="" fw(c)n(ount)25="" fo(is)19="" b(set)h(to)g="" fo(,)g(so)g(that)g(the)f(system)g(is)g(able)g(to)h(p)q(erform)f(up)h(to)f="" fj(c)14="" fi(\001)f="" fj(n)20="" fo(up)q(dates)46="" y(b)q(efore)g(the)f(b)q="" (eginning)h(of)g(the)g(next)f(phase.)33="" b(deletions)19="" b(are)h(p)q(erformed)e="" (b)o(y)i(attac)o(hing)g(a)g(tag)46="" y(\\deleted")c(to)i(the)e(table)h(en)="" o(try)f(to)i(b)q(e)f(erased;)g(only)g(when)g(a)g(new)g(lev)o(el-1)f(hash)h="" (function)g="" fj(h)g="" fo(or)46="" y(a)g(new)g(hash)h(function)f="" fj(h)523="" fl(j)559="" fo(for)g(the)g(subtable)g="" fj(t)944="" fl(j)979="" fo(is)g(c)o(hosen,)f(do)i(w)o(e)f(drop)g(the)g(elemen)="" o(ts)d(with)j(a)46="" y(tag)g(\\deleted")f(from)f="" fj(t)491="" fl(j)509="" fo(.)46="" y(let)k(us)h(\014rst)g(analyze)f(the)g="" (space)g(needed)g(b)o(y)g(the)g(sc)o(heme.)28="" b(by)19="" b(a)g="" fw(phase)24="" fo(w)o(e)18="" b(mean)h(the)g(time)46="" y(p)q(erio)q(d)d="" (during)g(whic)o(h)f(one)g(lev)o(el-1)g(hash)h(function)f="" fj(h)h="" fo(is)f(\\in)h(use":)21="" b(a)16="" b(phase)h(starts)f(when)g(some)46="" fj(h)e="" fo(is)g(c)o(hosen)g(and)h(ends)f(when)g(the)g(next)f(lev)o="" (el-1)g(function)h(is)g(c)o(hosen,)g(either)f(in)g(the)h(same)f(or)i(the)46="" y(subsequen)o(t)k(call)g(to)g="" fw(r)n(ehasha)o(l)r(l)5="" fo(.)32="" b(phases)20="" b(that)g(only)f(consist)h(of)g(c)o(ho)q(osing)g(an)g="" fo(to)f(\014nd)h(out)46="" y(that)g="" fo(do)q(es)g(not)h="" (satisfy)e(condition)h(\()p="" fo(\))f(are)h(called)f="" fw(de)n(gener)n(ate)t="" b(during)20="" b(an)o(y)g(phase)g(that)46="" y(starts)f(with)g="" fo(k)o(eys)e(b)q(eing)i(stored)g(in)f(the)h="" (dictionary)f(the)g(n)o(um)o(b)q(er)f(of)i(k)o(eys)f(will)g(nev)o(er)f(drop)="" y(b)q(elo)o(w)g(\(1)12="" fi(\000)g="" fo(,)18="" b(since)e(at)i(most)f="" fj(c)12="" fj(n)18="" fo(up)q(dates)g(are)f(made.)24="" b(th)o(us,)17="" b(the)g(follo)o(wing)g(lemma)e="" (is)46="" y(su\016cien)o(t)g(to)h(pro)o(v)o(e)g(the)g(claimed)e(space)i(b)q="" (ound.)46="" fh(lemm)o(a)g(2.1)24="" fw(the)h(memory)e(sp)n(ac)n(e)h(use)n="" (d)g(during)h(a)f(phase)g(that)h(starts)f(with)g="" fj(n)h="" fw(keys)f(b)n(eing)46="" y(stor)n(e)n(d)16="" b(in)i(the)g(dictionary)f(is)g="" fo(\))p="" fw(.)46="" y(pr)n(o)n(of:)67="" fo(the)12="" b(lemm)o(a)e(is)i(ob)o(viously)f(true)h(for)h(degenerate)e(phases,)j(since)d="" fo(\))14="" b(=")g(\002\()p" fo(\).)20="" b(th)o(us,)46="" y(w)o(e)i(assume)f(that)i(a)g(function)f="" fo(is)g(c)o(hosen)h(that)f(satis\014es)h(condition)f(\()p="" fo(\),)h(and)g(determine)46="" y(ho)o(w)h(big)g(the)g(table)f="" fj(t)30="" fo(has)25="" b(to)f(b)q(e)g(to)h(accommo)q(date)d(all)h(v)o(ersions)g="" (of)i(all)e(subtables.)44="" b(f)l(or)46="" y(0)14="" fi(\024)g="" fj(j)i(<)e(s)p="" fo(\),)14="" b(let)22="" b(\026)-33="" fj(m)477="" fl(j)508="" fo(denote)14="" b(the)f(\014nal)h(capacit)o(y)="" f(of)h="" fj(t)1122="" fl(j)1139="" fo(,)g(that)g(is,)f(the)h(v)m(alue)f="" (of)h="" fj(m)1631="" fl(j)1662="" fo(at)g(the)g(end)46="" y(of)h(the)g(phase,)g(and)h(let)g(\026)-26="" fj(s)517="" fl(j)549="" fo(=")14" b(2)9="" fj(m)668="" fl(j)686="" fo(\()9="" fj(m)748="" fl(j)774="" fi(\000)9="" fo(1\))15="" b(b)q(e)g(the)g(\014nal)g(size)f(of)h="" fj(t)1309="" fl(j)1327="" fo(.)21="" b(the)15="" b(previous)g(v)o(ersions)f="" (of)46="" fj(t)75="" fl(j)109="" fo(\(if)i(there)f(w)o(ere)h(an)o="" (y\))g(had)h(capacit)o(y)813="" fm(1)p="" a(2)845="" fo(\026)-33="" fj(m)879="" fl(j)897="" fo(,)931="" a(4)963="" fo(\026)g="" fj(m)997="" fl(j)1015="" fo(,)16="" fj(:)8="" b(:)g(:)15="" b(since,)15="" b(for)i="" fj(l)d="" fi(\025)g="" fo(0,)360="" y(2)d="" fo(\(2)463="" ff(\000)p="" fl(l)515="" fi(\001)20="" fj(m)583="" fl(j)601="" fo(\)\(2)663="" fl(l)715="" fj(m)783="" fl(j)812="" fi(\000)10="" fi(\024)e="" fo(4)995="" fl(l)1047="" fi(\001)e="" fo(2)f(\026)-34="" fj(m)1139="" fl(j)1157="" fj(m)1219="" fl(j)1248="" fi(\000)11="" fo(1\))j(=")g(4)1431" fl(l)1483="" fo(\026)-26="" fj(s)1531="" fl(j)1549="" fj(;)p="" eop="" %%page:="" bop="" fx(2)49="" b(d)o(ynamic)14="" b(perfect)j(hashing)989="" fo(6)p="" fh(pro)r(cedure)17="" fw(insert)5="" fj(x)p="" fo(\);)95="" fw(c)n(ount)19="" fi(="" )13="" fo(+)11="" b(1;)95="" fh(if)18="" fw(c)n(ount)h="" fj(="">)14 b(M)95 648 y Fh(then)143 708 y Fw(R)n(ehashA)o(l)r(l)5 b Fo(\()p Fj(x)p Fo(\);)95 768 y Fh(else)143 828 y Fj(j)17 b Fi( )d Fj(h)p Fo(\()p Fj(x)p Fo(\);)143 888 y Fh(if)19 b Fo(p)q(osition)e Fj(h)409 895 y Fl(j)427 888 y Fo(\()p Fj(x)p Fo(\))f(of)g(subtable)h Fj(T)789 895 y Fl(j)823 888 y Fo(con)o(tains)f Fj(x)192 948 y Fh(then)241 1009 y(if)i Fj(x)e Fo(is)g(mark)o(ed)f(\\deleted") h Fh(then)i Fo(remo)o(v)o(e)c(this)i(tag;)192 1069 y Fh(else)i Fo(\()p Fi(\003)e Fj(x)g Fo(is)g(new)g(for)h Fj(W)674 1076 y Fl(j)708 1069 y Fi(\003)p Fo(\))241 1129 y Fj(b)262 1136 y Fl(j)294 1129 y Fi( )c Fj(b)378 1136 y Fl(j)408 1129 y Fo(+)e(1;)241 1189 y Fh(if)18 b Fj(b)313 1196 y Fl(j)345 1189 y Fi(\024)c Fj(m)441 1196 y Fl(j)290 1249 y Fh(then)k Fo(\()p Fi(\003)e Fo(size)g(of)g Fj(T)655 1256 y Fl(j)689 1249 y Fo(su\016cien)o(t)f Fi(\003)p Fo(\))339 1310 y Fh(if)j Fo(p)q(osition)f Fj(h)604 1317 y Fl(j)622 1310 y Fo(\()p Fj(x)p Fo(\))f(of)h Fj(T)789 1317 y Fl(j)823 1310 y Fo(is)f(empt)o(y)387 1370 y Fh(then)436 1430 y Fo(store)h Fj(x)f Fo(in)f(p)q(osition)i Fj(h)870 1437 y Fl(j)889 1430 y Fo(\()p Fj(x)p Fo(\))f(of)g Fj(T)1055 1437 y Fl(j)1073 1430 y Fo(;)387 1490 y Fh(else)436 1550 y Fo(go)h(through)g(the)f (subtable)h Fj(T)994 1557 y Fl(j)1012 1550 y Fo(,)e(put)i(all)f(elemen)o(t)o (s)436 1611 y(not)h(mark)o(ed)d(\\deleted")i(in)o(to)g(a)h(list)e Fj(L)1167 1618 y Fl(j)1186 1611 y Fo(,)g(and)436 1671 y(mark)g(all)h(p)q (ositions)h(of)f Fj(T)919 1678 y Fl(j)954 1671 y Fo(empt)o(y;)436 1731 y(app)q(end)h Fj(x)f Fo(to)h(list)e Fj(L)826 1738 y Fl(j)845 1731 y Fo(;)h Fj(b)896 1738 y Fl(j)927 1731 y Fi( )e Fo(length)i(of)h Fj(L)1229 1738 y Fl(j)1247 1731 y Fo(;)436 1791 y Fh(rep)r(eat)g Fj(h)638 1798 y Fl(j)671 1791 y Fi( )c Fo(randomly)i(c)o(hosen)i(function)f (in)f Fi(H)1399 1798 y Fl(s)1415 1809 y(j)436 1858 y Fh(un)n(til)j Fj(h)596 1865 y Fl(j)631 1858 y Fo(is)e(injectiv)o(e)e(on)j(the)f(elemen)n (ts)e(of)i(list)g Fj(L)1395 1865 y Fl(j)1413 1858 y Fo(;)436 1918 y Fh(for)j Fo(all)d Fj(y)h Fo(on)g(list)f Fj(L)815 1925 y Fl(j)849 1918 y Fo(store)h Fj(y)h Fo(in)e(p)q(osition)g Fj(h)1281 1925 y Fl(j)1300 1918 y Fo(\()p Fj(y)r Fo(\))g(of)g Fj(T)1464 1925 y Fl(j)1482 1918 y Fo(;)290 1978 y Fh(else)h Fo(\()p Fi(\003)f Fj(T)486 1985 y Fl(j)520 1978 y Fo(is)h(to)q(o)g(small)d Fi(\003)p Fo(\))339 2038 y Fj(m)382 2045 y Fl(j)413 2038 y Fi( )g Fo(2)d Fi(\001)g Fo(max)o Fi(f)p Fo(1)p Fj(;)d(m)742 2045 y Fl(j)760 2038 y Fi(g)p Fo(;)16 b Fj(s)838 2045 y Fl(j)870 2038 y Fi( )e Fo(2)p Fj(m)1001 2045 y Fl(j)1019 2038 y Fo(\()p Fj(m)1081 2045 y Fl(j)1110 2038 y Fi(\000)d Fo(1\);)339 2099 y Fh(if)18 b Fo(condition)e(\()p Fi(\003\003)p Fo(\))g(is)g(still)f(satis\014ed)387 2159 y Fh(then)k Fo(\()p Fi(\003)d Fo(double)g(capacit)o(y)f(of)i Fj(T)1010 2166 y Fl(j)1044 2159 y Fi(\003)p Fo(\))436 2219 y(allo)q(cate)f(new)h(space,)e(namely)g Fj(s)1053 2226 y Fl(j)1087 2219 y Fo(cells,)g(for)h(new)h(subtable)f Fj(T)1606 2226 y Fl(j)1624 2219 y Fo(;)436 2279 y(go)h(through)g(old)g(subtable)f Fj(T)991 2286 y Fl(j)1009 2279 y Fo(,)g(put)g(all)g(elemen)o(ts)436 2339 y(not)h(mark)o(ed)d(\\deleted")i(in)o(to)g(a)h(list)e Fj(L)1167 2346 y Fl(j)1186 2339 y Fo(,)436 2399 y(and)i(mark)e(all)h(p)q (ositions)h(empt)o(y;)436 2460 y(app)q(end)g Fj(x)f Fo(to)h(list)e Fj(L)826 2467 y Fl(j)845 2460 y Fo(;)h Fj(b)896 2467 y Fl(j)927 2460 y Fi( )e Fo(length)i(of)h Fj(L)1229 2467 y Fl(j)1247 2460 y Fo(;)436 2520 y Fh(rep)r(eat)g Fj(h)638 2527 y Fl(j)671 2520 y Fi( )c Fo(randomly)i(c)o(hosen)i(function)f(in)f Fi(H)1399 2527 y Fl(s)1415 2538 y(j)436 2586 y Fh(un)n(til)j Fj(h)596 2593 y Fl(j)631 2586 y Fo(is)e(injectiv)o(e)e(on)j(the)f(elemen)n(ts)e(of)i (list)g Fj(L)1395 2593 y Fl(j)1413 2586 y Fo(;)436 2647 y Fh(for)j Fo(all)d Fj(y)h Fo(on)g(list)f Fj(L)815 2654 y Fl(j)849 2647 y Fo(store)h Fj(y)h Fo(in)e(p)q(osition)g Fj(h)1281 2654 y Fl(j)1300 2647 y Fo(\()p Fj(y)r Fo(\))g(of)g Fj(T)1464 2654 y Fl(j)1482 2647 y Fo(;)387 2707 y Fh(else)i Fo(\()p Fi(\003)e Fo(lev)o(el-1-function)f Fj(h)h Fo(\\bad")i Fi(\003)p Fo(\))436 2767 y Fw(R)n(ehashA)o(l)r(l)5 b Fo(\()p Fj(x)p Fo(\);)p 46 2984 V 729 3040 a Fh(Figure)18 b(1:)24 b(Insertion)p eop %%Page: 7 7 6 bop 46 188 a Fx(2)49 b(D)o(YNAMIC)14 b(PERFECT)j(HASHING)989 b Fo(7)p 46 323 1831 2 v 46 478 a Fh(pro)r(cedure)17 b Fw(R)n(ehashA)o(l)r(l) 5 b Fo(\()p Fj(x)p Fo(\);)143 538 y(\()p Fi(\003)17 b Fw(R)n(ehashA)o(l)r(l)5 b Fo(\()p Fj(x)p Fo(\))16 b(is)g(either)f(called)h(b)o(y)f Fw(Insert)5 b Fo(\()p Fj(x)p Fo(\),)16 b(and)h(then)f Fj(x)d Fi(2)h Fj(U)5 b Fo(,)241 598 y(or)17 b(b)o(y)e Fw(Delete)t Fo(\()p Fj(x)p Fo(\),)i(and)g(then)f Fj(x)e Fo(=)f Fi(\000)p Fo(1.)22 b Fw(R)n(ehashA)o(l)r(l)5 b Fo(\()p Fj(x)p Fo(\))16 b(builds)g(a)h(new)f(table)241 658 y(for)g(all)g(elemen)o(ts)e(curren)o(tly)g (in)i(the)g(table)g(plus)g Fj(x)g Fo(\(if)g Fj(x)e Fi(2)g Fj(U)5 b Fo(\).)316 b Fi(\003)p Fo(\))95 719 y(go)17 b(through)g(the)f(whole)g (table)g Fj(T)7 b Fo(,)15 b(put)i(all)e(elemen)o(ts)f(not)j(tagged)g (\\deleted")95 779 y(in)o(to)f(a)g(list)g Fj(L)p Fo(,)g(coun)o(t)g(them,)e (and)j(mark)e(all)h(p)q(ositions)h(in)f Fj(T)22 b Fo(\\empt)o(y";)95 839 y Fh(if)c Fj(x)c Fi(2)g Fj(U)21 b Fh(then)d Fo(app)q(end)g Fj(x)e Fo(to)g Fj(L)p Fo(;)95 899 y Fw(c)n(ount)j Fi( )13 b Fo(length)j(of)h(list)f Fj(L)p Fo(;)95 959 y Fj(M)j Fi( )14 b Fo(\(1)d(+)g Fj(c)p Fo(\))g Fi(\001)g Fo(max)o Fi(f)p Fw(c)n(ount)5 b Fj(;)j Fo(4)p Fi(g)p Fo(;)95 1019 y Fh(rep)r(eat)17 b Fj(h)d Fi( )g Fo(randomly)h(c)o(hosen)h(function)g(in)g Fi(H)1039 1027 y Fl(s)p Fm(\()p Fl(M)t Fm(\))1123 1019 y Fo(;)143 1080 y Fh(for)j Fo(all)d Fj(j)s Fo(,)g(0)e Fi(\024)f Fj(j)k(<)d(s)p Fo(\()p Fj(M)5 b Fo(\),)16 b Fh(do)j Fo(form)c(a)i(list)e Fj(L)1021 1087 y Fl(j)1056 1080 y Fo(of)i(all)e Fj(x)f Fi(2)g Fj(L)i Fo(with)g Fj(h)p Fo(\()p Fj(x)p Fo(\))e(=)g Fj(j)s Fo(;)143 1140 y Fh(for)19 b Fo(all)d Fj(j)s Fo(,)g(0)e Fi(\024)f Fj(j)k(<)d(s)p Fo(\()p Fj(M)5 b Fo(\),)16 b Fh(do)192 1200 y Fj(b)213 1207 y Fl(j)245 1200 y Fi( )30 b Fo(length)16 b(of)h(list)e Fj(L)644 1207 y Fl(j)663 1200 y Fo(;)29 b Fj(m)749 1207 y Fl(j)781 1200 y Fi( )14 b Fo(2)d Fi(\001)g Fj(b)926 1207 y Fl(j)944 1200 y Fo(;)30 b Fj(s)1011 1207 y Fl(j)1043 1200 y Fi( )13 b Fo(2)p Fj(m)1173 1207 y Fl(j)1192 1200 y Fo(\()p Fj(m)1254 1207 y Fl(j)1283 1200 y Fi(\000)d Fo(1\);)95 1260 y Fh(un)n(til)18 b Fo(condition)e(\()p Fi(\003\003)p Fo(\))g(is)g(satis\014ed;)95 1320 y Fh(for)i Fo(all)e Fj(j)s Fo(,)g(0)e Fi(\024)g Fj(j)i(<)e(s)p Fo(\()p Fj(M)5 b Fo(\),)16 b Fh(do)143 1381 y Fo(allo)q(cate)h(space)f Fj(s)477 1388 y Fl(j)511 1381 y Fo(for)h(subtable)f Fj(T)810 1388 y Fl(j)828 1381 y Fo(;)143 1441 y Fh(rep)r(eat)i Fj(h)346 1448 y Fl(j)378 1441 y Fi( )c Fo(randomly)h(c)o(hosen)h(function)g(in)g Fi(H)1106 1448 y Fl(s)1122 1459 y(j)143 1507 y Fh(un)n(til)j Fj(h)304 1514 y Fl(j)338 1507 y Fo(is)d(injectiv)o(e)e(on)j(the)f(elemen)o (ts)d(of)k(list)e Fj(L)1102 1514 y Fl(j)1121 1507 y Fo(;)143 1568 y Fh(for)k Fo(all)d Fj(x)g Fo(on)h(list)e Fj(L)524 1575 y Fl(j)559 1568 y Fh(do)k Fo(store)d Fj(x)g Fo(in)g(p)q(osition)h Fj(h)1071 1575 y Fl(j)1089 1568 y Fo(\()p Fj(x)p Fo(\))f(of)h Fj(T)1256 1575 y Fl(j)1274 1568 y Fo(;)46 1710 y Fh(pro)r(cedure)g Fw(Delete)6 b Fo(\()p Fj(x)p Fo(\);)95 1771 y Fw(c)n(ount)19 b Fi( )13 b Fw(c)n(ount)k Fo(+)11 b(1;)95 1831 y Fj(j)16 b Fi( )e Fj(h)p Fo(\()p Fj(x)p Fo(\);)95 1891 y Fh(if)k Fo(p)q(osition)f Fj(h)360 1898 y Fl(j)378 1891 y Fo(\()p Fj(x)p Fo(\))f(of)h(subtable)f Fj(T)740 1898 y Fl(j)774 1891 y Fo(con)o(tains)g Fj(x)143 1951 y Fh(then)j Fo(mark)c Fj(x)h Fo(as)h(\\deleted")143 2011 y Fh(else)h Fo(return\()p Fj(x)e Fo(is)g(not)g(a)h(mem)o(b)q(er)c(of)k Fj(S)s Fo(\);)95 2072 y Fh(if)h Fw(c)n(ount)h Fi(\025)14 b Fj(M)143 2132 y Fh(then)19 b Fo(\()p Fi(\003)d Fo(start)h(new)f(phase)h Fi(\003)p Fo(\))192 2192 y Fw(R)n(ehashA)o(l)r(l)5 b Fo(\()p Fi(\000)p Fo(1\);)46 2335 y Fh(pro)r(cedure)17 b Fw(L)n(o)n(okup)r Fo(\()p Fj(x)p Fo(\);)95 2395 y Fj(j)f Fi( )e Fj(h)p Fo(\()p Fj(x)p Fo(\);)95 2455 y Fh(if)k Fo(p)q(osition)f Fj(h)360 2462 y Fl(j)378 2455 y Fo(\()p Fj(x)p Fo(\))f(of)h(subtable)f Fj(T)740 2462 y Fl(j)774 2455 y Fo(con)o(tains)g Fj(x)g Fo(\(not)h(mark)o(ed)e (\\deleted"\))143 2515 y Fh(then)k Fo(return\(\\)p Fj(x)d Fo(is)g(a)h(mem)n (b)q(er)c(of)k Fj(S)s Fo("\))143 2576 y Fh(else)h Fo(return\(\\)p Fj(x)e Fo(is)g(not)h(a)f(mem)o(b)q(er)d(of)k Fj(S)s Fo("\);)46 2718 y Fh(pro)r(cedure)g Fw(Initialize)p Fo(;)95 2779 y Fj(T)j Fi( )13 b Fo(an)k(empt)o(y)d(table;)95 2839 y Fw(R)n(ehashA)o(l)r(l)5 b Fo(;)p 46 2973 V 341 3029 a Fh(Figure)18 b(2:)25 b(Setup,)18 b(Deletion,)e(Lo)r(okup,)h(and)j(Rehashing)p eop %%Page: 8 8 7 bop 46 188 a Fx(2)49 b(D)o(YNAMIC)14 b(PERFECT)j(HASHING)989 b Fo(8)46 342 y(the)16 b(total)g(n)o(um)o(b)q(er)f(of)h(cells)f(o)q(ccupied)h (b)o(y)g(all)g(v)o(ersions)g(of)g(all)g(subtables)g Fj(T)1474 349 y Fl(j)1509 342 y Fo(is)g(b)q(ounded)h(b)o(y)370 441 y Fg(X)315 535 y Fm(0)p Ff(\024)p Fl(j)r(<s)p 2="" 8="" 9="" 10="" 11="" 18="" 25="" 28="" 34="" 36="" 41="" 46="" 49="" 51="" 121="" 136="" 140="" 143="" 188="" 190="" 226="" 253="" 342="" 349="" 402="" 409="" 422="" 431="" 441="" 442="" 449="" 450="" 462="" 469="" 471="" 479="" 483="" 490="" 517="" 522="" 529="" 534="" 535="" 564="" 570="" 582="" 589="" 623="" 624="" 639="" 643="" 650="" 652="" 665="" 683="" 699="" 725="" 743="" 762="" 769="" 784="" 785="" 792="" 803="" 804="" 816="" 819="" 821="" 822="" 829="" 846="" 863="" 864="" 872="" 879="" 882="" 886="" 898="" 904="" 912="" 919="" 924="" 927="" 939="" 941="" 942="" 946="" 949="" 956="" 964="" 984="" 999="" 1003="" 1009="" 1011="" 1018="" 1026="" 1028="" 1040="" 1042="" 1044="" 1046="" 1048="" 1052="" 1059="" 1063="" 1070="" 1077="" 1079="" 1080="" 1104="" 1120="" 1123="" 1127="" 1131="" 1173="" 1200="" 1224="" 1239="" 1242="" 1299="" 1300="" 1303="" 1304="" 1359="" 1363="" 1371="" 1397="" 1419="" 1423="" 1457="" 1480="" 1483="" 1487="" 1491="" 1517="" 1540="" 1543="" 1544="" 1550="" 1571="" 1600="" 1626="" 1660="" 1667="" 1684="" 1690="" 1692="" 1704="" 1720="" 1721="" 1751="" 1797="" 1807="" 1811="" 1815="" 1837="" 1848="" 1855="" 1871="" 1881="" 1883="" 1901="" 1931="" 1941="" 1991="" 2005="" 2052="" 2065="" 2102="" 2112="" 2125="" 2154="" 2162="" 2172="" 2222="" 2232="" 2239="" 2283="" 2286="" 2290="" 2292="" 2310="" 2343="" 2350="" 2353="" 2354="" 2385="" 2393="" 2400="" 2403="" 2413="" 2415="" 2427="" 2461="" 2463="" 2480="" 2523="" 2530="" 2581="" 2584="" 2608="" 2638="" 2641="" 2644="" 2648="" 2651="" 2760="" 2763="" 2767="" 2770="" 2807="" 2820="" 2823="" 2830="" 2864="" 2867="" 2872="" 2880="" 2883="" 2887="" 2901="" 2927="" 2941="" 2987="" 3001="" 3003="" 3010="" 3048="" 3063="" 3070="" fm(\()p="" fl(m)t="" fm(\))507="" y="" fg(x)509="" fl(l)p="" ff(\025)p="" fm(0)575="" fo(4)599="" ff(\000)p="" fl(l)651="" fi(\001)c="" fo(\026)-26="" b="" fj(s)699="" fl(j)745="" fo(=")816" y(4)p="" v="" a(3)856="" fi(\001)936="" fg(x)881="" fm(0)p="" ff(\024)p="" fl(j)r(<s)p="" fm(\))1061="" fo(\026)g="" fj(s)1082="" fl(j)1128="" fi(\024)1200="" fo(4)p="" a(3)1240="" fi(\001)1265="" fg(\022)1300="" fo(32)p="" fj(m)1400="" fm(2)p="" a="" fj(s)p="" fo(\()p="" fj(m)5="" fo(\))1437="" y(+)11="" b(4)p="" fj(m)1562="" fg(\023)1594="" fj(:)46="" fo(the)18="" b(last)g(inequalit)o(y)e(holds)i(since)f(the)h(algorithm)e(mak)o(es)h(sure)g="" (that)i(condition)e(\()p="" fi(\003\003)p="" fo(\))h(remains)46="" y(v)m(alid)e(throughout)h(the)f(phase.)46="" y(the)j(space)g(required)e(b)o="" (y)i(the)g(header)f(table)h(is)g(at)g(most)f(5)p="" fo(\),)20="" b(since)e(the)h="" fj(j)s="" fo(th)g(en)o(try)f(of)h(the)46="" y(header)f(table)h(need)f(only)g(con)o(tain)h(a)g(p)q(oin)o(ter)f(to)h="" fj(t)1034="" fl(j)1052="" fo(,)f(the)h(v)m(ariables)f="" fj(s)1398="" fl(j)1416="" fo(,)h="" fj(b)1470="" fl(j)1488="" fo(,)g(and)g="" fj(m)1661="" fl(j)1679="" fo(,)g(and)g(the)46="" y(n)o(um)o(b)q(er)i="" fj(k)255="" fl(j)296="" fo(that)h(describ)q(es)="" h(the)f(hash)h(function)f="" fj(h)1057="" fl(j)1076="" fo(.)39="" b(if)22="" b(w)o(e)g(let)g="" fo(\))25="" b(=")1553" fm(8)p="" a(15)1584="" fi(p)p="" x="" fo(30)16="" fi(\001)f="" fo(,)24="" b(the)46="" y(space)f(needed)f(b)o(y)h(the)g(subtables)g(and)h(the)f(header)="" g(table)f(tak)o(en)h(together)g(is)g(b)q(ounded)h(b)o(y)51="" fm(4)p="" a(3)85="" fi(\001)110="" fg(\020)140="" fm(32)p="" fl(m)214="" fe(2)230="" ff(\001)p="" fm(15)p="" a(8)161="" ff(p)p="" fm(30)o="" fl(m)292="" fo(+)11="" fj(m)417="" fg(\021)455="" fo(+)g(5)h="" fi(\001)579="" a(15)611="" fo(30)h="" fi(\001)e="" fj(m)20="" fm(16)p="" a(3)904="" fo(\()923="" fo(30)13="" b(+)e(1\))h="" fj(m)21="" b(<)15="" fo(35)d="" fi(\001)g="" fo(\(1)g(+)f="" fj(c)p="" fo(\))h="" fj(n)p="" fo(,)17="" b(whic)o(h)f(pro)o(v)o(es)46="" y(the)g(lemm)o(a.)1552="" fd(2)46="" fo(no)o(w)16="" b(w)o(e)g(turn)h="" (to)g(the)f(time)e(b)q(ounds.)23="" b(note)16="" b(\014rst)h(that)g(mem)o(b)q="" (ership)c(queries)j(do)h(not)f(in)o(terfere)46="" y(with)11="" b(the)h(time)d(analysis,)k(since)d(they)i(are)f(executed)f(in)i(constan)o(t)g="" (time)d(in)j(the)f(w)o(orst)h(case.)20="" b(th)o(us,)46="" y(there)13="" b(is)g(no)i(harm)e(in)g(assuming)g(that)h(there)g(are)f(no)i(mem)n(b)q="" (ership)c(queries)i(at)h(all.)20="" b(note)13="" b(further)46="" y(that)20="" b(instructions)g(that)h(are)f(executed)f(in)h(constan)o(t)g(time)e="" (\(i.)8="" b(e.,)20="" b(deletions)f(in)h(an)o(y)g(case)g(and)46="" y(insertions)d(if)g(they)h(do)g(not)g(cause)g(a)g(subtable)f="" fj(t)981="" fl(j)1017="" fo(to)h(b)q(e)g(rearranged\))g(can)g(b)q(e)g="" (safely)f(ignored,)46="" y(since)12="" b(they)g(will)g(not)h(in)o(v)m(alidate)="" e(an)i(o)o(v)o(erall)f(linear)g(time)e(b)q(ound.)21="" b(th)o(us,)14="" b(w)o(e)e(only)g(need)h(to)g(w)o(orry)46="" y(ab)q(out)20="" b(the)e(time)f(sp)q(en)o(t)i(for)g(installing)f(new)g(lev)o(el-1)g(functions)="" g(at)h(the)g(b)q(eginning)g(of)g(a)g(phase)46="" y(\(in)14="" fw(r)n(ehasha)o(l)r(l)5="" fo(\),)16="" b(and)g(for)f(constructing)g(new)g(v)o="" (ersions)g(of)g(the)g(subtables)g="" fj(t)1493="" fl(j)1526="" fo(\(in)g="" fw(r)n(ehasha)o(l)r(l)20="" fo(or)46="" y(in)c="" fw(insert)5="" fo(\).)46="" fh(lemm)o(a)16="" b(2.2)24="" fw(the)18="" b(exp)n(e)n(cte)n(d)g(time)g(for)f(a)g(phase)h(that)g(starts)f(with)h="" fj(n)f="" fw(keys)h(b)n(eing)h(stor)n(e)n(d)d(in)i(the)46="" y(dictionary)f(is)g="" fj(o)q="" fo(\))p="" fw(.)46="" y(pr)n(o)n(of:)74="" fo(consider)20="" b(the)f(call)f(to)h="" fw(r)n(ehasha)o(l)r(l)="" fo(in)19="" b(whic)o(h)f(the)h(phase)h(starts.)31="" b(clearing)19="" b(the)g(old)46="" y(table)13="" b(\(header)g(table)f(and)i(the)f(subtables\))g="" (and)h(building)f(up)g(the)g(list)f="" fj(l)i="" fo(tak)o(es)e(time)f="" fo(\),)j(since)46="" y(b)o(y)k(2.1)g(the)h(old)f="" (table)g(o)q(ccupies)g(only)g(space)h="" fo(\).)28="" b(time)17="" b(linear)g(in)h="" fj(n)h="" fo(su\016ces)f(to)g(construct)46="" y(the)f(sublists)g="" fj(l)339="" fl(j)357="" fo(,)g(to)h(compute)e="" (the)h(v)m(alues)g="" fj(b)900="" fl(j)918="" fo(,)g="" fj(m)992="" fl(j)1010="" fj(s)1159="" fl(j)1178="" fo(,)g(0)e="" fi(\024)h="" fj(j)i(<)d(s)p="" fo(\),)18="" b(and)f(to)h(compute)46="" fg(p)90="" fm(\))270="" fj(s)293="" fl(j)311="" fo(.)j(th)o(us,)14="" b(if)g(the)h(phase)f(is)h="" (degenerate,)f(it)g(tak)o(es)g="" fo(\))h(time)d(in)i(the)="" g(w)o(orst)h(case.)21="" b(in)46="" y(a)13="" b(non-degenerate)g(phase)g="" fj(h)f="" fo(initially)f(satis\014es)i(\()p="" fo(\).)19="" b(by)12="" b(the)g(remarks)f(immediatel)o(y)e(preceding)46="" y(lemma)14="" b(2.2,)i(w)o(e)g(only)f(ha)o(v)o(e)h(to)g(estimate)f(the)g(time)f="" (sp)q(en)o(t)j(for)f(installing)f(new)i(hash)g(functions)46="" fj(h)74="" fl(j)111="" fo(for)i(the)f(subtables.)29="" b(fix)17="" b(some)h="" fo(,)h(and)g(split)f(the)g(phase,)i(as)f(far)g(as)g="" fj(t)1462="" fl(j)1498="" fo(is)g(concerned,)f(in)o(to)46="" y(subphases,)d(one)f(subphase)h(b)q(eing)f(de\014ned)g(as)h(a)f(maximal)="" d(time)h(p)q(erio)q(d)j(in)e(whic)o(h)h(the)f(capacit)o(y)46="" fj(m)89="" fl(j)123="" fo(and)k(hence)e(the)h(size)g="" fj(s)553="" fl(j)587="" fo(of)h="" fj(t)672="" fl(j)706="" fo(ha)o(v)o(e)e(a)i(\014xed)f(v)m(alue.)21="" b(w)l(e)16="" b(need)g(the)g(follo)o(wing)g(observ)m(ation:)46="" fx(claim:)68="" fo(assume)14="" b(a)h(hash)i(function)e="" fj(h)808="" fl(j)842="" fo(is)g(c)o(hosen)g(for)h="" fj(t)1148="" fl(j)1181="" fo(at)g(the)f(b)q(eginning)h(or)f(in)g(the)h(course)46="" y(of)g(a)g(subphase.)22="" b(then)15="" b(the)h(probabilit)o(y)e(that)i="" fj(h)960="" fl(j)994="" fo(sta)o(ys)g(in)f(use)h(un)o(til)f(the)g="" (end)g(of)h(the)g(subphase)46="" y(exceeds)226="" fm(1)p="" a(2)249="" fw(pr)n(o)n(of)k(of)h="" (claim:)35="" fo(let)21="" b(the)f(capacit)o(y)g(of)h="" fj(t)872="" fl(j)911="" fo(during)g(the)g(phase)g(b)q(e)g="" fj(m)1413="" fl(j)1431="" fo(.)35="" b(let)21="" fj(w)1618="" fl(j)1657="" fo(b)q(e)g(the)g(set)46="" y(of)g(k)o(eys)f="" fj(x)h="" fo(con)o(tained)f(in)h(the)f(list)h="" fj(l)762="" fl(j)801="" fo(when)g="" fj(h)961="" fl(j)1000="" fo(is)g(c)o(hosen.)35="" fj(v)1363="" fl(j)1403="" fo(b)q(e)g(the)g(set)f(of)i(the)e(\014rst)p="" eop="" %%page:="" bop="" fx(2)49="" b(d)o(ynamic)14="" b(perfect)j(hashing)989="" fo(9)46="" fl(j)120="" fi(\000)13="" b(j)p="" fj(w)232="" fl(j)250="" fi(j)19="" fw(di\013er)n(ent)25="" fo(k)o(eys)18="" fo(in)h(the)f(sequence)f(of)i(the)f="" fw(insert)24="" fo(instructions)c(to)f(b)q(e)h(executed)46="" y(next)e(that)h(satisfy)g="" fj(h)p="" fj(x)p="" fo(\))e(=")h" fj(j)k="" fo(and)d(do)g(not)g(o)q(ccur)g(in)f="" fj(w)1125="" fl(j)1143="" fo(.)28="" b(then,)19="" b(b)o(y)f(relation)h(\()p="" fi(\003)p="" fo(\),)f(table)g(size)46="" fj(s)69="" fl(j)101="" fj(m)220="" fl(j)238="" fj(m)300="" fl(j)328="" fi(\000)c="" fo(1\))17="" b(\(for)f="" fj(m)573="" fl(j)604="" fi(6)p="" (y)f(exceeding)1571="" a(2)1610="" fo(the)g(elemen)o(ts)46="" y(of)i="" fj(w)148="" fl(j)177="" fi([)12="" fj(v)250="" fl(j)285="" fo(will)k(b)q(e)g(mapp)q(ed)g="" (b)o(y)g="" fj(h)726="" fl(j)761="" fo(to)h(di\013eren)o(t)f(lo)q(cations)="" h(in)g="" fj(t)1306="" fl(j)1323="" fo(.)23="" b(if)16="" b(this)g(happ)q(ens,)i="" (the)e(w)o(a)o(y)46="" fj(b)67="" fl(i)97="" fo(is)g(c)o(hanged)h="" (and)f(rep)q(eated)h(k)o(eys)e(are)h(treated)g(in)g="" fw(insert)22="" fo(and)17="" fw(delete)k="" fo(implies)14="" b(that)j="" fj(h)1736="" fl(j)1770="" fo(sta)o(ys)46="" y(in)f(use)g(un)o(til)f="" fj(b)323="" fo(gro)o(ws)j(b)q(ey)o(ond)e="" fj(m)707="" fl(j)725="" fo(;)g(that)g(is,)g(un)o(til)f(the)h(end)g(of)h="" (the)f(subphase.)279="" fo(by)19="" b(the)h(claim,)f(the)h="" (probabilit)o(y)f(that)i="" fj(u)f="" fo(or)h(more)e(hash)i(functions)f="" fj(h)1412="" fl(j)1451="" fo(are)g(used)g(in)g(a)h(single)46="" y(subphase)g(is)f(at)h(most)e(2)523="" fl(u)p="" fm(1\))646="" fo(,)i(and)g(hence)e(the)h(exp)q(ected)g(n)="" o(um)o(b)q(er)e(of)j(functions)f="" fj(h)1698="" fl(j)1737="" fo(c)o(hosen)46="" y(during)c(the)g(subphase)h(is)f(b)q(ounded)h(b)o="" (y)f(2.)21="" b(th)o(us,)16="" b(the)g(exp)q(ected)f(cost)i(for)f(installing)g(new)="" g(hash)46="" y(functions)21="" fj(h)289="" fl(j)329="" fo(during)g(a)h(subphase)g(in)f(whic)o(h)g="" fj(t)984="" fl(j)1023="" fo(has)h(size)f="" fj(s)1236="" fl(j)1275="" fo(is)g="" fj(s)1409="" fl(j)1428="" fo(\).)36="" b(exactly)21="" b(as)h(in)f(the)46="" y(pro)q(of)e(of)g(lemma)e(2.1)i(w)o(e)f="" (get)g(an)h(o)o(v)o(erall)e(b)q(ound)j(of)f="" fo(\))18="" fo(\))h(for)g(the)f(exp)q="" (ected)g(time)46="" y(for)d(installing)f(new)h(hash)g(functions)g="" fj(h)777="" fl(j)795="" fo(,)g(0)f="" fi(\024)f="" fj(j)k(<)d(s)p="" fo(\),)15="" b(b)o(y)f(summing)e(o)o(v)o(er)i(all)g(subtable)h="" (sizes)46="" y(and)i(all)e="" fo(,)h(and)h(using)g(\()p="" fo(\).)1253="" fo(in)20="" b(order)h(to)h(\014nish)f="" (the)f(time)f(analysis,)j(w)o(e)e(will)g(sho)o(w)i(in)e(the)h(follo)o(wing)f="" (t)o(w)o(o)h(lemmas)d(that)46="" y(there)f(will)g(not)h(b)q(e)g(to)q(o)h="" (man)o(y)d(phases.)27="" b(some)g(phase,)h(and)g(let)f="" fj(s)k="" fo(b)q(e)d(the)g(set)f(of)h(elemen)o(ts)46="" y(stored)h(in)f(the)h="" (table)f(at)i(the)e(b)q(eginning)h(of)g(the)g(phase)g(\(whose)g(n)o(um)o(b)q="" (er)e(is)i="" fo(\))g(together)g(with)46="" y(those)14="" b(that)g(o)q(ccur)f(in)h(the)f(next)g="" fj(c)6="" fj(n)13="" fo(up)q(date)h(instructions)f(to)h(b)q(e)g(executed)e(\(rep)q(eated)h="" (elemen)o(ts)46="" y(are)k(only)g(coun)o(ted)g(once\).)24="" b(let)18="" fj(m)j="" b(\(1)e(+)e="" fo(;)17="" b(clearly)l(,)f="" fi(j)p="" fj(s)s="" fi(j)f(\024)g="" fo(.)25="" b(f)l(or)18="" fj(h)d="" fi(2)h(h)1636="" fl(s)p="" fm(\))1737="" y(at)g(random,)g(de\014ne)g="" fj(w)485="" fl(j)517="" fi(f)p="" fj(x)f="" fi(2)h="" fj(s)j="" fi(j)d="" fo(\))f(=")h" fi(g)p="" fo(,)i(0)e="" b(2.3)24="" fw(\(a\))67="" b(with)18="" b(pr)n(ob)n(ability)e(exc)n(e)n(e)n(ding)1046="" a(2)1086="" fw(we)i(have)577="" fg(x)522="" fm(\))700="" fj(w)784="" fl(j)803="" fo(\(2)p="" fj(w)920="" fl(j)938="" fi(j)11="" b(\000)g="" fo(1\))j="" fj(<)1127="" fj(m)1227="" fo(\))1264="" fj(m)r(:)46="" fw(\(b\))63="" b(if)12="" b(the)h(ine)n(quality)h(in)f(\(a\))g(is)f(satis\014e)n(d)h(for)f(the)="" i(level-1)h(function)f="" fw(chosen)h(at)f(the)g(b)n(e)n(ginning)46="" y(of)j(the)h(phase,)g(then)g(the)g(phase)f(ends)h(with)g(the)g(variable)="" b(c)n(ount)k(r)n(e)n(aching)16="" fw(;)17="" b(i.)9="" b(e.,)17="" b(the)g(phase)46="" y(c)n(omprises)g="" fj(c)11="" fj(n)17="" fw(up)n(dates.)46="" y(pr)n(o)n(of:)23="" fo(\(a\))16="" b(in)g(the)g(situation)h(just)f(describ)q(ed,)f(relation)h(\()p="" fo(\))g(reads)509="" y(pr)561="" fg(="" )657="" y(x)602="" fm(\))780="" )813="" fj(w)873="" fl(j)891="" fi(j)847="" fo(2)905="" fg(!)952="" fi(\024)1009="" fo(2)p="" fj(m)17="" fi(\000)11="" fo(1\))p="" fo(\))1267="" fg(!)1313="" fi(\025)1371="" fo(1)p="" a(2)1400="" fo(the)16="" b(claim)e(follo)o="" (ws)i(b)o(y)g(a)h(simple)d(transformation,)h(using)i(the)f(ob)o(vious)h="" (inequalit)o(y)46="" fj(w)330="" fl(j)348="" fi(j)f(\024)h="" fo(\(b\))k(it)f(is)g(immediate)d(from)j(the)g(w)o(a)o(y)h(the)f(v)m="" (ariables)h="" fj(b)1086="" fl(j)1122="" fo(and)g="" fj(m)1261="" fl(j)1297="" fo(are)f(initialized)f(in)h="" fw(r)n(ehasha)o(l)r(l)46="" fo(and)d(up)q(dated)h(in)e="" fo(,)14="" b(and)h(from)d(the)i="" (fact)g(that)g(only)f(k)o(eys)g(from)g="" fo(can)d(o)q(ccur)g(in)f(the)h="" (phase,)46="" y(that)i="" fj(b)172="" fl(j)204="" fi(\024)e(j)p="" fj(w)317="" fl(j)335="" fo(,)h(and)i(hence)e="" fj(m)651="" fl(j)683="" fi(\024)e="" fj(w)819="" fl(j)838="" fo(,)i(throughout)j(the)e(phase.)21="" b(since)15="" fj(s)1524="" fl(j)1556="" fj(m)1675="" fl(j)1693="" fj(m)1755="" fl(j)1784="" fo(1\))46="" y(throughout)15="" b(the)f(algorithm,)f(w)o(e)g="" (see)h(that)g(the)f(inequalit)o(y)f(in)i(\(a\))g(en)o(tails)f(that)h(\()p="" fo(\))g(sta)o(ys)g(v)m(alid)46="" y(throughout)k(the)e="" (phase.)1321="" fd(2)p="" fx(3)49="" b(higher)15="" b(order)i(hash)e(functions)849="" fo(10)46="" b(2.4)24="" fw(supp)n(ose)18="" b(that)36="" b(r)n(ehasha)o(l)r(l)42="" b(is)18="" b(c)n(al)r(le)n(d)h(at)f(a)f(time)i="" (when)g="" fj(n)14="" fi(\025)h="" fo(1)j="" fw(keys)g(ar)n(e)g(stor)n(e)n(d)46="" y(in)j(the)h(dictionary.)34="" b(then)22="" b(the)g(\(exp)n(e)n(cte)n(d\))f="" (time)h(ne)n(e)n(de)n(d)f(until)i(the)f(\014rst)f(c)n(al)r(l)h(to)43="" b(r)n(ehasha)o(l)r(l)46="" y(after)17="" fj(n)18="" fw(up)n(dates)f(have)h(b)n(e)n(en)g(p)n(erforme)n(d)e(is)i="" fw(pr)n(o)n(of:)79="" b(an)g(arbitrary)g(phase)g(that)g(starts)h(b)q(efore)f(the)g="" (next)f="" fj(c)13="" fi(\001)h="" fj(n)20="" fo(up)q(dates)g(ha)o(v)o(e)46="" y(b)q(een)d(pro)q(cessed.)25="" b(the)17="" b(n)o(um)o(b)q(er)e(of)j(k)o(eys)e="" (in)h(the)g(table)g(at)g(the)g(b)q(eginning)h(of)f(this)h(phase)f(is)g="" fj(n)1851="" ff(0)1863="" fo(,)46="" y(where)g(\(1)12="" fi(\000)g="" fo(\))f="" fj(n)k(<)g(n)499="" ff(0)526="" fj(<)g="" fo(\(1)c(+)g="" fo(\))g="" fj(n:)17="" fo(by)g(lemma)e(2.3\(a\)\(b\),)j(the)f(probabilit)o(y)g(that)g="" (during)46="" y(this)d(phase)g="" fj(n)348="" ff(0)374="" fo(up)q(dates)14="" b(are)g(p)q(erformed)f(exceeds)1042="" a(2)1064="" fo(.)21="" b(since)13="" fj(n)g="" fi(\000)g(j)p="" fj(n)1394="" ff(0)1412="" fi(j)13="" b(\024)h="" fj(n)1642="" ff(0)1668="" fo(no)14="" b(matter)46="" y(if)19="" fj(n)123="" ff(0)154="" fj(<)g(n)g="" fo(or)h="" fj(n)351="" ff(0)382="" fi(\025)f="" fo(,)h(this)f(means)g(that)g="" (the)h(probabilit)o(y)e(that)i(this)f(phase)h(extends)f(further)46="" y(than)e(the)g="" fj(c)12="" fo(up)q(dates)h(w)o(e)e(are)="" h(considering)g(is)g(at)g(least)1173="" a(2)1195="" fo(.)24="" b(th)o(us,)17="" b(the)f(exp)q(ected)g(n)o(um)="" o(b)q(er)f(of)46="" y(phases)g(needed)f(to)h(p)q(erform)f(these)g="" fj(c)8="" fo(up)q(dates)i(is)e(not)h(more)e(than)j(2.)21="" b(eac)o(h)14="" b(phase)h(o)q(ccurring)46="" y(starts)j(with)g="" fj(n)325="" ff(0)354="" fo(k)o(eys,)e(\(1)d="" fi(\000)e="" fj(n)k(<)g(n)788="" ff(0)816="" fj(n;)17="" fo(and)h(tak)o(es)g="" fo(\))g(steps)g(\(exp)q(ected\))e(b)o(y)46="" y(2.2.)21="" b(this)c(\014nishes)f(the)g(pro)q(of)h(of)g(2.4)g(and)f(the)g="" (time)e(analysis.)597="" fo(lemmas)14="" b(2.1{2.4)j(tak)o(en)f="" (together)h(yield)e(the)h(follo)o(wing)f(result.)46="" fh(theorem)h(2.5)24="" fw(dynamic)17="" b(p)n(erfe)n(ct)g(hashing,)g(as)g="" (describ)n(e)n(d)g(by)g(the)h(algorithm)f(in)g(this)g(se)n(ction,)46="" y(uses)i(line)n(ar)g(sp)n(ac)n(e,)f(ne)n(e)n(ds)g(c)n(onstant)i(time)f="" (for)f(memb)n(ership)g(queries,)h(and)g(has)f="" fo(\(1\))i="" fw(exp)n(e)n(cte)n(d)46="" y(amortize)n(d)c(insertion)j(and)e(deletion)i(c)="" n(ost.)990="" fh(remark)16="" b(2.6)24="" fo(ob)o(viously)l(,)16="" b(the)g(space)h(b)q(ound)h(35\(1)13="" b(+)e="" fo(pro)o(v)o(ed)f(in)h(2.1)g(is)g(not)g(satisfactory)46="" y(from)c(a)j(practical)e(p)q(oin)o(t)g(of)h(view.)20="" b(there)15="" b(are)f(man)o(y)g(conceiv)m(able)f(w)o(a)o(ys)i(of)g(reducing)f(the)g(space)="" y(b)q(ound,)22="" b(b)o(y)e(v)m(arying)h(the)f(parameters)f(\014xed)h="" (in)g(the)h(algorithm,)f(b)o(y)f(using)i(sligh)o(tly)e(di\013eren)o(t)46="" y(hash)c(functions,)f(or)g(b)o(y)f(adapting)i(more)e(in)o(v)o(olv)o(ed)f="" (sc)o(hemes,)g(e.)c(g.,)13="" b(that)h(describ)q(ed)g(in)f([fks84],)46="" y(whic)o(h)g(ac)o(hiev)o(es)f(an)j="" fj(n)6="" fo(+)g="" fj(o)p="" fo(\))14="" b(space)g(b)q(ound)h(in)f(the)f(static)h(case.)20="" b(most)15="" b(of)f(these)f(v)m(ariations)i(will)46="" y(increase)i(the)h(b)q="" (ounds)h(on)f(the)g(exp)q(ected)f(computation)g(time,)f(but)i(this)g(do)q(es)="" g(not)h(necessarily)46="" y(mean)10="" b(that)i(the)f(time)e(requiremen)o(ts)g="" (observ)o(ed)i(in)g(practice)f(will)h(gro)o(w)h(signi\014can)o(tly)l(.)18="" b(m.)11="" b(w)l(enzel)46="" y([w90])h(has)i(implem)o(en)n(ted)c(a)j(v)m="" (arian)o(t)f(of)h(the)g(sc)o(heme)d(describ)q(ed)i(ab)q(o)o(v)o(e.)20="" b(in)12="" b(his)h(implem)o(e)o(n)o(tation)46="" y(the)i(univ)o(erse)e="" fj(u)20="" fo(is)15="" fo(0)p="" fj(;)8="" fj(;)g(:)g(:)g(:)g(;)g="" fo(2)647="" fm(31)693="" fo(;)16="" b(the)e(space)h(requiremen)o(ts)d(are)j(k)o(ept)f(small)g(b)o(y)g(a)o(v)o="" (oiding)46="" y(the)20="" b(use)h(of)g(subtables)g(if)g="" fj(w)612="" fl(j)630="" fi(j)f="" fo(is)h(small.)33="" b(he)20="" b(rep)q(orts)h(that)h(the)e(space)h(requiremen)o(ts)d(of)j(his)46="" y(implem)o(en)n(tation)16="" b(are)h(comparable)g(to)h(those)g(of)g="" (balanced)g(trees)f(and)h(that)g(the)g(running)g(time)46="" y(is)i(sup)q(erior)g(to)g(searc)o(h)g(trees)g(pro)o(vided)f="" fo(is)g(mo)q(derately)e(large)i(\()p="" fo(1000\).)34="" b(w)l(e)20="" b(refer)f(the)46="" y(reader)d(to)h([w90])f(for)g="" (details.)46="" fk(3)83="" b(higher)26="" b(order)i(hash)f(functions)46="" fo(in)12="" b(this)g(section)g(w)o(e)g(generalize)f(inequalit)o(y)f(\()p="" fi(y)p="" fo(\))i(from)g(section)f(2)i(\(whic)o(h)f(originated)g(in)g([fks84]\))="" y(to)20="" b(p)q(olynomials)f(of)h(degree)f(larger)h(than)h(1,)f(and)h="" (note)f(some)f(consequences)g(of)h(this)g(general-)46="" y(ization.)29="" b(these)19="" b(extensions)f(ha)o(v)o(e)g(pro)o(v)o(ed)h(useful)f="" (since)g(the)h(\014rst)g(v)o(ersion)f(of)i(this)e(pap)q(er)i(ap-)46="" y(p)q(eared)i(as)g([dkm88)q(],)g(see,)g(e.)8="" b(g.,)22="" b([dm89)q(,)f(dm90a)q(,)h(dm90b].)38="" b(in)22="" b(order)g(to)g(form)o(ulate)e="" (the)46="" y(result)d(in)h(a)g(sligh)o(tly)f(more)g(general)h(w)o(a)o(y)f="" (than)i(just)f(for)g(p)q(olynomials,)f(w)o(e)h(recall)f(a)h(de\014nition)p="" fo(11)46="" y(giv)o(en)14="" b(originally)h(in)f([w)o(c79)q(],)g(and)i="" (studied)f(further)g(\(with)g(v)m(arying)g(notation\))h(for)g(example)d(in)46="" y([mv84)o(,)j(s89)q(].)46="" fh(de\014nition)h(3.1)24="" fo(\([w)o(c79)q(]\))e="" fw(l)n(et)h="" fi(h)g="" fw(b)n(e)g(a)g(c)n(ol)r(le)n="" (ction)i(of)e(functions)h="" fw(with)g(domain)g="" fj(d)h="" fw(and)46="" y(r)n(ange)g="" fj(r)p="" fw(.)42="" b(l)n(et)23="" fj(c)j(="">)g Fo(0)e Fw(and)g Fj(k)k Fi(2)e Fj(I)-10 b(N)t Fw(.)42 b(The)24 b(class)g Fi(H)h Fw(is)f(c)n(al)r(le)n(d)g Fo(\()p Fj(c;)8 b(k)r Fo(\)-univ)o(ersal)23 b Fw(if)h(for)f(al)r(l)46 684 y(se)n(quenc)n(es)c Fj(x)292 691 y Fm(1)312 684 y Fj(;)8 b(:)g(:)g(:)f(;)h(x)449 691 y Fl(k)488 684 y Fw(of)18 b(di\013er)n(ent)g(elements)i(of)e Fj(D)q Fw(,)g(al)r(l)i(se)n(quenc)n(es)f Fj(y)1386 691 y Fm(1)1406 684 y Fj(;)8 b(:)g(:)g(:)f(;)h(y)1539 691 y Fl(k)1578 684 y Fw(of)18 b(elements)i(of)46 744 y Fj(R)p Fw(,)e(and)f(r)n(andomly)g(chosen)h Fj(h)c Fi(2)g(H)574 853 y Fo(Pr\()p Fj(h)p Fo(\()p Fj(x)720 860 y Fl(i)734 853 y Fo(\))g(=)g Fj(y)843 860 y Fl(i)874 853 y Fw(for)j Fo(1)d Fi(\024)g Fj(i)f Fi(\024)h Fj(k)r Fo(\))g Fi(\024)1276 819 y Fj(c)p 1243 841 87 2 v 1243 887 a Fi(j)p Fj(R)p Fi(j)1308 872 y Fl(k)1335 853 y Fj(:)46 980 y Fw(\(A)o(lternatively,)i (such)e(classes)h(have)f(b)n(e)n(en)g(c)n(al)r(le)n(d)h Fj(c)f Fw(str)n(ongly)f Fj(k)j Fw(universal)f(or)e Fo(\()p Fj(k)r Fo(\))1554 987 y Fl(c)1571 980 y Fw(-indep)n(endent.\))46 1142 y Fh(Examples:)19 b Fo(\(a\))d([W)o(C79)q(])f(If)h Fj(F)23 b Fo(is)16 b(a)h(\014nite)e(\014eld,)g(w)o(e)h(ma)o(y)f(let)g Fj(D)h Fo(=)e Fj(R)g Fo(=)g Fj(F)7 b Fo(;)15 b(then)395 1242 y Fi(H)g Fo(=)e Fi(f)p Fj(h)h Fi(j)g Fj(h)p Fo(\()p Fj(x)p Fo(\))f(=)779 1200 y Fg(X)757 1293 y Fm(0)p Ff(\024)p Fl(i<k)869 2="" 11="" 12="" 13="" 14="" 15="" 16="" 17="" 18="" 21="" 22="" 24="" 30="" 36="" 45="" 46="" 51="" 65="" 70="" 85="" 88="" 89="" 90="" 109="" 111="" 134="" 146="" 164="" 188="" 283="" 294="" 309="" 313="" 315="" 324="" 326="" 329="" 334="" 335="" 336="" 342="" 343="" 349="" 351="" 352="" 354="" 358="" 361="" 363="" 370="" 371="" 380="" 383="" 385="" 397="" 402="" 404="" 409="" 411="" 429="" 438="" 461="" 462="" 471="" 493="" 517="" 521="" 522="" 525="" 528="" 536="" 540="" 541="" 543="" 548="" 562="" 566="" 580="" 581="" 582="" 588="" 589="" 592="" 640="" 642="" 643="" 655="" 662="" 669="" 672="" 674="" 677="" 686="" 702="" 709="" 713="" 718="" 721="" 722="" 725="" 727="" 729="" 740="" 754="" 756="" 762="" 763="" 764="" 765="" 769="" 771="" 774="" 780="" 781="" 785="" 792="" 798="" 803="" 804="" 808="" 817="" 822="" 827="" 829="" 831="" 834="" 840="" 842="" 849="" 858="" 861="" 882="" 889="" 893="" 895="" 901="" 903="" 909="" 913="" 920="" 921="" 928="" 930="" 932="" 938="" 942="" 943="" 947="" 949="" 954="" 969="" 972="" 973="" 979="" 981="" 988="" 992="" 1002="" 1003="" 1009="" 1015="" 1022="" 1033="" 1040="" 1050="" 1052="" 1062="" 1063="" 1073="" 1086="" 1091="" 1094="" 1096="" 1097="" 1102="" 1109="" 1110="" 1117="" 1122="" 1123="" 1135="" 1136="" 1139="" 1142="" 1153="" 1160="" 1163="" 1166="" 1167="" 1169="" 1178="" 1181="" 1182="" 1183="" 1189="" 1190="" 1195="" 1196="" 1200="" 1213="" 1218="" 1221="" 1224="" 1228="" 1230="" 1232="" 1236="" 1237="" 1240="" 1242="" 1243="" 1249="" 1250="" 1262="" 1264="" 1269="" 1273="" 1280="" 1284="" 1288="" 1291="" 1293="" 1297="" 1303="" 1304="" 1306="" 1308="" 1310="" 1313="" 1324="" 1326="" 1336="" 1338="" 1339="" 1353="" 1363="" 1384="" 1396="" 1409="" 1414="" 1421="" 1427="" 1439="" 1444="" 1449="" 1451="" 1479="" 1496="" 1503="" 1504="" 1511="" 1533="" 1536="" 1541="" 1543="" 1546="" 1549="" 1559="" 1564="" 1571="" 1591="" 1598="" 1599="" 1602="" 1603="" 1606="" 1610="" 1620="" 1624="" 1631="" 1632="" 1642="" 1644="" 1651="" 1659="" 1685="" 1691="" 1711="" 1729="" 1745="" 1747="" 1752="" 1758="" 1760="" 1770="" 1771="" 1778="" 1803="" 1813="" 1815="" 1822="" 1824="" 1832="" 1838="" 1842="" 1843="" 1847="" 1863="" 1866="" 1870="" 1873="" 1876="" 1878="" 1883="" 1888="" 1890="" 1892="" 1895="" 1898="" 1899="" 1914="" 1915="" 1918="" 1926="" 1928="" 1930="" 1952="" 1959="" 1986="" 1991="" 1997="" 1998="" 2012="" 2015="" 2017="" 2019="" 2040="" 2047="" 2057="" 2058="" 2075="" 2079="" 2094="" 2105="" 2107="" 2116="" 2117="" 2127="" 2131="" 2134="" 2135="" 2139="" 2144="" 2145="" 2147="" 2161="" 2178="" 2185="" 2188="" 2191="" 2195="" 2199="" 2205="" 2218="" 2220="" 2226="" 2238="" 2241="" 2245="" 2248="" 2251="" 2255="" 2260="" 2286="" 2311="" 2315="" 2320="" 2342="" 2346="" 2357="" 2361="" 2371="" 2373="" 2375="" 2380="" 2382="" 2403="" 2407="" 2417="" 2421="" 2432="" 2433="" 2435="" 2440="" 2447="" 2458="" 2467="" 2476="" 2477="" 2483="" 2495="" 2500="" 2502="" 2507="" 2517="" 2527="" 2555="" 2574="" 2587="" 2595="" 2616="" 2620="" 2623="" 2634="" 2647="" 2670="" 2676="" 2680="" 2689="" 2691="" 2702="" 2707="" 2709="" 2723="" 2731="" 2740="" 2743="" 2768="" 2769="" 2777="" 2784="" 2787="" 2793="" 2794="" 2800="" 2828="" 2848="" 2860="" 2887="" 2892="" 2905="" 2924="" 2945="" 2947="" 2954="" 2958="" 2959="" 2966="" 2971="" 2973="" 2974="" 2977="" 2981="" 2996="" 2998="" 3007="" 3014="" 3015="" 3018="" 3025="" 3027="" 3031="" 3041="" 3067="" 3068="" 3069="" 3074="" y="" fj(a)895="" fl(i)917="" fj(x)945="" fl(i)975="" fo(for)k="" fj(x)c="" fi(2)h="" fj(f)q(;)j(a)1228="" fm(0)1247="" fj(;)8="" b(:)g(:)g(:)f(;)h(a)1382="" fl(k)1417="" fi(2)14="" b="" fj(f)7="" fi(g)46="" fo(is)20="" b(\(1)p="" b(k)r="" fo(\)-univ)o(ersal.)33="" b(this)20="" b(holds)h(since)e(for)i(eac)o(h)e="" (sequence)h(of)g="" fj(k)j="" fo(di\013eren)o(t)c(argumen)o(ts)g(in)h="" fj(f)46="" fo(and)e="" fj(k)i="" fo(prescrib)q(ed)e(v)m(alues)f(there)h(is)f="" (exactly)g(one)h(p)q(olynomial)f(of)h(degree)f(at)i(most)e="" fj(k)d="" fi(\000)e="" fo(1)18="" b(that)46="" y(in)o(terp)q(olates)e(through)h="" (these)f(argumen)o(t-v)m(alue)f(pairs.)46="" y(\(b\))h([w)o(c79,)g(mv84])f="" (if)h="" fi(h)g="" fo(is)g(\()p="" fj(c;)8="" fo(\)-univ)o(ersal)15="" b(and)i="" fj(r)e="" fo(:)e="" fj(r)i="" fi(!)e="" fj(r)1259="" ff(0)1288="" fo(is)i(suc)o(h)h(that)g="" fi(j)p="" fj(r)1588="" ff(\000)p="" fm(1)1636="" fo(\()p="" fj(j)s="" fo(\))p="" fi(j)d(\024)h="" fj(d)i="" fo(for)46="" y(all)e="" fj(j)j="" fo(then)d(the)h(\(m)o(ulti\)set)c="" fi(h)608="" ff(0)633="" fo(=")j" fi(f)p="" fj(r)9="" fi(\016)e="" fj(h)14="" fi(j)g="" fj(h)g="" fi(2)g(hg)g="" fo(is)h(\()q(^)-25="" fo(\)-univ)o(ersal,)13="" b(for)j(^)-25="" fj(c)14="" fj(c)7="" fi(\001)g="" fj(d)p="" fj(r)1706="" ff(0)1719="" fj(=")p" fj(r)p="" fo(\))1841="" fl(k)1863="" fj(:)46="" fo(\(c\))23="" b(a)g(direct)g(consequence)f(of)i="" (\(a\))g(and)g(\(b\):)36="" b(if)23="" fj(p)h="" fo(is)g(prime,)e(and)j(1)h="" fi(\024)g="" fj(s)g="" fj(p)p="" fo(,)g(then)d(for)46="" fj(d)16="" fo(0)p="" b(:)g(:)g(:)g(;)g(p)j="" fi(\000)g="" fo(1)p="" fi(g)17="" fo(and)g="" fj(r)d="" b(:)g(:)g(:)f(;)h(s)j="" fo(the)f(set)125="" fi(h)168="" fl(k)168="" y(s)203="" fg(n)296="" fj(h)e="" fo(:)f="" fj(d)j="" fj(r)536="" fg(\014)536="" y(\014)536="" y(\014)g="" fj(h)p="" fj(x)p="" fo(\))f(=")723" fg(\022)784="" y(x)762="" fm(0)p="" ff(\024)p="" fl(i<k)888="" fj(a)914="" fl(i)936="" fj(x)964="" fl(i)992="" fo(mo)q(d)g="" fj(p)1123="" fg(\023)1168="" fj(s;)j="" fo(0)e="" fj(a)1445="" fm(0)1464="" b(:)g(:)g(:)g(;)g(a)1600="" fl(k)q="" fm(1)1680="" fj(<)13="" b(p)1770="" fg(o)46="" fo(is)j(\()p="" fo(\)-univ)o(ersal,)15="" b(for)h="" fj(c)e="" fi(d)p="" fj(p="s)p" fi(e)e(\001)f="" fj(s="p)p" fo(\))862="" fl(k)897="" fi(\024)i="" fo(\(1)f(+)f="" fo(\))1143="" fl(k)1165="" fo(\(d\))20="" b(f)l(or)h(the)f(\014nite)g(\014eld)g="" fj(r)g="" fw(gf)p="" fj(p)902="" fl(l)915="" fo(\),)g="" fj(p)g="" fo(prime,)e="" fj(l)j="" fi(\025)f="" fo(1,)g(w)o(e)f(obtain)h(\(1)p="" fo(\)-univ)o(ersal)46="" y(classes)13="" b(with)g="" fj(r)357="" ff(0)369="" fj(p)472="" fl(l)483="" fc(0)497="" fo(,)f(1)h="" fj(l)631="" ff(0)656="" fj(l)q="" fo(,)e(b)o(y)h(com)o(bining)="" e(\(a\))j(with)e(a)i(suitable)e(function)h="" fi(!)g="" fj(r)1851="" ff(0)1863="" fo(.)46="" y(\(see)i([mv84)o(])g="" (for)g(further)g(examples.\))46="" y(in)i(the)h(follo)o(wing,)g(w)o(e)f="" (assume)h(that)g="" b(:)g(:)g(:)g(;)g(s)13="" fi(g)19="" fo(are)g(\014xed,)f(and)i(that)f="" fi(h)h="" fo(is)46="" y(a)f(class)h(of)f="" (functions)g(from)g="" fo(to)e="" fo(.)31="" b(let)19="" b(a)h(set)f="" fj(s)j="" fi(\022)c="" fo(b)q(e)f(\014xed,)f="" fj(s)s="" fi(j)f="" fj(n)p="" fo(,)h(and)f(let)g="" fj(x)1788="" fm(0)1826="" fo(b)q(e)46="" y(an)g(elemen)o(t)c(of)j="" fj(d)c="" fi(\000)f="" fo(.)27="" b(f)l(or)18="" fj(h)f="" fi(2)g(h)i="" fo(and)g(0)e="" fj(j)k(<)c(s)h="" fo(w)o(e)g(de\014ne)f="" fj(b)1367="" fl(h)1364="" y(j)1406="" fj(x)e="" fi(2)i="" fj(s)i="" fi(j)d="" fo(\))g(=")g" fo(and)k="" fj(b)166="" fl(h)166="" y(j)210="" fj(b)324="" fl(h)321="" y(j)345="" fo(;)h(further,)e(w)o(e)f(de\014ne)h="" fj(b)841="" fl(h)838="" y(x)858="" fm(0)901="" fj(x)f="" fi(j)e="" fj(x)1430="" fm(0)1449="" fi(g)f="" fj(b)1634="" fl(h)1634="" y(x)1654="" fm(0)1697="" fj(b)1811="" fl(h)1808="" y(x)1828="" fm(0)1849="" y(assume)15="" b(that)j="" fo(is)g(c)o(hosen)g(uniformly)e(at)j(random)="" e(from)h="" fi(h)p="" fo(.)24="" b(\(in)17="" b(the)g(notation,)g(w)o(e)g(drop)h(the)46="" y(sup)q(erscript)g="" fo(.\))26="" b(arbitrary)g="" fj(z)g="" fi(2)f="" fj(i)-10="" b(r)p="" fo(,)18="" fj(k)h="" fi(\025)d="" fo(0,)j(w)o(e)e(let)g="" (\()p="" fj(z)r="" fo(\))1220="" fl(k)1259="" fo(denote)h(the)g(\\falling)f="" (factorial")46="" fj(z)c="" fi(\000)d="" fo(1\))e="" fj(:)g(:)g(:)h="" fj(z)k="" fj(k)k="" fo(+)d(1\))p="" fh(lemm)o(a)16="" b(3.2)24="" fw(if)18="" fw(is)f="" fw(-universal)19="" b(for)e="" fw(and)e="" fw(,)h(then)79="" y(\(a\))24="" fj(e)s="" fo(\(\()p="" fj(b)266="" fl(j)284="" fo(\))303="" fl(k)324="" fo(\))14="" fi(\024)f="" fj(c)f="" fi(\001)471="" fo(\))538="" fl(k)p="" v="" a="" fj(s)516="" fl(k)579="" fi(\024)h="" fi(\001)688="" fg(\020)718="" a(s)752="" fg(\021)777="" fl(k)798="" fw(,)18="" fo(0)f="" fi(\024)e="" fj(j)k(<)d(s)p="" fo(;)p="" eop="" %%page:="" bop="" fx(3)49="" b(higher)15="" b(order)i(hash)e(functions)849="" fo(12)81="" fw(\(b\))25="" fl(x)286="" fm(0)307="" fo(\))326="" fm(1)393="" fo(\))13="" fj(c)d="" fi(\001)540="" fo(\))607="" fm(1)p="" fj(s)585="" fm(1)692="" fi(\024)j="" fi(\001)802="" fg(\020)831="" a(s)866="" fg(\021)890="" fm(1)957="" fw(pr)n(o)n(of:)69="" fo(f)l(or)16="" fj(l)e="" fi(\025)g="" fo(1,)i(let)g(\()p="" fo(\))610="" fl(l)639="" fo(denote)g(the)g(set)g="" fj(x)1028="" fm(1)1047="" b(:)g(:)g(:)g(;)g(x)1185="" fl(l)1197="" fi(2)g="" fj(s)1310="" fl(l)1337="" fj(x)1393="" fm(1)1412="" b(:)g(:)g(:)f(;)h(x)1549="" fl(l)1578="" fo(di\013eren)o(t)p="" fi(g)p="" y(\(a\))16="" b(fix)g="" fo(,)g(and)g(de\014ne)g(random)g(v)m(ariables)g="" fj(x)921="" fl(x)941="" fm(1)961="" fl(;:::;x)1031="" y(k)1054="" fj(x)1123="" fm(1)1142="" fj(;)g(:)g(:)g(:)f(;)h(x)1279="" fl(k)1300="" fo(\))1451="" fl(k)1472="" fo(,)i(b)o(y)488="" fj(x)528="" fl(x)548="" fm(1)568="" fl(;:::)o(;x)637="" y(k)674="" fg(\032)765="" fj(;)49="" fo(if)16="" fj(x)972="" fm(1)991="" fo(\))e(=")g" fj(:)8="" b(:)g(:)13="" fj(x)1274="" fl(k)1295="" fj(j;)765="" fo(otherwise.)46="" y(then)17="" fj(x)272="" fl(x)292="" fm(1)312="" fl(;:::)o(;x)381="" y(k)404="" fj(x)605="" fl(x)625="" fm(1)645="" fl(;:::)o(;x)714="" y(k)737="" fj(c="s)969" fl(k)991="" fo(,)i(since)g="" fo(is)f(\()p="" fo(\)-univ)o(ersal.)21="" b(on)c(the)f(other)46="" y(hand,)g(it)g(is)g(clear)g(that)656="" y(\()p="" fj(b)696="" fl(j)714="" fo(\))733="" fl(k)768="" fg(x)820="" fm(\()p="" fl(x)854="" fm(1)873="" fl(;:::)o(;x)942="" y(k)963="" fm(\))p="" ff(2)p="" fl(s)r="" fm(\))1052="" fl(k)1081="" fj(x)1121="" fl(x)1141="" fm(1)1161="" fl(;:::)o(;x)1230="" y(k)1253="" fo(consequen)o(tly)l(,)318="" fj(b)416="" fl(j)434="" fo(\))453="" fl(k)475="" fo(\))d(=")656" fg(x)559="" fl(x)593="" fm(1)613="" fl(;:::)o(;x)682="" y(k)703="" fm(\))792="" fl(k)820="" fj(x)918="" fl(x)938="" fm(1)958="" fl(;:::;x)1028="" y(k)1051="" fo(\))h="" fi(\024)f(j)p="" fo(\))1221="" fl(k)1242="" fi(j)e(\001)1309="" fj(c)p="" a(s)1320="" fl(k)1360="" fo(\))1479="" fl(k)1511="" fi(\001)1553="" a(s)1564="" fl(k)1590="" fo(\(b\))19="" b(the)g(pro)q(of)i(is)e(similar)e(to)i(the)g(one)g(giv)o(en)g(in)="" g(\(a\).)30="" b(de\014ne)19="" b(random)g(v)m(ariables)g="" fj(y)1697="" fl(j)1686="" y(x)1706="" fm(1)1726="" fl(;:::)o(;x)1795="" y(k)q="" fm(1)1863="" fo(,)46="" fj(x)93="" fm(1)112="" b(:)g(:)g(:)g(;)g(x)250="" fm(1)316="" fo(\))467="" fm(1)533="" fo(,)h(0)e="" fo(,)i(b)o(y)359="" fj(y)399="" fl(j)387="" y(x)407="" fm(1)427="" fl(;:::;x)497="" fm(1)579="" fg(\032)670="" fj(;)48="" fj(x)876="" fm(1)896="" fj(x)1178="" fm(1)1244="" fj(x)1404="" fm(0)1423="" fj(j;)670="" y(then)16="" fj(y)270="" fl(j)259="" y(x)279="" fm(1)299="" fl(;:::)o(;x)368="" fm(1)436="" fo(\))e="" fl(k)612="" fo(,)h(since)h="" fo(\)-univ)o(ersal.)20="" b(f)l(urther,)489="" fj(b)529="" fl(x)549="" fm(0)571="" fo(\))590="" fm(1)670="" fg(x)722="" fl(x)756="" fm(1)775="" fl(;:::)o(;x)844="" fm(1)910="" y(\))p="" fm(\))999="" fm(1)1110="" fg(x)1087="" fl(j)r(<s)1214="" fj(y)1253="" fl(j)1242="" y(x)1262="" fm(1)1282="" fl(;:::)o(;x)1351="" fm(1)1419="" fo(t)l(aking)c(exp)q="" (ected)g(v)m(alues,)f(w)o(e)h(get)475="" fj(b)573="" fm(0)614="" fo(\))633="" fm(1)700="" fo(\))870="" fm(1)936="" fi(j)e(\001)g="" fi(\001)1062="" a(s)1073="" fl(k)1113="" fo(\))1232="" fm(1)1310="" fi(\001)1374="" a(s)1362="" fm(1)1434="" fj(;)46="" fo(as)j(claimed.)1554="" fd(2)46="" fo(a)15="" b(hash)h(function)f="" fj(h)h="" fo(is)f(called)g="" fw(-p)n(erfe)n(ct)20="" fo(for)15="" fj(s)k="" fo(if)c="" fj(b)1010="" fl(h)1010="" y(j)1046="" fo(for)g(all)f="" fo(,)g(0)f="" fj(j)j(<)d(s)p="" fo(,)h(i.)8="" b(e.,)13="" b(if)i(no)h(blo)q(c)o(k)46="" fj(b)86="" fl(h)83="" y(j)124="" fo(has)h(size)e(exceeding)g="" fh(corollary)j(3.3)24="" fw(in)d(the)g(situation)g(of)g(l)n(emma)e(3.2,)j="" (if)e(we)i(further)e(assume)h(that)g="" fj(s)f="" fw(,)j(we)46="" y(have:)79="" y(\(a\))i="" fo(pr\()p="" fj(h)16="" fj(k)g="" fo(1\)-p)q(erfect\))20="" fo(1)14="" fo(!\))g="" fi(\001)h="" fj(n)f="" fj(n="s)p" fo(\))1187="" fm(1)1253="" fw(.)33="" b(in)21="" b(c)n(ase)g="" fj(n)1606="" fm(1+1)p="" fl(=")p" fm(1\))1801="" fw(this)168="" y(pr)n(ob)n(ability)d(exc)n(e)n(e)n(ds)g="" fo(1)12="" fo(!)p="" fj(:)81="" fj(e)207="" fg(\022)268="" y(x)246="" fl(j)r(<s)351="" fj(b)391="" fl(j)409="" fo(\))428="" fl(k)449="" fg(\023)493="" fi(\024)14="" fj(c)567="" fl(k)599="" fi(\001)d="" fw(,)22="" b(for)e(some)h(c)n(onstant)h="" fj(c)1118="" fl(k)1140="" fw(.)32="" b(\(her)n(e)21="" fl(j)1385="" fo(\))1404="" fl(k)1446="" fw(is)g(the)h="" fj(k)r="" fw(-th)f(p)n(ower)g(of)168="" fj(b)189="" fl(j)207="" fw(.\))p="" fo(13)81="" fw(\(c\))25="" fo(pr)220="" fg(\022)281="" y(x)259="" fl(j)r(<s)364="" fj(b)404="" fl(j)422="" fo(\))441="" fl(k)476="" fo(2)p="" fj(c)574="" fl(k)606="" fj(n)660="" fg(\023)705="" fi(\025)762="" a(2)785="" fw(,)17="" b(for)g="" fj(c)916="" fl(k)955="" fw(as)g(in)h(\(b\).)46="" fh(corollary)g(3.4)24="" fw(in)17="" b(the)h(situation)g(of)g(l)n(emma)e(3.2)h(we)i(have:)46="" y(\(a\))e(f)l(or)g="" fo(0)d="" fj(j)j(<)c(s)18="" fw(arbitr)n(ary:)="" fj(b)477="" fl(j)509="" fi(\025)c="" fj(u)p="" fo(\))f="" fi(\024)675="" fg(\()716="" fj(e)815="" fl(u)p="" fm(1)883="" fl(u)957="" fo(\))g="" fo(\))1126="" fl(u)1149="" fw(for)17="" fo(1)d="" fj(u)g(<)f(k)r="" fo(;)716="" fm(1)881="" fo(\))1124="" fl(k)1146="" fj(;)52="" fj(k)f="" fj(u:)46="" fw(in)j(p)n(articular,)g(for)g="" fj(s)d="" fj(n)j="" fw(and)h="" fj(u)13="" fi(\025)h="" fw(,)j(we)i(have)f="" fj(b)1086="" fl(j)1118="" fi(\025)13="" fo(\))h(=")g" fj(o)q="" fj(u)1368="" fw(\(b\))287="" fj(b)379="" fl(x)399="" fm(0)435="" fi(\024)600="" fg(\()642="" fj(e)741="" fm(1)808="" fl(u)883="" fo(\))1052="" fl(u)1075="" fj(;)136="" fj(u)h(<)g(k)f="" fo(1;)642="" fj(c)g="" fm(2)807="" fm(1)926="" fo(\))1095="" fj(;)50="" fj(k)c="" fo(1)j="" fw(in)k(p)n(articular,)g(for)g="" b(we)g(have)g="" fj(b)1171="" fl(x)1191="" fm(0)1226="" fj(u)1476="" fm(1\))1598="" fh(note:)20="" fo(the)d(sp)q(ecial)e(case)i="" fj(k)e="" fj(u)i="" fo(in)g(3.4\(a\))h(has)g(already)f(b)q(een)g(analyzed)g="" (in)g([mv84].\))46="" fw(pr)n(o)n(of)g(of)h(cor)n(ol)r(lary)g(3.3:)46="" fo(\(a\))f(w)l(e)g(estimate)f(the)h(probabilit)o(y)f(that)i="" fo(is)g(not)h(\()p="" fo(1\)-p)q(erfect.)46="" y(clearly)168="" y(pr\()p="" fi(9)p="" fj(j)17="" fo(:)c="" fj(b)352="" fl(j)384="" fo(:)c(\()p="" fj(b)751="" fl(j)769="" fo(\))788="" fl(k)823="" fo(!\))f="" fo(max)o="" fo(pr\(\()p="" fj(b)1288="" fl(j)1306="" fo(\))1325="" fi(\025)i="" fo(!\))h="" fo(0)i="" fo(by)20="" b(3.2\(a\))i(and)g(the)f(mark)o(o)o(v)g(inequalit)o="" (y)e(the)i(last)h(term)d(is)i(b)q(ounded)h(ab)q(o)o(v)o(e)g(b)o(y)f="" fj(s)14="" fi(\001)46="" fo(\))160="" fl(k)195="" fo(!\))d="" fj(n)g="" fo(\))586="" fm(1)653="" y(\(b\))19="" b(let)h="" fi(f)g="" fj(b)468="" fl(j)505="" b(since)879="" fg(p)923="" fl(j)r="" fl(j)995="" fj(b)1016="" fl(j)1053="" fi(\024)19="" b(j)p="" fo(,)g(it)g(follo)o(ws)g(from)g(elemen)o(t)o(ary)46="" y(considerations)14="" b(that)465="" fg(p)509="" fl(j)573="" fj(b)613="" fl(j)631="" fo(\))650="" fl(k)685="" fi(\024)g(b)787="" fl(n)p="" a(k)q="" fm(1)834="" fi(c)7="" b(\001)g="" fo(1\))1027="" fl(k)1055="" fo(+)g(\()p="" fi(\000)g(b)1250="" fm(1)1298="" fi(c)g(\001)g="" fo(1\)\))1509="" fl(k)1545="" fi(\024)13="" fj(n)7="" fo(1\))1796="" fo(w)l(e)16="" b(need)g(the)g(follo)o(wing)g(simple)e="" (fact:)46="" fx(claim:)19="" fo(if)d="" fo(,)j(then)g="" fj(z)541="" fl(k)563="" fj(<)e(e)760="" fm(1)826="" fw(pr)n(o)n(of)i(of)h="" (claim:)70="" fj(z)95="" fo(\))114="" fl(k)154="" fm(1)211="" fg(y)207="" fm(="1)325" fj(z)p="" a(z)c="" fj(j)411="" fi(\024)463="" fm(1)469="" fg(y)465="" fj(k)p="" a(k)i="" fj(j)671="" fj(k)790="" fi(\000)c="" fo(1\)!)910="" y(=")962" fm(1)964="" fg(x)966="" fl(l)p="" fg(="" )1067="" fo(1)1116="" fj(l)1180="" fg(!)1213="" fi(\001)1232="" fo(1\))1381="" fo(1\)!)1414="" fj(<)1466="" fm(1)1468="" fg(x)1470="" fo(1\))1693="" fo(!)1725="" fj(<)j(e)1800="" fm(1)1866="" fj(:)p="" fo(\))46="" y(th)o(us,)i(w)o(e)g(ma)o(y)e(write:)174="" fj(e)213="" fg(\022)275="" y(x)252="" fl(j)r(<s)365="" fj(b)386="" fl(k)386="" y(j)408="" fg(\023)452="" fj(e)543="" fg(\022)575="" y(x)574="" fl(j)644="" fj(b)665="" fl(k)665="" y(j)698="" fo(+)748="" fg(x)747="" ff(62)p="" fl(j)817="" fj(b)838="" fl(k)838="" y(j)859="" fg(\023)903="" fj(n)d="" fo(1\))1171="" fm(1)1249="" fo(+)g="" fj(e)1321="" fm(1)1398="" fj(e)1462="" fg(\022)1523="" y(x)1501="" fl(j)r(<s)1606="" fj(b)1646="" fl(j)1664="" fo(\))1683="" fl(k)1704="" fg(\023)1735="" fx(4)49="" b(optimal)16="" b(lo)o(wer)g(bounds)g(f)o(or)f(the)h="" (deterministic)g(case)232="" fo(14)46="" y(by)15="" b(3.2,)i(w)o(e)e(obtain)i="" (for)g="" fj(c)534="" fl(k)569="" fo(1\))770="" fm(1)848="" fo(+)h="" fj(ce)941="" fm(1)1024="" fo(that)16="" fo(\()1187="" fg(p)1231="" fl(j)r(<s)1346="" fl(k)1367="" y(j)1389="" fo(\))d="" fj(c)1495="" fl(k)1527="" fo(,)16="" b(as)h(claimed.)46="" y(\(c\))f(is)g(imme)o(diate)d(from)i(\(b\).)="" fw(pr)n(o)n(of)16="" b(of)h(cor)n(ol)r(lary)g(3.4:)46="" fo(\(a\))f(assume)f(\014rst)i(that)f="" fj(u)e="" fo(.)21="" b(then,)16="" b(b)o(y)f(3.2\(a\),)269="" fj(b)361="" fl(j)393="" fi(\025)e="" fo(\))594="" fl(k)629="" fj(b)792="" fl(j)810="" fo(\))829="" fl(k)864="" fo(\))982="" fl(k)1003="" fl(k)1159="" fj(b)1310="" fl(j)1328="" fo(\))1347="" fl(k)1368="" fi(\001)1510="" fg(\022)1546="" a(s)1580="" fg(\023)1611="" fl(k)1640="" fo(whence)16="" b(w)o(e)f(get)678="" fj(b)770="" fl(j)802="" fi(\024)1006="" fo(\))1039="" fl(k)1076="" fi(\001)1101="" fg(\022)1136="" a(s)1170="" fg(\023)1201="" fl(k)1231="" fo(by)i(the)h(claim)e(in)i(the)g(pro)q="" (of)i(of)e(3.3\(b\),)g(this)h(implies)649="" fj(b)741="" fl(j)773="" fi(\024)943="" fj(e)1023="" fj(u)1020="" fl(k)1105="" fi(\001)1130="" fg(\022)1166="" a(s)1200="" fg(\023)1231="" fl(k)1260="" fo(in)19="" b(case)h(1)f="" fj(u)g="" fj(k)15="" fo(1)20="" b(it)f(is)g(easily)g(seen)g(that)h="" fo(is)f(also)h(\()p="" fo(\)-univ)o(ersal.)30="" b(applying)19="" b(the)46="" y(ab)q(o)o(v)o(e)d(result)g(yields)f(the)h="" (desired)f(estimate)g(pr\()p="" fj(b)994="" fl(j)1026="" fj(e)1272="" fm(1)1350="" fo(\))1489="" fl(u)1512="" fl(u)1586="" fo(\(b\))16="" b(the)g(argumen)o(t)g(is)g="" (exactly)f(the)h(same)f(as)i(in)f(\(a\).)21="" b(just)c(use)f(3.2\(b\))h="" (instead)f(of)g(3.2\(a\).)60="" fk(4)83="" b(optimal)18="" b(lo)n(w)n(er)i(b)r(ounds)h(for)g(the)g(deterministic)c(case)46="" fo(in)12="" b(this)g(and)h(the)f(follo)o(wing)g(section)g(w)o(e)g="" (consider)g="" fw(deterministic)17="" fo(algorithms)11="" b(for)i(the)f(dictionary)="" y(problem)18="" b(that)i(are)g(based)h(on)f(hashing,)h(and)g(lo)o(w)o="" (er)e(b)q(ounds)i(on)f(their)f(p)q(erformance.)31="" b(it)20="" b(will)46="" y(turn)14="" b(out)h(that)g(suc)o(h)f(deterministic)d(algorithms)j(m)o(ust)="" f(b)q(e)h(m)o(uc)o(h)e(slo)o(w)o(er)i(than)h(the)f(randomized)46="" y(algorithms)h(describ)q(ed)h(in)g(the)g(preceding)f(sections.)46="" y(as)24="" b(a)g(basis)h(for)f(our)h(lo)o(w)o(er)e(b)q(ound)i(pro)q(ofs)g="" (w)o(e)f(in)o(tro)q(duce)f(a)i(simpli\014ed,)e(abstract)i(t)o(yp)q(e)e(of)46="" y(algorithm.)37="" b(suc)o(h)21="" b(algorithms)g(main)o(tain)g(the)g(follo)o="" (wing)h(data)g(structure)g="" fj(d)q="" fo(.)39="" b(if)21="" fj(s)26="" fi(\022)e="" fj(u)j="" y(the)20="" b(set)h(of)g(elemen)o(ts)d(in)j="" (the)g(dictionary)l(,)g(then)f="" fo(consists)e(of)g(a)h(ro)q(oted)f="" (tree)g(whose)g(lea)o(v)o(es)46="" y(are)c(lab)q(elled)e(with)i(the)g="" (elemen)n(ts)d(of)k="" fo(.)k(the)17="" b(inner)f(no)q(des)i(are)f(lab)q="" (elled)e(with)i(hash)h(functions)46="" y(whose)d(v)m(alues)g(corresp)q(ond)="" g(to)h(the)e(edges)h(lea)o(ving)e(the)i(no)q(de.)21="" b(in)15="" b(order)f(to)h(access)g(a)g(k)o(ey)f="" y(one)j(starts)h(at)f(the)g(ro)q(ot)h(and)f(rep)q(eatedly)g(ev)m="" (aluates)g(the)f(hash)i(function)f(at)g(the)g(curren)o(t)f(no)q(de)46="" y(\(with)21="" fj(x)g="" fo(as)h(argumen)o(t\))e(to)h(determine)e(the)i="" (edge)g(to)g(b)q(e)h(follo)o(w)o(ed)e(out)i(of)f(the)g(no)q(de)h(un)o(til)e="" (a)46="" y(leaf)d(is)g(reac)o(hed.)24="" b(this)18="" b(leaf)f(has)h(lab)q(el)f="" fo(.)25="" b(data)g(structure)f(generalizes)f(the)i(one)f(used)="" h(in)46="" y(section)c(2,)g(where)g(t)o(w)o(o)g(hash)i(functions)e(had)h="" (to)g(b)q(e)f(ev)m(aluated)g(to)h(access)f(a)h(k)o(ey)l(.)k(w)l(e)14="" b(coun)o(t)g(one)46="" y(step)i(for)h(the)f(ev)m(aluation)g(of)h(a)f(hash)h="" (function.)46="" y(in)h(more)g(detail,)g(the)g(data)i(structure)e(can)h(b)q="" (e)g(describ)q(ed)f(as)i(follo)o(ws.)28="" fj(d)21="" fo(is)d(a)h(ro)q(oted)h="" (tree)e(in)46="" y(whic)o(h)e(eac)o(h)h(inner)f(no)q(de)i="" fj(v)g="" fo(is)f(lab)q(elled)f(with)g(a)i(hash)g(function)e="" fj(h)1301="" fl(v)1337="" fj(u)20="" fi(!)15="" b(f)p="" fj(;)g(:)g(:)g(:)g(;)g(m)1732="" fl(v)1763="" fi(\000)k="" y(with)18="" fj(m)202="" fl(v)239="" fo(2,)j(and)f(has)h="" fj(m)579="" fl(v)617="" fo(c)o(hildren,)e(one)h(for)g(eac)o(h)g(v)m(alue)f(of)="" i="" fj(h)1311="" fl(v)1331="" b(eac)o(h)18="" fj(u)23="" fo(determines)46="" y(a)c(path)g(from)e(the)h(ro)q="" (ot)h(to)g(a)g(leaf.)27="" b(this)19="" b(path)g(is)f(giv)o(en)f(b)o(y)h="" fj(w)1250="" fm(0)1270="" b(w)1327="" fm(1)1346="" fj(;)g(:)g(:)g(:)f(;)h(w)1490="" fl(r)1509="" fo(,)19="" b(where)f="" fj(w)1720="" fm(0)1758="" fo(is)g(the)p="" fo(15)46="" y(ro)q(ot,)18="" fj(w)200="" fl(t)p="" fm(+1)276="" fo(is)f(the)g="" fj(h)439="" fl(w)464="" y(t)481="" fo(\)-th)g(c)o(hild)f(of)h="" fj(w)836="" fl(t)851="" fo(,)f(for)i(0)d="" fj(<)g(t)g(<)g(r)q="" fo(,)i(and)h="" fj(w)1320="" fl(r)1355="" fo(is)f(a)h(leaf.)23="" b(w)l(e)16="" b(sa)o(y)h(that)h="" fj(d)46="" fo(is)d(a)h(dictionary)f(for)h="" fj(s)h="" fj(x)591="" fm(0)610="" b(:)g(:)g(:)f(;)h="" (x)747="" fl(n)771="" fi(g)13="" b(\022)h="" fj(u)21="" fo(if)15="" b(eac)o(h)g(leaf)g(con)o(tains)h(exactly)e(one)i(of)g(the)f="" fj(x)1772="" fl(i)1786="" b(t)l(o)46="" y(eac)o(h)d(no)q(de)h="" fj(v)h="" fo(of)f="" fo(w)o(e)d(asso)q(ciate)i(the)e(set)h="" fj(a)p="" fj(v)r="" fi(\022)h="" fj(u)24="" fo(of)19="" b(k)o(eys)e(that)j(are)e(\\sen)o(t)h(to")h="" fo(b)o(y)e(the)46="" y(hash)f(functions)f(on)h(the)f(path)h(from)f(the)g(ro)q(ot)h(to)g="" fo(.)k(w)l(e)16="" b(de\014ne)g(inductiv)o(ely:)j="" b(=")h" fj(u)22="" fo(for)16="" fj(v)46="" fo(the)h(ro)q(ot,)i(and)f="" fj(v)427="" fl(q)445="" fi(j)h="" fj(h)826="" fl(v)847="" fj(q)r="" fi(g)h="" fo(for)h(0)e="" fj(q)i(<)e(m)1357="" fl(v)1394="" fo(where)i="" fj(v)1561="" fl(q)1579="" fo(,)g(0)e="" fj(q)i(<)e(m)1843="" fl(v)1863="" y(are)g(the)g(c)o(hildren)f(of)h="" y(f)l(or)23="" b(our)h(lo)o(w)o(er)f(b)q(ound)h(argumen)o(ts,)g(w)o(e)f="" (will)f(consider)h(only)g(insertions.)42="" b(t)l(o)24="" b(insert)f(a)h(k)o(ey)46="" fj(x)74="" fm(+1)156="" fo(in)o(to)10="" b(a)i(dictionary)e="" fo(for)e="" fo(,)g(w)o(e)g(follo)o(w)f(the)h(path)h="" fj(w)1306="" fm(0)1325="" fj(;)c(w)1382="" fm(1)1402="" fj(;)g(:)g(:)g(:)f(;)h="" (w)1546="" fl(r)1576="" fo(determined)g(b)o(y)46="" fo(,)13="" b(and)h(for)f(some)f(no)q(de)i="" fo(on)g(this)f(path)h="" (\(determined)c(b)o(y)j(the)g(algorithm\))f(p)q(erform)g(a)i="" fw(r)n(ehashing)46="" fo(at)h="" fo(,)g(whic)o(h)f(means)g(that)i(w)o="" (e)e(c)o(ho)q(ose)i(a)f(new)g(p)q(erfect)g(hash)h(function)f="" fj(h)1405="" fl(v)1440="" fo(for)h="" fo(\))7="" fi(\\)i="" fi([)d(f)p="" fo(th)o(us,)k(all)e="" fo(\))s="" fi(\\)s="" fj(s)6="" fi([)s(f)p="" fi(j)k="" fo(c)o(hildren)h(of)i="" fo(b)q(ecome)e(lea)o(v)o(es,)g(and)i(to)g="" (eac)o(h)e(of)i(them)d(corresp)q(onds)46="" y(exactly)h(one)h(elemen)o(t)d="" (of)j="" fo(\))r="" fi(\\)r="" fj(s)5="" fi([)r(f)p="" fo(\).)20="" b(suc)o(h)11="" b(a)h(rehashing)h="" fw(must)k="" fo(b)q(e)12="" b(p)q(erformed)f(for)h(exactly)46="" y(one)j(no)q(de)h="" fo(on)f(the)g(path.)21="" b(the)15="" b(cost)h(of)f(suc)o="" (h)f(an)i(insertion)e(is)h(depth\()p="" fo(\))8="" b(+)g="" fi([)e(f)p="" fo(the)17="" b(cost)g(of)h="" (inserting)f="" fj(x)535="" fm(1)554="" b(x)604="" fm(2)623="" fj(;)g(:)g(:)g(:)g(;)g(x)761="" fl(n)799="" fi(2)16="" fo(in)o(to)17="" b(a)h(dictionary)e="" fo(is)e(the)g(sum)f(of)i(the)f(costs)g(of)46="" y(the)f(single)f="" (insertions.)21="" b(note)16="" b(that)h(w)o(e)e(assume)h(that)h="" fj(d)h="" fo(initially)c(con)o(tains)i(one)g(elemen)o(t)d="" fj(x)1800="" fm(0)1836="" fo(in)46="" y(a)k(leaf,)e(with)h(no)h(ro)q(ot.)46="" fh(remark)f(4.1)24="" fo(when)16="" b(a)h(rehashing)f(at)g="" fj(v)i="" fo(is)d(p)q(erformed,)g(a)h(p)q(erfect)f(hash)i(function)f(for)g="" fo(\))9="" fi(\\)46="" b(is)g(giv)o(en)f(at)i(linear)e(cost;)h(in)="" g(addition,)g(setting)g(up)g(the)g(hash)h(table,)f(i.)8="" b(e.,)k(the)i="" (subtree)46="" y(of)k(depth)f(1,)h(for)g(this)g(set)f(has)i(linear)e(cost)h="" (as)g(w)o(ell.)24="" b(assumption)f(excludes)g(searc)o(h)g(trees)46="" y(that)e(use)g(an)g(order)g(on)g(the)f(univ)o(erse)g="" fo(to)15="" b(de\014ne)f(the)h(w)o(a)o(y)f(k)o(eys)g(are)g(distributed)g(at)i="" (no)q(des,)f(as)46="" y(w)o(ell)e(as)j(other)e(sc)o(hemes)f(in)o(v)o="" (olving)g(clev)o(erly)f(c)o(hosen)i(hash)i(functions)f(that)g(can)g(b)q(e)f="" (extended)g(to)46="" y(additional)i(k)o(eys)f(at)i(lo)o(w)f(cost)h(while)e="" (k)o(eeping)g(the)h(function)g(injectiv)o(e.)46="" fh(remark)g(4.2)24="" fo(\(a\))15="" b(w)l(e)g(require)e(that)j(collisions)e(are)g(resolv)o(ed)g="" (immedi)o(ately)d(b)o(y)k(rehashing.)21="" b(in)46="" y(particular,)14="" b(w)o(e)g(do)h(not)h(allo)o(w)e(forming)g(c)o(hains,)g(i.)8="" b(link)o(ed)g(lists,)h(at)h(the)f(lea)o(v)o(es)g(of)h(the)f(tree)g="" (as)46="" y(is)g(done)g(in)g(man)o(y)f(hashing)i(sc)o(hemes.)k(but)14="" b(the)g(absence)g(of)g(this)g(restriction)g(w)o(ould)g(not)h(c)o(hange)46="" y(the)j(lo)o(w)o(er)g(b)q(ounds)h(b)o(y)f(m)o(uc)o(h.)26="" b(if)18="" b(w)o(e)g(w)o(ere)f(to)i(allo)o(w)f(c)o(haining,)g(inserting)g="" fj(n)h="" fo(elemen)o(ts)d(w)o(ould)46="" y(cost)c="" fo(steps,)h(b)q="" (ecause)f(w)o(e)g(could)f(insert)h(eac)o(h)f(elemen)o(t)e(at)k(the)e(head)h="" (of)h(the)e(c)o(hain,)h(whic)o(h)g(w)o(ould)46="" y(mean)g(constan)o(t)j="" (time)c(p)q(er)j(insertion.)20="" b(t)l(o)14="" b(justify)g(our)g(mo)q(del,)e(w)o="" (e)i(ha)o(v)o(e)f(to)h(consider)f(tasks)i(with)46="" y(insertions)h="" fw(and)21="" fo(lo)q(okups.)h(if)16="" b(after)g(inserting)g="" fo(w)o(e)g(include)e(a)j(lo)q(okup)g(for)f(the)g(elemen)o(t)d(at)k(the)46="" y(end)f(of)h(the)f(c)o(hain)g(in)o(to)g(whic)o(h)g="" fo(w)o(as)h(inserted,)e(then)i(this)f(lo)q(okup)h(costs)g(essen)o(tially)e="" (as)i(m)o(uc)o(h)46="" y(as)e(rehashing)f(at)h(the)f(leaf)g(to)g(whic)o(h)g="" (the)g(c)o(hain)f(b)q(elongs.)22="" b(th)o(us)14="" b(algorithms)g(for)g="" (insertions)g(and)46="" y(lo)q(okups,)20="" b(with)e(c)o(haining)g(allo)o(w)o="" (ed,)g(are)h(as)h(least)e(as)h(costly)g(as)g(algorithms)f(without)h(c)o="" (haining)46="" y(for)d(insertions)g(only)l(.)46="" y(\(b\))f(one)f(could)="" h(ask)g(if)f(it)h(w)o(ould)g(b)q(e)g(adv)m(an)o(tageous)h(to)f(also)h(allo)o="" (w)e(rehashings)i(at)f(no)q(des)h="" fo(that)46="" y(do)f(not)g(lie)e="" (on)i(the)g(path)g(determined)d(b)o(y)i(the)g="" fo(just)h(b)q(eing)g="" (inserted.)20="" b(but)14="" b(it)g(is)g(easily)g(c)o(hec)o(k)o(ed)46="" y(that)f(the)g(algorithm)f(do)q(es)i(not)f(b)q(ecome)f(slo)o(w)o(er)g="" (if)g(suc)o(h)h(rehashings)h(are)f(p)q(erformed)f(at)h(the)g(time)46="" y(when)j(the)g(last)g(elemen)o(t)e(of)i="" fj(s)e="" fi(\\)d="" fo(\))k(is)h(inserted)g(in)o(to)g="" fo(.)22="" b(th)o(us)16="" b(it)g(is)g(justi\014ed)g(not)g(to)h(admit)46="" y(suc)o(h)f(\\sp)q(on)o(taneous")j(rehashings.)46="" fh(remark)d(4.3)24="" fw(the)15="" b(r)n(ole)g(of)g(sp)n(ac)n(e)g(limitations.)="" fo(in)13="" b(the)h(description)e(of)i(the)g(data)g(structure)f="" y(w)o(e)i(ha)o(v)o(e)g(not)h(in)o(tro)q(duced)g(the)f="" (concept)h(of)g(the)f(space)h(used)g(b)o(y)f="" b(on)16="" b(the)g(other)g(hand,)g(some)p="" fo(16)46="" y(space)16="" b(restriction)f(is)h="" (necessary)l(,)f(since)h(using)g(the)g(iden)o(tit)o(y)e(function)i(as)h(the)e="" (hash)i(function)f(at)46="" y(the)g(ro)q(ot)h(w)o(ould)f(mak)o(e)f(all)h="" (rehashing)g(sup)q(er\015uous.)46="" y(if)k(w)o(e)g(assume)g(that)h(storing)="" g(a)g(hash)g(function)g="" fj(h)1012="" fl(v)1053="" fo(together)g(with)f="" (the)g(corresp)q(onding)i(table)46="" y(tak)o(es)c(space)h="" fj(m)406="" fl(v)427="" fj(r)q(ang)r(e)p="" fj(h)765="" fl(v)786="" fo(\),)f(then)984="" fg(p)1028="" fl(v)7="" fm(no)q(de)e(in)g="" fl(d)1207="" fj(m)1250="" fl(v)1289="" fo(is)19="" b(a)g(reasonable)h(measure)d(for)46="" y(the)c(space)h(used)g(b)o(y)f="" b(in)13="" b(our)h="" (description)f(of)h(the)g(data)g(structure)f="" fo(w)o(e)d(assumed)g="" (that)h(ev)o(ery)46="" y(leaf)19="" b(of)i="" fo(con)o(tained)d(an)i="" (elemen)o(t)c(of)j="" fo(,)h(so)f(for)h(ev)o(ery)d="" fj(h)1171="" fl(v)1212="" fo(and)i(ev)o(ery)f="" fj(j)k="" fi(2)e(f)p="" b(:)g(:)g(:)f(;)h(m)1741="" fl(v)1775="" fi(\000)13="" fo(there)k(is)g(some)f="" fo(with)d="" fj(h)627="" fl(v)648="" b(since)16="" b(in)h(ev)o(ery)f(ro)q="" (oted)i(tree)f(with)g="" fj(n)12="" fo(+)f(1)18="" b(lea)o(v)o(es)e(and)46="" y(outdegree)j(at)h(least)f(t)o(w)o(o)g(the)g(n)o(um)o(b)q(er)f(of)i="" (edges)f(is)g(b)q(ounded)i(b)o(y)d(2)p="" fo(,)j(our)e(data)i(structure)e="" fo(satis\014es)226="" fg(p)269="" fl(v)298="" fj(m)341="" fl(v)375="" fo(,)i(whic)o(h)f(means)h(that)g(it)g(needs)g(linear)g(space.)46="" y(if)d(the)g(algorithm)f(w)o(ere)g(allo)o(w)o(ed)h(to)g(use)g(hash)h="" (functions)g="" fj(h)1156="" fl(v)1189="" fo(with)f(range)h(larger)f="" (than)h="" fo(\))5="" fi(\\)g="" fi(j)46="" fo(when)17="" b(rehashing)h(at)g(no)q(de)g="" fo(,)f(then)g(the)h(lo)o(w)o(er)e(b)q(ounds)j(giv)o(en)e(in)g(the)g(theorems)="" f(b)q(elo)o(w)i(w)o(ould)46="" y(still)i(hold,)j(with)f(constan)o(ts)g="" (smaller)e(b)o(y)h(a)h(factor)g(of)1117="" a(4)1161="" fo(than)g(those)g(in)f(the)h(theorems.)="" b(w)l(e)46="" y(only)23="" b(ha)o(v)o(e)g(to)h(assume)e(that)i(the)f(space)="" h(used)f(b)o(y)g="" fo(is)d(not)h(to)q(o)g(large)g(in)f(relation)g(to)h="" (the)46="" y(size)18="" b(of)h(the)f(univ)o(erse)g="" fo(\(namely)17="" fj(u)5="" fi(j)18="" b(\025)g="" fo(\)2)8="" b(log)h="" fo(\))1137="" fm(2)d(log)g="" fl(n)1249="" fi(\001)12="" fo(+)g(1\))19="" b(in)f(theorem)g(4.4)h(and)46="" fi(j)14="" b(\025)f="" fo(\))367="" fl(k)403="" fo(in)g(theorem)g(4.6)h(for)g="" fg(p)1019="" fl(v)1048="" fj(m)1091="" fl(v)1111="" fj(:)f="" fo(w)l(e)h(shall)f="" (commen)o(t)e(on)k(this)e(in)h(more)46="" y(detail)h(b)q(elo)o(w)i(when)f="" (the)g(adv)o(ersary)g(strategies)g(for)h(the)f(lo)o(w)o(er)f(b)q(ound)j(pro)q="" (ofs)f(are)f(discussed.)46="" y(w)l(e)g(w)o(an)o(t)g(to)h(study)f(the)g="" (follo)o(wing)g(quan)o(tities.)71="" fj(t)7="" fo(\))110="" b(w)o(orst-case)13="" b(\(amortized\))e(cost)i(incurred)f(b)o="" (y)g(an)h(optimal)e(algorithm)h(to)h(insert)372="" fo(elemen)o(ts.)71="" fj(t)100="" fm(max)167="" fo(\))50="" b(cost)h(needed)e(for)h(a)g="" (single)g(insertion)f(or)i(mem)n(b)q(ership)c(query)i(in)372="" y(a)f(sequence)g(of)g="" fo(instructions.)71="" fl(k)121="" fo(\))96="" b(w)o(orst-case)16="" b(amortized)d(cost)j(needed)e(b)o(y)h(an)h(optimal)d="" (algorithm)i(to)g(insert)g="" fj(n)372="" fo(elemen)o(ts,)i(if)i(the)g="" (depth)h(of)g(the)f(tree)g(is)g(not)h(allo)o(w)o(ed)f(to)h(exceed)e="" fo(,)i(i.)8="" b(e.,)19="" b(if)372="" y(the)d(w)o(orst)g(case)h(lo)q="" (okup)g(time)d(is)i="" y(the)f(follo)o(wing)f(three)g="" (theorems)f(sum)h(up)h(the)f(results)h(\(upp)q(er)g(and)g(lo)o(w)o(er)f(b)q="" (ounds\))i(concerning)46="" y(these)h(three)h(quan)o(tities.)25="" b(theorem)16="" b(4.4)i(sho)o(ws)h(that)f(amortized)e(time)g="" fo(\))j(for)f="" fo(insertions)46="" y(cannot)f(b)q(e)g="" (ac)o(hiev)o(ed)d(in)j(the)f(deterministic)d(case,)j(but)h(that)g(a)g(slo)o="" (wdo)o(wn)g(b)o(y)f(a)h(factor)g(log)9="" fj(n)17="" y(una)o(v)o(oidable.)j(theorem)14="" b(4.5)h(sho)o(ws)h(that)g(in)e(an)o(y)="" h(case)g(there)g(will)f(b)q(e)h(single)f(instructions)h(that)46="" y(are)20="" b(v)o(ery)f(costly)l(.)31="" b(if)20="" b(w)o(e)f(demand)g(constan)o="" (t)i(lo)q(okup)f(time)e(to)j(b)q(e)f(guaran)o(teed,)h(theorem)d(4.6)46="" y(sho)o(ws)f(that)g(this)f(can)g(only)g(b)q(e)h(ac)o(hiev)o(ed)d(b)o(y)i="" (man)o(y)f(costly)g(rehashings.)p="" fx(5)49="" b(pr)o(oofs)16="" b(of)g(the)g(lo)o(wer)h(bounds)869="" fo(17)46="" fh(theorem)16="" b(4.4)79="" fw(\(a\))24="" fo(+)g(1\))g="" fo(log)q(\()p="" fo(+)g(1\))p="" b(if)f="" fo(log)i="" fo(\))1151="" fm(2)5="" fl(n)1260="" fi(\001)11="" fo(+)f(1\))p="" fo(3\()p="" fo(+)g(1\))d(log)r(\()p="" b(4.5)79="" fj(t)197="" fm(max)264="" fi(p)p="" fw(,)k(if)f="" fo(2\()723="" fj(n)q="" fo(\))813="" ff(p)p="" x="" fl(n)865="" b(if)d(only)h(algorithms)g(with)g(a)g(total)g(c)n(ost)g(smal)r(ler)h(than)f="" fj(f)5="" fo(\))15="" fj(n)23="" fw(for)f="" fw(insertions)g(ar)n(e)168="" y(c)n(onsider)n(e)n(d,)16="" fo(\()693="" a(f)t="" fm(\))743="" fo(\))762="" fm(2)p="" fl(f)t="" fm(\))862="" b(then)g="" fj(t)1208="" fm(max)1276="" fo(\))c(=")f(\012\()p" fo(\)\))p="" b(4.6)79="" fl(k)218="" fj(k)r(="e)p" fj(n)528="" fj(e)826="" fl(k)847="">)g Fo(\(2)p Fj(n=k)r Fo(\))1201 1142 y Fl(k)1223 1160 y Fj(:)81 1317 y Fw(\(b\))25 b Fj(T)197 1324 y Fl(k)218 1317 y Fo(\()p Fj(n)p Fo(\))f Fi(\024)g Fj(d)397 1324 y Fl(k)434 1317 y Fi(\001)15 b Fj(n)492 1299 y Fm(1+1)p Fl(=k)617 1317 y Fw(for)23 b(al)r(l)h(su\016ciently)h(lar)n(ge)e Fj(n)p Fw(,)h(wher)n(e)g(the)f(c)n(onstants)h Fj(d)1694 1324 y Fl(k)1739 1317 y Fw(c)n(an)f(b)n(e)168 1377 y(chosen)18 b(to)g(satisfy)f Fj(d)563 1384 y Fl(k)598 1377 y Fi(\030)d Fj(k)r(=e:)j Fw(\(Her)n(e)g Fj(e)d Fo(=)f(2)p Fj(:)p Fo(71828)8 b Fj(:)g(:)g(:)16 b Fo(=)h Fw(Euler's)h(c)n(onstant.\))46 1529 y Fo(The)e Fw(pr)n(o)n(ofs)j Fo(of)d(the)g(theorems)f(will)g(b)q(e)i(giv)o(en)e(in)h(the)g(next)g (section.)46 1681 y Fh(Remark)g(4.7)24 b Fo(If)d(w)o(e)f(reconsider)g(the)h (randomized)e(algorithm)h(presen)o(ted)g(in)g(Section)g(2,)i(w)o(e)46 1742 y(see)13 b(that)h(randomization)f(is)h(only)f(used)h(for)g(constructing) f(p)q(erfect)g(hash)i(functions)f(at)g(exp)q(ected)46 1802 y(linear)i(cost.)25 b(Th)o(us,)17 b(if)g(w)o(e)g(giv)o(e)f(suc)o(h)h(hash)i (functions)e(at)h(guaran)o(teed)f(linear)g(cost,)g(w)o(e)g(should)46 1862 y(obtain)h(a)f(deterministic)d(algorithm)j(that)g(is)g(as)h(least)g(not) f(slo)o(w)o(er)g(than)h(the)f(randomized)f(one.)46 1922 y(This)d(seems)f(to)h (con)o(tradict)f(our)i(lo)o(w)o(er)e(b)q(ounds!)22 b(T)l(o)13 b(resolv)o(e)f(this)h(parado)o(x,)h(consider)f(adv)o(ersary)46 1982 y(strategies)g(for)g(the)g(randomized)e(computation)h(mo)q(del.)19 b(Here)12 b(the)h(adv)o(ersary)f(has)i(to)f(determine)46 2043 y(the)18 b(mo)o(v)o(es)f(of)h(the)g(strategy)h(without)g(kno)o(wledge)f(of)h (the)f(outcomes)f(of)i(the)f(coin)h(\015ips)f(of)h(the)46 2103 y(algorithm)e(to)h(b)q(e)g(executed.)24 b(This)18 b(means)f(that)i(the)e (data)i(structure)e(pro)q(duced)i(b)o(y)e(the)h(algo-)46 2163 y(rithm)d(cannot)j(b)q(e)f(tak)o(en)g(in)o(to)g(consideration)g(b)o(y)g(the)f (adv)o(ersary)l(.)24 b(But)17 b(this)g(is)g(what)h(happ)q(ens)46 2223 y(in)f(the)h(deterministic)c(case)k(and)g(what)g(mak)o(es)e(the)i(adv)o (ersary)f(as)i(strong)f(as)h(indicated)d(in)i(the)46 2283 y(lo)o(w)o(er)d(b)q (ounds)j(for)e(the)g(deterministic)d(mo)q(del.)46 2435 y Fh(Remark)j(4.8)24 b Fo(Theorem)16 b(4.5)i(giv)o(es)e(a)h(lo)o(w)o(er)g(b)q(ound)h(for)f(our)h (mo)q(del)e(that)h(is)g(bigger)g(than)h(the)46 2496 y Fj(O)q Fo(\(log)10 b Fj(n)p Fo(\))22 b(w)o(orst-case)g(b)q(ound)h(for)f(single)g (instructions)g(guaran)o(teed)g(b)o(y)f(implem)o(en)o(tations)e(of)46 2556 y(dictionaries)f(as)i(balanced)f(searc)o(h)g(trees.)29 b(This)19 b(is)g(an)h(e\013ect)e(of)i(the)f(quite)f(sev)o(ere)g(restriction) 46 2616 y(that)g(rehashing)g(at)f(a)h(no)q(de)g Fj(v)h Fo(has)f(cost)g (linear)e(in)h(the)g(size)g(of)h(the)f(subtree)g(ro)q(oted)h(at)g Fj(v)r Fo(.)24 b(\(Cf.)46 2676 y(Remark)14 b(1.1.\))46 2899 y Fk(5)83 b(Pro)r(ofs)28 b(of)g(the)f(lo)n(w)n(er)g(b)r(ounds)46 3067 y Fo(This)16 b(section)g(con)o(tains)g(the)g(pro)q(ofs)i(of)f(the)f (theorems)e(stated)j(in)f(Section)g(4.)p eop %%Page: 18 18 17 bop 46 188 a Fx(5)49 b(PR)o(OOFS)16 b(OF)g(THE)g(LO)o(WER)h(BOUNDS)869 b Fo(18)46 342 y Fb(5.1)69 b(The)23 b(adv)n(ersary)i(strategy)46 493 y Fo(F)l(or)19 b(pro)o(ving)f(the)g(lo)o(w)o(er)g(b)q(ounds,)i(w)o(e)e (apply)g(an)h(adv)o(ersary)g(argumen)o(t)e(in)h(eac)o(h)g(case.)29 b(Let)18 b(us)46 553 y(\014rst)e(giv)o(e)g(a)g(general)g(description)f(of)i (the)f(adv)o(ersary)g(strategy)l(.)21 b(Initially)l(,)14 b(the)i(tree)f Fj(D)k Fo(con)o(tains)46 614 y(one)h(elemen)o(t)d Fj(x)351 621 y Fm(0)371 614 y Fo(.)33 b(The)20 b(adv)o(ersary)g(c)o(ho)q(oses,)i(step) e(b)o(y)g(step,)g(the)g(elemen)o(t)d Fj(x)1540 621 y Fl(i)1574 614 y Fo(to)k(b)q(e)f(inserted)46 674 y(next.)j(Basically)l(,)15 b Fj(x)422 681 y Fl(i)453 674 y Fo(is)i(alw)o(a)o(ys)f(c)o(hosen)h(in)g(suc)o (h)g(a)g(w)o(a)o(y)g(that)g(it)g(has)g(to)h(follo)o(w)e(a)i(longest)f(path)46 734 y(in)f Fj(D)q Fo(.)46 853 y(In)g(order)g(to)g(alw)o(a)o(ys)g(b)q(e)h (able)e(to)i(\014nd)f(suc)o(h)g(an)h(elemen)o(t)c Fj(x)1161 860 y Fl(i)1175 853 y Fo(,)i(w)o(e)h(m)o(ust)f(mak)o(e)f(sure)i(that)h(the)f (set)46 913 y(of)k(elemen)o(ts)e(of)j Fj(U)k Fo(that)c(b)q(elongs)g(to)g(suc) o(h)f(a)h(longest)f(path)h(is)f(not)h(empt)o(y)l(.)31 b(The)20 b(aim)f(of)i(the)46 974 y(adv)o(ersary)e(is)h(to)g(build)f(up)h(long)g(paths) g Fj(w)868 981 y Fm(0)888 974 y Fj(;)8 b(w)945 981 y Fm(1)964 974 y Fj(;)g(w)1021 981 y Fm(2)1041 974 y Fj(;)g(:)g(:)g(:)19 b Fo(in)g(the)g(tree)g(and)i(to)f(mak)o(e)d(sure)j(that)46 1034 y(the)e(sets)h Fj(A)p Fo(\()p Fj(w)321 1041 y Fm(0)340 1034 y Fo(\))p Fj(;)8 b(A)p Fo(\()p Fj(w)472 1041 y Fm(1)491 1034 y Fo(\))p Fj(;)g(A)p Fo(\()p Fj(w)623 1041 y Fm(2)642 1034 y Fo(\))p Fj(;)g(:)g(:)g(:)17 b Fo(are)i(as)g(large)f(as)i(p)q(ossible.) 28 b(Th)o(us,)19 b(if)f(a)h(decision)e(is)i(to)f(b)q(e)46 1094 y(made)d(whic)o(h)i(path)g(to)g(c)o(ho)q(ose,)g(the)g(adv)o(ersary)f(will,)g (at)h(eac)o(h)f(no)q(de)h Fj(v)r Fo(,)f(c)o(ho)q(ose)h(that)h(c)o(hild)d Fj(q)j Fo(of)46 1154 y Fj(v)g Fo(that)f(maximi)o(zes)d Fi(j)p Fj(h)474 1136 y Ff(\000)p Fm(1)474 1166 y Fl(v)521 1154 y Fo([)p Fj(q)r Fo(])c Fi(\\)h Fj(A)p Fo(\()p Fj(v)r Fo(\))p Fi(j)p Fo(,)k(0)g Fi(\024)f Fj(q)i(<)f(m)997 1161 y Fl(v)1017 1154 y Fo(.)22 b(\(If)16 b(there)g(is)h(a)g(tie,)e(the)h(smallest)f(suc)o(h)i Fj(q)h Fo(is)46 1214 y(c)o(hosen.\))24 b(F)l(or)18 b(the)f(sak)o(e)g(of)h (simplicit)n(y)c(of)k(notation,)g(w)o(e)f(will)f(assume)h(that)h Fj(q)f 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Fj(n)d Fo(+)g(1,)17 b(i.)8 b(e.,)14 b Fi(j)p Fj(U)5 b Fi(j)15 b(\025)46 1222 y Fo(\()96 1202 y Fl(n)p 70 1210 73 2 v 70 1239 a Fm(log)5 b Fl(n)148 1222 y Fo(\))167 1204 y Fm(2)g(log)h Fl(n)276 1222 y Fi(\001)11 b Fo(\()p Fj(n)g Fo(+)g(1\))p Fj(:)46 1341 y Fo(De\014ne)131 1451 y Fj(L)j Fo(=)g Fi(f)8 b Fj(x)291 1458 y Fl(i)347 1451 y Fi(j)22 b Fo(1)14 b Fi(\024)f Fj(i)h Fi(\024)g Fj(n;)41 b Fo(depth\()p Fj(v)r Fo(\))13 b Fi(\025)g Fo(2)8 b(log)i Fj(n)17 b Fo(for)f(the)g(v)o(ertex)f Fj(v)j Fo(in)d Fj(D)641 1511 y Fo(at)h(whic)o(h)g(rehashing)h(is)f(p)q(erformed)f(when)h Fj(x)1497 1518 y Fl(i)1527 1511 y Fo(is)g(inserted)8 b Fi(g)p Fj(:)46 1621 y Fo(Clearly)l(,)20 b(for)g(eac)o(h)g Fj(x)450 1628 y Fl(i)484 1621 y Fi(2)h Fj(L)f Fo(the)g(cost)h(of)f(ev)m(aluating)g(the)g (hash)h(functions)g(on)f(the)g(w)o(a)o(y)g(do)o(wn)46 1682 y(to)f Fj(v)h Fo(alone)e(is)h(at)f(least)h(2)8 b(log)i Fj(n)p Fo(.)28 b(W)l(e)18 b(determine)e(a)j(lo)o(w)o(er)f(b)q(ound)h(for)g (inserting)f(the)g(elemen)o(ts)46 1742 y(in)k Fi(f)p Fj(x)162 1749 y Fl(i)174 1754 y Fe(1)193 1742 y Fj(;)8 b(x)243 1749 y Fl(i)255 1754 y Fe(2)274 1742 y Fj(;)g(:)g(:)g(:)f(;)h(x)411 1749 y Fl(i)423 1759 y Fa(n)442 1752 y Fc(0)457 1742 y Fi(g)24 b Fo(=)g Fi(f)p Fj(x)621 1749 y Fm(1)640 1742 y Fj(;)8 b(:)g(:)g(:)g(;)g(x) 778 1749 y Fl(n)801 1742 y Fi(g)15 b(\000)g Fj(L)22 b Fo(in)o(to)g(the)g (tree)g(as)h(follo)o(ws.)39 b(\(Here,)22 b Fj(n)1696 1724 y Ff(0)1732 1742 y Fo(=)i Fj(n)15 b Fi(\000)46 1802 y(j)p Fj(L)p Fi(j)p Fo(.\))32 b(Observ)o(e)18 b(that)j(if)e(w)o(e)g(disregard)i(all)e (elemen)o(ts)e Fj(x)1113 1809 y Fl(i)1147 1802 y Fi(2)j Fj(L)g Fo(and)g(all)g(inner)f(no)q(des)h(at)h(depth)46 1862 y Fi(\025)f Fo(2)8 b(log)i Fj(n)20 b Fo(in)f(the)h(computation)f(for)h Fj(x)800 1869 y Fm(1)820 1862 y Fj(;)8 b(:)g(:)g(:)f(;)h(x)957 1869 y Fl(n)980 1862 y Fo(,)21 b(then)e(w)o(e)h(obtain)g(a)h(computation)e (in)g(whic)o(h)46 1922 y Fj(x)74 1929 y Fl(i)86 1934 y Fe(1)105 1922 y Fj(;)8 b(:)g(:)g(:)f(;)h(x)242 1929 y Fl(i)254 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Fj(T)7 b Fo(\()p Fj(n)p Fo(\))14 b Fi(\024)f Fo(2)f(+)f(2)442 1000 y Fl(t)p Ff(\000)p Fm(1)513 1021 y Fo(+)g Fj(n)g Fo(+)g(2)h Fi(\001)e Fo(3)i Fi(\001)f Fo(2)796 1000 y Fl(t)p Ff(\000)p Fm(1)864 1021 y Fo(log)q(\(2)970 1000 y Fl(t)p Ff(\000)p Fm(1)1031 1021 y Fo(\))g(+)g(3)g Fi(\001)g Fo(\()p Fj(n)g Fi(\000)g Fo(2)1303 1000 y Fl(t)1329 1021 y Fo(+)g(1\))d(log)r(\()p Fj(n)j Fi(\000)g Fo(2)1626 1000 y Fl(t)1652 1021 y Fo(+)g(1\))p Fj(:)46 1117 y Fo(With)16 b(2)11 b(+)g(2)280 1099 y Fl(t)p Ff(\000)p Fm(1)352 1117 y Fo(+)g Fj(n)j Fi(\024)f Fo(3)f Fi(\001)f Fo(2)581 1099 y Fl(t)612 1117 y Fo(it)16 b(follo)o(ws)g(that)408 1201 y(~)399 1213 y Fj(T)6 b Fo(\()p Fj(n)p Fo(\))14 b Fi(\024)f Fo(3)f Fi(\001)f Fo(2)652 1193 y Fl(t)675 1213 y Fo(log)q(\(2)781 1193 y Fl(t)796 1213 y Fo(\))g(+)g(3)h Fi(\001)f Fo(\()p Fj(n)g Fi(\000)g Fo(2)1069 1193 y Fl(t)1095 1213 y Fo(+)g(1\))d(log)r(\()p Fj(n)j Fi(\000)g Fo(2)1392 1193 y Fl(t)1418 1213 y Fo(+)g(1\);)46 1310 y(hence,)17 b(b)o(y)g(the)h(con)o(v)o(exit)o(y)d(of)j(the)g(function)f Fj(y)10 b Fo(log)g Fj(y)r Fo(,)17 b(w)o(e)g(get)1234 1297 y(~)1224 1310 y Fj(T)6 b Fo(\()p Fj(n)p Fo(\))17 b Fi(\024)f Fo(3\()p Fj(n)d Fo(+)f(1\))c(log)r(\()p Fj(n)k Fo(+)g(1\),)18 b(as)46 1370 y(desired.)46 1489 y(This)e(\014nishes)g(the)g(pro)q(of)i(of)e (4.4\(b\).)46 1691 y Fb(5.3)69 b(Pro)r(of)24 b(of)g(Theorem)19 b(4.5)46 1842 y Fo(\(a\))c(Apply)f(the)g(simple)e(adv)o(ersary)j(strategy)g (from)e(Section)i(5.1.)21 b(If)14 b(at)h(some)e(time)g(the)h(leftmost)46 1902 y(path)g(in)f(the)h(tree)f(b)q(ecomes)f(longer)i(than)834 1867 y Fi(p)p 876 1867 30 2 v 35 x Fj(n)g Fo(then)f(at)h(least)g(one)g (insertion)f(had)h(cost)1671 1867 y Fi(p)p 1713 1867 V 35 x Fj(n)p Fo(.)20 b(Oth-)46 1962 y(erwise,)c(the)g(assumption)h Fi(j)p Fj(U)5 b Fi(j)15 b(\025)g Fo(2\()728 1927 y Fi(p)p 770 1927 V 35 x Fj(n)p Fo(\))818 1920 y Ff(p)p 847 1920 22 2 v 24 x Fl(n)887 1962 y Fo(guaran)o(tees,)j(b)o(y)e(Lemma)f(5.3,)i(that)g(the)g (adv)o(ersary)46 2023 y(strategy)k(can)h(b)q(e)f(carried)g(out.)36 b(Only)21 b(no)q(des)h(on)f(the)g(leftmost)f(path)i(ha)o(v)o(e)e(c)o (hildren,)g(hence)46 2083 y(there)15 b(m)o(ust)g(b)q(e)h(one)g(no)q(de)g(on)h (the)e(leftmost)g(path)h(that)g(has)h(at)f(least)1381 2047 y Fi(p)p 1423 2047 30 2 v 36 x Fj(n)g Fo(c)o(hildren.)j(Th)o(us,)d(the)46 2143 y(cost)g(of)h(the)f(last)g(rehashing)h(at)g(this)f(no)q(de)h(w)o(as)f (at)h(least)1138 2107 y Fi(p)p 1180 2107 V 36 x Fj(n)p Fo(.)46 2262 y(\(b\))k(Apply)f(the)g(mo)q(di\014ed)g(adv)o(ersary)h(strategy)g(from)f (the)g(pro)q(of)i(of)f(Theorem)f(4.4,)i(for)f Fj(r)1805 2244 y Ff(0)1839 2262 y Fo(=)46 2322 y(min)n Fi(f)p Fj(r)o(;)8 b Fo(2)p Fj(f)d Fo(\()p Fj(n)p Fo(\))p Fi(g)p Fo(.)21 b(A)o(t)13 b(most)g Fj(n=)p Fo(2)h(k)o(eys)f(can)h(b)q(e)g(inserted)f(b)q(elo)o(w)h(lev) o(el)d(2)p Fj(f)5 b Fo(\()p Fj(n)p Fo(\),)15 b(b)o(y)e(the)g(o)o(v)o(erall)g (time)46 2383 y(b)q(ound;)18 b(hence)f(at)g(least)g Fj(n=)p Fo(2)h(will)e(b)q(e)i(ab)q(o)o(v)o(e)f(that)g(lev)o(el.)22 b(In)17 b(lev)o(els)e(smaller)g(than)j(2)p Fj(f)5 b Fo(\()p Fj(n)p Fo(\),)18 b(only)46 2443 y(no)q(des)i(on)h(the)e(leftmost)f(path)j (can)f(ha)o(v)o(e)f(c)o(hildren;)g(as)h(in)g(\(a\))g(it)f(follo)o(ws)h(that)g (one)g(insertion)46 2503 y(m)o(ust)15 b(ha)o(v)o(e)g(had)i(cost)g(at)f(least) g Fj(n=)p Fo(2)p Fj(f)5 b Fo(\()p Fj(n)p Fo(\))p Fj(:)46 2704 y Fb(5.4)69 b(Pro)r(of)24 b(of)g(Theorem)19 b(4.6)46 2856 y Fh(5.4.1)55 b(The)18 b(lo)n(w)n(er)i(b)r(ound)46 3007 y Fo(Let)d(an)g (arbitrary)g(algorithm)e(for)i(inserting)f Fj(x)936 3014 y Fm(1)956 3007 y Fj(;)8 b(:)g(:)g(:)f(;)h(x)1093 3014 y Fl(n)1133 3007 y Fo(\(in)o(to)16 b(a)h(table)g(that)g(initially)e(con)o(tains)46 3067 y(one)22 b(elemen)o(t)d Fj(x)355 3074 y Fm(0)375 3067 y Fo(\))j(b)q(e)h(giv)o(en.)38 b(W)l(e)22 b(use)g(the)g(simple)e(adv)o (ersary)i(strategy)h(from)e(Section)h(5.1.)p eop %%Page: 22 22 21 bop 46 188 a Fx(5)49 b(PR)o(OOFS)16 b(OF)g(THE)g(LO)o(WER)h(BOUNDS)869 b Fo(22)46 342 y(F)l(rom)18 b(Lemma)e(5.3)k(w)o(e)e(kno)o(w)h(that)g(the)g (assumption)g Fi(j)p Fj(U)5 b Fi(j)18 b(\025)g Fo(\(2)p Fj(n=k)r Fo(\))1378 324 y Fl(k)1419 342 y Fo(is)h(su\016cien)o(t)f(to)h(ensure)46 402 y(that)i Fi(j)p Fj(A)p Fo(\()p Fj(v)r Fo(\))p Fi(j)f(\025)i Fo(2)f(for)g Fj(v)i Fo(the)d(leftmost)f(leaf)i(of)g Fj(D)q Fo(,)h(and)g(hence)e(that)h(the)g(strategy)g(is)g(alw)o(a)o(ys)46 462 y(applicable)15 b(under)h(this)g(assumption.)46 581 y(F)l(or)e Fj(k)i Fi(\025)e Fo(1,)h Fj(n)f Fi(\025)f Fo(1)i(de\014ne)561 569 y(^)551 581 y Fj(T)580 588 y Fl(k)601 581 y Fo(\()p Fj(n)p Fo(\))f(=)g(the)g(minim)o(al)e(n)o(um)o(b)q(er)g(of)j(steps)g(needed)e(b)o(y) h(an)o(y)h(algorithm)46 642 y(to)f(insert)f Fj(x)264 649 y Fm(1)283 642 y Fj(;)8 b(:)g(:)g(:)g(;)g(x)421 649 y Fl(n)457 642 y Fo(c)o(hosen)13 b(according)h(to)g(the)f(adv)o(ersary)g(strategy)l(.)21 b(Clearly)l(,)13 b Fj(T)1592 649 y Fl(k)1613 642 y Fo(\()p Fj(n)p Fo(\))g Fi(\025)1756 629 y Fo(^)1746 642 y Fj(T)1775 649 y Fl(k)1796 642 y Fo(\()p Fj(n)p Fo(\).)46 702 y(T)l(rivially)l(,)261 689 y(^)252 702 y Fj(T)281 709 y Fl(k)302 702 y Fo(\(0\))h(=)g(0)i(for)h(all) f Fj(k)f Fi(\025)f Fo(1.)46 875 y Fh(Lemm)o(a)i(5.5)350 862 y Fo(^)340 875 y Fj(T)369 882 y Fl(k)390 875 y Fo(\()p Fj(n)p Fo(\))i Fw(satis\014es)g(the)g(fol)r(lowing)i(ine)n(qualities.)79 1036 y(\(a\))177 1023 y Fo(^)168 1036 y Fj(T)197 1043 y Fm(1)216 1036 y Fo(\()p Fj(n)p Fo(\))14 b(=)g(\()p Fj(n)d Fo(+)g(1\)\()p Fj(n)g Fo(+)g(2\))p Fj(=)p Fo(2)i Fi(\000)d Fo(1)p Fw(,)18 b(for)f Fj(n)d Fi(\025)g Fo(0)p Fw(.)81 1175 y(\(b\))177 1162 y Fo(^)168 1175 y Fj(T)197 1182 y Fl(k)218 1175 y Fo(\()p Fj(n)p Fo(\))g Fi(\025)f Fo(min)441 1127 y Fg(n)477 1175 y Fj(l)e Fo(+)578 1121 y Fl(l)553 1133 y Fg(X)552 1225 y Fl(j)r Fm(=1)614 1175 y Fo(\()p Fj(j)s(a)682 1182 y Fl(j)711 1175 y Fo(+)769 1162 y(^)760 1175 y Fj(T)789 1182 y Fl(k)q Ff(\000)p Fm(1)855 1175 y Fo(\()p Fj(a)900 1182 y Fl(j)929 1175 y Fi(\000)f Fo(1\)\))k Fi(j)g Fj(l)h Fi(\025)e Fo(1)p Fj(;)8 b(a)1236 1182 y Fm(1)1256 1175 y Fj(;)g(:)g(:)g(:)f(;)h(a)1391 1182 y Fl(l)1417 1175 y Fi(2)15 b Fj(I)-10 b(N)t(;)1567 1121 y Fl(l)1543 1133 y Fg(X)1542 1225 y Fl(j)r Fm(=1)1612 1175 y Fj(a)1638 1182 y Fl(j)1669 1175 y Fo(=)14 b Fj(n)1758 1127 y Fg(o)1786 1175 y Fw(,)168 1274 y(for)j Fj(n)d Fi(\025)f Fo(2)p Fw(,)18 b Fj(k)e Fi(\025)d Fo(2)p Fw(.)46 1447 y(Pr)n(o)n(of:)22 b Fo(\(a\))c(If)f Fj(k)h Fo(=)d(1,)j(then)f(ev)o(ery)f(elemen)o(t)e Fj(x)931 1454 y Fl(i)962 1447 y Fo(is)k(inserted)e(b)o(y)h(rehashing)h(at)g(the)f(ro)q(ot,)h (whic)o(h)46 1507 y(has)f(cost)f Fj(i)11 b Fo(+)g(1.)22 b(Th)o(us)503 1495 y(^)493 1507 y Fj(T)522 1514 y Fm(1)542 1507 y Fo(\()p Fj(n)p Fo(\))13 b(=)674 1474 y Fg(P)718 1487 y Fl(n)718 1520 y(i)p Fm(=1)777 1507 y Fo(\()p Fj(i)e Fo(+)g(1\))j(=)g(\()p Fj(n)d Fo(+)g(1\)\()p Fj(n)h Fo(+)f(2\))p Fj(=)p Fo(2)h Fi(\000)f Fo(1.)46 1652 y(\(b\))j(Let)g Fj(x)238 1659 y Fm(1)257 1652 y Fj(;)8 b(:)g(:)g(:)g(;)g(x)395 1659 y Fl(n)432 1652 y Fo(b)q(e)14 b(inserted,)f(c)o(hosen)h(according)g(to)h(the)e(simple)f(adv)o(ersary)i (strategy)l(.)21 b(Con-)46 1712 y(sider)14 b(an)h(algorithm)f(that)h(for)g (inserting)f(these)h(elemen)o(ts)d(needs)1287 1699 y(^)1277 1712 y Fj(T)1306 1719 y Fl(k)1327 1712 y Fo(\()p Fj(n)p Fo(\))j(steps.)21 b(Let)15 b Fj(x)1664 1719 y Fl(i)1676 1724 y Fe(0)1695 1712 y Fj(;)8 b(x)1745 1719 y Fl(i)1757 1724 y Fe(1)1776 1712 y Fj(;)g(:)g(:)g(:)f(;)46 1772 y(x)74 1779 y Fl(i)86 1785 y Fa(l)p Fc(\000)p Fe(1)159 1772 y Fo(b)q(e)20 b(those)g(elemen)o(ts)d(that)j(are)g (inserted)f(b)o(y)g(global)i(rehashing,)f(i.e.,)f(b)o(y)g(constructing)h(a)46 1832 y(new)h(p)q(erfect)f(hash)i(function)e(at)h(the)g(ro)q(ot.)36 b(\(F)l(or)21 b Fj(x)1061 1839 y Fm(1)1101 1832 y Fo(this)g(is)g(forced,)g (hence)f Fj(i)1576 1839 y Fm(0)1617 1832 y Fo(=)i(1.\))35 b(Also,)46 1892 y(let)15 b Fj(i)133 1899 y Fl(l)160 1892 y Fo(=)f Fj(n)d Fo(+)h(1.)22 b(Note)16 b(that)g(b)q(et)o(w)o(een)g(global)h(rehashings)g(the) f(elemen)o(ts)d Fj(x)1468 1899 y Fl(i)1480 1904 y Fa(j)q Fc(\000)p Fe(1)1536 1899 y Fm(+1)1583 1892 y Fj(;)8 b(:)g(:)g(:)f(;)h(x)1720 1899 y Fl(i)1732 1904 y Fa(j)1748 1899 y Ff(\000)p Fm(1)1812 1892 y Fo(are)46 1952 y(c)o(hosen)13 b(so)h(that)g(they)f(are)g(all)g(sen)o (t)g(to)g(the)h(subtree)f(ro)q(oted)h(at)f(the)h(leftmost)d(c)o(hild)h(of)i (the)f(ro)q(ot)h(of)46 2013 y Fj(D)q Fo(,)k(and)f(that)h(insertions)f(in)o (to)f(this)h(subtree)g(are)g(p)q(erformed)f(according)h(to)h(some)e(strategy) h(for)46 2073 y Fj(i)63 2080 y Fl(j)90 2073 y Fi(\000)10 b Fj(i)156 2080 y Fl(j)r Ff(\000)p Fm(1)228 2073 y Fi(\000)f Fo(1)16 b(elemen)o(ts)d(and)j(depth)g Fj(k)11 b Fi(\000)f Fo(1;)15 b(further,)g(after)h(the)f(insertion)g(of)h Fj(x)1543 2080 y Fl(i)1555 2085 y Fa(j)q Fc(\000)p Fe(1)1627 2073 y Fo(this)g(subtree)46 2142 y(already)h(has)g(one)h(elemen)o(t.)i(By)c(the)h(de\014nition)g(of)1044 2129 y(^)1034 2142 y Fj(T)1063 2149 y Fl(k)1084 2142 y Fo(\()p Fj(n)p Fo(\),)g(inserting)f(these)h(elemen)o(ts)d(in)o(to)j(the)46 2202 y(subtree)g(has)h(cost)g(at)g(least)594 2189 y(^)584 2202 y Fj(T)613 2209 y Fl(k)q Ff(\000)p Fm(1)679 2202 y Fo(\()p Fj(i)715 2209 y Fl(j)745 2202 y Fi(\000)12 b Fj(i)813 2209 y Fl(j)r Ff(\000)p Fm(1)888 2202 y Fi(\000)g Fo(1\).)25 b(In)17 b(addition,)h(for)g(these)f(elemen)o(ts)d(the)k(hash)46 2262 y(function)h(at)g(the)f(ro)q(ot)i(has)g(to)f(b)q(e)g(ev)m(aluated,)g(whic)o (h)f(has)i(cost)f Fj(i)1303 2269 y Fl(j)1334 2262 y Fi(\000)13 b Fj(i)1403 2269 y Fl(j)r Ff(\000)p Fm(1)1478 2262 y Fi(\000)g Fo(1.)29 b(Inserting)19 b Fj(x)1833 2269 y Fl(i)1845 2274 y Fa(j)1863 2262 y Fo(,)46 2322 y Fj(j)e Fo(=)c(0)p Fj(;)8 b Fo(1)p Fj(;)g(:)g(:)g(:)g(;)g(l)k Fi(\000)f Fo(1,)16 b(has)h(cost)g Fj(i)650 2329 y Fl(j)679 2322 y Fo(+)11 b(1.)21 b(Th)o(us)c(the)f(total)g (cost)h(is)295 2459 y(^)286 2471 y Fj(T)315 2478 y Fl(k)336 2471 y Fo(\()p Fj(n)p Fo(\))41 b Fi(\025)550 2417 y Fl(l)525 2430 y Fg(X)525 2521 y Fl(j)r Fm(=1)586 2471 y Fo(\()p Fj(i)622 2478 y Fl(j)651 2471 y Fi(\000)11 b Fj(i)718 2478 y Fl(j)r Ff(\000)p Fm(1)792 2471 y Fi(\000)g Fo(1\))g(+)954 2459 y(^)945 2471 y Fj(T)974 2478 y Fl(k)q Ff(\000)p Fm(1)1040 2471 y Fo(\()p Fj(i)1076 2478 y Fl(j)1105 2471 y Fi(\000)g Fj(i)1172 2478 y Fl(j)r Ff(\000)p Fm(1)1246 2471 y Fi(\000)g Fo(1\)\))g(+)1421 2417 y Fl(l)p Ff(\000)p Fm(1)1419 2430 y Fg(X)1418 2521 y Fl(j)r Fm(=1)1479 2471 y Fo(\()p Fj(i)1515 2478 y Fl(j)1544 2471 y Fo(+)g(1\))445 2629 y(=)550 2575 y Fl(l)525 2588 y Fg(X)525 2679 y Fl(j)r Fm(=1)604 2617 y Fo(^)594 2629 y Fj(T)623 2636 y Fl(k)q Ff(\000)p Fm(1)689 2629 y Fo(\()p Fj(i)725 2636 y Fl(j)754 2629 y Fi(\000)g Fj(i)821 2636 y Fl(j)r Ff(\000)p Fm(1)895 2629 y Fi(\000)g Fo(1\))g(+)1073 2575 y Fl(l)1049 2588 y Fg(X)1048 2679 y Fl(j)r Fm(=1)1118 2629 y Fj(i)1135 2636 y Fl(j)1153 2629 y Fj(:)46 2787 y Fo(Let)21 b Fj(a)164 2794 y Fl(l)p Fm(+1)p Ff(\000)p Fl(j)288 2787 y Fo(=)h Fj(i)365 2794 y Fl(j)397 2787 y Fi(\000)15 b Fj(i)468 2794 y Fl(j)r Ff(\000)p Fm(1)552 2787 y Fo(for)21 b(1)i Fi(\024)f Fj(j)j Fi(\024)d Fj(l)q Fo(.)36 b(Then)1043 2753 y Fg(P)1087 2766 y Fl(l)1087 2799 y(j)r Fm(=1)1159 2787 y Fj(a)1185 2794 y Fl(j)1225 2787 y Fo(=)22 b Fj(i)1302 2794 y Fl(l)1329 2787 y Fi(\000)14 b Fj(i)1399 2794 y Fm(0)1441 2787 y Fo(=)22 b Fj(n)p Fo(,)g(and)1666 2753 y Fg(P)1710 2766 y Fl(l)1710 2799 y(j)r Fm(=1)1781 2787 y Fj(i)1798 2794 y Fl(j)1839 2787 y Fo(=)46 2853 y Fj(l)12 b Fi(\001)e Fj(i)114 2860 y Fm(0)145 2853 y Fo(+)194 2819 y Fg(P)238 2833 y Fl(l)238 2865 y(j)r Fm(=1)301 2853 y Fo(\()p Fj(l)i Fo(+)f(1)g Fi(\000)g Fj(j)s Fo(\))g Fi(\001)g Fj(a)585 2860 y Fl(l)p Fm(+1)p Ff(\000)p Fl(j)686 2853 y Fo(;)16 b(hence)561 2994 y(^)552 3007 y Fj(T)581 3014 y Fl(k)602 3007 y Fo(\()p Fj(n)p Fo(\))25 b Fi(\025)782 2953 y Fl(l)758 2965 y Fg(X)757 3056 y Fl(j)r Fm(=1)819 3007 y Fo(\()p Fj(j)14 b Fi(\001)c Fj(a)922 3014 y Fl(j)951 3007 y Fo(+)1010 2994 y(^)1000 3007 y Fj(T)1029 3014 y Fl(k)q Ff(\000)p Fm(1)1095 3007 y Fo(\()p Fj(a)1140 3014 y Fl(j)1169 3007 y Fi(\000)h Fo(1\)\))h(+)f Fj(l)q(:)p eop %%Page: 23 23 22 bop 46 188 a Fx(5)49 b(PR)o(OOFS)16 b(OF)g(THE)g(LO)o(WER)h(BOUNDS)869 b Fo(23)46 342 y(This)16 b(pro)o(v)o(es)g(part)h(\(b\).)1346 b Fd(2)46 461 y Fo(The)16 b(pro)q(of)h(of)g(Theorem)e(4.6)i(is)f(completed)e (b)o(y)h(the)h(follo)o(wing)g(lemma.)46 625 y Fh(Lemm)o(a)g(5.6)770 673 y Fo(^)760 685 y Fj(T)789 692 y Fl(k)810 685 y Fo(\()p Fj(n)p Fo(\))25 b Fi(\025)g Fj(g)989 692 y Fl(k)1011 685 y Fo(\()p Fj(n)11 b Fo(+)g(1\))46 769 y Fw(for)17 b(al)r(l)h Fj(k)e Fi(\025)e Fo(1)p Fw(,)k Fj(n)c Fi(\025)f Fo(0)p Fw(,)18 b(wher)n(e)558 955 y Fj(g)581 962 y Fl(k)603 955 y Fo(\()p Fj(y)r Fo(\))13 b(=)732 830 y Fg(8)732 868 y(>)732 880 y(>)732 893 y(>)732 905 y(<)732 980 y(>)732 992 y(>)732 1005 y(>)732 1017 y(:)777 859 y Fo(0)p Fj(;)226 b Fw(for)17 b Fj(y)e Fo(=)f(0)p Fw(;)777 919 y Fj(y)c Fo(ln)e Fj(y)r(;)141 b Fw(for)17 b Fo(0)d Fj(<)g(y)h Fi(\024)e Fj(e)1323 901 y Fl(k)1344 919 y Fw(;)782 993 y Fj(k)p 782 1015 28 2 v 784 1060 a(e)825 1026 y Fi(\001)e Fj(y)876 1006 y Fm(1+1)p Fl(=k)977 1026 y Fj(;)50 b Fw(for)17 b Fj(e)1142 1008 y Fl(k)1176 1026 y Fj(<)d(y)r Fw(.)46 1203 y Fo(F)l(or)i(the)g Fw(pr)n(o)n(of)24 b Fo(of)16 b(this)g(lemma)d(see)i(the)h (App)q(endix.)21 b(It)15 b(is)h(a)g(tec)o(hnical)e(argumen)o(t)h(based)h (solely)46 1263 y(on)h(the)f(inequalities)e(of)i(Lemma)f(5.5.)46 1451 y Fh(5.4.2)55 b(The)18 b(upp)r(er)h(b)r(ound)46 1602 y Fo(W)l(e)j(will)e(describ)q(e)i(an)g(algorithm)f(for)h(inserting)g Fj(n)g Fo(elemen)o(t)o(s)e Fj(x)1300 1609 y Fm(1)1319 1602 y Fj(;)8 b(:)g(:)g(:)f(;)h(x)1456 1609 y Fl(n)1503 1602 y Fi(2)24 b Fj(U)j Fo(in)o(to)22 b(a)g(table)46 1663 y(\(whic)o(h)16 b(initially)f(con)o(tains)i(one)g(elemen)o(t)d Fj(x)877 1670 y Fm(0)896 1663 y Fo(\))j(so)h(that)f(the)f(depth)h(of)g(the)g(resulting)g (tree)f(nev)o(er)46 1723 y(b)q(ecomes)d(larger)h(than)g Fj(k)r Fo(.)21 b(As)14 b(in)g(the)g(pro)q(of)h(of)f(Theorem)f(4.4\(b\),)h(the)g (hash)h(function)f Fj(h)1702 1730 y Fl(v)1737 1723 y Fo(c)o(hosen)46 1783 y(for)22 b(a)h(v)o(ertex)e Fj(v)j Fo(alw)o(a)o(ys)e(satis\014es)g Fi(j)p Fj(h)764 1765 y Ff(\000)p Fm(1)764 1795 y Fl(v)812 1783 y Fo([)p Fj(q)r Fo(])13 b Fi(\\)j Fj(A)p Fo(\()p Fj(v)r Fo(\))p Fi(j)23 b Fo(=)h(1)e(for)h(all)f Fj(q)j(>)f Fo(0.)40 b(This)23 b(means)e(that)46 1843 y(subsequen)o(t)14 b(elemen)o(ts)e(that)j(are)g (inserted)f(in)g(the)h(subtree)f(ro)q(oted)i(at)f Fj(v)h Fo(are)f(alw)o(a)o (ys)f(sen)o(t)h(to)g(the)46 1903 y(leftmost)f(subtree)i(of)h Fj(v)r Fo(.)k(Let)391 2036 y Fj(d)416 2043 y Fm(1)450 2036 y Fo(=)13 b(1)p Fj(;)57 b(d)621 2043 y Fl(k)657 2036 y Fo(=)14 b Fj(k)f Fi(\001)777 2002 y Fj(k)g Fo(+)e(1)p 777 2024 112 2 v 777 2070 a Fj(k)i Fo(+)e(2)904 2036 y Fi(\001)929 1963 y Fg( )977 2002 y Fj(d)1002 2009 y Fl(k)q Ff(\000)p Fm(1)p 967 2024 113 2 v 967 2070 a Fj(k)i Fi(\000)e Fo(1)1084 1963 y Fg(!)1117 1974 y Fm(\()p Fl(k)q Ff(\000)p Fm(1\))p Fl(=k)1256 2036 y Fj(;)57 b Fo(for)16 b Fj(k)g(>)e Fo(1.)46 2193 y(Then)i Fj(d)198 2200 y Fl(k)234 2193 y Fo(=)e Fj(k)f Fi(\001)349 2145 y Fg(\020)373 2160 y(Q)413 2173 y Fl(k)413 2205 y(q)q Fm(=2)477 2193 y Fo(\(\()p Fj(q)f Fo(+)f(1\))p Fj(=)p Fo(\()p Fj(q)i Fo(+)e(2\))q(\))831 2169 y Fl(q)850 2145 y Fg(\021)874 2156 y Fm(1)p Fl(=k)931 2193 y Fo(.)21 b(As)16 b(an)h(abbreviation,)f(let)f Fj(b)1494 2200 y Fl(k)1529 2193 y Fo(=)f Fj(d)1606 2200 y Fl(k)1628 2193 y Fj(=k)r Fo(.)46 2364 y Fw(A)o(lgorithm)j(for)g(a)h(table)g(of)g(depth) g(at)f(most)g Fj(k)j Fw(\(Inductive)f(description\))r Fo(:)46 2483 y Fj(k)d Fo(=)e(1:)21 b(Insert)16 b(eac)o(h)g(elemen)o(t)d(b)o(y)j (global)g(rehashing.)46 2602 y Fj(k)24 b(>)d Fo(1:)31 b(Let)21 b Fj(i)332 2609 y Fl(t)368 2602 y Fo(=)g Fi(d)458 2569 y Fg(P)502 2582 y Fl(t)502 2614 y(s)p Fm(=1)565 2602 y Fo(\()p Fj(s=)p Fo(\()p Fj(k)c Fi(\001)d Fj(b)741 2609 y Fl(k)q Ff(\000)p Fm(1)807 2602 y Fo(\)\))845 2578 y Fl(k)q Ff(\000)p Fm(1)920 2602 y Fi(e)p Fo(,)21 b(for)g Fj(t)h Fo(=)f(0)p Fj(;)8 b Fo(1)p Fj(;)g Fo(2)p Fj(;)g Fo(3)p 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Ff(\000)p Fm(1)705 2843 y Fi(\000)g Fo(1)16 b(elemen)o(ts.)46 3007 y(It)k(is)g(ob)o(vious)g (that)h(this)f(algorithm)g(alw)o(a)o(ys)g(main)o(tains)f(a)i(tree)e(of)i (depth)f(at)h(most)e Fj(k)r Fo(,)i(hence)46 3067 y(a)j(lo)q(okup)f(time)f(of) h Fj(k)i Fo(is)e(guaran)o(teed.)43 b(W)l(e)23 b(only)g(ha)o(v)o(e)g(to)g (analyze)g(the)g(time)e(required)h(for)p eop %%Page: 24 24 23 bop 46 188 a Fx(5)49 b(PR)o(OOFS)16 b(OF)g(THE)g(LO)o(WER)h(BOUNDS)869 b Fo(24)46 342 y(insertions.)21 b(F)l(or)16 b Fj(k)g Fi(\025)e Fo(1,)i Fj(n)e Fi(\025)f Fo(0,)k(let)256 428 y(~)246 441 y Fj(T)275 448 y Fl(k)296 441 y Fo(\()p Fj(n)p Fo(\))42 b(=)57 b(cost)17 b(of)f(inserting)g Fj(x)885 448 y Fm(1)905 441 y Fj(;)8 b(:)g(:)g(:)f(;)h(x)1042 448 y Fl(n)1081 441 y Fo(in)o(to)16 b(a)h(table,)e(whic)o(h)h(initially)500 501 y(has)h(one)g(elemen)o(t,)c (using)j(the)g(algorithm)f(just)i(describ)q(ed.)46 600 y(\(Note)24 b(that)h(for)f(the)g(cost)h(of)g(the)f(algorithm)f(it)h(is)g(irrelev)m(an)o 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b(Ob)o(viously)l(,)733 1374 y(~)723 1387 y Fj(T)752 1394 y Fm(1)771 1387 y Fo(\()p Fj(n)p Fo(\))14 b(=)g(\()p Fj(n)d Fo(+)g(1\)\()p Fj(n)h Fo(+)f(2\))p Fj(=)p Fo(2)h Fi(\000)f Fo(1)j Fi(\024)f Fj(n)1436 1369 y Fm(2)1473 1387 y Fo(for)j Fj(n)e Fi(\025)g Fo(3.)46 1520 y(Induction)19 b(step)h(\()p Fj(k)j(>)d Fo(1\):)29 b(Assume)782 1507 y(~)772 1520 y Fj(T)801 1527 y Fl(k)q Ff(\000)p Fm(1)887 1520 y Fi(\024)20 b Fj(d)971 1527 y Fl(k)q Ff(\000)p Fm(1)1051 1520 y Fi(\001)14 b Fj(n)1108 1502 y Fl(k)q(=)p Fm(\()p Fl(k)q Ff(\000)p Fm(1\))1259 1520 y Fo(for)20 b(all)f Fj(n)i Fi(\025)f Fj(n)1546 1527 y Fl(k)q Ff(\000)p Fm(1)1612 1520 y Fo(.)32 b(No)o(w)20 b(let)f Fj(n)46 1580 y Fo(b)q(e)f(\014xed,)f Fj(n)h Fo(large)g(enough.)26 b(De\014ne)18 b Fj(t)777 1587 y Fm(0)812 1580 y Fo(=)f(min)n Fi(f)p Fj(t)f Fi(\025)g Fo(1)h Fi(j)f Fj(i)1150 1587 y Fl(t)1181 1580 y Fj(>)g(n)p Fi(g)p Fo(,)i(for)g(the)g(sequence)e Fj(i)1703 1587 y Fl(t)1718 1580 y Fo(,)h Fj(t)f Fi(\025)h Fo(1,)46 1640 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2441 y Fl(t)p Ff(\000)p Fm(1)1223 2434 y Fi(\000)g Fo(1\)\))g(+)1395 2379 y Fl(t)1408 2384 y Fe(0)1426 2379 y Ff(\000)p Fm(1)1403 2392 y Fg(X)1404 2483 y Fl(t)p Fm(=1)1471 2434 y Fo(\()p Fj(i)1507 2441 y Fl(t)1532 2434 y Fo(+)g(1\))p Fj(;)46 2565 y Fo(or,)16 b(after)g(a)h(trivial)e (transformation,)362 2684 y(~)352 2697 y Fj(T)381 2704 y Fl(k)402 2697 y Fo(\()p Fj(n)p Fo(\))25 b Fi(\024)573 2642 y Fl(t)586 2647 y Fe(0)558 2655 y Fg(X)559 2746 y Fl(t)p Fm(=1)626 2697 y Fo(\(\()p Fj(t)682 2704 y Fm(0)713 2697 y Fo(+)11 b(1)g Fi(\000)g Fj(t)p Fo(\)\()p Fj(i)920 2704 y Fl(t)945 2697 y Fi(\000)g Fj(i)1012 2704 y Fl(t)p Ff(\000)p Fm(1)1071 2697 y Fo(\))g(+)1160 2684 y(~)1150 2697 y Fj(T)1179 2704 y Fl(k)q Ff(\000)p Fm(1)1245 2697 y Fo(\()p Fj(i)1281 2704 y Fl(t)1307 2697 y Fi(\000)g Fj(i)1374 2704 y Fl(t)p Ff(\000)p Fm(1)1444 2697 y Fi(\000)g Fo(1\)\))p Fj(:)46 2839 y Fo(Substituting)17 b(the)g(induction)g(h)o(yp)q (othesis)879 2826 y(~)870 2839 y Fj(T)899 2846 y Fl(k)q Ff(\000)p Fm(1)965 2839 y Fo(\()p Fj(n)1013 2821 y Ff(0)1025 2839 y Fo(\))e Fi(\024)g Fj(d)1138 2846 y Fl(k)q Ff(\000)p Fm(1)1216 2839 y Fi(\001)d Fo(\()p Fj(n)1290 2821 y Ff(0)1301 2839 y Fo(\))1320 2821 y Fl(k)q(=)p Fm(\()p Fl(k)q Ff(\000)p Fm(1\))1451 2839 y Fo(,)17 b(for)g Fj(n)1586 2821 y Ff(0)1613 2839 y Fi(\025)f Fj(n)1697 2846 y Fl(k)q Ff(\000)p Fm(1)1763 2839 y Fo(,)h(in)o(to)46 2899 y(this)f(inequalit)o(y)e(yields)66 3018 y(~)56 3031 y Fj(T)85 3038 y Fl(k)106 3031 y Fo(\()p Fj(n)p Fo(\))g Fi(\024)254 2976 y Fl(t)267 2981 y Fe(0)239 2989 y Fg(X)240 3080 y Fl(t)p Fm(=1)308 3031 y Fo(\(\()p Fj(t)364 3038 y Fm(0)394 3031 y Fo(+)d(1)g Fi(\000)g Fj(t)p Fo(\)\()p Fj(i)601 3038 y Fl(t)626 3031 y Fi(\000)g Fj(i)693 3038 y Fl(t)p Ff(\000)p Fm(1)764 3031 y Fi(\000)g Fo(1\))g(+)g Fj(d)942 3038 y Fl(k)q Ff(\000)p Fm(1)1009 3031 y Fo(\()p Fj(i)1045 3038 y Fl(t)1070 3031 y Fi(\000)g Fj(i)1137 3038 y Fl(t)p Ff(\000)p Fm(1)1208 3031 y Fi(\000)g Fo(1\))1301 3010 y Fl(k)q(=)p Fm(\()p Fl(k)q Ff(\000)p Fm(1\))1432 3031 y Fo(\))g(+)1516 3011 y Fm(1)p 1516 3019 18 2 v 1516 3048 a(2)1538 3031 y Fj(t)1556 3038 y Fm(0)1576 3031 y Fo(\()p Fj(t)1613 3038 y Fm(0)1643 3031 y Fo(+)g(1\))g(+)g Fj(E)1831 3038 y Fl(k)1853 3031 y Fj(;)p eop %%Page: 25 25 24 bop 46 188 a Fx(5)49 b(PR)o(OOFS)16 b(OF)g(THE)g(LO)o(WER)h(BOUNDS)869 b Fo(25)46 342 y(for)18 b(some)e(constan)o(t)i Fj(E)479 349 y Fl(k)518 342 y Fo(\(needed)f(to)g(mak)o(e)f(up)i(for)f(the)h(error)f (caused)h(b)o(y)f(replacing)1690 329 y(~)1680 342 y Fj(T)1709 349 y Fl(k)q Ff(\000)p Fm(1)1776 342 y Fo(\()p Fj(i)1812 349 y Fl(t)1838 342 y Fi(\000)46 402 y Fj(i)63 409 y Fl(t)p Ff(\000)p Fm(1)134 402 y Fi(\000)11 b Fo(1\))18 b(b)o(y)f Fj(d)339 409 y Fl(k)q Ff(\000)p Fm(1)405 402 y Fo(\()p Fj(i)441 409 y Fl(t)467 402 y Fi(\000)12 b Fj(i)535 409 y Fl(t)p Ff(\000)p Fm(1)606 402 y Fi(\000)g Fo(1\))700 384 y Fl(k)q(=)p Fm(\()p Fl(k)q Ff(\000)p Fm(1\))848 402 y Fo(for)17 b Fj(t)g Fo(so)h(small)d(that)j Fj(i)1271 409 y Fl(t)1297 402 y Fi(\000)11 b Fj(i)1364 409 y Fl(t)p Ff(\000)p Fm(1)1436 402 y Fi(\000)g Fo(1)16 b Fj(<)f(n)1608 409 y Fl(k)q Ff(\000)p Fm(1)1675 402 y Fo(\).)23 b(By)17 b(the)46 462 y(de\014nition)i(of)h Fj(i)342 469 y Fl(t)376 462 y Fo(w)o(e)g(clearly)e (ha)o(v)o(e)h Fj(i)744 469 y Fl(t)772 462 y Fi(\000)13 b Fj(i)841 469 y Fl(t)p Ff(\000)p Fm(1)914 462 y Fi(\000)g Fo(1)21 b Fi(\024)e Fo(\()p Fj(t=)p Fo(\()p Fj(k)r(b)1197 469 y Fl(k)q Ff(\000)p Fm(1)1263 462 y Fo(\)\))1301 438 y Fl(k)q Ff(\000)p Fm(1)1368 462 y Fo(;)i(furthermore,)d(from)h(the)46 522 y(b)q(ounds)e(on)g Fj(t)301 529 y Fm(0)337 522 y Fo(in)f(\()p Fi(z)p Fo(\))g(it)g(follo)o(ws)g (that)h Fj(t)809 504 y Fm(2)803 535 y(0)842 522 y Fo(=)c Fj(O)q Fo(\()q(\()p Fj(n)999 504 y Fm(1)p Fl(=k)1056 522 y Fo(\))1075 504 y Fm(2)1094 522 y Fo(\))h(=)g Fj(O)q Fo(\()p Fj(n)p Fo(\).)22 b(Th)o(us,)251 651 y(~)241 663 y Fj(T)270 670 y Fl(k)291 663 y Fo(\()p Fj(n)p Fo(\))42 b Fi(\024)495 608 y Fl(t)508 613 y Fe(0)480 622 y Fg(X)481 712 y Fl(t)p Fm(=1)540 603 y Fg(\024\022)646 630 y Fj(t)p 598 652 115 2 v 598 697 a(k)r(b)646 704 y Fl(k)q Ff(\000)p Fm(1)717 603 y Fg(\023)747 614 y Fl(k)q Ff(\000)p Fm(1)825 663 y Fi(\001)11 b Fo(\()p Fj(t)887 670 y Fm(0)917 663 y Fo(+)g(1)g Fi(\000)g Fj(t)p Fo(\))g(+)g Fj(d)1173 670 y Fl(k)q Ff(\000)p Fm(1)1251 663 y Fi(\001)1276 603 y Fg(\022)1360 630 y Fj(t)p 1311 652 V 1311 697 a(k)r(b)1359 704 y Fl(k)q Ff(\000)p Fm(1)1430 603 y Fg(\023)1461 614 y Fl(k)1493 603 y Fg(\025)1526 663 y Fo(+)g Fj(O)q Fo(\()p Fj(n)p Fo(\))400 813 y(=)42 b(\()p Fj(k)r(b)547 820 y Fl(k)q Ff(\000)p Fm(1)613 813 y Fo(\))632 792 y Fm(1)p Ff(\000)p Fl(k)710 813 y Fi(\001)735 752 y Fg(\022)780 758 y Fl(t)793 763 y Fe(0)765 771 y Fg(X)766 862 y Fl(t)p Fm(=1)825 813 y Fo(\()p Fj(t)862 820 y Fm(0)893 813 y Fo(+)11 b(1\))p Fj(t)1003 792 y Fl(k)q Ff(\000)p Fm(1)1080 813 y Fi(\000)1145 758 y Fl(t)1158 763 y Fe(0)1130 771 y Fg(X)1131 862 y Fl(t)p Fm(=1)1198 813 y Fj(t)1216 792 y Fl(k)1237 813 y Fj(=k)1288 752 y Fg(\023)1330 813 y Fo(+)g Fj(O)q Fo(\()p Fj(n)p Fo(\))p Fj(:)46 958 y Fo(W)l(e)j(substitute)h(the)g(t)o(w)o(o)f (inequalities)783 925 y Fg(P)827 937 y Fl(t)840 942 y Fe(0)827 969 y Fl(t)p Fm(=1)895 958 y Fj(t)913 940 y Fl(k)q Ff(\000)p Fm(1)993 958 y Fi(\024)g Fo(\()p Fj(t)1083 965 y Fm(0)1110 958 y Fo(+)8 b(1\))1199 940 y Fl(k)1221 958 y Fj(=k)17 b Fo(and)1381 925 y Fg(P)1424 937 y Fl(t)1437 942 y Fe(0)1424 969 y Fl(t)p Fm(=1)1493 958 y Fj(t)1511 940 y Fl(k)1560 958 y Fi(\025)d Fj(t)1637 938 y Fl(k)q Fm(+1)1631 969 y(0)1703 958 y Fj(=)p Fo(\()p Fj(k)f Fo(+)e(1\))46 1019 y(\(obtained)17 b(b)o(y)f(replacing)g(the)h (sums)f(b)o(y)g(in)o(tegrals\),)g(and)i(simplify)l(,)13 b(noting)18 b(that)f(\()p Fj(t)1635 1026 y Fm(0)1665 1019 y Fo(+)11 b(1\))1757 1001 y Fl(k)q Fm(+1)1839 1019 y Fo(=)46 1079 y Fj(t)70 1058 y Fl(k)q Fm(+1)64 1090 y(0)147 1079 y Fo(+)g Fj(O)q Fo(\()p Fj(t)277 1061 y Fl(k)271 1091 y Fm(0)298 1079 y Fo(\).)21 b(In)16 b(this)h(w)o(a)o(y)e(w)o(e)h(get)437 1193 y(~)427 1206 y Fj(T)456 1213 y Fl(k)477 1206 y Fo(\()p Fj(n)p Fo(\))25 b Fi(\024)681 1172 y Fo(1)p 638 1194 112 2 v 638 1240 a Fj(k)13 b Fo(+)e(1)765 1206 y Fi(\001)g Fo(\()p Fj(k)r(b)857 1213 y Fl(k)q Ff(\000)p Fm(1)923 1206 y Fo(\))942 1185 y Fm(1)p Ff(\000)p Fl(k)1020 1206 y Fi(\001)g Fj(t)1069 1185 y Fl(k)q Fm(+1)1063 1218 y(0)1146 1206 y Fo(+)g Fj(O)q Fo(\()p Fj(t)1276 1185 y Fl(k)1270 1218 y Fm(0)1297 1206 y Fo(\))g(+)g Fj(O)q Fo(\()p Fj(n)p Fo(\))p Fj(:)46 1343 y Fo(By)18 b(\()p Fi(z)p Fo(\),)h(w)o(e)g(ha)o(v)o(e)f Fj(t)432 1325 y Fl(k)426 1356 y Fm(0)472 1343 y Fo(=)g Fj(O)q Fo(\()q(\()p Fj(n)634 1325 y Fm(1)p Fl(=k)690 1343 y Fo(\))709 1325 y Fl(k)731 1343 y Fo(\))g(=)h Fj(O)q Fo(\()p Fj(n)p Fo(\))g(and)h (furthermore)d(that)j Fj(t)1452 1323 y Fl(k)q Fm(+1)1446 1354 y(0)1536 1343 y Fo(=)f Fj(k)1620 1325 y Fl(k)q Fm(+1)1699 1343 y Fi(\001)13 b Fj(b)1747 1318 y Fl(k)q Ff(\000)p Fm(1)p Fl(=k)1747 1356 y(k)q Ff(\000)p Fm(1)1863 1343 y Fi(\001)46 1404 y Fj(n)75 1386 y Fm(1+1)p Fl(=k)188 1404 y Fo(+)e Fj(O)q Fo(\()p Fj(t)318 1386 y Fl(k)312 1416 y Fm(0)339 1404 y Fo(\).)21 b(Hence)352 1518 y(~)343 1531 y Fj(T)372 1538 y Fl(k)393 1531 y Fo(\()p Fj(n)p Fo(\))41 b Fi(\024)630 1497 y Fo(1)p 587 1519 V 587 1565 a Fj(k)13 b Fo(+)e(1)714 1531 y Fi(\001)g Fo(\()p Fj(k)r(b)806 1538 y Fl(k)q Ff(\000)p Fm(1)872 1531 y Fo(\))891 1510 y Fm(1)p Ff(\000)p Fl(k)969 1531 y Fi(\001)g Fj(k)1021 1510 y Fl(k)q Fm(+1)1098 1531 y Fi(\001)g Fj(b)1144 1505 y Fl(k)q Ff(\000)p Fm(1)p Fl(=k)1144 1543 y(k)q Ff(\000)p Fm(1)1258 1531 y Fi(\001)g Fj(n)1312 1510 y Fm(1+1)p Fl(=k)1425 1531 y Fo(+)g Fj(O)q Fo(\()p Fj(n)p Fo(\))502 1659 y(=)619 1625 y Fj(k)646 1607 y Fm(2)p 587 1647 V 587 1693 a Fj(k)i Fo(+)e(1)714 1659 y Fi(\001)g Fj(b)760 1633 y Fm(\()p Fl(k)q Ff(\000)p Fm(1\))p Fl(=k)760 1671 y(k)q Ff(\000)p Fm(1)902 1659 y Fi(\001)f Fj(n)955 1638 y Fm(1+1)p Fl(=k)1068 1659 y Fo(+)h Fj(O)q Fo(\()p Fj(n)p Fo(\))p Fj(:)46 1780 y Fo(F)l(or)16 b Fj(n)h Fo(large)f(enough,)g(this)h(implies)441 1895 y(~)432 1907 y Fj(T)461 1914 y Fl(k)482 1907 y Fo(\()p Fj(n)p Fo(\))25 b Fi(\024)f Fj(k)13 b Fi(\001)705 1873 y Fj(k)g Fo(+)e(1)p 705 1896 V 705 1941 a Fj(k)i Fo(+)e(2)833 1907 y Fi(\001)g Fj(b)879 1882 y Fm(\()p Fl(k)q Ff(\000)p Fm(1\))p Fl(=k)879 1920 y(k)q Ff(\000)p Fm(1)1020 1907 y Fi(\001)g Fj(n)1074 1887 y Fm(1+1)p Fl(=k)1201 1907 y Fo(=)25 b Fj(d)1289 1914 y Fl(k)1321 1907 y Fi(\001)11 b Fj(n)1375 1887 y Fm(1+1)p Fl(=k)1477 1907 y Fj(;)46 2029 y Fo(and)17 b(this)f(is)g(what)h(w)o(e)f(w)o(an)o(ted)g (to)g(sho)o(w.)46 2173 y(\(b\))g(By)g(de\014nition,)439 2139 y Fj(d)464 2146 y Fl(k)p 439 2161 47 2 v 449 2207 a Fj(k)504 2173 y Fo(=)556 2113 y Fg(\022)608 2119 y Fl(k)591 2132 y Fg(Y)587 2222 y Fl(q)q Fm(=2)649 2113 y Fg(\022)684 2139 y Fj(q)d Fo(+)e(1)p 684 2161 108 2 v 684 2207 a Fj(q)i Fo(+)e(2)797 2113 y Fg(\023)828 2124 y Fl(q)847 2113 y Fg(\023)877 2124 y Fm(1)p Fl(=k)934 2173 y Fo(.)21 b(Recall)15 b(that)608 2292 y Fg( )646 2331 y Fj(q)d Fo(+)f(1)p 646 2353 V 646 2399 a Fj(q)h Fo(+)f(2)759 2292 y Fg(!)791 2303 y Fl(q)q Fm(+2)880 2365 y Fi(\024)949 2331 y Fo(1)p 949 2353 25 2 v 950 2399 a Fj(e)1003 2365 y Fi(\024)1067 2292 y Fg( )1105 2331 y Fj(q)h Fo(+)f(1)p 1105 2353 108 2 v 1105 2399 a Fj(q)h Fo(+)f(2)1218 2292 y Fg(!)1251 2303 y Fl(q)q Fm(+1)46 2498 y Fo(for)16 b(all)g Fj(q)r Fo(,)f(and)i(hence)292 2604 y Fj(d)317 2611 y Fl(k)p 292 2626 47 2 v 302 2672 a Fj(k)355 2638 y Fi(\001)380 2577 y Fg(\022)432 2584 y Fl(k)415 2597 y Fg(Y)411 2687 y Fl(q)q Fm(=2)473 2577 y Fg(\022)508 2604 y Fj(q)c Fo(+)e(1)p 508 2626 108 2 v 508 2672 a Fj(q)i Fo(+)e(2)621 2577 y Fg(\023)652 2589 y Fm(2)680 2577 y Fg(\023)710 2589 y Fm(1)p Fl(=k)792 2638 y Fi(\024)855 2577 y Fg(\022)891 2604 y Fo(1)p 891 2626 25 2 v 892 2672 a Fj(e)920 2577 y Fg(\023)951 2589 y Fm(\()p Fl(k)q Ff(\000)p Fm(1\))p Fl(=k)1106 2638 y Fi(\024)1175 2604 y Fj(d)1200 2611 y Fl(k)p 1175 2626 47 2 v 1185 2672 a Fj(k)1238 2638 y Fi(\001)1263 2577 y Fg(\022)1315 2584 y Fl(k)1298 2597 y Fg(Y)1293 2687 y Fl(q)q Fm(=2)1355 2577 y Fg(\022)1391 2604 y Fj(q)h Fo(+)f(1)p 1391 2626 108 2 v 1391 2672 a Fj(q)h Fo(+)f(2)1504 2577 y Fg(\023)o(\023)1565 2589 y Fm(1)p Fl(=k)1621 2638 y Fj(:)46 2781 y Fo(Clearly)l(,)525 2876 y(lim)514 2906 y Fl(k)q Ff(!1)604 2816 y Fg(\022)656 2822 y Fl(k)639 2835 y Fg(Y)635 2925 y Fl(q)q Fm(=2)710 2843 y Fj(q)h Fo(+)f(1)p 710 2865 V 710 2911 a Fj(q)h Fo(+)f(2)823 2816 y Fg(\023)853 2827 y Fm(1)p Fl(=k)924 2876 y Fo(=)25 b(lim)976 2906 y Fl(k)q Ff(!1)1065 2816 y Fg(\022)1145 2843 y Fo(3)p 1101 2865 112 2 v 1101 2911 a Fj(k)13 b Fo(+)e(1)1217 2816 y Fg(\023)1248 2827 y Fm(1)p Fl(=k)1318 2876 y Fo(=)j(1)p Fj(;)46 3033 y Fo(and)j(th)o(us)27 b(lim)248 3063 y Fl(k)q Ff(!1)351 3000 y Fj(d)376 3007 y Fl(k)p 351 3022 47 2 v 361 3067 a Fj(k)417 3033 y Fo(=)473 3000 y(1)p 473 3022 25 2 v 474 3067 a Fj(e)503 3033 y Fo(,)16 b(as)g(claimed.)1068 b Fd(2)p eop %%Page: 26 26 25 bop 46 188 a Fx(REFERENCES)1447 b Fo(26)46 342 y Fk(References)46 472 y Fo([AL86])101 b(Aho,)15 b(H.)f(V.,)h(and)h(Lee,)f(D.,)g(Storing)h(a)g (dynamic)d(sparse)j(table,)f Fx(Pro)q(c.)h(of)g(the)f(27th)290 532 y(IEEE)h(F)o(OCS)p Fo(,)g(1986,)h(pp.)f(55{60.)46 634 y([BK88])95 b(Brassard,)17 b(G.,)f(and)h(Kannan,)g(S.,)e(The)i(generation)g(of)f(random)h (p)q(erm)o(utations)e(on)290 694 y(the)h(\015y)l(,)f Fx(Information)h(Pro)q (cessing)h(Letters)h Fh(28)e Fo(\(1988\))i(207{212.)46 796 y([CW79])83 b(Carter,)24 b(J.)e(L.,)i(and)g(W)l(egman,)f(M.)f(N.,)h(Univ)o (ersal)e(classes)i(of)g(hash)h(functions,)290 856 y Fx(J.)16 b(Comput.)f(Syst.)h(Sci.)f Fh(18)h Fo(\(1979\))i(143{154.)46 958 y([DKM88])48 b(Dietzfelbinger,)23 b(M.,)h(Karlin,)h(A.,)f(Mehlhorn,)h (K.,)g(Mey)o(er)d(auf)i(der)g(Heide,)g(F.,)290 1018 y(Rohnert,)13 b(H.,)g(and)h(T)l(arjan,)h(R.)e(E.,)g(Dynamic)f(p)q(erfect)g(hashing:)21 b(Upp)q(er)13 b(and)h(lo)o(w)o(er)290 1078 y(b)q(ounds,)25 b Fx(Pro)q(c.)e(of)h(the)e(29th)i(IEEE)f(F)o(OCS)p Fo(,)g(1988,)i(pp.)e (524{531;)29 b(also:)35 b(T)l(ec)o(h.)290 1138 y(Rep)q(ort)16 b(No.)g(282,)h(F)l(ac)o(h)o(b)q(ereic)o(h)d(Informatik,)g(Univ)o(ersit\177) -24 b(at)15 b(Dortm)o(und,)g(1988.)46 1240 y([DM89])86 b(Dietzfelbinger,)11 b(M.,)i(and)h(Mey)o(er)e(auf)h(der)h(Heide,)e(F.,)h(An)g(optimal)f(parallel)g (dictio-)290 1300 y(nary)l(,)i Fx(Pro)q(c.)g(of)g(A)o(CM)f(Symp.)f(on)j(P)o (arallel)e(Algorithms)f(and)j(Arc)o(hitectures)p Fo(,)c(1989,)290 1360 y(pp.)16 b(360{368.)46 1462 y([DM90a])62 b(Dietzfelbinger,)12 b(M.,)i(Mey)o(er)f(auf)i(der)g(Heide,)e(F.,)h(Ho)o(w)g(to)h(distribute)f(a)h (dictionary)290 1522 y(in)h(a)g(complete)e(net)o(w)o(ork,)h Fx(Pro)q(c.)i(of)f(the)g(22nd)i(A)o(CM)d(STOC)p Fo(,)h(1990,)i(pp.)e (117{127.)46 1624 y([DM90b])59 b(Dietzfelbinger,)16 b(M.,)h(and)i(Mey)o(er)e (auf)h(der)g(Heide,)f(F.,)h(A)g(new)g(univ)o(ersal)f(class)h(of)290 1684 y(hash)i(functions,)g(and)h(dynamic)d(hashing)i(in)f(real)h(time,)e Fx(Pro)q(c.)h(of)h(17th)h(ICALP)p Fo(,)290 1744 y(Springer)16 b(LNCS)g(443,)h(1990,)h(pp.)e(6{19.)46 1846 y([FKS84])71 b(F)l(redman,)12 b(M.)h(L.,)h(Koml\023)-24 b(os,)13 b(J.,)g(and)h(Szemer)o(\023)-23 b(edi,)11 b(E.,)j(Storing)g(a)g(sparse)g(table)g(with)290 1906 y Fj(O)q Fo(\(1\))j(w)o(orst)g(case)f(access)g(time,)e Fx(J.)i(A)o(CM)k Fh(31)p Fo(\(3\),)c(1984,)h(538{544.)46 2008 y([G81])130 b(Gonnet,)15 b(Gaston)i(H.,)d(Exp)q(ected)g(length)h(of)h(the)f(longest)g(prob)q(e)h (sequence)e(in)h(hash)290 2068 y(co)q(de)h(searc)o(hing,)g Fx(J.)g(A)o(CM)k Fh(28)p Fo(\(3\))d(\(1981\))g(289{304.)46 2170 y([M84])123 b(Mehlhorn,)23 b(K.,)h Fw(Data)f(Structur)n(es)h(and)g(A)o (lgorithms)p Fo(,)g(V)l(ol.)d(1,)k(Springer)e(V)l(erlag,)290 2230 y(Berlin,)14 b(1984.)46 2332 y([MNR90])50 b(Mehlhorn,)22 b(K.,)g(N\177)-24 b(aher,)22 b(S.,)h(and)f(Rauc)o(h,)h(M.,)f(On)f(the)h (complexit)o(y)c(of)k(a)g(game)290 2392 y(related)13 b(to)h(the)f(dictionary) g(problem,)f Fx(SIAM)h(J.)g(Comput.)g Fh(19)p Fo(\(5\))h(\(1990\))h(902{906.) 46 2494 y([MV84])86 b(Mehlhorn,)15 b(K.,)f(and)j(Vishkin,)d(U.,)g(Randomized) h(and)h(deterministic)c(sim)o(ulations)290 2554 y(of)h(PRAMs)f(b)o(y)h (parallel)f(mac)o(hines)f(with)i(restricted)f(gran)o(ularit)o(y)g(of)h (parallel)f(mem-)290 2614 y(ory)l(,)k Fx(Acta)f(Informatica)g Fh(21)i Fo(\(1984\))g(339{374.)46 2716 y([S89])141 b(Siegel,)24 b(A.,)h(On)f(univ)o(ersal)f(classes)h(of)g(fast)h(hash)g(functions,)g(their)e (time-space)290 2776 y(tradeo\013,)12 b(and)g(their)e(applications,)h Fx(Pro)q(c.)g(of)g(the)g(30th)h(IEEE)f(F)o(OCS)p Fo(,)f(1989,)j(pp.)d(20{)290 2836 y(25.)46 2938 y([T83])133 b(T)l(routman,)18 b(J.)g(L.,)h Fw(V)l(ariational)h(c)n(alculus)h(with)f(elementary)h(c)n(onvexity)p Fo(,)f(Springer)290 2998 y(V)l(erlag,)15 b(New)h(Y)l(ork,)f(1983.)p eop %%Page: 27 27 26 bop 46 188 a Fx(REFERENCES)1447 b Fo(27)46 342 y([W)o(C79])84 b(W)l(egman,)19 b(M.)g(N.,)g(and)h(Carter,)g(J.)f(L.,)h(New)f(classes)g(and)h (applications)g(of)g(hash)290 402 y(functions,)c Fx(Pro)q(c.)g(of)g(the)g (20th)i(IEEE)e(F)o(OCS)s Fo(,)f(1979,)j(175{182.)46 504 y([W90])118 b(W)l(enzel,)13 b(M.,)h(Eine)g(Impleme)o(n)o(ti)o(erung)e(v)o(on)j(Dynamic)e (P)o(erfect)h(Hashing,)h(Diplom-)290 564 y(arb)q(eit,)h(Univ)o(ersit\177)-24 b(at)14 b(des)i(Saarlandes,)h(1990.)p eop %%Page: 28 28 27 bop 46 188 a Fx(A)48 b(APPENDIX)1432 b Fo(28)46 342 y Fk(A)83 b(App)r(endix)46 525 y Fb(A.1)69 b(Pro)r(of)24 b(of)f(Lemm)n(a)d(5.6)46 722 y Fo(W)l(e)g(sho)o(w)i(the)e(follo)o(wing:)30 b(If)20 b(the)h(functions)f Fj(T)967 729 y Fl(k)988 722 y Fj(;)8 b(k)24 b Fi(\025)d Fo(1,)h(satisfy)f (the)f(inequalities)f(stated)i(in)46 782 y(Lemma)14 b(5.5,)i(that)h(is,)e Fj(T)509 789 y Fl(k)530 782 y Fo(\(0\))f(=)g(0)j(for)f(all)g Fj(k)g Fi(\025)e Fo(1,)i(and)81 936 y(\(a\))25 b Fj(T)197 943 y Fm(1)216 936 y Fo(\()p Fj(n)p Fo(\))14 b Fi(\025)g Fo(\()p Fj(n)d Fo(+)g(1\)\()p Fj(n)g Fo(+)g(2\))p Fj(=)p Fo(2)h Fi(\000)f Fo(1)p Fj(;)30 b Fo(for)17 b(all)f Fj(n)d Fi(\025)h Fo(1)p Fj(;)78 1073 y Fo(\(b\))25 b Fj(T)197 1080 y Fl(k)218 1073 y Fo(\()p Fj(n)p Fo(\))14 b Fi(\025)i Fo(min)437 1025 y Fg(n)471 1073 y Fo(1)11 b(+)581 1019 y Fl(l)556 1032 y Fg(X)555 1123 y Fl(j)r Fm(=1)617 1073 y Fo(\()p Fj(j)s(a)685 1080 y Fl(j)713 1073 y Fo(+)g Fj(T)791 1080 y Fl(k)q Ff(\000)p Fm(1)857 1073 y Fo(\()p Fj(a)902 1080 y Fl(j)931 1073 y Fi(\000)g Fo(1\)\))1057 1023 y Fg(\014)1057 1048 y(\014)1057 1073 y(\014)j Fj(l)h Fi(\025)e Fo(1)p Fj(;)8 b(a)1239 1080 y Fm(1)1259 1073 y Fj(;)g(:)g(:)g(:)f(;)h(a)1394 1080 y Fl(l)1420 1073 y Fi(2)14 b Fj(I)-10 b(N)5 b(;)1570 1019 y Fl(l)1546 1032 y Fg(X)1545 1123 y Fl(j)r Fm(=1)1606 1073 y Fj(a)1632 1080 y Fl(j)1664 1073 y Fo(=)14 b Fj(n)1753 1025 y Fg(o)1781 1073 y Fj(;)168 1172 y Fo(for)i(all)g Fj(n)e Fi(\025)g Fo(1)p Fj(;)8 b(k)16 b Fi(\025)d Fo(2)p Fj(;)46 1326 y Fo(then)j(the)g (functions)g Fj(T)480 1333 y Fl(k)517 1326 y Fo(satisfy)h(the)f(assertion)g (of)h(Lemma)d(5.6;)i(that)h(is,)771 1428 y Fj(T)800 1435 y Fl(k)821 1428 y Fo(\()p Fj(n)p Fo(\))d Fi(\025)g Fj(g)978 1435 y Fl(k)999 1428 y Fo(\()p Fj(n)e Fo(+)f(1\))46 1531 y(for)16 b(all)g Fj(k)g Fi(\025)e Fo(1)p Fj(;)8 b(n)14 b Fi(\025)f Fo(0,)k(where,)e (for)i Fj(k)e Fi(\025)f Fo(1,)530 1686 y Fj(g)553 1693 y Fl(k)575 1686 y Fo(\()p Fj(y)r Fo(\))f(=)704 1587 y Fg(8)704 1624 y(>)704 1637 y(<)704 1711 y(>)704 1724 y(:)749 1625 y Fo(0)p Fj(;)300 b Fo(if)15 b Fj(y)h Fo(=)e(0;)749 1685 y Fj(y)c Fo(ln)e Fj(y)r(;)215 b Fo(if)15 b(0)g Fj(<)e(y)j Fi(\024)d Fj(e)1336 1667 y Fl(k)1357 1685 y Fo(;)749 1745 y(\()p Fj(k)r(=e)p Fo(\))e Fi(\001)g Fj(y)923 1727 y Fm(1+1)p Fl(=k)1024 1745 y Fj(;)49 b Fo(if)15 b Fj(e)1154 1727 y Fl(k)1189 1745 y Fj(<)f(y)r(:)46 1845 y Fo(W)l(e)23 b(pro)q(ceed)g(b)o(y)g(induction)g(on)h Fj(k)r Fo(.)42 b(F)l(or)24 b Fj(k)k Fo(=)d(1,)h(it)c(is)i(easily)e(c)o(hec)o(k)o(ed)f(that)j Fj(g)1631 1852 y Fm(1)1651 1845 y Fo(\()p Fj(n)16 b Fo(+)g(1\))26 b Fi(\024)46 1905 y Fo(\()p Fj(n)12 b Fo(+)g(1\)\()p Fj(n)g Fo(+)g(2\))p Fj(=)p Fo(2)h Fi(\000)f Fo(1)18 b(for)g(all)f Fj(n)f Fi(\025)g Fo(0.)26 b(Th)o(us,)17 b(let)g Fj(k)h(>)e Fo(1,)i(and)h(assume)d(the)i(claim)d(to)j(b)q(e)g(true)46 1965 y(for)f Fj(k)d Fi(\000)d Fo(1;)18 b(that)f(is,)g Fj(T)465 1972 y Fl(k)q Ff(\000)p Fm(1)531 1965 y Fo(\()p Fj(n)p Fo(\))e Fi(\025)g Fj(g)690 1972 y Fl(k)q Ff(\000)p Fm(1)757 1965 y Fo(\()p Fj(n)d Fo(+)f(1\))16 b(=)f Fj(g)r Fo(\()p Fj(n)d Fo(+)g(1\),)17 b(for)g(all)g Fj(n)e Fi(\025)g Fo(0)p Fj(:)i Fo(\(F)l(rom)f(here)h(on,)g(w)o(e)46 2025 y(will)h(write)g Fj(g)j Fo(for)e Fj(g)412 2032 y Fl(k)q Ff(\000)p Fm(1)479 2025 y Fj(:)p Fo(\))f(F)l(or)h Fj(n)g Fo(=)f(0,)i(the)f (claim)d(is)j(trivially)e(satis\014ed.)29 b(Let)19 b Fj(n)g Fi(\025)f Fo(1)h(b)q(e)g(\014xed.)46 2085 y(By)d(assumption)h(\(b\))g(ab)q(o) o(v)o(e)h(and)f(the)g(induction)g(h)o(yp)q(othesis,)g(w)o(e)g(ma)o(y)e(\014x) i(some)g Fj(l)f Fi(\025)f Fo(1)i(and)h(a)46 2146 y(sequence)d Fj(a)e Fo(=)h(\()p Fj(a)384 2153 y Fm(1)404 2146 y Fj(;)8 b(:)g(:)g(:)f(;)h (a)539 2153 y Fl(l)551 2146 y Fo(\))17 b(of)f(natural)h(n)o(um)o(b)q(ers)e (with)1121 2112 y Fg(P)1164 2126 y Fl(l)1164 2158 y(j)r Fm(=1)1236 2146 y Fj(a)1262 2153 y Fl(j)1294 2146 y Fo(=)f Fj(n)i Fo(and)376 2292 y Fj(T)405 2299 y Fl(k)426 2292 y Fo(\()p Fj(n)p Fo(\))11 b Fi(\000)g Fo(1)j Fi(\025)670 2238 y Fl(l)646 2250 y Fg(X)645 2341 y Fl(j)r Fm(=1)706 2292 y Fo(\()p Fj(j)s(a)774 2299 y Fl(j)803 2292 y Fo(+)d Fj(T)881 2299 y Fl(k)q Ff(\000)p Fm(1)947 2292 y Fo(\()p Fj(a)992 2299 y Fl(j)1021 2292 y Fi(\000)g Fo(1\)\))j Fi(\025)1225 2238 y Fl(l)1200 2250 y Fg(X)1200 2341 y Fl(j)r Fm(=1)1261 2292 y Fo(\()p Fj(j)s(a)1329 2299 y Fl(j)1358 2292 y Fo(+)d Fj(g)r Fo(\()p Fj(a)1477 2299 y Fl(j)1495 2292 y Fo(\)\))p Fj(:)267 b Fo(\(1\))46 2436 y(W)l(e)23 b(w)o(an)o(t)g(to)g(\014nd)g(a)h(lo)o (w)o(er)e(b)q(ound)i(on)g(the)e(last)i(sum)e(in)g(\(1\).)42 b(The)23 b(\014rst)h(step)f(w)o(e)f(tak)o(e)h(is)46 2496 y(to)f(transform)f (sums)h(to)g(in)o(tegrals)f(and)i(sequences)e(of)h(natural)g(n)o(um)o(b)q (ers)e(to)j(real)e(functions.)46 2556 y(The)j(sequence)f Fj(a)h Fo(ma)o(y)f(b)q(e)h(regarded)g(as)h(equiv)m(alen)o(t)e(to)i(the)f(piecewise)e (constan)o(t)j(function)46 2616 y Fj(f)70 2623 y Fl(a)91 2616 y Fo(:)8 b Fj(I)-10 b(R)162 2595 y Fm(+)162 2629 y(0)205 2616 y Fi(!)14 b Fj(I)-10 b(R)318 2595 y Fm(+)318 2629 y(0)364 2616 y Fo(de\014ned)16 b(b)o(y)481 2740 y Fj(f)505 2747 y Fl(a)526 2740 y Fo(\()p Fj(x)p Fo(\))e(=)657 2679 y Fg(\032)697 2709 y Fj(a)723 2716 y Fl(j)741 2709 y Fj(;)48 b Fo(if)16 b Fj(j)e Fi(\000)d Fo(1)j Fi(\024)f Fj(x)h(<)g(j)s Fo(,)i Fj(j)g Fo(=)e(1)p Fj(;)8 b(:)g(:)g(:)g(;)g(l)q(;)697 2770 y Fo(0)p Fj(;)68 b Fo(if)16 b Fj(l)e Fi(\024)g Fj(x)f(<)h Fi(1)p Fo(.)46 2869 y(The)k(condition)365 2835 y Fg(P)409 2849 y Fl(l)409 2881 y(j)r Fm(=1)480 2869 y Fj(a)506 2876 y Fl(j)542 2869 y Fo(=)g Fj(n)h Fo(translates)g(to)933 2833 y Fg(R)960 2846 y Ff(1)952 2881 y Fm(0)998 2869 y Fj(f)1022 2876 y Fl(a)1043 2869 y Fo(\()p Fj(x)p Fo(\))8 b Fj(dx)18 b Fo(=)f Fj(n)p Fo(,)i(and)g(the)g(sum)e(in)i (\(1\))g(can)f(b)q(e)46 2929 y(expressed)e(as)454 2970 y Fl(l)429 2982 y Fg(X)429 3073 y Fl(j)r Fm(=1)490 3024 y Fo(\()p Fj(j)s(a)558 3031 y Fl(j)587 3024 y Fo(+)11 b Fj(g)r Fo(\()p Fj(a)706 3031 y Fl(j)724 3024 y Fo(\)\))j(=)827 2965 y Fg(Z)869 2978 y Ff(1)851 3060 y Fm(0)906 2976 y Fg(\020)931 3024 y Fj(xf)983 3031 y Fl(a)1004 3024 y Fo(\()p Fj(x)p Fo(\))c(+)h Fj(g)r Fo(\()p Fj(f)1197 3031 y Fl(a)1218 3024 y Fo(\()p Fj(x)p Fo(\)\))1303 2976 y Fg(\021)1336 3024 y Fj(dx)g Fo(+)1454 2990 y Fj(n)p 1454 3012 30 2 v 1456 3058 a Fo(2)1488 3024 y Fj(:)312 b Fo(\(2\))p eop %%Page: 29 29 28 bop 46 188 a Fx(A)48 b(APPENDIX)1432 b Fo(29)46 342 y(Our)15 b(aim)e(is)h(no)o(w)h(to)h(\014nd)f(a)g(lo)o(w)o(er)f(b)q(ound)h(on)h(the)e (in)o(tegral)g(in)g(\(2\).)21 b(T)l(o)16 b(this)e(end,)h(w)o(e)f(transform)46 402 y(the)g(minimiz)o(ation)e(problem)i(a)h(little)e(further:)20 b(instead)15 b(of)g(piecewise)f(constan)o(t)h(functions)g(suc)o(h)46 462 y(as)i Fj(f)130 469 y Fl(a)167 462 y Fo(w)o(e)f(will)f(consider)h(con)o (tin)o(uous)g(functions.)46 623 y Fh(De\014nition)h(A.1)79 726 y Fw(\(a\))24 b(L)n(et)17 b Fi(D)i 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Fo(\()p Fj(f)5 b Fo(\))13 b(=)633 954 y Fg(R)661 967 y Ff(1)653 1002 y Fm(0)698 989 y Fj(G)p Fo(\()p Fj(x;)8 b(f)d Fo(\()p Fj(x)p Fo(\)\))j Fj(dx)p Fw(.)168 1049 y(\(Note)18 b(that)g(the)f(c)n(ondition)h Fo(lim)766 1056 y Fl(x)p Ff(!1)867 1049 y Fj(e)890 1031 y Fl(x)911 1049 y Fj(f)5 b Fo(\()p Fj(x)p Fo(\))14 b Fj(>)g Fo(0)k Fw(ensur)n(es)g(that)f(the)h(inte)n(gr)n(al)g (exists.\))46 1210 y Fo(It)h(is)h(easy)g(to)g(see)g(that)g(for)g(an)o(y)g (giv)o(en)f Fj(")h(>)g Fo(0)g(the)f(piecewise)g(constan)o(t)h(function)g Fj(f)1693 1217 y Fl(a)1733 1210 y Fo(can)g(b)q(e)46 1270 y(appro)o(ximated)15 b(b)o(y)h(some)f Fj(f)569 1277 y Fl(a;")630 1270 y Fi(2)f(D)k Fo(in)e(suc)o(h)g(a)g(w)o(a)o(y)g(that)557 1398 y Fj(I)t Fo(\()p Fj(f)626 1405 y Fl(a;")672 1398 y Fo(\))e Fj(<)g(")d Fo(+)840 1339 y Fg(Z)881 1353 y Ff(1)863 1434 y Fm(0)919 1398 y Fj(xf)971 1405 y Fl(a)991 1398 y Fo(\()p Fj(x)p Fo(\))g(+)g Fj(g)r Fo(\()p Fj(f)1185 1405 y Fl(a)1206 1398 y Fo(\()p Fj(x)p Fo(\)\))d Fj(dx:)448 b Fo(\(3\))46 1525 y(No)o(w)16 b(it)g(follo)o(ws)g(from)f(\(1\),)h (\(2\),)g(\(3\),)g(and)h(the)f(fact)g(that)h Fj(f)1162 1532 y Fl(a;")1223 1525 y Fi(2)d(D)k Fo(for)f(all)e Fj(")f(>)g Fo(0)i(that)531 1655 y Fj(T)560 1662 y Fl(k)581 1655 y Fo(\()p Fj(n)p Fo(\))11 b Fi(\000)709 1595 y Fg(\022)745 1622 y Fj(n)p 745 1644 30 2 v 747 1689 a Fo(2)790 1655 y(+)g(1)863 1595 y Fg(\023)921 1655 y Fi(\025)28 b Fo(inf)s Fi(f)8 b Fj(I)t Fo(\()p Fj(f)d Fo(\))14 b Fi(j)f Fj(f)20 b Fi(2)14 b(D)c(g)p Fj(:)422 b Fo(\(4\))46 1781 y(The)18 b(follo)o(wing)g(prop)q(osition)h(establishes)f(the)f (existence)g(of)h(a)h(function)f Fj(f)1470 1788 y Fm(0)1506 1781 y Fi(2)f(D)j Fo(that)f(realizes)46 1841 y(this)e(in\014m)o(um;)e(moreo)o (v)o(er)g(it)i(pro)o(vides)g(an)h(equation)g(for)g Fj(f)1174 1848 y Fm(0)1211 1841 y Fo(that)g(will)f(enable)g(us)h(to)g(calculate)46 1902 y Fj(f)70 1909 y Fm(0)105 1902 y Fo(explicitly)l(.)g(Then)d(w)o(e)g(ma)o (y)f(ev)m(aluate)h Fj(I)t Fo(\()p Fj(f)892 1909 y Fm(0)911 1902 y Fo(\))h(to)g(obtain)f(the)h(desired)e(lo)o(w)o(er)h(b)q(ound)h(on)g Fj(T)1775 1909 y Fl(k)1796 1902 y Fo(\()p Fj(n)p Fo(\).)46 1962 y(The)21 b(prop)q(osition)h(is)f(pro)o(v)o(ed)f(b)o(y)g(reducing)h(the)g (problem)e(of)i(minimiz)o(ing)d Fj(I)t Fo(\()p Fj(f)5 b Fo(\))21 b(o)o(v)o(er)f Fi(D)j Fo(to)e(a)46 2022 y(standard)h(situation)e(treated)h (in)f(the)g(Calculus)g(of)h(V)l(ariations.)34 b(\(The)21 b(details)f(of)h (this)f(pro)q(of,)46 2082 y(whic)o(h)e(will)g(b)q(e)h(giv)o(en)g(in)f(the)h (second)g(part)h(of)f(the)g(app)q(endix,)g(are)h(irrelev)m(an)o(t)d(for)j (the)e(rest)h(of)46 2142 y(the)d(argumen)o(t.\))46 2303 y Fh(Prop)r(osition)i (A.2)24 b Fw(Ther)n(e)17 b(is)g(a)h(unique)h(function)f Fj(f)1066 2310 y Fm(0)1100 2303 y Fi(2)c(D)19 b Fw(so)f(that)662 2413 y Fj(I)t Fo(\()p Fj(f)731 2420 y Fm(0)750 2413 y Fo(\))c(=)g(min)n Fi(f)8 b Fj(I)t Fo(\()p Fj(f)d Fo(\))14 b Fi(j)g Fj(f)19 b Fi(2)14 b(D)c(g)p Fj(:)553 b Fo(\(5\))46 2523 y Fw(Mor)n(e)n(over,)16 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(\(6\))i(b)q(ecomes)640 3026 y Fj(x)10 b Fo(+)h Fj(g)752 3005 y Ff(0)764 3026 y Fo(\()p Fj(f)807 3033 y Fm(0)827 3026 y Fo(\()p Fj(x)p Fo(\)\))i(=)h Fj(A)8 b(;)40 b Fo(for)17 b Fj(x)c Fi(\025)h Fo(0)p Fj(:)531 b Fo(\(7\))p eop %%Page: 30 30 29 bop 46 188 a Fx(A)48 b(APPENDIX)1432 b Fo(30)46 342 y(It)16 b(follo)o(ws)g(easily)f(from)g(the)h(de\014nition)g(of)h Fj(g)f Fo(=)d Fj(g)978 349 y Fl(k)q Ff(\000)p Fm(1)1062 342 y Fo(that)474 483 y Fj(g)499 463 y Ff(0)511 483 y Fo(\()p Fj(y)r Fo(\))g(=)640 410 y Fg(\()694 446 y Fo(1)f(+)f(ln)c Fj(y)r(;)201 b Fo(if)16 b(0)e Fj(<)g(y)h Fi(\024)f Fj(e)1318 428 y Fl(k)q Ff(\000)p Fm(1)1384 446 y Fj(;)694 520 y Fo(\()p Fj(k)r(=e)p Fo(\))d Fi(\001)g Fj(y)868 502 y Fm(1)p Fl(=)p Fm(\()p Fl(k)q Ff(\000)p Fm(1\))997 520 y Fj(;)57 b Fo(if)16 b Fj(e)1136 502 y Fl(k)q Ff(\000)p Fm(1)1216 520 y Fi(\024)d Fj(y)j(<)d Fi(1)p Fo(.)46 626 y(Ob)o(viously)l(,)i Fj(g)310 608 y Ff(0)321 626 y Fo(\()p Fj(y)r Fo(\))h(is)g(a)h(strictly)e(increasing)h(function)g(of)g 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y Fx(Case)17 b(1:)22 b Fj(n)15 b Fi(\025)f Fj(e)343 1955 y Fl(k)q Ff(\000)p Fm(1)409 1973 y Fj(:)i Fo(Then)h Fj(A)10 b Fi(\000)i Fj(k)k Fi(\025)e Fo(0,)j(and)g(w)o(e)f (get)g(from)g(\(12\),)h(b)o(y)f(substituting)h(\(8\))g(and)g(the)46 2034 y(de\014nition)f(of)g Fj(g)g Fo(=)e Fj(g)432 2041 y Fl(k)q Ff(\000)p Fm(1)499 2034 y Fo(,)i(that)298 2164 y Fj(I)t Fo(\()p Fj(f)367 2171 y Fm(0)386 2164 y Fo(\))42 b(=)526 2106 y Fg(Z)568 2119 y Fl(A)p Ff(\000)p Fl(k)549 2200 y Fm(0)643 2164 y Fj(x)11 b Fi(\001)g Fo(\()p Fj(e)p Fo(\()p Fj(A)f Fi(\000)h Fj(x)p Fo(\))p Fj(=k)r Fo(\))982 2144 y Fl(k)q Ff(\000)p Fm(1)1059 2164 y Fo(+)1113 2131 y Fj(k)i Fi(\000)e Fo(1)p 1113 2153 113 2 v 1158 2199 a Fj(e)1242 2164 y Fi(\001)f Fo(\()p Fj(e)p Fo(\()p Fj(A)h Fi(\000)f Fj(x)p Fo(\))p Fj(=k)r Fo(\))1541 2144 y Fl(k)1571 2164 y Fj(dx)575 2293 y Fo(+)621 2235 y Fg(Z)663 2248 y Ff(1)644 2329 y Fl(A)p Ff(\000)p Fl(k)720 2293 y Fj(x)g Fi(\001)h Fj(e)806 2273 y Fl(A)p Ff(\000)p Fl(x)p Ff(\000)p Fm(1)938 2293 y 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Fo(=)41 b Fj(A)11 b Fi(\001)g Fj(e)498 2684 y Fl(k)q Ff(\000)p Fm(1)575 2705 y Fi(\001)617 2671 y Fo(1)p 605 2693 49 2 v 605 2739 a Fj(k)632 2725 y Fl(k)669 2705 y Fi(\001)g Fo(\()p Fj(A)750 2684 y Fl(k)782 2705 y Fi(\000)g Fj(k)859 2684 y Fl(k)880 2705 y Fo(\))g Fi(\000)965 2671 y Fj(e)988 2653 y Fl(k)q Ff(\000)p Fm(1)p 965 2693 90 2 v 985 2739 a Fj(k)1012 2725 y Fl(k)1070 2705 y Fi(\001)1144 2671 y Fo(1)p 1100 2693 112 2 v 1100 2739 a Fj(k)i Fo(+)e(1)1216 2705 y(\()p Fj(A)1272 2684 y Fl(k)q Fm(+1)1349 2705 y Fi(\000)g Fj(k)1426 2684 y Fl(k)q Fm(+1)1493 2705 y Fo(\))323 2840 y(=)41 b Fj(A)439 2819 y Fl(k)q Fm(+1)516 2840 y Fi(\001)11 b Fj(e)564 2819 y Fl(k)q Ff(\000)p Fm(1)641 2840 y Fi(\001)683 2806 y Fo(1)p 671 2828 49 2 v 671 2874 a Fj(k)698 2860 y Fl(k)736 2840 y Fi(\001)808 2806 y Fj(k)p 765 2828 112 2 v 765 2874 a(k)j Fo(+)d(1)893 2840 y Fi(\000)g Fj(A)f Fi(\001)h Fj(e)1038 2819 y Fl(k)q Ff(\000)p Fm(1)1115 2840 y Fo(+)g Fj(e)1187 2819 y Fl(k)q Ff(\000)p Fm(1)1265 2840 y Fi(\001)1337 2806 y Fj(k)p 1294 2828 V 1294 2874 a(k)j Fo(+)d(1)1411 2840 y Fj(:)46 2964 y Fo(Altogether,)493 3046 y Fj(I)t Fo(\()p Fj(f)562 3053 y Fm(0)581 3046 y Fo(\))i(=)h Fj(A)702 3026 y Fl(k)q Fm(+1)768 3046 y Fj(e)791 3026 y Fl(k)q Ff(\000)p Fm(1)868 3046 y Fi(\001)910 3012 y Fo(1)p 898 3035 49 2 v 898 3080 a Fj(k)925 3066 y Fl(k)963 3046 y Fi(\001)1035 3012 y Fj(k)p 993 3035 112 2 v 993 3080 a(k)f Fo(+)e(1)1120 3046 y Fi(\000)1218 3012 y Fo(1)p 1175 3035 V 1175 3080 a Fj(k)i Fo(+)e(1)1302 3046 y Fi(\001)g Fj(e)1350 3026 y Fl(k)q Ff(\000)p Fm(1)1416 3046 y Fj(:)p eop %%Page: 31 31 30 bop 46 188 a Fx(A)48 b(APPENDIX)1432 b Fo(31)46 342 y(Substituting)21 b(the)f(v)m(alue)h Fj(A)g Fo(=)h(\()p Fj(n=e)762 324 y Fl(k)q Ff(\000)p Fm(1)828 342 y Fo(\))847 324 y Fm(1)p Fl(=k)918 342 y Fi(\001)14 b Fj(k)23 b Fo(giv)o(en)d(b)o(y)g(\(11\))h(in)o(to)g(the)f(last) h(equation)g(and)46 402 y(using)16 b(the)g(fact)h(that)487 382 y Fl(k)p 464 390 65 2 v 464 419 a(k)q Fm(+1)548 402 y Fi(\025)c Fj(e)623 384 y Ff(\000)p Fm(1)p Fl(=k)723 402 y Fo(results)j(in)281 551 y Fj(I)t Fo(\()p Fj(f)350 558 y Fm(0)369 551 y Fo(\))e(=)501 518 y Fj(k)p 459 540 112 2 v 459 586 a(k)f Fo(+)e(1)586 551 y Fi(\001)g Fj(e)634 531 y Ff(\000)p Fm(1+1)p Fl(=k)774 551 y Fi(\001)g Fj(k)i Fi(\001)e Fj(n)891 531 y Fm(1+1)p Fl(=k)1004 551 y Fi(\000)1070 518 y Fj(e)1093 500 y Fl(k)q Ff(\000)p Fm(1)p 1059 540 V 1059 586 a Fj(k)i Fo(+)e(1)1189 551 y Fi(\025)1247 518 y Fj(k)p 1247 540 28 2 v 1249 586 a(e)1290 551 y Fi(\001)g Fj(n)1344 531 y Fm(1+1)p Fl(=k)1456 551 y Fi(\000)1523 518 y Fj(e)1546 500 y Fl(k)q Ff(\000)p Fm(1)p 1511 540 112 2 v 1511 586 a Fj(k)i Fo(+)e(1)1628 551 y Fj(:)46 681 y Fo(In)16 b(com)o(bination)f(with)h(\(4\))g(and)h(\(5\))g(this)f(yields)578 822 y Fj(T)607 829 y Fl(k)628 822 y Fo(\()p Fj(n)p Fo(\))e Fi(\025)766 788 y Fj(k)p 766 810 28 2 v 768 856 a(e)809 822 y Fi(\001)d Fj(n)863 801 y Fm(1+1)p Fl(=k)976 822 y Fi(\000)1042 788 y Fj(e)1065 770 y Fl(k)q Ff(\000)p Fm(1)p 1031 810 112 2 v 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Ff(\000)p Fl(x)1040 2585 y Fo(,)16 b(for)g(0)f Fi(\024)e Fj(x)h(<)f Fi(1)p Fo(.)p eop %%Page: 32 32 31 bop 46 188 a Fx(A)48 b(APPENDIX)1432 b Fo(32)46 342 y Fh(De\014nition)17 b(A.3)79 445 y Fw(\(a\))24 b(L)n(et)13 b Fi(E)18 b Fw(b)n(e)c(the)g(class)g (of)g(al)r(l)h(functions)f Fj(')p Fo(:)8 b([0)p Fj(;)g Fo(1])14 b Fi(!)f Fj(I)-10 b(R)1144 424 y Fm(+)1144 457 y(0)1187 445 y Fw(that)13 b(have)i(a)e(c)n(ontinuous)h(derivative)179 485 y Fl(d)p 173 493 31 2 v 173 522 a(dt)209 505 y Fj(')p Fo(\()p Fj(t)p Fo(\))f(=)h Fj(')394 487 y Ff(0)405 505 y Fo(\()p Fj(t)p Fo(\))g Fj(>)g Fo(0)j Fw(in)h Fo([0)p Fj(;)8 b Fo(1])17 b Fw(and)h(satisfy)f Fj(')p Fo(\(0\))d(=)g(0)k Fw(and)f Fj(')p Fo(\(1\))d(=)g Fj(n)p Fw(.)81 606 y(\(b\))25 b(L)n(et)17 b Fj(H)t Fo(:)8 b([0)p Fj(;)g Fo(1])j Fi(\002)g Fj(I)-10 b(R)526 586 y Fm(+)526 619 y(0)569 606 y Fi(!)14 b Fj(I)-10 b(R)17 b Fw(b)n(e)h(de\014ne)n(d)g(by)290 752 y Fj(H)t Fo(\()p Fj(t;)8 b(z)r Fo(\))14 b(=)503 679 y Fg(\()582 695 y Fm(1)p 562 703 58 2 v 562 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y Fg(8)667 1743 y(>)667 1756 y(<)667 1830 y(>)667 1843 y(:)725 1721 y(Z)767 1735 y Ff(\000)6 b Fm(ln)o(\(1)p Ff(\000)p Fl(t)p Fm(\))748 1816 y(0)908 1780 y Fj(f)f Fo(\()p Fj(\030)r Fo(\))j Fj(d\030)13 b(;)58 b Fw(if)18 b Fo(0)c Fi(\024)g Fj(t)f(<)h Fo(1)8 b(;)876 1858 y Fj(n)g(;)210 b Fw(if)18 b Fj(t)13 b Fo(=)h(1)8 b Fj(;)46 1946 y Fw(and)356 2006 y Fj(f)380 2013 y Fl(')405 2006 y Fo(\()p Fj(x)p Fo(\))13 b(=)551 1986 y Fl(d)p 541 1994 39 2 v 541 2023 a(dx)584 2006 y Fj(')p Fo(\(1)e Fi(\000)g Fj(e)743 1985 y Ff(\000)p Fl(x)792 2006 y Fo(\))j(=)g Fj(')909 1985 y Ff(0)920 2006 y Fo(\(1)e Fi(\000)f Fj(e)1048 1985 y Ff(\000)p Fl(x)1097 2006 y Fo(\))g Fi(\001)g Fj(e)1175 1985 y Ff(\000)p Fl(x)1232 2006 y Fj(;)25 b Fw(if)17 b Fo(0)e Fi(\024)e Fj(x)h Fi(\024)f(1)p Fj(:)46 2093 y Fw(Mor)n(e)n(over,)j(we)i(have) g Fj(I)t Fo(\()p Fj(f)5 b Fo(\))14 b(=)g Fj(J)5 b Fo(\()p Fj(')701 2100 y Fl(f)723 2093 y Fo(\))p Fw(,)17 b(for)g(al)r(l)i Fj(f)g Fi(2)14 b(D)r Fw(.)46 2195 y(Pr)n(o)n(of:)48 b 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