%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: paper.dvi %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips -Ppips -o paper.ps paper.dvi %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1997.04.18:1452 %%BeginProcSet: texc.pro /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 /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies 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Fo(\024)h Fp(s)g Fo(jAj)1662 1807 y Fl(q)1699 1837 y Fm(,)30 b(then)1909 1807 y Fj(3)397 1991 y Fm(1.)42 b Fp(P)12 b(r)r Fv([)p Fp(M)714 2003 y Fl(j)750 1991 y Fv([)p Fp(i)p Fv(])22 b Fo(\025)h Fp(s)p Fv(])g Fp(<)g(b)1144 1961 y Fk(\000)p Fj(1)1232 1991 y Fp(l)1259 1961 y Fk(\000)p Fj(2)1348 1991 y Fm(,)30 b(and)397 2091 y(2.)42 b(the)30 b(dynamic)h(pr)l(o)l(gr) l(amming)g(phase)g(of)g(LSS2)e(runs)g(in)h Fp(O)r Fv(\()2419 2058 y Fl(mn)p 2420 2072 100 4 v 2434 2120 a(h)2473 2103 y Fc(2)2530 2091 y Fv(\))g Fm(exp)l(e)l(cte)l(d)g(time.)365 2259 y(Pr)l(o)l(of.)43 b Fv(\(1\))22 b(Let)f Fp(X)964 2271 y Fl(c)1019 2259 y Fv(b)r(e)h(the)g(random)e(v)-5 b(ariable)20 b Fp(')p Fv(\()p Fp(T)12 b(sam)p Fv(\()p Fp(j)f Fo(\000)6 b Fp(c)p Fv(\))23 b Fo(2)g Fp(Q)2552 2271 y Fl(i)p Fk(\000)p Fl(c)2661 2259 y Fv(\),)f(where)e(0)j Fo(\024)g Fp(c)g Fo(\024)365 2359 y Fp(r)15 b Fo(\000)d Fv(1)24 b(and)h Fp(r)g Fv(=)e Fp(max)1033 2371 y Fl(eop)1148 2359 y Fv(+)12 b Fp(s)p Fv(.)25 b(So)f Fp(X)1493 2371 y 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Fl(c)1638 2774 y Fv(=)30 b(1])g Fo(\024)1933 2740 y Fl(h)p Fj(+)p Fl(max)2156 2749 y Fg(indel)p 1933 2755 365 4 v 2052 2803 a Fk(jAj)2145 2786 y Fg(q)2307 2774 y Fv(.)i(Our)f(goal)g(is)h(to)g(deriv)n(e)f(an) 365 2888 y(upp)r(er)d(b)r(ound)g(for)f Fp(P)12 b(r)r Fv([)p Fp(X)30 b Fo(\025)23 b Fp(s)p Fv(].)k(Clearly)-7 b(,)716 3134 y Fp(P)12 b(r)r Fv([)p Fp(X)30 b Fo(\025)23 b Fp(s)p Fv(])g(=)1202 3021 y Fl(max)1335 3029 y Fg(eop)1427 3021 y Fj(+)p Fl(s)1296 3055 y Fi(X)1302 3229 y Fl(t)p Fj(=)p Fl(s)1523 3134 y Fp(P)12 b(r)r Fv([)p Fp(X)30 b Fv(=)23 b Fp(t)p Fv(])g Fo(\024)f Fv(\()p Fp(max)2196 3146 y Fl(eop)2318 3134 y Fv(+)c(1\))23 b Fp(P)12 b(r)r Fv([)p Fp(X)30 b Fv(=)23 b Fp(s)p Fv(])p Fp(:)365 3367 y Fv(With)29 b(\(1)18 b Fo(\000)g Fp(P)12 b(r)r Fv([)p Fp(X)951 3379 y Fl(c)1008 3367 y Fv(=)23 b(1]\))g Fo(\024)g Fv(1,)k(w)n(e)g(\014rst)g(deriv)n(e)g(an)g(upp)r(er)h(b)r(ound)g(for)f Fp(P)12 b(r)r Fv([)p Fp(X)30 b Fv(=)23 b Fp(s)p Fv(].)634 3576 y Fp(P)12 b(r)r Fv([)p Fp(X)30 b Fv(=)22 b Fp(s)p Fv(])h(=)1120 3459 y Fi(\022)1181 3520 y Fp(max)1345 3532 y Fl(eop)1467 3520 y Fv(+)18 b Fp(s)1366 3633 y(s)1589 3459 y Fi(\023)1673 3576 y Fp(P)12 b(r)r Fv([)p Fp(X)1869 3588 y Fl(c)1926 3576 y Fv(=)23 b(1])2079 3542 y Fl(s)2137 3576 y Fv(\(1)18 b Fo(\000)g Fp(P)12 b(r)r Fv([)p Fp(X)2508 3588 y Fl(c)2566 3576 y Fv(=)22 b(1]\))2750 3542 y Fl(max)2883 3550 y Fg(eop)1032 3813 y Fo(\024)1120 3696 y Fi(\022)1181 3757 y Fp(max)1345 3769 y Fl(eop)1467 3757 y Fv(+)c Fp(s)1366 3870 y(s)1589 3696 y Fi(\023)c(\022)1735 3757 y Fp(h)k Fv(+)g Fp(max)2048 3769 y Fl(indel)p 1735 3794 470 4 v 1895 3870 a Fo(jAj)2007 3846 y Fl(q)2214 3696 y Fi(\023)2275 3713 y Fl(s)2325 3813 y Fp(:)365 4059 y Fv(It)27 b(can)f(b)r(e)h(sho)n (wn)e(b)n(y)h(using)g(Stirling's)g(form)n(ula)1925 4029 y Fj(4)1988 4059 y Fv(that)2166 3992 y Fi(\000)2204 4022 y Fl(max)2337 4030 y Fg(eop)2429 4022 y Fj(+)p Fl(s)2342 4088 y(s)2512 3992 y Fi(\001)2573 4059 y Fo(\024)2660 3967 y Fi(\020)2720 4017 y Fl(e)19 b Fj(\()p 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2154 y Fl(j)2870 2142 y Fv([)p Fp(i)p Fv(])23 b Fo(\025)g Fp(s)p Fv(])g Fo(2)g Fp(O)r Fv(\(1\))p Fp(:)681 2340 y Fv(The)j(fact)g(that)g(the)g(exp)r (ected)g(total)f(w)n(ork)g(for)g(the)h(dynamic)g(programming)e(phase)h (is)g(in)681 2440 y Fp(O)r Fv(\()788 2407 y Fl(mn)p 789 2421 100 4 v 803 2468 a(h)842 2452 y Fc(2)899 2440 y Fv(\))j(\014nally)f(follo)n(ws)g(from)g Fp(m)1752 2410 y Fk(0)1798 2440 y Fo(2)c Fp(O)r Fv(\()1983 2407 y Fl(m)p 1984 2421 59 4 v 1994 2468 a(h)2053 2440 y Fv(\),)28 b Fp(n)2186 2410 y Fk(0)2232 2440 y Fo(2)c Fp(O)r Fv(\()2418 2407 y Fl(n)p 2419 2421 42 4 v 2420 2468 a(h)2470 2440 y Fv(\),)k(and,)f(for)g(some)g(constan)n(t)g Fp(c)p Fv(,)695 2663 y Fp(E)5 b Fv([total)28 b(w)n(ork)n(])c Fo(\024)e Fp(E)5 b Fv([)1438 2584 y Fi(X)1391 2757 y Fc(1)p Fe(\024)p Fg(i)p Fe(\024)p Fg(m)1582 2745 y Fe(0)1396 2812 y Fc(1)p Fe(\024)p Fg(j)r Fe(\024)p Fg(n)1577 2800 y Fe(0)1629 2663 y Fv(w)n(ork)26 b(at)h Fp(M)2014 2675 y Fl(j)2049 2663 y Fv([)p Fp(i)p Fv(]])c Fo(\024)2314 2584 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Fv(whic)n(h)f(again)f(dep)r(ends)681 3317 y(on)j Fp(l)r Fv(,)f Fp(l)j Fv(also)d(m)n(ust)i(b)r(e)f(relativ)n(ely)f(small)h (compared)f(to)h Fo(jAj)2540 3287 y Fl(q)2577 3317 y Fv(.)g(Finally)-7 b(,)24 b Fp(s)f Fv(pla)n(ys)f(a)h(role)f(here)681 3417 y(as)i(w)n(ell)i(b)r(ecause)f(the)g(larger)f(it)h(is,)h(the)g (more)e(lik)n(ely)h Fp(s)g Fv(out)h(of)f Fp(max)2840 3429 y Fl(eop)2957 3417 y Fv(+)13 b Fp(s)26 b(q)s Fv(-samples)e(do)681 3516 y(not)i(o)r(ccur)g(\(in)h(the)g(same)f(order\))g(in)g Fp(max)2035 3528 y Fl(eop)2154 3516 y Fv(+)16 b Fp(s)27 b Fv(blo)r(c)n(ks)f(\(hence)g(the)h(p)r(ermitted)h(size)e(of)681 3616 y Fp(b)h Fv(and)h Fp(l)h Fv(also)d(hinges)h(on)h Fp(s)p Fv(\).)805 3720 y(So,)k(if)g(the)h(parameters)d(satisfy)h(the)i (ab)r(o)n(v)n(e)d(theorem,)i(then)g(LSS2)g(runs)f(in)i Fp(O)r Fv(\()3420 3687 y Fl(mn)p 3420 3701 100 4 v 3435 3749 a(h)3474 3732 y Fc(2)3530 3720 y Fv(\))681 3820 y(time.)k(The)g(same)g(is)g(true)g(for)f(LSS1)h(b)r(ecause)f(it)i(uses) 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Fv(\))d(=)f Fp(O)r Fv(\()p Fp(n)p Fv(\))32 b(time)e(and)g(the)h(dynamic)f(programming)e(phase)681 4664 y(requires)21 b Fp(O)r Fv(\()p Fp(n)p Fv(\))i(time)f(on)g(the)g(a) n(v)n(erage,)e(pro)n(vided)h(that)h Fp(e)h(l)2535 4634 y Fj(1+)2628 4612 y Fc(3)p 2628 4621 29 4 v 2628 4654 a Fg(s)2694 4664 y Fp(b)2730 4634 y Fj(1+)2823 4612 y Fc(1)p 2823 4621 V 2823 4654 a Fg(s)2889 4664 y Fo(\024)f Fp(s)h Fo(jAj)3150 4634 y Fl(q)3187 4664 y Fv(.)g(Ho)n(w)n(ev)n(er,)681 4764 y Fp(h)g Fo(2)g Fp(\012)t Fv(\()930 4704 y Fo(p)p 1000 4704 73 4 v 1000 4764 a Fp(m)p Fv(\))j(implies)g Fp(r)f Fv(=)e Fp(max)1725 4776 y Fl(eop)1842 4764 y Fv(+)14 b Fp(s)23 b Fo(2)h Fp(O)r Fv(\()p Fp(l)r(=)2228 4704 y Fo(p)p 2297 4704 V 60 x Fp(m)p Fv(\).)i(So)f(the)h(algorithms)e(run)i (in)f(linear)681 4863 y(exp)r(ected)j(time)g(only)f(for)g(stringen)n(t) g(threshold)g(v)-5 b(alues)27 b(\(infrequen)n(t)h(errors,)e(that)h (is\).)p eop %%Page: 13 13 13 12 bop 365 387 a Fq(4)112 b(Exp)s(erimen)m(tal)36 b(results)365 598 y Fv(In)30 b(order)f(to)h(compare)e(our)i(algorithms) e(with)j(My)n(ers')d(algorithm,)h(their)h(resp)r(ectiv)n(e)f(\014l-)365 698 y(tration)k(e\016ciencies)g(ha)n(v)n(e)g(b)r(een)h(ev)-5 b(aluated.)33 b(The)g(\014ltration)h(e\016ciency)f(quan)n(ti\014es)g (the)365 798 y(n)n(um)n(b)r(er)i(of)g(en)n(tries)g(in)g(matrices)g (that)g(ha)n(v)n(e)f(to)h(b)r(e)h(computed)f(in)h(the)f(dynamic)g(pro-) 365 897 y(gramming)40 b(phase.)h(A)g(\014ltration)g(e\016ciency)g(of)g (90\045)f(means)h(that)g(only)g(10\045)f(of)h(the)365 997 y Fp(l)21 b Fo(\001)f Fp(m)f Fo(\001)h Fv(\()p Fp(n)f Fo(\000)g Fp(l)i Fv(+)e(1\))29 b(en)n(tries)f(ha)n(v)n(e)g(to)h(b)r(e)h (computed)f(\(cf.)h(in)n(tro)r(duction\).)f(In)g(the)h(exp)r(eri-)365 1096 y(men)n(ts,)c(b)r(oth)f Fp(P)37 b Fv(and)25 b Fp(T)36 b Fv(w)n(ere)24 b(uniformly)h(random)g(strings.)f(In)h(the)h(\014rst)f (exp)r(erimen)n(t)g(w)n(e)365 1196 y(used)g(the)g(Lev)n(ensh)n(tein)f (distance)g(and)g(an)g(alphab)r(et)h(of)f(size)h(20.)e(The)i(length)g 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V 105 w(6)p 1660 4391 V 105 w(5)p 1803 4391 V 144 w(4)p 1985 4391 V 144 w(3)p 2166 4391 V 143 w(2)p 2348 4391 V 144 w(2)p 2530 4391 V 143 w(1)p 2711 4391 V 144 w(1)p 2893 4391 V 144 w(1)p 3075 4391 V 143 w(0)p 3256 4391 V 984 4482 V 1050 4455 a(LSS1)26 b(\()p Fs(q)e Fu(=)d(4\))p 1517 4482 V 67 w(14)p 1660 4482 V 67 w(14)p 1803 4482 V 106 w(14)p 1985 4482 V 105 w(14)p 2166 4482 V 106 w(14)p 2348 4482 V 105 w(14)p 2530 4482 V 106 w(14)p 2711 4482 V 106 w(14)p 2893 4482 V 105 w(14)p 3075 4482 V 106 w(14)p 3256 4482 V 984 4574 V 1050 4546 a(LSS2)26 b(\()p Fs(q)e Fu(=)d(2\))p 1517 4574 V 105 w(8)p 1660 4574 V 105 w(8)p 1803 4574 V 144 w(8)p 1985 4574 V 144 w(8)p 2166 4574 V 143 w(8)p 2348 4574 V 144 w(8)p 2530 4574 V 143 w(8)p 2711 4574 V 144 w(8)p 2893 4574 V 144 w(8)p 3075 4574 V 143 w(8)p 3256 4574 V 986 4577 2273 4 v 1214 4807 a Ft(T)-7 b(able)29 b(5.)c Fu(Maxim)n(um)f Fs(k)k Fu(with)e(\014ltration)g(e\016ciency)g Fd(\025)21 b Fu(95\045)p eop %%Page: 15 15 15 14 bop 365 387 a Ff(Ac)m(kno)m(wledgemen)m(ts:)31 b Fv(I)i(am)f(indebted)i(to)e(Mic)n(hael)g(Jandrey)f(who)i(pro)n(vided) e(the)i(C)365 487 y(implemen)n(tation)i(of)f(the)h(algorithms)e(and)h (suggested)f(man)n(y)h(impro)n(v)n(emen)n(ts.)f(Thanks)365 587 y(go)28 b(to)h(Stefan)g(Kurtz)f(for)h(useful)g(commen)n(ts)f(on)h (a)f(previous)g(v)n(ersion)f(of)i(the)g(pap)r(er)f(and)365 686 y(to)d(Gene)g(My)n(ers)e(for)i(fruitful)g(discussions)f(on)h(appro) n(ximate)e(string)h(searc)n(hing.)f(F)-7 b(urther-)365 786 y(more,)35 b(I)g(am)g(grateful)f(to)h(P)n(eter)f(Bruns)g(for)h(a)f (discussion)h(on)f(the)i(probabilit)n(y)e(of)h(the)365 886 y(ev)n(en)n(t)27 b Fp(X)653 898 y Fl(c)710 886 y Fv(=)c(1.)365 1155 y Fq(References)365 1345 y Fu([CL92])105 b(W.I.)29 b(Chang)h(and)f(J.)d(Lamp)r(e.)45 b(Theoretical)32 b(and)d(Empirical)h(Comparisons)g(of)g(Ap-)691 1437 y(pro)n(ximate)18 b(String)g(Matc)n(hing)i(Algorithms.)31 b(In)19 b Fr(Pr)l(o)l(c.)i(CPM) p Fu(,)e(pages)h(175{184.)h(LNCS)691 1528 y Ft(644)p Fu(,)26 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y(X)g(macro)g(pac)n(k)l(age)h(with)g(LLNCS)f(st)n(yle)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF