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%   email = "kb@@mail.tug.org",
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2646 y(in)n(termitten)n(tly)f(reev)l(aluates)i(the)g(p)r(ossibilit)n(y)
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2829 y(\014ltering)d(information)h(ab)r(out)e(the)h(region)h(y)n(et)e
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Fu(1)112 b(In)m(tro)s(duction)523 3469 y Ft(The)28 b(problem)f(of)h
(appro)n(ximate)d(string)i(matc)n(hing)h(is)f(stated)h(as)f(follo)n
(ws:)36 b(giv)n(en)27 b(a)g(data-)523 3568 y(base)34
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b Ft(,)35 b(a)f(threshold)g Fs(k)s Ft(,)i(\014nd)e(all)g(appro)n
(ximate)f(matc)n(hes,)523 3668 y(i.e.,)h(all)f(sub)n(w)n(ords)e
Fs(v)36 b Ft(of)d Fs(T)44 b Ft(whose)32 b(edit)i(distance)e(to)h
Fs(P)45 b Ft(is)32 b(at)h(most)g Fs(k)s Ft(.)52 b(The)33
b(dynamic)523 3768 y(programming)f(approac)n(h)f([8])j(pro)n(vides)e
(the)i(general)f(solution,)i(and)e(is)h(still)g(un)n(b)r(eaten)523
3867 y(w.r.t.)d(its)g(v)n(ersatilit)n(y)-7 b(.)46 b(Its)31
b(running)g(time,)i(ho)n(w)n(ev)n(er,)d(is)h Fr(O)r Ft(\()p
Fs(mn)p Ft(\),)h(where)f Fs(m)e Ft(=)f Fr(j)p Fs(P)12
b Fr(j)31 b Ft(and)523 3967 y Fs(n)23 b Ft(=)g Fr(j)p
Fs(T)12 b Fr(j)p Ft(.)33 b(This)21 b(is)g(impractical)f(for)g
(large-scale)e(applications)i(lik)n(e)g(biosequence)g(analysis,)523
4066 y(where)i(the)g(size)g(of)g(the)g(gene)f(and)h(protein)g
(databases)e(\(i.e.)j Fs(T)12 b Ft(\))21 b(gro)n(ws)f(exp)r(onen)n
(tially)i(due)523 4166 y(to)30 b(the)h(adv)-5 b(ances)29
b(in)i(sequencing)f(tec)n(hnology)-7 b(.)43 b(Recen)n(t)30
b(\014lter)h(tec)n(hniques)f(for)g(the)g(unit)523 4266
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(10)o(]\))h(reduce)f(the)h(a)n(v)n(erage)c(case)j(complexit)n(y)523
4365 y(to)i(at)f(least)h(linear)f(time.)56 b(Filter)34
b(algorithms)e(are)h(based)h(on)f(the)h(observ)-5 b(ation)33
b(that|)523 4465 y(giv)n(en)38 b(that)i Fs(k)i Ft(is)d(small)g(suc)n(h)
g(that)h(appro)n(ximate)d(matc)n(hes)i(are)f(rare|the)g(dynamic)523
4565 y(programming)d(computation)i(sp)r(ends)g(most)g(of)g(its)h(time)f
(v)n(erifying)f(that)i(there)f(is)g FA(no)523 4664 y
Ft(appro)n(ximate)d(matc)n(h)h(in)h(a)f(giv)n(en)g(region)f(of)i
Fs(T)12 b Ft(.)60 b(A)35 b(\014lter)h(rules)f(out)h(regions)e(of)h
Fs(T)47 b Ft(in)523 4764 y(whic)n(h)36 b(appro)n(ximate)f(matc)n(hes)h
FA(c)l(annot)44 b Ft(o)r(ccur,)38 b(and)f(then)g(a)f(dynamic)g
(programming)523 4863 y(computation)27 b(is)h(applied)f(only)h(to)f
(the)h(remaining)f(regions.)p Black Black eop
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2 1 bop Black Black 648 387 a Ft(By)19 b(static)i(\014lter)f(tec)n
(hniques)g(w)n(e)f(refer)h(to)g(the)h(approac)n(h)d(as)h(describ)r(ed)h
(b)r(efore.)35 b(There)523 487 y(is)f(a)f(strict)h(separation)e(b)r(et)
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(hec)n(king)523 587 y(phase)19 b(via)g(a)g(dynamic)g(programming)e
(computation.)34 b(Dynamic)19 b(\014ltering)h(as)e(in)n(tro)r(duced)523
686 y(in)28 b(this)h(pap)r(er)f FA(impr)l(oves)36 b Ft(on)28
b(this)g(b)n(y)g(merging)f(the)i(\014ltering)f(and)g(the)g(c)n(hec)n
(king)f(phase.)523 786 y(It)k(ev)-5 b(aluates)29 b(the)i(statically)f
(deriv)n(ed)f(\014lter)h(information)g(during)g(the)g(c)n(hec)n(king)f
(phase,)523 886 y(strengthening)39 b(it)i(b)n(y)f(information)f
(determined)i(dynamically)-7 b(.)73 b(Rather)40 b(than)g(using)523
985 y(static)27 b(information)f(once)g(to)h(decide)f(whether)h(a)f
(\(complete\))i(region)d(m)n(ust)i(b)r(e)g(c)n(hec)n(k)n(ed,)523
1085 y(it)g(is)f(consulted)h(in)n(termitten)n(tly)-7
b(,)27 b(in)g(order)e(to)h(immediately)h(abandon)f(a)g(region)f(as)h
(so)r(on)523 1184 y(it)32 b(b)r(ecomes)f(clear)g(that,)h(due)g(to)f(a)h
(p)r(o)r(or)e(start,)i(an)g(appro)n(ximate)d(matc)n(h)j(is)f(no)g
(longer)523 1284 y(p)r(ossible.)648 1384 y(T)-7 b(o)29
b(understand)g(ho)n(w)g(this)h(w)n(orks,)f(recall)g(that)h(a)f(dynamic)
h(programming)d(compu-)523 1483 y(tation)f(giv)n(es)e(us)i(information)
f(ab)r(out)h FA(actual)35 b Ft(di\013erences)25 b(of)h(the)g(sub)n(w)n
(ords)e(of)i Fs(T)37 b Ft(and)25 b Fs(P)523 1583 y Ft(under)h
(consideration.)35 b(As)26 b(opp)r(osed)g(to)f(this,)i(the)g
(statically)e(deriv)n(ed)g(information)h(tells)523 1683
y(us)d(the)g(righ)n(tmost)g(p)r(ositions)f(where)h(actual)f
(di\013erences)h(migh)n(t)g(o)r(ccur.)35 b(These)22 b(p)r(ositions)523
1782 y(are)27 b(referred)g(to)h(as)g(the)h(p)r(ositions)f(of)g(the)g
FA(guar)l(ante)l(e)l(d)37 b Ft(di\013erences.)i(By)28
b(de\014nition,)h(the)523 1882 y(actual)k(di\013erences)h(o)r(ccur)f
FA(b)l(efor)l(e)41 b Ft(or)33 b FA(at)42 b Ft(the)34
b(guaran)n(teed)e(di\013erences.)55 b(No)n(w)33 b(supp)r(ose)523
1981 y(w)n(e)26 b(ha)n(v)n(e)g(a)g(sub)n(w)n(ord)f Fs(s)i
Ft(passing)e(the)i(static)f(\014lter,)h(and)g(w)n(e)f(kno)n(w)g(that,)h
(after)f(reading)f(a)523 2081 y(pre\014x)31 b Fs(v)800
2093 y Fq(1)869 2081 y Ft(of)h Fs(s)p Ft(,)g Fs(d)g Ft(actual)f
(di\013erences)g(ha)n(v)n(e)g(o)r(ccurred.)47 b(No)n(w,)32
b(if)h(the)e(remaining)g(su\016x)523 2181 y Fs(v)563
2193 y Fq(2)627 2181 y Ft(of)c Fs(s)g Ft(con)n(tains)f(more)g(than)h
Fs(k)19 b Fr(\000)e Fs(d)27 b Ft(guaran)n(teed)e(di\013erences,)h(w)n
(e)h(kno)n(w)f(that)h Fs(s)f Ft(cannot)523 2280 y(con)n(tain)h(an)g
(appro)n(ximate)f(matc)n(h.)648 2380 y(T)-7 b(o)30 b(further)g(clarify)
g(the)h(idea)f(of)g(dynamic)h(\014ltering,)g(let)f(us)h(sk)n(etc)n(h)f
(ho)n(w)f(it)i(applies)523 2480 y(to)25 b(Chang)g(and)g(La)n(wler's)f
(linear)h(exp)r(ected)g(time)h(algorithm)e(\(LET)h(for)g(short\))g
([2].)36 b(This)523 2579 y(algorithm)d(divides)i Fs(T)45
b Ft(=)34 b Fs(w)1447 2591 y Fq(1)1485 2579 y Fs(c)1521
2591 y Fq(1)1572 2579 y Fs(:)14 b(:)g(:)g(w)1742 2591
y Fp(r)1779 2579 y Fs(c)1815 2591 y Fp(r)1852 2579 y
Fs(w)1911 2591 y Fp(r)r Fq(+1)2067 2579 y Ft(in)n(to)34
b(sub)n(w)n(ords)f Fs(w)2669 2591 y Fq(1)2706 2579 y
Fs(;)14 b(:)g(:)g(:)g(;)g(w)2950 2591 y Fp(r)2987 2579
y Fs(;)g(w)3083 2591 y Fp(r)r Fq(+1)3239 2579 y Ft(of)34
b Fs(P)523 2679 y Ft(and)c(c)n(haracters)d Fs(c)1120
2691 y Fq(1)1157 2679 y Fs(;)14 b(:)g(:)g(:)g(;)g(c)1378
2691 y Fp(r)1414 2679 y Ft(.)44 b(The)29 b(sub)n(w)n(ords)f(are)h(of)h
(maximal)f(length)h(\(i.e.)g Fs(w)3078 2691 y Fp(i)3106
2679 y Fs(c)3142 2691 y Fp(i)3199 2679 y Ft(is)g(not)523
2778 y(a)40 b(sub)n(w)n(ord)e(of)i Fs(P)12 b Ft(\),)44
b(and)c(so)f(the)i(c)n(haracters)d Fs(c)2104 2790 y Fq(1)2141
2778 y Fs(;)14 b(:)g(:)g(:)f(;)h(c)2361 2790 y Fp(r)2438
2778 y Ft(mark)39 b(the)i(p)r(ositions)f(of)g(the)523
2878 y(guaran)n(teed)30 b(di\013erences.)49 b(In)32 b(particular,)f
(for)g(eac)n(h)g(sub)n(w)n(ord)f(of)i Fs(T)43 b Ft(b)r(eginning)31
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2978 y Fs(c)808 2990 y Fp(h)851 2978 y Ft(,)i(the)g Fs(i)p
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(haracter)d Fs(c)3010 2990 y Fp(h)p Fq(+)p Fp(i)3127
2978 y Ft(,)j(or)e(left)523 3077 y(to)g(it.)46 b(Therefore,)30
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(\014rst)f Fs(k)23 b Ft(+)d(1)30 b(of)h(the)523 3177
y(c)n(haracters)j Fs(c)963 3189 y Fp(h)1006 3177 y Fs(;)14
b(c)1079 3189 y Fp(h)p Fq(+1)1205 3177 y Fs(;)g(:)g(:)g(:)p
Ft(,)38 b(i.e.,)h(it)d(is)g(a)f(sub)n(w)n(ord)g(of)h
Fs(s)h Ft(=)f Fs(w)2504 3189 y Fp(h)2547 3177 y Fs(c)2583
3189 y Fp(h)2640 3177 y Fs(:)14 b(:)g(:)g(w)2810 3189
y Fp(h)p Fq(+)p Fp(k)2941 3177 y Fs(c)2977 3189 y Fp(h)p
Fq(+)p Fp(k)3108 3177 y Fs(w)3167 3189 y Fp(h)p Fq(+)p
Fp(k)q Fq(+1)3382 3177 y Ft(.)523 3277 y(Since)39 b(an)g(appro)n
(ximate)f(matc)n(h)h(is)g(of)g(length)h Fr(\025)i Fs(m)26
b Fr(\000)g Fs(k)s Ft(,)42 b(LET)d(discards)f Fs(s)h
Ft(if)h Fr(j)p Fs(s)p Fr(j)i Fs(<)523 3376 y(m)23 b Fr(\000)h
Fs(k)s Ft(.)59 b(Otherwise,)36 b(LET)f(applies)g(a)g(dynamic)g
(programming)e(computation)i(to)g(the)523 3476 y(en)n(tire)24
b(sub)n(w)n(ord)f Fs(s)p Ft(.)36 b(This)25 b(is)f(where)g(the)h
(dynamic)g(v)n(ersion)e(of)h(LET)g(b)r(eha)n(v)n(es)g(di\013eren)n(tly)
-7 b(.)523 3575 y(It)37 b(susp)r(ends)g(the)g(dynamic)g(programming)d
(computation)j(at)f Fs(c)2638 3587 y Fp(h)p Fq(+)p Fp(i)2756
3575 y Ft(,)j(whenev)n(er)d Fs(d)i(>)g(i)523 3675 y Ft(di\013erences)e
(ha)n(v)n(e)e(o)r(ccurred)h(already)-7 b(,)37 b(and)e(an)h(extension)f
(up)i(to)e(length)h Fs(m)24 b Fr(\000)g Fs(k)38 b Ft(adds)523
3775 y(more)21 b(than)i Fs(k)10 b Fr(\000)e Fs(d)22 b
Ft(guaran)n(teed)e(di\013erences.)35 b(This)22 b(is)h(the)f(in)n
(tuitiv)n(e)g(idea|its)g(realization)523 3874 y(is)28
b(m)n(uc)n(h)f(more)g(complicated)g(since)g(partial)g(matc)n(hes)g(ma)n
(y)g(o)n(v)n(erlap.)648 3974 y(One)j(of)h(the)g(main)g(c)n
(haracteristics)e(of)h(a)h(\014lter)g(algorithm)e(is)i(its)g(critical)g
(threshold)523 4074 y Fs(k)566 4094 y Fq(max)693 4074
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b(of)g Fs(k)i Ft(suc)n(h)e(that)g(the)g(\014lter)g(algorithm)f(is)h
(still)g(linear)523 4173 y(or)39 b(sublinear,)j(resp)r(ectiv)n(ely)-7
b(.)72 b(Strictly)40 b(sp)r(eaking,)i Fs(k)2278 4194
y Fq(max)2444 4173 y Ft(is)e(the)g(v)-5 b(alue)40 b(for)f(whic)n(h)g(w)
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b(in)g(an)g(exp)r(ected)g(case)523 4372 y(analysis,)25
b Fs(k)900 4393 y Fq(max)1053 4372 y Ft(is)h(obtained)g(b)n(y)g
(roughly)f(estimating)h(the)g(probabilit)n(y)g(of)g(appro)n(ximate)523
4472 y(matc)n(hes.)34 b(In)20 b(practice,)h(ho)n(w)n(ev)n(er,)e(one)h
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4572 y(those)26 b(theoretically)f(obtained.)36 b(In)26
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4671 y(obtained)h(v)-5 b(alues.)648 4771 y(In)30 b(this)g(pap)r(er,)g
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(ell-kno)n(wn)f(static)g(\014lter)h(algorithms)f(LET)523
686 y(and)20 b(SET)h(devised)f(b)n(y)g(Chang)g(and)h(La)n(wler)e([2)o
(])i(and)g(the)g(more)e(sophisticated)h(algorithms)523
786 y(LEQ)36 b(and)h(LA)n(Q)f(recen)n(tly)h(presen)n(ted)f(b)n(y)h
(Sutinen)g(and)g(T)-7 b(arhio)36 b(in)h([9,)g(10)o(].)65
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(core)523 1184 y(of)32 b(a)g(m)n(uc)n(h)g(wider)g(rep)r(ort)f([5],)i
(where)f(w)n(e)g(clarify)f(the)i(pro)r(ofs)e(b)n(y)h(Chang)f(and)h(La)n
(wler,)523 1284 y(describ)r(e)37 b(some)g(further)h(impro)n(v)n(emen)n
(ts)e(also)h(for)g(static)h(\014lters,)i(giv)n(e)c(new)i(complete)523
1384 y(pro)r(ofs)23 b(for)h(exp)r(ected)g(time)g(complexit)n(y)-7
b(,)25 b(and)e(presen)n(t)h(extensiv)n(e)f(exp)r(erimen)n(tal)g
(results.)523 1641 y Fu(2)112 b(Basic)37 b(Notions)f(and)i
(De\014nitions)523 1829 y Ft(W)-7 b(e)25 b(assume)f(that)i(the)f
(reader)e(is)i(familiar)f(with)i(the)f(standard)f(terminology)f(on)i
(strings,)523 1929 y(as)i(used,)g(e.g.,)g(in)h([2)o(].)37
b(Let)28 b Fs(u)f Ft(and)g Fs(v)k Ft(b)r(e)c(strings.)36
b(An)28 b FA(alignment)36 b Fs(A)27 b Ft(of)h Fs(u)f
Ft(and)g Fs(v)k Ft(is)c(a)g(list)523 2028 y([)p Fs(a)590
2040 y Fq(1)636 2028 y Fr(!)9 b Fs(b)764 2040 y Fq(1)802
2028 y Fs(;)14 b(:)g(:)g(:)f(;)h(a)1030 2040 y Fp(p)1078
2028 y Fr(!)9 b Fs(b)1206 2040 y Fp(p)1244 2028 y Ft(])24
b(of)g(edit)h(op)r(erations)e(\(i.e.,)j(deletions)e Fs(a)9
b Fr(!)g Fs(")p Ft(,)25 b(replacemen)n(ts)e Fs(a)9 b
Fr(!)g Fs(b)p Ft(,)523 2128 y(and)21 b(insertions)g Fs(")9
b Fr(!)g Fs(b)21 b Ft(of)h(single)e(c)n(haracters\))g(suc)n(h)h(that)h
Fs(u)g Ft(=)h Fs(a)2536 2140 y Fq(1)2587 2128 y Fs(:)14
b(:)g(:)g(a)2742 2140 y Fp(p)2801 2128 y Ft(and)22 b
Fs(v)k Ft(=)c Fs(b)3146 2140 y Fq(1)3197 2128 y Fs(:)14
b(:)g(:)g(b)3344 2140 y Fp(p)3382 2128 y Ft(.)523 2227
y(If)34 b Fs(A)h Ft(is)e(an)h(alignmen)n(t,)h(then)f
Fs(\016)s Ft(\()p Fs(A)p Ft(\))h(denotes)f(the)g(n)n(um)n(b)r(er)f(of)h
(edit)g(op)r(erations)f Fs(a)20 b Fr(!)f Fs(b)p Ft(,)523
2327 y Fs(a)k Fr(6)p Ft(=)g Fs(b)p Ft(,)i(in)g Fs(A)p
Ft(.)36 b Fs(edist)p Ft(\()p Fs(u;)14 b(v)s Ft(\))25
b(denotes)f(the)i FA(e)l(dit)h(distanc)l(e)32 b Ft(b)r(et)n(w)n(een)25
b Fs(u)f Ft(and)h Fs(v)j Ft(and)d(is)g(de\014ned)523
2427 y(b)n(y)30 b Fs(edist)p Ft(\()p Fs(u;)14 b(v)s Ft(\))28
b(=)g(min)p Fr(f)p Fs(\016)s Ft(\()p Fs(A)p Ft(\))g Fr(j)g
Fs(A)g Ft(is)g(an)f(alignmen)n(t)g(of)h Fs(u)f Ft(and)g
Fs(v)s Fr(g)p Ft(.)46 b(An)31 b(alignmen)n(t)f Fs(A)h
Ft(of)523 2526 y Fs(u)c Ft(and)h Fs(v)j Ft(is)c FA(optimal)p
Ft(,)i(if)f Fs(\016)s Ft(\()p Fs(A)p Ft(\))c(=)f Fs(edist)p
Ft(\()p Fs(u;)14 b(v)s Ft(\).)648 2626 y(Supp)r(ose)27
b(w)n(e)g(are)g(giv)n(en)g(a)g FA(thr)l(eshold)k(value)j
Fs(k)26 b Fr(2)e Fo(N)2308 2638 y Fq(0)2345 2626 y Ft(,)k(a)f
FA(p)l(attern)i(string)35 b Fs(P)k Ft(of)28 b(length)523
2726 y Fs(m)p Ft(,)35 b(and)e(a)f FA(text)i(string)41
b Fs(T)j Ft(of)33 b(length)g Fs(n)p Ft(.)54 b(An)33 b
FA(appr)l(oximate)k(match)j Ft(is)33 b(a)g(sub)n(w)n(ord)f
Fs(v)k Ft(of)523 2825 y Fs(T)43 b Ft(s.t.)32 b Fs(edist)p
Ft(\()p Fs(P)r(;)14 b(v)s Ft(\))31 b Fr(\024)f Fs(k)s
Ft(.)50 b(The)32 b Fs(k)s Ft(-)p FA(di\013er)l(enc)l(es)i(pr)l(oblem)39
b Ft(is)32 b(to)g(en)n(umerate)f(all)g(p)r(ositions)523
2925 y(in)c Fs(T)38 b Ft(where)26 b(an)h(appro)n(ximate)e(matc)n(h)h
(ends.)37 b(A)27 b(generalization)e(of)i(this)g(problem)f(is)h(the)523
3024 y FA(appr)l(oximate)c(string)f(matching)h(pr)l(oblem)j
Ft(whic)n(h)19 b(additionally)f(asks)g(for)g(a)h(corresp)r(onding)523
3124 y(appro)n(ximate)34 b(matc)n(h.)63 b(Sellers')35
b(algorithm)g([8])h(solv)n(es)e(the)j Fs(k)s Ft(-di\013erences)e
(problem)h(in)523 3224 y Fr(O)r Ft(\()p Fs(mn)p Ft(\))27
b(time)f(and)g Fr(O)r Ft(\()p Fs(m)p Ft(\))g(space)f(b)n(y)h(ev)-5
b(aluating)25 b(an)h(\()p Fs(m)15 b Ft(+)f(1\))h Fr(\002)f
Ft(\()p Fs(n)h Ft(+)g(1\))25 b(table)h Fs(D)r Ft(\()p
Fs(i;)14 b(j)5 b Ft(\))23 b(=)523 3323 y(min)p Fr(f)p
Fs(edist)p Ft(\()p Fs(P)12 b Ft([1)i Fs(:)g(:)g(:)f(i)p
Ft(])p Fs(;)h(s)p Ft(\))31 b Fr(j)i Fs(s)27 b Ft(is)h(a)f(su\016x)g(of)
h Fs(T)12 b Ft([1)i Fs(:)g(:)g(:)e(j)5 b Ft(])p Fr(g)33
b Ft(using)f(dynamic)h(programming.)523 3423 y(If)28
b Fs(D)r Ft(\()p Fs(m;)14 b(j)5 b Ft(\))23 b Fr(\024)g
Fs(k)s Ft(,)28 b(then)g(there)f(is)h(an)f(appro)n(ximate)f(matc)n(h)h
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b(follo)n(wing)f(de\014nitions)h(are)f(motiv)-5 b(ated)37
b(b)n(y)g(Ehrenfeuc)n(h)n(t)g(and)f(Haussler's)g([4])523
3622 y(notion)h(of)h FA(c)l(omp)l(atible)i(markings)p
Ft(.)67 b(A)38 b FA(p)l(artition)45 b Ft(of)37 b Fs(v)k
Ft(w.r.t.)c Fs(u)g Ft(is)h(a)f(list)g([)p Fs(w)3086 3634
y Fq(1)3124 3622 y Fs(;)14 b(c)3197 3634 y Fq(1)3234
3622 y Fs(;)g(:)g(:)g(:)g(;)523 3722 y(w)582 3734 y Fp(r)619
3722 y Fs(;)g(c)692 3734 y Fp(r)729 3722 y Fs(;)g(w)825
3734 y Fp(r)r Fq(+1)946 3722 y Ft(])37 b(of)g(sub)n(w)n(ords)f
Fs(w)1540 3734 y Fq(1)1578 3722 y Fs(;)14 b(:)g(:)g(:)f(;)h(w)1821
3734 y Fp(r)1858 3722 y Fs(;)g(w)1954 3734 y Fp(r)r Fq(+1)2113
3722 y Ft(of)37 b Fs(u)g Ft(and)g(c)n(haracters)e Fs(c)2914
3734 y Fq(1)2951 3722 y Fs(;)14 b(:)g(:)g(:)f(;)h(c)3171
3734 y Fp(r)3245 3722 y Ft(suc)n(h)523 3821 y(that)23
b Fs(v)k Ft(=)22 b Fs(w)911 3833 y Fq(1)949 3821 y Fs(c)985
3833 y Fq(1)1036 3821 y Fs(:)14 b(:)g(:)g(w)1206 3833
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Fs(w)1375 3833 y Fp(r)r Fq(+1)1496 3821 y Ft(.)35 b(Let)23
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3833 y Fp(r)2384 3821 y Fs(;)g(c)2457 3833 y Fp(r)2494
3821 y Fs(;)g(w)2590 3833 y Fp(r)r Fq(+1)2711 3821 y
Ft(])23 b(b)r(e)h(a)e(partition)h(of)g Fs(v)523 3921
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b Fs(w)2170 3933 y Fq(1)2208 3921 y Fs(;)14 b(:)g(:)g(:)f(;)h(w)2451
3933 y Fp(r)2488 3921 y Fs(;)g(w)2584 3933 y Fp(r)r Fq(+1)2727
3921 y Ft(are)21 b(the)i FA(submatches)523 4021 y Ft(in)33
b(\011.)50 b Fs(c)799 4033 y Fq(1)837 4021 y Fs(;)14
b(:)g(:)g(:)f(;)h(c)1057 4033 y Fp(r)1126 4021 y Ft(are)32
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b(If)33 b(for)f(all)g Fs(h;)14 b Ft(1)30 b Fr(\024)h
Fs(h)g Fr(\024)f Fs(r)r Ft(,)35 b Fs(w)3283 4033 y Fp(h)3326
4021 y Fs(c)3362 4033 y Fp(h)523 4120 y Ft(is)27 b(not)h(a)f(sub)n(w)n
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(p)l(artition)35 b Ft(of)27 b Fs(v)j Ft(w.r.t.)e Fs(u)p
Ft(,)f(denoted)523 4220 y(b)n(y)e(\011)701 4232 y Fp(lr)758
4220 y Ft(\()p Fs(u;)14 b(v)s Ft(\).)37 b Fr(j)p Ft(\011)1098
4232 y Fp(lr)1156 4220 y Ft(\()p Fs(u;)14 b(v)s Ft(\))p
Fr(j)25 b Ft(is)g(the)g FA(maximal)k(matches)f(distanc)l(e)k
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4320 y Fs(mmdist)p Ft(\()p Fs(u;)14 b(v)s Ft(\).)1025
4289 y Fq(1)1099 4320 y Ft(It)27 b(can)f(b)r(e)h(computed)g(in)g
Fr(O)r Ft(\()p Fr(j)p Fs(u)p Fr(j)17 b Ft(+)f Fr(j)p
Fs(v)s Fr(j)p Ft(\))27 b(time)g(and)g Fr(O)r Ft(\()p
Fr(j)p Fs(u)p Fr(j)p Ft(\))g(space,)f(using)523 4419
y(the)i(su\016x)f(tree)g(for)f Fs(u)h Ft(\(see)g([7]\).)37
b(The)27 b(follo)n(wing)f(lemma)h(states)g(an)g(imp)r(ortan)n(t)g
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(partitions.)52 b(Using)33 b(this)g(lemma,)i(it)e(is)g(easy)f(to)h(sho)
n(w)f(that)523 4619 y FA(mmdist)8 b Ft(\()p Fs(u;)14
b(v)s Ft(\))23 b Fr(\024)g Fs(edist)p Ft(\()p Fs(u;)14
b(v)s Ft(\).)p Black 523 4687 1153 4 v 627 4740 a Fn(1)661
4772 y Fz(Note)23 b(that)g Fv(mmdist)30 b Fz(is)24 b(not)e(a)i
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h(not)523 4863 y(symmetric,)g(see)j([4].)p Black Black
1943 5112 a Ft(3)p Black eop
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4 3 bop Black Black Black 523 387 a Fm(Lemma)14 b(1.)p
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Fs(r)n(;)14 b Ft(0)35 b Fr(\024)g Fs(r)j Fr(\024)e Fs(\016)s
Ft(\()p Fs(A)p Ft(\))p FA(,)523 487 y(and)26 b(a)g(p)l(artition)h
Ft([)p Fs(w)1167 499 y Fq(1)1204 487 y Fs(;)14 b(c)1277
499 y Fq(1)1314 487 y Fs(;)g(:)g(:)g(:)g(;)g(w)1558 499
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587 y Fs(w)582 599 y Fp(r)r Fq(+1)733 587 y FA(is)30
b(a)g(su\016x)f(of)i Fs(u)p FA(.)523 841 y Fu(3)112 b(Dynamic)37
b(Filtering)e(applied)i(to)g(LET)523 1025 y Ft(The)g
FA(line)l(ar)i(exp)l(e)l(cte)l(d)g(time)g(algorithm)45
b Ft(\(LET)37 b(for)f(short\))h(of)g(Chang)g(and)g(La)n(wler)e([2])523
1124 y(is)j(based)g(on)h(the)f(observ)-5 b(ation)38 b(that)g(an)h
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Fs(P)r(;)14 b(T)e Ft(\))23 b(=)523 1324 y([)p Fs(w)605
1336 y Fq(1)643 1324 y Fs(;)14 b(c)716 1336 y Fq(1)753
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1324 y Fs(;)g(c)1107 1336 y Fp(r)1143 1324 y Fs(;)g(w)1239
1336 y Fp(r)r Fq(+1)1360 1324 y Ft(].)37 b(Th)n(us,)25
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y(whic)n(h)35 b(con)n(tains)e Fs(k)26 b Ft(+)d(1)34 b(consecutiv)n(e)g
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1523 y(do)r(es)28 b(not)h(con)n(tain)f(an)g(appro)n(ximate)f(matc)n(h.)
39 b(The)29 b(remaining)e(sub)n(w)n(ords)g(ma)n(y)h(con)n(tain)523
1622 y(appro)n(ximate)i(matc)n(hes.)47 b(Hence,)32 b(they)g(are)e
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1722 y(pro)r(cessed)d(b)n(y)i(a)f(dynamic)g(programming)f(computation.)
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2589 1898 y Fp(r)2626 1886 y Fs(;)g(c)2699 1898 y Fp(r)2736
1886 y Fs(;)g(w)2832 1898 y Fp(r)r Fq(+1)2953 1886 y
Ft(].)83 b(F)-7 b(or)42 b(eac)n(h)523 1986 y Fs(h;)14
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Fp(p)p Fl(\000)p Fq(1)2306 1986 y Fs(w)2365 1998 y Fp(h)p
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1998 y Fp(h;k)q Fq(+1)2872 1986 y Fr(j)i(\025)g Fs(m)t
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2284 y(using)j(a)g(dynamic)h(programming)d(computation.)648
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(ximate)e(matc)n(h)523 2510 y(within)38 b Fs(w)850 2522
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y Fq(4)2532 3511 y Fs(c)2568 3523 y Fq(4)2715 3511 y
Fs(w)2774 3523 y Fq(5)523 3659 y Ft(Ob)n(viously)-7 b(,)38
b Fs(s)980 3671 y Fq(1)p Fp(;k)q Fq(+1)1196 3659 y Ft(=)g
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y(and)33 b Fs(i)22 b Ft(+)g Fr(j)p Fs(s)890 3970 y Fq(2)p
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1665 y Fp(h;k)q Fq(+1)1087 1653 y Fr(j)23 b(\025)g Fs(m)18
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1623 y Fq(3)1768 1653 y FA(.)648 1836 y Ft(In)35 b(the)i(string)e
(searc)n(hing)f(literature,)j(this)f(result)f(is)h(often)g(cited)g(but)
g(few)g(p)r(eople)523 1935 y(seem)26 b(to)g(kno)n(w)f(ho)n(w)g(these)h
(constan)n(ts)f(can)g(actually)h(b)r(e)g(computed.)36
b(In)27 b([5)o(],)g(it)f(is)g(sho)n(wn)523 2035 y(ho)n(w)h
Fs(c)732 2047 y Fq(1)797 2035 y Ft(and)g Fs(c)994 2047
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(b)r(elo)n(w.)p Black 523 2218 a Fm(Theorem)15 b(6.)p
Black 40 w FA(If)36 b Fs(k)g Fr(\024)1357 2176 y Fp(m)p
Fl(\000)p Fq(6)11 b(log)1598 2192 y Fj(b)1640 2176 y
Fp(m)p Fq(+3)p 1308 2199 524 4 v 1308 2246 a(log)1394
2262 y Fj(b)1436 2246 y Fp(m)p Fq(+2+2)g(log)1760 2262
y Fj(b)1802 2246 y Fp(c)1864 2218 y Fr(\000)23 b Ft(2)p
FA(,)36 b(wher)l(e)g Fs(c)e Ft(=)2463 2155 y Fk(P)2550
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2174 y Fp(m)2550 2243 y(d)p Fq(=0)2852 2218 y Ft(\()2920
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(and)i(\014ltering)523 2721 y(phase)31 b(requires)e Fr(O)r
Ft(\()p Fs(n)p Ft(\))j(time,)h(it)f(follo)n(ws)e(that)h(the)h(exp)r
(ected)f(running)g(time)g(of)h(LET)e(is)523 2820 y Fr(O)r
Ft(\()p Fs(n)p Ft(\).)37 b(The)25 b(same)g(holds)g(for)g(the)h(impro)n
(v)n(ed)e(v)n(ersion)g(DLET,)h(but)h(the)g(b)r(ound)f(for)g
Fs(k)j Ft(can)523 2920 y(b)r(e)g(w)n(eak)n(ened;)e(see)i([5)o(].)523
3169 y Fm(3.4)95 b(Exp)s(erimen)m(tal)30 b(Results)523
3335 y Ft(In)j(our)e(exp)r(erimen)n(ts)h(w)n(e)g(v)n(eri\014ed)f(that)i
(our)f(dynamic)g(\014ltering)g(tec)n(hnique,)h(when)g(ap-)523
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(and)h(SET,)f(leads)h(to)f(an)h(impro)n(v)n(ed)f(critical)523
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(series)g(w)n(e)g(used)g(random)g(text)g(strings)g Fs(T)41
b Ft(of)31 b(length)f Fs(n)e Ft(=)f(500)p Fs(;)p Ft(000)h(o)n(v)n(er)
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b Ft(4)p Fs(;)g Ft(10)31 b(and)i(40.)54 b(W)-7 b(e)33
b(c)n(hose)g(patterns)g(of)g(a)g(\014xed)g(length)h Fs(m)f
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b(Figures)26 b(2)g(and)g(3)g(sho)n(w)g(the)h(e\013ect)g(of)f(a)g(v)-5
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(programming.)52 b(It)34 b(can)f(b)r(e)g(seen)h(that)f(in)h(order)e(to)
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-7 b(,)31 b(algorithms)d(DLET)i(and)f(DSET)h(allo)n(w)f(for)523
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y(alphab)r(et)20 b(size)g(w)n(as)f Fr(jAj)k Ft(=)g(80.)33
b(W)-7 b(e)21 b(c)n(hose)e(pattern)h(of)g(length)g Fs(m)j
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Ft(128)p Fr(g)h Ft(o)n(v)n(er)523 3663 y(the)31 b(same)f(alphab)r(et.)
46 b(Figures)29 b(5)h(and)h(6)f(sho)n(w)f(ho)n(w)h(a)g(v)-5
b(arying)30 b(pattern)g(length)h(e\013ects)523 3763 y(the)39
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(resp)r(ectiv)n(ely)-7 b(.)38 b(The)29 b(larger)d(the)j(pattern,)f(the)
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Fs(k)3103 4082 y Fq(max)3230 4062 y Ft(.)44 b(All)523
4161 y(algorithms)23 b(ac)n(hiev)n(e)g(a)h(larger)f(v)-5
b(alue)24 b(of)h Fs(k)1861 4182 y Fq(max)2012 4161 y
Ft(with)g Fs(m)g Ft(b)r(ecoming)f(larger.)34 b(The)24
b(dynamic)523 4261 y(\014lter)k(algorithms)e(are)g(alw)n(a)n(ys)g(sup)r
(erior)h(to)g(the)h(static)f(\014lter)h(algorithms.)648
4361 y(There)23 b(is)h(virtually)f(no)h(time)g(p)r(enalt)n(y)g(for)g
(the)g(complex)g(dynamic)f(\014lter)h(algorithms,)523
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35 b(When)26 b(the)g(static)g(\014lter)f(lo)r(oses)g(its)g(e\013ect,)
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4680 y Fq(max)1137 4659 y Ft(,)g(then)f(the)g(dynamic)g(\014lter)g(has)
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(preferable)g(an)n(yw)n(a)n(y)-7 b(.)p Black 1922 5112
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Fy(j)p Fi(A)p Fy(j)22 b Fz(=)f(80)26 b(\(SET)g(and)f(DSET\))644
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 @endspecial 1223 2381 a Fw(Fig.)30 b(7.)j Fi(k)1544
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%%Page: 1 1
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%%Page: 12 12
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(area)h(is)h(su\016cien)n(tly)g(large)e(\(see)i(Theorem)g(3)f(in)i([9)o
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Black eop
%%Page: 13 13
13 12 bop Black Black 648 387 a Ft(Notice)32 b(that)h(for)g
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b Fr(\024)f Ft(\()p Fs(j)755 1346 y Fp(b)811 1334 y Ft(+)22
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Fs(j)1587 1346 y Fp(b)1643 1334 y Ft(+)22 b Fs(k)j Ft(+)d
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Fs(j)2370 1346 y Fp(b)2426 1334 y Ft(+)22 b(1\))p Fs(h)g
Fr(\000)g Fs(q)j Ft(+)d Fs(m)h Ft(+)e Fs(k)s Ft(.)55
b Fs(v)37 b Ft(will)523 1433 y(b)r(e)c(detected)g(if)g(DLEQ)f(computes)
g Fs(D)r Ft(\()p Fs(i;)14 b(r)r Ft(\))34 b(for)e(all)g
Fs(i;)14 b(r)35 b Ft(with)e(0)e Fr(\024)f Fs(i)h Fr(\024)g
Fs(m)p Ft(,)j Fs(b)d Fr(\024)f Fs(r)k Fr(\024)d Fs(e)p
Ft(.)523 1533 y(Therefore)26 b(w)n(e)i(ha)n(v)n(e)e(to)i(sho)n(w:)p
Black 596 1699 a(\()p Fs(i)p Ft(\))p Black 42 w FA(BPM)q
Ft(\()p Fs(j)986 1711 y Fp(b)1020 1699 y Ft(\))g(is)f(true,)p
Black 567 1865 a(\()p Fs(ii)p Ft(\))p Black 42 w FA(CPM)q
Ft(\()p Fs(j)5 b Ft(\))28 b(ist)g(true)f(for)g(eac)n(h)g
Fs(j;)14 b(j)1762 1877 y Fp(b)1814 1865 y Ft(+)k(1)23
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1865 y Ft(+)c Fs(k)j Ft(+)d Fs(s)p Ft(.)p Black 539 2031
a(\()p Fs(iii)p Ft(\))p Black 41 w(If)28 b Fs(j)g(>)22
b(j)997 2043 y Fp(b)1049 2031 y Ft(+)c Fs(k)k Ft(+)c
Fs(s)27 b Ft(and)h FA(CPM)q Ft(\()p Fs(j)5 b Ft(\))28
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Fs(T)n(sam)p Ft(\()p Fs(j)2601 2209 y Fp(b)2653 2197
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Fs(l)h Fr(\024)f Fs(k)e Ft(+)d Fs(s)p Ft(,)28 b Fs(T)n(sam)p
Ft(\()p Fs(j)2033 2309 y Fp(b)2084 2297 y Ft(+)18 b Fs(l)r
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Fs(ii)p Ft(\))i(Consider)f(an)h(arbitrary)d(but)k(\014xed)f
Fs(j;)14 b(j)1954 2409 y Fp(b)2007 2397 y Ft(+)19 b(1)25
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y(By)h(construction)f(w)n(e)h(ha)n(v)n(e)f Fs(A)1519
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b Fs(l)e Ft(=)2546 2927 y Fk(j)2599 2979 y Fl(j)p Fp(W)9
b Fq(\()p Fp(i;j)s(h)p Fq(\))p Fl(j\000)p Fp(q)p 2599
3000 361 4 v 2760 3048 a(h)2988 3019 y Ft(+)18 b(1)3113
2927 y Fk(k)3156 3019 y Ft(.)37 b(Since)523 3177 y Fs(b)23
b Fr(\024)f Ft(\()p Fs(j)735 3189 y Fp(b)769 3177 y Ft(+1\))p
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Fs(l)24 b Ft(=)1643 3085 y Fk(j)1697 3140 y Fp(j)s(h)p
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4 v 1837 3206 a(h)2045 3177 y Ft(+)18 b(1)2170 3085 y
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3137 y Fq(+1\))p Fp(h)p Fl(\000)p Fp(q)r Fq(+1\)+1)p
Fl(\000)p Fp(q)p 2377 3158 744 4 v 2730 3206 a(h)3149
3177 y Ft(+)g(1)3274 3085 y Fk(k)3340 3177 y Ft(=)523
3243 y Fk(j)577 3294 y Fp(j)s(h)p Fl(\000)p Fq(\()p Fp(j)751
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Fs(A)947 3471 y Fq(2)985 3459 y Ft(\))g(=)g Fs(\016)s
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Fs(b)p Ft(.)37 b(Eac)n(h)26 b(of)h(these)g(can)g(prev)n(en)n(t)g(at)g
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Ft(\()p Fs(j)18 b Ft(+)c(1\))p Fs(;)g(:)g(:)g(:)f(;)h(T)n(sam)p
Ft(\()p Fs(j)k Ft(+)c Fs(k)j Ft(+)c Fs(s)h Fr(\000)g
Ft(\()p Fs(j)19 b Fr(\000)14 b Fs(j)1981 3670 y Fp(b)2014
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Fp(b)1097 3957 y Ft(,)30 b(w)n(e)f(ha)n(v)n(e)g FA(GD)p
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FA(CPM)q Ft(\()p Fs(j)5 b Ft(\))28 b(holds.)523 4157
y(\()p Fs(iii)p Ft(\))33 b(Let)h Fs(j)k(>)33 b(j)1066
4169 y Fp(b)1122 4157 y Ft(+)22 b Fs(k)j Ft(+)d Fs(s)34
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4256 y(a)k(con)n(tradiction.)68 b(There)38 b(is)g(an)g
Fs(i;)14 b Ft(0)40 b Fr(\024)h Fs(i)f Fr(\024)h Fs(m)d
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Ft(\))41 b Fr(\024)f Fs(k)i Ft(and)c Fs(b)i Ft(=)523
4356 y Fs(j)5 b(h)24 b Fr(\000)g(j)p Fs(W)12 b Ft(\()p
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(the)523 4455 y Fs(q)s Ft(-samples)1077 4555 y Fs(T)n(sam)p
Ft(\()p Fs(j)1346 4567 y Fp(b)1398 4555 y Ft(+)18 b(1\))p
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y Fp(b)2059 4555 y Ft(+)k Fs(k)j Ft(+)d Fs(s)p Ft(\))p
Fs(;)c(:)g(:)g(:)g(;)g(T)n(sam)p Ft(\()p Fs(j)5 b Ft(\))p
Black 448 w(\(2\))p Black 523 4705 a(Since)35 b Fs(j)k(>)c(j)954
4717 y Fp(b)1010 4705 y Ft(+)23 b Fs(k)j Ft(+)d Fs(s)p
Ft(,)36 b(the)f(sequence)f(\(2\))h(is)f(of)h(length)f(at)h(least)f
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Fr(j)26 b(\025)f Ft(\()p Fs(k)e Ft(+)c Fs(s)p Ft(\))p
Fs(h)h Ft(+)f Fs(q)s Ft(.)42 b(Let)29 b Fs(l)f Ft(=)1795
4737 y Fk(j)1849 4789 y Fl(j)p Fp(W)9 b Fq(\()p Fp(i;j)s(h)p
Fq(\))p Fl(j\000)p Fp(q)p 1849 4810 361 4 v 2010 4858
a(h)2237 4829 y Ft(+)18 b(1)2362 4737 y Fk(k)2406 4829
y Ft(.)42 b(Then)29 b Fs(l)f Fr(\025)d Fs(k)d Ft(+)e
Fs(s)f Ft(+)g(1)29 b(whic)n(h)p Black 1922 5112 a(13)p
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%%Page: 14 14
14 13 bop Black Black 523 387 a Ft(implies)21 b FA(GD)p
Ft(\()p Fs(j;)14 b Ft(1)p Fs(;)g(k)7 b Ft(+)e Fs(s)g
Fr(\000)g Fs(l)r Ft(\))22 b(=)h(0.)34 b(Th)n(us)20 b(w)n(e)h(conclude)f
Fs(D)r Ft(\()p Fs(i;)14 b(j)5 b(h)p Ft(\))g(+)g FA(GD)o
Ft(\()p Fs(j;)14 b Ft(1)p Fs(;)g(k)7 b Ft(+)e Fs(s)g
Fr(\000)g Fs(l)r Ft(\))22 b Fr(\024)h Fs(k)s Ft(,)523
487 y(whic)n(h)k(means)g(that)g FA(CPM)q Ft(\()p Fs(j)5
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b Fb(2)648 670 y Ft(Lik)n(e)19 b(LEQ)g(\(see)h([9]\),)i(an)e(e\016cien)
n(t)g(implemen)n(tation)g(of)g(DLEQ)g(utilizes)g(the)h(shift-add)523
769 y(tec)n(hnique)28 b(of)f([1]:)37 b(for)27 b(eac)n(h)g
Fs(j;)14 b Ft(0)22 b Fr(\024)h Fs(j)28 b Fr(\024)1826
702 y Fk(\004)1875 737 y Fp(n)p 1875 751 42 4 v 1876
798 a(h)1926 702 y Fk(\005)1993 769 y Ft(a)f(v)n(ector)f
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1015 y Fs(M)1078 1027 y Fp(j)1113 1015 y Ft(\()p Fs(i)p
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Fs(j)1420 1069 y Ft(0)1075 b(otherwise)523 1253 y(One)40
b(easily)f(sho)n(ws)g(that)h Fs(M)1476 1265 y Fp(j)s
Fq(+1)1595 1253 y Ft(\()p Fs(i)27 b Ft(+)f(1\))44 b(=)f
Fs(M)2081 1265 y Fp(j)2115 1253 y Ft(\()p Fs(i)p Ft(\))27
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1353 y FA(GD)p Ft(\()p Fs(j;)14 b Ft(1)p Fs(;)g(r)r Ft(\))24
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Fp(r)1330 1353 y Ft(\()p Fs(r)r Ft(\))23 b(hold.)35 b(As)21
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1452 y Fs(M)604 1464 y Fp(j)664 1452 y Ft(b)n(y)26 b(some)f(simple)g
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Fs(j)5 b Ft(\))523 1651 y(can)32 b(b)r(e)i(decided)f(in)g
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b(the)f(dynamic)f(c)n(hec)n(king)g(in)h(DLEQ)f(requires)523
1751 y Fr(O)r Ft(\()p Fs(mn=h)p Ft(\))c(time)g(in)g(the)g(w)n(orst)e
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b Ft(+)20 b Fs(l)r Ft(\))28 b Fr(62)g Fs(Q)3192 2278
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(y)g(Chang)g(and)g(Marr)f([3])523 2465 y(in)h(order)e(to)h(obtain)g(a)g
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%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF