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[(.)-679.3(W)82.6(e)-418(denote)]TJ
/F7 1 Tf
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[(the)-406.7(length)-406(of)]TJ
/F12 1 Tf
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[(and)-409.9(b)30.1(y)]TJ
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[(the)-406.7(empt)30.1(y)-409(string.)-687.5(F)76.9(o)0(r)-404.4(a)0(n)30.1(y)-407(indices)]TJ
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[(,)-328(w)30.1(e)-333.2(denote)-333.2(b)30.1(y)]TJ
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(i)Tj
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[(-th)-332(letter)-335.1(of)]TJ
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[(,)-340.8(and)-329.6(b)30.1(y)]TJ
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/F2 1 Tf
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[(])-330.3(the)-336.4(substring)]TJ
/F12 1 Tf
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(i)Tj
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()Tj
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(a)Tj
/F14 1 Tf
7.9701 0 0 7.9701 157.71 584.7 Tm
(j)Tj
/F2 1 Tf
11.9552 0 0 11.9552 166.23 586.5 Tm
[(that)-342.9(starts)-335.9(with)]TJ
/F12 1 Tf
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(i)Tj
/F2 1 Tf
0.6725 0 TD
[(and)-349.7(ends)-339.1(with)]TJ
/F12 1 Tf
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(j)Tj
/F2 1 Tf
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[(.)-480.8(F)76.9(or)-334.2(a)-343.5(set)]TJ
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/F12 1 Tf
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(A)Tj
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()Tj
/F2 1 Tf
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[(,)-350.3(w)30.1(e)-343.2(denote)-343.2(b)30.1(y)]TJ
/F7 1 Tf
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(|)Tj
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[(the)-326.4(cardinalit)30.1(y)-321.9(o)0(f)]TJ
/F12 1 Tf
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[(and)-319.6(b)30.1(y)]TJ
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(||)Tj
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(S)Tj
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(s)Tj
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(S)Tj
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(|)Tj
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[(b)-30.1(e)-325.7(the)-326.4(total)-326.1(size)-328.4(of)-325.4(the)-326.4(strings)-322.3(in)]TJ
/F12 1 Tf
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(S)Tj
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(.)Tj
-33.987 -1.7666 TD
[(F)76.9(o)0(r)-434.5(strings)]TJ
/F12 1 Tf
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[(u,)-174.5(v)-40.1(,)-168.1(w)]TJ
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(A)Tj
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()Tj
/F2 1 Tf
11.9552 0 0 11.9552 221.43 549.42 Tm
[(,)-470.8(w)30.1(e)-443.6(s)0(a)30.1(y)-444.3(that)]TJ
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(v)Tj
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[(,)-470.8(and)]TJ
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(w)Tj
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[(,)-460.8(resp)-30.1(ectiv)30.1(ely)111.6(,)-500.9(are)-451(a)]TJ
/F11 1 Tf
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[(pr)49.3(ex)]TJ
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[(,)-440.7(and)-420(a)]TJ
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(sux)Tj
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[(of)-425.8(a)-413.8(string)]TJ
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[(uv)-40.1(w)]TJ
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[(.)-701.7(Then,)-448.8(the)-416.7(substring)]TJ
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(v)Tj
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[(o)-30.1(ccurs)-410.6(in)]TJ
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(at)Tj
-35.4023 -1.335 TD
[(p)-30.1(osition)]TJ
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(p)Tj
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(|)Tj
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[(+)-222.1(1)0(.)-473.1(Equiv)66.5(alen)30.1(tly)89.7(,)-360.4(i)0(f)]TJ
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([1])Tj
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([)Tj
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(n)Tj
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[(],)-336.6(then)]TJ
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(v)Tj
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[(o)-30.1(ccurs)-340.4(in)]TJ
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(s)Tj
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[(at)-334.2(p)-30.1(osition)]TJ
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(i)Tj
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(i)Tj
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(v)Tj
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([)Tj
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(])Tj
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([)Tj
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(i)Tj
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(|)Tj
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(v)Tj
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(|)Tj
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[(1].)-421.9(The)]TJ
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[(left)-321.2(p)48.3(osition)]TJ
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[(and)-299.5(the)]TJ
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[(right)-321.2(p)48.3(osition)]TJ
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(v)Tj
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[(corresp)-30.1(onding)]TJ
-30.3133 -1.325 TD
[(to)-354.2(this)-354.3(o)-30.1(ccurrence)-343.2(of)-355.5(in)]TJ
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(t)Tj
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(i)Tj
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(and)Tj
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(i)Tj
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(|)Tj
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(v)Tj
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(|)Tj
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[(1,)-352.6(resp)-30.1(ectiv)30.1(ely)111.6(.)-541.1(The)]TJ
/F11 1 Tf
8.8631 0 TD
[(r)51.1(eversal)]TJ
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3.734 0 TD
[(of)-355.5(a)-353.6(string)]TJ
/F12 1 Tf
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(x)Tj
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(=)Tj
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(a)Tj
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(1)Tj
/F7 1 Tf
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0.1739 Tc
()Tj
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(a)Tj
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(m)Tj
/F2 1 Tf
11.9552 0 0 11.9552 148.95 469.74 Tm
[(is)-325.5(dened)-326.4(b)30.1(y)]TJ
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(x)Tj
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(R)Tj
/F2 1 Tf
11.9552 0 0 11.9552 234.03 469.74 Tm
(=)Tj
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(a)Tj
/F14 1 Tf
7.9701 0 0 7.9701 252.51 467.94 Tm
(n)Tj
/F7 1 Tf
11.9552 0 0 11.9552 260.19 469.74 Tm
0.1739 Tc
()Tj
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1.335 0 TD
0 Tc
(a)Tj
/F10 1 Tf
7.9701 0 0 7.9701 282.27 467.94 Tm
(1)Tj
/F2 1 Tf
11.9552 0 0 11.9552 287.07 469.74 Tm
(.)Tj
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14.3462 0 0 14.3462 94.95 440.7 Tm
[(2.1.)-500.1(T)88.1(ext)-370.9(and)-369.5(P)33.5(atterns)]TJ
/F2 1 Tf
11.9552 0 0 11.9552 94.95 416.58 Tm
[(In)-327.4(this)-334.2(subsection,)-339.8(w)30.1(e)-333.2(in)30.1(tro)-30.1(duce)-330.9(the)-336.4(class)]TJ
/F7 1 Tf
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[(ER)94.5(P)]TJ
/F2 1 Tf
2.5796 0 TD
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[(of)-325.4(erasing)-335.2(regular)-330.6(patterns)-333.3(of)-335.4(Shi-)]TJ
-21.8215 -1.335 TD
[(nohara)-465.3([12)10(],)-503.5(also)-470.5(kno)30.1(wn)-460.8(as)-469.2(exible)-464(patterns)-463.8([9)10(].)-852.5(Let)-469.3()-507.4(=)]TJ
/F7 1 Tf
26.3986 0 TD
({)Tj
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[(a,)-169.5(b,)-164.9(c,)-169.8(.)-169.7(.)-169.7(.)]TJ
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3.8161 0 TD
(})Tj
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[(b)-30.1(e)-456.2(a)-464(nite)]TJ
-31.6784 -1.335 TD
[(alphab)-30.1(et)-275.9(of)]TJ
/F11 1 Tf
5.079 0 TD
[(c)48.5(onstant)-313(symb)47.2(ols)]TJ
/F2 1 Tf
7.1768 0 TD
[(.)-410.6(W)82.6(e)-287.5(assume)-275.4(a)-283.3(s)0(p)-30.1(ecial)-277.6(sym)30.1(b)-30.1(o)0(l)]TJ
/F7 1 Tf
12.8881 0 TD
()Tj
0.7729 0 TD
()Tj
T*
()Tj
/F2 1 Tf
0.9435 0 TD
[()-286.5(called)-282.4(a)]TJ
/F11 1 Tf
4.4968 0 TD
(variable)Tj
/F2 1 Tf
3.6637 0 TD
(\(or)Tj
/F11 1 Tf
-35.0208 -1.335 TD
[(variable-length)-407.2(don)94.5(t)-398.8(c)48.5(a)0(r)49.3(e)0(s)]TJ
/F2 1 Tf
11.2721 0 TD
(,)Tj
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(VLDC)Tj
/F2 1 Tf
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[(\),)-381.1(whic)30.1(h)-388.6(represen)30.1(ts)-389.3(arbitrary)-390.8(long)-393.1(p)-30.1(ossibly-empt)30.1(y)]TJ
-14.8756 -1.325 TD
[(nite)-293.4(string)-297(in)-298.2()]TJ
/F13 1 Tf
7.9701 0 0 7.9701 163.83 357.18 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 168.63 352.86 Tm
[(.)-430.6(Then,)-298.2(an)]TJ
/F11 1 Tf
4.8381 0 TD
[(er)49.5(asing)-319.4(r)51.1(e)48.5(gular)-331.8(p)48.3(attern)]TJ
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(\(or)Tj
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[(p)48.3(attern)]TJ
/F2 1 Tf
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[(,)-310.2(for)-295.9(short\))-298.6(o)30.1(v)30.1(e)0(r)-305.7()-296.6(is)-295.4(a)]TJ
-26.8503 -1.335 TD
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/F12 1 Tf
2.7101 0 TD
(P)Tj
/F7 1 Tf
1.0439 0 TD
()Tj
/F2 1 Tf
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(\()Tj
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[()-86.1({})]TJ
/F2 1 Tf
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7.9701 0 0 7.9701 181.59 341.22 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 189.39 336.9 Tm
[(consisting)-257.2(of)-255.1(constan)30.1(t)-264.6(sym)30.1(b)-30.1(ols)-263.1(and)-259.4(v)55(ariables.)-413.6(An)-257.8(erasing)-264.9(regular)]TJ
-9.1743 -1.335 TD
(pattern)Tj
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(P)Tj
/F2 1 Tf
1.0841 0 TD
[(is)-315.4(said)-305.7(to)-314.1(b)-30.1(e)]TJ
/F11 1 Tf
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[(in)-334.4(c)48.5(anonic)39.5(al)-324(form)]TJ
/F2 1 Tf
7.8694 0 TD
[(if)-312.1(it)-310.8(is)-315.4(written)-312.1(as)]TJ
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(P)Tj
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(=)Tj
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1.0339 0 TD
(w)Tj
/F10 1 Tf
7.9701 0 0 7.9701 415.71 319.14 Tm
(0)Tj
/F7 1 Tf
11.9552 0 0 11.9552 422.79 320.94 Tm
()Tj
/F12 1 Tf
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(w)Tj
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7.9701 0 0 7.9701 439.47 319.14 Tm
(1)Tj
/F7 1 Tf
11.9552 0 0 11.9552 446.43 320.94 Tm
0.1739 Tc
[()-18.7()-10()]TJ
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(w)Tj
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7.9701 0 0 7.9701 487.71 319.14 Tm
(m)Tj
/F2 1 Tf
11.9552 0 0 11.9552 499.47 320.94 Tm
(for)Tj
-35.1112 -1.335 TD
[(some)-292.1(in)30.1(teger)]TJ
/F12 1 Tf
5.611 0 TD
(m)Tj
/F7 1 Tf
1.1342 0 TD
()Tj
/F2 1 Tf
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[(0)-293.3(and)-289.5(some)-292.1(non-empt)30.1(y)-291.8(strings)]TJ
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13.0488 0 TD
(w)Tj
/F10 1 Tf
7.9701 0 0 7.9701 337.35 303.18 Tm
(0)Tj
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11.9552 0 0 11.9552 342.03 304.98 Tm
0.1697 Tc
(,w)Tj
/F10 1 Tf
7.9701 0 0 7.9701 355.71 303.18 Tm
0 Tc
(1)Tj
/F12 1 Tf
11.9552 0 0 11.9552 360.39 304.98 Tm
0.1697 Tc
[(,...)10.1(,w)]TJ
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7.9701 0 0 7.9701 395.07 303.18 Tm
0 Tc
(m)Tj
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11.9552 0 0 11.9552 406.35 304.98 Tm
()Tj
/F2 1 Tf
0.9435 0 TD
()Tj
/F10 1 Tf
7.9701 0 0 7.9701 426.03 309.3 Tm
(+)Tj
/F2 1 Tf
11.9552 0 0 11.9552 433.11 304.98 Tm
[(.)-420.6(Eac)30.1(h)-294.2(constan)30.1(t)]TJ
-29.5604 -1.335 TD
(string)Tj
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(w)Tj
/F14 1 Tf
7.9701 0 0 7.9701 121.95 287.22 Tm
(i)Tj
/F2 1 Tf
11.9552 0 0 11.9552 129.87 289.02 Tm
[(is)-385.7(called)-382.7(a)]TJ
/F11 1 Tf
4.7477 0 TD
[(se)47(gment)]TJ
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(of)Tj
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(P)Tj
/F2 1 Tf
0.7729 0 TD
[(.)-591.2(A)0(n)-388.2(erasing)-375.4(regular)-380.8(pattern)]TJ
/F11 1 Tf
12.8179 0 TD
[(in)-394.7(c)48.5(anonic)39.5(al)-394.3(form)]TJ
/F2 1 Tf
8.0702 0 TD
(is)Tj
-35.6231 -1.335 TD
[(also)-259.7(called)-272.3(as)-258.4(a)]TJ
/F11 1 Tf
6.5043 0 TD
[(exible)-297.1(p)48.3(attern)]TJ
/F2 1 Tf
6.5746 0 TD
[(or)-263.9(a)]TJ
/F11 1 Tf
1.8871 0 TD
[(VLDC)-292.8(p)48.3(attern)]TJ
/F2 1 Tf
6.424 0 TD
[([9].)-410.9(W)82.6(e)-267.4(denote)-262.9(b)30.1(y)]TJ
/F7 1 Tf
7.7289 0 TD
0.0945 Tc
[(ER)94.5(P)]TJ
/F2 1 Tf
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7.9701 0 0 7.9701 159.51 717.42 Tm
(1)Tj
/F12 1 Tf
11.9552 0 0 11.9552 164.19 719.22 Tm
0.1697 Tc
(,M)Tj
/F10 1 Tf
7.9701 0 0 7.9701 180.75 717.42 Tm
0 Tc
(0)Tj
/F12 1 Tf
11.9552 0 0 11.9552 185.55 719.22 Tm
0.1697 Tc
(,N)Tj
/F10 1 Tf
7.9701 0 0 7.9701 200.19 717.42 Tm
0 Tc
(1)Tj
/F12 1 Tf
11.9552 0 0 11.9552 204.87 719.22 Tm
0.1697 Tc
(,N)Tj
/F10 1 Tf
7.9701 0 0 7.9701 219.51 717.42 Tm
0 Tc
(0)Tj
/F2 1 Tf
11.9552 0 0 11.9552 224.19 719.22 Tm
[(\))-362(corresp)-30.1(onding)-358.9(to)-364.3(a)-353.6(sample)]TJ
/F7 1 Tf
12.4565 0 TD
(S)Tj
/F2 1 Tf
0.6725 0 TD
[(,)-370.4(a)0(n)-361.7(o)0(b)-50.5(jectiv)30.1(e)-367.5(function)]TJ
/F12 1 Tf
10.1078 0 TD
(F)Tj
/F2 1 Tf
0.7729 0 TD
(,)Tj
-36.0145 -1.325 TD
[(and)-279.4(pattern)]TJ
/F12 1 Tf
5.2998 0 TD
(P)Tj
/F2 1 Tf
0.7729 0 TD
[(.)-410.6(F)76.9(or)-284(an)30.1(y)-276.5(patterns)]TJ
/F12 1 Tf
8.02 0 TD
0.0301 Tc
[(P,)-140.8(Q)]TJ
/F2 1 Tf
1.8796 0 TD
0.2875 Tc
[(,i)287.5(f)]TJ
/F12 1 Tf
1.4127 0 TD
0 Tc
(LO)Tj
/F2 1 Tf
1.4454 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
0.0301 Tc
[(P,)-141.5(S)]TJ
/F2 1 Tf
1.7673 0 TD
0.2616 Tc
(\)=)Tj
/F12 1 Tf
1.6863 0 TD
0 Tc
(LO)Tj
/F2 1 Tf
1.4454 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
[(Q,)-167.7(S)]TJ
/F2 1 Tf
1.8777 0 TD
[(\))-271.7(then)]TJ
/F12 1 Tf
2.8406 0 TD
(P)Tj
/F2 1 Tf
1.0439 0 TD
(and)Tj
/F12 1 Tf
1.867 0 TD
(Q)Tj
/F2 1 Tf
1.0539 0 TD
(realizes)Tj
-33.1739 -1.335 TD
[(the)-336.4(same)-332.3(classication)-325.2(function)]TJ
/F12 1 Tf
13.7213 0 TD
(P)Tj
/F2 1 Tf
1.0539 0 TD
(:)Tj
/F7 1 Tf
0.5521 0 TD
0.358 Tc
[(S)73.2({)]TJ
/F2 1 Tf
2.7484 0 TD
0 Tc
(0)Tj
/F12 1 Tf
0.4896 0 TD
(,)Tj
/F2 1 Tf
0.4458 0 TD
(1)Tj
/F7 1 Tf
0.4896 0 TD
(})Tj
/F2 1 Tf
0.5 0 TD
[(,)-324.4(and)-339.7(th)30.1(us)-333.3(giv)30.1(es)-329.4(the)-336.4(same)-322.2(con)30.1(tingency)]TJ
-20.0006 -1.335 TD
[(table.)-437.1(Hence,)-326.4(the)-326.4(result)-331.3(follo)30.1(ws.)]TJ
/F21 1 Tf
35.5328 -0.0502 TD
()Tj
ET
q
409.2 0 0 -0.48 90.018 651.288 cm
/Im1 Do
Q
BT
/F3 1 Tf
11.9552 0 0 11.9552 90.03 630.42 Tm
(Algorithm)Tj
/F5 1 Tf
5.3901 0 TD
(FindOptMax)Tj
/F2 1 Tf
6.4612 0 TD
(\()Tj
/F7 1 Tf
0.3808 0 TD
(S)Tj
/F12 1 Tf
0.6761 0 TD
0.1697 Tc
(,F)Tj
/F2 1 Tf
1.2145 0 TD
0 Tc
(\))Tj
/F11 1 Tf
-14.1228 -1.2045 TD
(input)Tj
/F2 1 Tf
2.2 0 TD
[(:)-438.9(A)-330(sample)]TJ
/F7 1 Tf
5.0371 0 TD
(S)Tj
/F2 1 Tf
0.6725 0 TD
[(,)-340.3(a)0(n)-321.5(o)0(b)-50.5(jectiv)30.1(e)-337.4(function)]TJ
/F12 1 Tf
9.9672 0 TD
(F)Tj
/F2 1 Tf
1.0439 0 TD
(:)Tj
/F7 1 Tf
0.5521 0 TD
0.348 Tc
[(S)63.2({)]TJ
/F2 1 Tf
2.7384 0 TD
0 Tc
(0)Tj
/F12 1 Tf
0.4896 0 TD
(,)Tj
/F2 1 Tf
0.4458 0 TD
(1)Tj
/F7 1 Tf
0.4896 0 TD
(})Tj
/F2 1 Tf
0.5 0 TD
[(,)-314.3(and)-329.6(an)-321.5(in)30.1(teger)]TJ
/F12 1 Tf
7.0906 0 TD
(K)Tj
/F7 1 Tf
1.1844 0 TD
()Tj
/F2 1 Tf
1.0539 0 TD
(1.)Tj
/F11 1 Tf
-33.465 -1.2045 TD
(output)Tj
/F2 1 Tf
2.7 0 TD
[(:)-440.8(The)-321.2(top-)]TJ
/F12 1 Tf
4.461 0 TD
(K)Tj
/F2 1 Tf
1.2304 0 TD
[(optimal)-329.6(patterns)]TJ
/F12 1 Tf
7.4579 0 TD
(P)Tj
/F10 1 Tf
7.9701 0 0 7.9701 287.07 599.82 Tm
(1)Tj
/F12 1 Tf
11.9552 0 0 11.9552 291.87 601.62 Tm
0.1697 Tc
[(,...)20.1(,P)]TJ
/F14 1 Tf
7.9701 0 0 7.9701 325.59 599.82 Tm
0 Tc
(K)Tj
/F2 1 Tf
11.9552 0 0 11.9552 337.71 601.62 Tm
[(minimizing)-326.4(the)-326.4(cost)]TJ
/F12 1 Tf
8.7627 0 TD
(G)Tj
/F13 1 Tf
7.9701 0 0 7.9701 451.71 599.82 Tm
(S)Tj
/F14 1 Tf
0.7227 0 TD
(,F)Tj
/F2 1 Tf
11.9552 0 0 11.9552 466.83 601.62 Tm
(\()Tj
/F12 1 Tf
0.3808 0 TD
(P)Tj
/F2 1 Tf
0.7735 0 TD
(\).)Tj
-32.6721 -1.2145 TD
[(1)-985.9(Let)]TJ
/F7 1 Tf
3.222 0 TD
(Q)Tj
/F2 1 Tf
1.0941 0 TD
(:=)Tj
/F7 1 Tf
1.3149 0 TD
(\015)Tj
/F2 1 Tf
0.8231 0 TD
[(b)-30.1(e)-325.7(an)-331.6(empt)30.1(y)-328.7(priorit)30.1(y)-326.7(queue)-325.4(with)-324.4(the)-326.4(cost)-325.8(as)-328.7(the)-326.4(k)30.1(e)0(y)93.6(.)]TJ
-6.4541 -1.2045 TD
(2)Tj
/F3 1 Tf
1.4755 0 TD
[(foreac)30.1(h)]TJ
/F2 1 Tf
3.8444 0 TD
[(maximal)-323.2(pattern)]TJ
/F12 1 Tf
7.4779 0 TD
(P)Tj
/F7 1 Tf
1.0439 0 TD
0.2769 Tc
(M)Tj
/F2 1 Tf
2.4692 0 TD
0 Tc
(in)Tj
/F7 1 Tf
1.1443 0 TD
(S)Tj
/F3 1 Tf
1.0037 0 TD
[(do)-378.4(b)-30.1(egin:)]TJ
/F2 1 Tf
-18.459 -1.2045 TD
[(3)-2451.4(Let)]TJ
/F12 1 Tf
4.6976 0 TD
(LO)Tj
/F2 1 Tf
1.4454 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
0.0301 Tc
(P,)Tj
/F7 1 Tf
1.1048 0 TD
0 Tc
(S)Tj
/F2 1 Tf
0.6725 0 TD
[(\))-321.8(b)-30.1(e)-325.7(the)-326.4(lo)-30.1(cation)-321.2(list)-325.2(of)]TJ
/F12 1 Tf
10.1981 0 TD
(P)Tj
/F2 1 Tf
0.7729 0 TD
(;)Tj
-19.272 -1.2145 TD
[(4)-2451.4(Compute)-329.3(the)-326.4(con)30.1(tingency)-324.1(table)]TJ
/F12 1 Tf
16.7024 0 TD
()Tj
/F2 1 Tf
0.813 0 TD
0.2823 Tc
(=\()Tj
/F12 1 Tf
1.4247 0 TD
0 Tc
(M)Tj
/F10 1 Tf
7.9701 0 0 7.9701 327.75 541.98 Tm
(1)Tj
/F12 1 Tf
11.9552 0 0 11.9552 332.43 543.78 Tm
0.1697 Tc
(,M)Tj
/F10 1 Tf
7.9701 0 0 7.9701 348.99 541.98 Tm
0 Tc
(0)Tj
/F12 1 Tf
11.9552 0 0 11.9552 353.79 543.78 Tm
0.1697 Tc
(,N)Tj
/F10 1 Tf
7.9701 0 0 7.9701 368.43 541.98 Tm
0 Tc
(1)Tj
/F12 1 Tf
11.9552 0 0 11.9552 373.11 543.78 Tm
0.1697 Tc
(,N)Tj
/F10 1 Tf
7.9701 0 0 7.9701 387.75 541.98 Tm
0 Tc
(0)Tj
/F2 1 Tf
11.9552 0 0 11.9552 392.43 543.78 Tm
[(\))-331.9(from)]TJ
/F12 1 Tf
3.0213 0 TD
(LO)Tj
/F2 1 Tf
1.4454 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
0.0301 Tc
(P,)Tj
/F7 1 Tf
1.1048 0 TD
0 Tc
(S)Tj
/F2 1 Tf
0.6725 0 TD
(\);)Tj
-31.9193 -1.2948 TD
(5)Tj
/F7 1 Tf
2.941 0 TD
(Q)Tj
/F2 1 Tf
1.0941 0 TD
(:=)Tj
/F7 1 Tf
1.3149 0 TD
0.2172 Tc
[(Q)-9.5({)]TJ
/F2 1 Tf
2.4272 0 TD
0 Tc
(\()Tj
/F12 1 Tf
0.3808 0 TD
0.0301 Tc
[(P,)-143.3(G)]TJ
/F14 1 Tf
7.9701 0 0 7.9701 210.03 533.94 Tm
0 Tc
()Tj
/F13 1 Tf
0 -1.084 TD
(S)Tj
/F14 1 Tf
0.7227 0 TD
(,F)Tj
/F2 1 Tf
11.9552 0 0 11.9552 224.91 528.3 Tm
(\()Tj
/F12 1 Tf
0.3808 0 TD
(P)Tj
/F2 1 Tf
0.7735 0 TD
(\)\))Tj
/F7 1 Tf
0.7615 0 TD
(})Tj
/F2 1 Tf
0.5 0 TD
(;)Tj
-13.698 -1.2246 TD
(6)Tj
/F3 1 Tf
1.4755 0 TD
(end)Tj
/F2 1 Tf
-1.4755 -1.2045 TD
(7)Tj
/F3 1 Tf
1.4755 0 TD
(output)Tj
/F2 1 Tf
3.6336 0 TD
[(the)-326.4(top)]TJ
/F12 1 Tf
3.4328 0 TD
(K)Tj
/F2 1 Tf
1.2246 0 TD
(patterns)Tj
/F12 1 Tf
3.8644 0 TD
(P)Tj
/F2 1 Tf
1.0941 0 TD
(in)Tj
/F7 1 Tf
1.1443 0 TD
(Q)Tj
/F2 1 Tf
1.1443 0 TD
[(in)-328.3(terms)-321.3(of)]TJ
/F12 1 Tf
4.9886 0 TD
(G)Tj
/F13 1 Tf
7.9701 0 0 7.9701 362.31 497.46 Tm
(S)Tj
/F14 1 Tf
0.7227 0 TD
(,F)Tj
/F2 1 Tf
11.9552 0 0 11.9552 377.31 499.26 Tm
(\()Tj
/F12 1 Tf
0.3808 0 TD
(P)Tj
/F2 1 Tf
0.7735 0 TD
(\);)Tj
ET
q
409.2 0 0 -0.48 90.018 484.728 cm
/Im1 Do
Q
BT
11.9552 0 0 11.9552 80.67 465.06 Tm
[(Figure)-371.2(2:)-513.2(An)-368.2(algorithm)-362.7(for)-366.2(the)-366.5(optimal)-369.8(pattern)-368.2(disco)30.1(v)30.1(ery)-369.9(via)-367.8(maximal)-363.3(pattern)-368.2(en)30.1(u-)]TJ
0 -1.2145 TD
(meration.)Tj
1.2748 -3.1518 TD
[(F)76.9(rom)-240.9(Theorem)-247.4(6,)-262.3(w)30.1(e)-242.8(kno)30.1(w)-251.9(that)-242.5(the)-246.1(algorithm)]TJ
/F5 1 Tf
19.4226 0 TD
(FindOptMax)Tj
/F2 1 Tf
6.705 0 TD
[(correctly)-238.7(nds)-250.4(top-)]TJ
/F12 1 Tf
7.974 0 TD
(K)Tj
/F2 1 Tf
-35.3764 -1.335 TD
[(optimal)-349.7(patterns)-353.3(in)]TJ
/F7 1 Tf
8.6724 0 TD
(S)Tj
/F2 1 Tf
1.0339 0 TD
[(minimizing)-356.5(the)-346.5(cost)]TJ
/F12 1 Tf
8.843 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 311.67 402.54 Tm
()Tj
/F13 1 Tf
0 -1.069 TD
(S)Tj
/F14 1 Tf
0.7227 0 TD
(,F)Tj
/F2 1 Tf
11.9552 0 0 11.9552 326.67 396.9 Tm
(\()Tj
/F12 1 Tf
0.3808 0 TD
(P)Tj
/F2 1 Tf
0.7735 0 TD
[(\).)-511.6(The)-351.3(eciency)-346.5(of)-355.5(this)-344.3(heuristics)]TJ
-21.7312 -1.335 TD
[(algorithm)-493.2(hea)30.1(vily)-487(dep)-30.1(ends)-482.2(on)-492.2(that)-483.4(of)-496(en)30.1(umeration)-490.9(of)-486(maximal)-483.8(patterns)-493.9(in)]TJ
/F7 1 Tf
33.716 0 TD
(M)Tj
/F2 1 Tf
1.6963 0 TD
(at)Tj
-35.4123 -1.3249 TD
[(Line)-285.1(2.)-412.8(Therefore,)-288.5(our)-282(goal)-276.8(in)-278.1(the)-276.2(remainder)-279.8(of)-275.2(this)-274(pap)-30.1(er)-276.5(is)-275.3(to)-284(dev)30.1(elop)-276.5(a)-273.3(memory)-279.8(and)]TJ
0 -1.335 TD
[(time)-354.6(ecien)30.1(t)-355.5(e)0(n)30.1(umeration)-360.4(algorithm)-352.6(for)-346.1(maximal)-353.3(patterns)-353.3(in)]TJ
/F7 1 Tf
26.6596 0 TD
(M)Tj
/F2 1 Tf
1.5458 0 TD
[(for)-356.2(the)-356.5(class)]TJ
/F7 1 Tf
5.5608 0 TD
0.0945 Tc
[(ER)94.5(P)]TJ
/F2 1 Tf
2.2384 0 TD
0 Tc
(.)Tj
-36.0045 -1.335 TD
[(Since)-369.5(the)-376.6(n)30.1(u)0(m)30.1(b)-30.1(er)]TJ
/F7 1 Tf
7.9397 0 TD
(|M|)Tj
/F2 1 Tf
2.128 0 TD
[(of)-365.6(maximal)-373.3(patterns)-363.4(is)-375.7(exp)-30.1(onen)30.1(tial)-366.9(in)-368.5(the)-366.5(total)-376.3(input)-365.2(size,)-377.6(it)]TJ
-10.0676 -1.335 TD
[(is)-335.5(natural)-332.2(to)-334.1(consider)-337.5(output-sensitiv)30.1(e)-339.2(algorithm)-332.6(for)]TJ
/F7 1 Tf
22.5743 0 TD
(M)Tj
/F2 1 Tf
1.2009 0 TD
[(.)-464.4(I)0(n)-327.4(what)-337.7(follo)30.1(ws,)-345.6(w)30.1(e)-333.2(refer)-337.4(to)]TJ
-23.7752 -1.335 TD
[(the)-306.3(maxim)30.1(um)-296.9(computation)-306(time)-294.3(of)]TJ
/F7 1 Tf
15.1867 0 TD
(A)Tj
/F2 1 Tf
1.0941 0 TD
[(b)-30.1(e)0(t)30.1(w)30.1(een)-302.8(consecutiv)30.1(e)-305.4(outputs)-304.1(as)-298.6(the)]TJ
/F11 1 Tf
15.3172 0 TD
(delay)Tj
/F2 1 Tf
2.5596 0 TD
[(of)-295.3(an)]TJ
-34.1576 -1.335 TD
[(en)30.1(umeration)-330.3(algorithm)]TJ
/F7 1 Tf
10.0977 0 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(.)Tj
/F3 1 Tf
14.3462 0 0 14.3462 95.91 258.3 Tm
[(4.)-480.1(T)88.1(ree-shap)-33.5(ed)-330.7(Searc)33.5(h)-343.8(Route)-343.3(for)-332.3(P)33.5(osition-Maximal)-341.9(Flexible)]TJ
1.3551 -1.2463 TD
[(P)33.5(atterns)]TJ
/F2 1 Tf
11.9552 0 0 11.9552 95.91 224.58 Tm
[(In)-267.1(this)-274(section,)-288.3(w)30.1(e)-273(will)-263.6(presen)30.1(t)-276.6(our)-272(algorithm)-272.3(for)-265.8(nding)-274.6(p)-30.1(osition-maximal)-272.4(patterns)]TJ
-1.2748 -1.335 TD
[(in)-368.5(a)-363.6(giv)30.1(en)-372.4(input)-365.2(string)-367.3(for)-366.2(the)-356.5(class)]TJ
/F7 1 Tf
15.6987 0 TD
0.0945 Tc
[(ER)94.5(P)]TJ
/F2 1 Tf
2.6098 0 TD
0 Tc
[(of)-365.6(exible)-363.6(patterns.)-553.1(First,)-379.4(w)30.1(e)-363.3(in)30.1(tro)-30.1(duce)-371.1(a)]TJ
-18.3084 -1.335 TD
[(tree-shap)-30.1(ed)-418.7(searc)30.1(h)-420.7(route)]TJ
/F7 1 Tf
11.0714 0 TD
(T)Tj
/F2 1 Tf
1.2145 0 TD
[(for)-426.4(the)-416.7(space)-420(of)-425.8(all)-421.9(maximal)-413.5(patterns)-423.6(in)]TJ
/F7 1 Tf
18.0374 0 TD
(M)Tj
/F2 1 Tf
1.2009 0 TD
[(.)-725.4(Then,)-438.7(in)]TJ
-31.5242 -1.335 TD
[(the)-376.6(next)-371.7(section,)-378.6(w)30.1(e)-383.4(giv)30.1(e)-374.3(a)-363.6(memory)-380.1(ecien)30.1(t)-375.6(algorithm)-372.7(for)-366.2(en)30.1(umerating)-380.5(all)-371.7(maximal)]TJ
0 -1.3249 TD
[(patterns)-313.2(based)-301.2(on)-311.5(the)-296.3(depth-rst)-311.9(searc)30.1(h)-310.3(o)30.1(v)30.1(e)0(r)]TJ
/F7 1 Tf
19.3523 0 TD
(T)Tj
/F2 1 Tf
0.803 0 TD
[(.)-430.6(Out)-301.1(strategy)-304.4(is)-315.4(explained)-303.7(as)-308.6(follo)30.1(ws:)]TJ
-20.1553 -1.335 TD
[(First)-239.8(w)30.1(e)-232.8(dene)-238(a)-233.1(binary)-234.1(relation)-239.3(b)-30.1(e)0(t)30.1(w)30.1(een)-232.5(maximal)-242.9(patterns,)-252(called)-232.2(the)]TJ
/F11 1 Tf
29.5303 0 TD
[(p)48.3(a)0(r)49.3(ent)-260.9(function)]TJ
/F2 1 Tf
6.4742 0 TD
(,)Tj
-36.0045 -1.335 TD
[(whic)30.1(h)-248(indicates)-236.1(a)-233.1(rev)30.1(erse)-242(edge)-234.1(from)-242.8(a)-233.1(c)30.1(hild)-241.6(to)-243.8(its)-235.8(paren)30.1(t.)-406.7(Next,)-261(w)30.1(e)-232.8(rev)30.1(erse)-242(the)-236(direction)]TJ
T*
[(of)-325.4(the)-326.4(edges)-329.7(to)-324.1(obtain)-331.6(a)-323.5(spanning)-322.3(tree)-335.5(for)]TJ
/F7 1 Tf
18.0775 0 TD
(M)Tj
/F2 1 Tf
1.2009 0 TD
(.)Tj
-18.0037 -1.6662 TD
[(W)82.6(e)-217.3(start)-220.3(with)-224.1(tec)30.1(hnical)-217.3(lemmas.)-397.7(Let)]TJ
/F12 1 Tf
15.4678 0 TD
0.0301 Tc
[(P,)-140.8(Q)]TJ
/F7 1 Tf
2.148 0 TD
0.2769 Tc
[(E)182.4(R)276.9(P)]TJ
/F2 1 Tf
3.4127 0 TD
0 Tc
[(b)-30.1(e)-215.3(patterns)-212.8(o)30.1(v)30.1(e)0(r)-225.4()0(.)-396(Recall)-226.2(that)]TJ
/F12 1 Tf
13.1993 0 TD
(Q)Tj
/F2 1 Tf
-35.5026 -1.335 TD
[(is)-275.3(a)-283.3(s)0(p)-30.1(ecialization)-275.3(of)]TJ
/F12 1 Tf
8.7929 0 TD
(P)Tj
/F2 1 Tf
0.7729 0 TD
[(,)-280.1(denoted)-281.1(b)30.1(y)]TJ
/F12 1 Tf
5.5106 0 TD
(P)Tj
/F7 1 Tf
1.0439 0 TD
()Tj
/F12 1 Tf
1.064 0 TD
(Q)Tj
/F2 1 Tf
0.7755 0 TD
0.2875 Tc
[(,i)287.5()]TJ
/F12 1 Tf
1.6737 0 TD
0 Tc
(Q)Tj
/F2 1 Tf
1.0539 0 TD
(=)Tj
/F12 1 Tf
1.0439 0 TD
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[(are)-370.7(subsets)-380.3(of)]TJ
/F12 1 Tf
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[(.)-571.2(Using)-374.7(these)-368.2(op)-30.1(erators,)-378.9(w)30.1(e)-373.3(can)-368.2(describ)-30.1(e)-371.8(the)-376.6(lo)-30.1(cation)-361.4(lists)]TJ
-11.0714 -1.335 TD
[(of)-385.6(a)-393.7(comp)-30.1(osite)-378.9(pattern)-388.2(of)-385.6(the)-386.6(form)]TJ
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[(,)-390.5(where)]TJ
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[(E)282.7(R)377.2(P)]TJ
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[(follo)30.1(ws.)]TJ
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[(Lemma)-373.8(11)]TJ
/F11 1 Tf
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[(Le)-51.3(t)]TJ
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()Tj
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()Tj
/F11 1 Tf
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[(b)48.5(e)-453.4(any)-452.4(input)-449.9(string,)]TJ
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[(b)48.5(e)-453.4(any)-452.4(non-empty)-458.6(c)48.5(onstant)]TJ
-24.4915 -1.335 TD
[(string,)-354.1(and)]TJ
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[(E)182.4(R)276.9(P)]TJ
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-8.9133 -2.409 TD
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/F12 1 Tf
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[(w,)-142.9(T)]TJ
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()Tj
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[(P,)-141.5(T)]TJ
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-19.2279 -2.1279 TD
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/F12 1 Tf
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/F12 1 Tf
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(\)=)Tj
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(\()Tj
/F12 1 Tf
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[(w,)-142.9(T)]TJ
/F2 1 Tf
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0 Tc
(\))Tj
/F21 1 Tf
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()Tj
/F13 1 Tf
7.9701 0 0 7.9701 241.95 399.42 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 250.35 401.22 Tm
(\()Tj
/F12 1 Tf
0.3808 0 TD
(LO)Tj
/F2 1 Tf
1.4461 0 TD
(\()Tj
/F12 1 Tf
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0.0301 Tc
[(P,)-141.5(T)]TJ
/F2 1 Tf
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0 Tc
(\))Tj
/F7 1 Tf
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(|)Tj
/F12 1 Tf
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(w)Tj
/F7 1 Tf
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(|)Tj
/F2 1 Tf
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(\))Tj
/F11 1 Tf
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(.)Tj
/F2 1 Tf
-20.2016 -2.5194 TD
(Let)Tj
/F12 1 Tf
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(T)Tj
/F7 1 Tf
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()Tj
/F2 1 Tf
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()Tj
/F13 1 Tf
7.9701 0 0 7.9701 150.63 375.54 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 159.87 371.1 Tm
[(b)-30.1(e)-365.9(an)30.1(y)-366.9(input)-375.3(string)-367.3(of)-365.6(length)]TJ
/F12 1 Tf
12.9383 0 TD
(n)Tj
/F7 1 Tf
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()Tj
/F2 1 Tf
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[(0.)-563.4(A)-370.1(maximal)-373.4(pattern)]TJ
/F12 1 Tf
9.9974 0 TD
(P)Tj
/F2 1 Tf
1.1342 0 TD
[(is)-375.7(a)]TJ
/F11 1 Tf
1.8871 0 TD
-0.0498 Tc
[(roo)-49.8(t)]TJ
-34.6394 -1.3249 TD
0 Tc
[(p)48.3(attern)]TJ
/F2 1 Tf
3.4128 0 TD
(in)Tj
/F12 1 Tf
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(T)Tj
/F2 1 Tf
1.0339 0 TD
(if)Tj
/F12 1 Tf
0.9034 0 TD
(O)Tj
/F2 1 Tf
0.7829 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
0.0301 Tc
[(P,)-141.5(T)]TJ
/F2 1 Tf
1.8174 0 TD
0.2616 Tc
(\)=)Tj
/F7 1 Tf
1.6763 0 TD
0 Tc
({)Tj
/F2 1 Tf
0.5 0 TD
(1)Tj
/F12 1 Tf
0.4896 0 TD
0.1707 Tc
[(,...)11.1(,)]TJ
/F7 1 Tf
2.2024 0 TD
0 Tc
(|)Tj
/F12 1 Tf
0.2778 0 TD
(T)Tj
/F7 1 Tf
0.7159 0 TD
(|})Tj
/F2 1 Tf
0.7778 0 TD
[(.)-425.8(N)0(o)30.1(w)0(,)-326.7(w)30.1(e)-323.1(sho)30.1(w)-332.3(the)-326.4(main)-327.7(result)-331.3(of)-325.4(this)-324.2(section.)]TJ
/F3 1 Tf
-16.1151 -2.5194 TD
[(Theorem)-374.4(12)-380.6(\(rev)30.1(erse)-372.2(searc)30.1(h)-374.6(prop)-30.1(ert)30.1(y)-371.4(o)0(f)]TJ
/F7 1 Tf
19.9144 0 TD
(M)Tj
/F3 1 Tf
1.2009 0 TD
(\))Tj
/F11 1 Tf
0.9271 0 TD
-0.0513 Tc
[(Le)-51.3(t)]TJ
/F12 1 Tf
1.6863 0 TD
0 Tc
(Q)Tj
/F7 1 Tf
1.0539 0 TD
0.2769 Tc
(M)Tj
/F11 1 Tf
2.4893 0 TD
0 Tc
[(b)48.5(e)-353(a)-343.2(maximal)-344.8(p)48.3(attern)]TJ
-27.2719 -1.335 TD
(in)Tj
/F12 1 Tf
1.1945 0 TD
(T)Tj
/F11 1 Tf
1.0539 0 TD
[(that)-337.4(is)-343.9(not)-341.4(a)-343.2(r)51.2(o)48.3(o)0(t)-349.4(p)48.3(attern.)-439.9(Then,)]TJ
/F7 1 Tf
14.183 0 TD
(P)Tj
/F2 1 Tf
0.7729 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
(Q)Tj
/F2 1 Tf
0.7755 0 TD
(\))Tj
/F11 1 Tf
0.7208 0 TD
[(is)-343.9(also)-347.5(a)-343.1(maximal)-344.8(p)48.3(attern)-338.4(in)]TJ
/F12 1 Tf
12.3762 0 TD
(T)Tj
/F11 1 Tf
0.7127 0 TD
[(.)-442.8(That)-343.9(is,)]TJ
-32.1702 -1.335 TD
(if)Tj
/F12 1 Tf
0.9536 0 TD
(Q)Tj
/F7 1 Tf
1.0539 0 TD
0.2769 Tc
(M)Tj
/F11 1 Tf
2.4893 0 TD
0 Tc
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/F7 1 Tf
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(P)Tj
/F2 1 Tf
0.7729 0 TD
(\()Tj
/F12 1 Tf
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(Q)Tj
/F2 1 Tf
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(\))Tj
/F7 1 Tf
0.6605 0 TD
0.2869 Tc
(M)Tj
/F11 1 Tf
2.4993 0 TD
0 Tc
[(holds.)-450.9(F)66.3(urthermor)51.4(e,)]TJ
/F7 1 Tf
8.8531 0 TD
(|P)Tj
/F2 1 Tf
1.0539 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
(Q)Tj
/F2 1 Tf
0.7755 0 TD
(\))Tj
/F7 1 Tf
0.3808 0 TD
(|)Tj
/F12 1 Tf
0.5508 0 TD
(<)Tj
/F7 1 Tf
1.0439 0 TD
(|)Tj
/F12 1 Tf
0.2778 0 TD
(Q)Tj
/F7 1 Tf
0.7755 0 TD
(|)Tj
/F11 1 Tf
0.623 0 TD
(holds.)Tj
/F3 1 Tf
-26.4789 -1.9172 TD
[(Pro)-30.1(of:)]TJ
/F2 1 Tf
3.7942 0 TD
(Let)Tj
/F12 1 Tf
1.7164 0 TD
(Q)Tj
/F2 1 Tf
1.064 0 TD
[(b)-30.1(e)-285.6(a)-283.3(maximal)-293(pattern)-287.9(that)-292.7(is)-285.3(not)-292.1(a)-283.3(r)0(o)-30.1(o)0(t)-286.9(pattern,)-296.9(and)-289.5(let)]TJ
/F12 1 Tf
24.0097 0 TD
(P)Tj
/F2 1 Tf
1.0439 0 TD
(=)Tj
/F7 1 Tf
1.0439 0 TD
(P)Tj
/F2 1 Tf
0.7729 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
(Q)Tj
/F2 1 Tf
0.7755 0 TD
0.2898 Tc
[(\)b)259.7(e)]TJ
-34.6012 -1.325 TD
0 Tc
[(the)-286.2(paren)30.1(t)-287.2(o)0(f)]TJ
/F12 1 Tf
5.7515 0 TD
(Q)Tj
/F2 1 Tf
0.7755 0 TD
[(.)-428(I)0(n)-287.2(what)-277.5(follo)30.1(ws,)-305.4(for)-285.9(an)30.1(y)-286.6(pattern)]TJ
/F12 1 Tf
14.4514 0 TD
(R)Tj
/F2 1 Tf
0.7528 0 TD
[(,)-290.1(w)30.1(e)-293(write)]TJ
/F12 1 Tf
4.4265 0 TD
(LO)Tj
/F2 1 Tf
1.4454 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
(R)Tj
/F2 1 Tf
0.7535 0 TD
[(\))-281.7(for)]TJ
/F12 1 Tf
2.1179 0 TD
(LO)Tj
/F2 1 Tf
1.4454 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
0.01 Tc
[(R,)-157.8(T)]TJ
/F2 1 Tf
1.9078 0 TD
0.2717 Tc
[(\)b)301.8(y)]TJ
-34.5892 -1.335 TD
0 Tc
(omitting)Tj
/F12 1 Tf
3.9949 0 TD
(T)Tj
/F2 1 Tf
1.0539 0 TD
[(for)-336.1(simplicit)30.1(y)101.6(.)-521(Supp)-30.1(ose)-337.5(to)-344.2(con)30.1(tradict)-345.8(that)]TJ
/F12 1 Tf
18.208 0 TD
(P)Tj
/F2 1 Tf
1.1142 0 TD
[(is)-345.6(not)-342.3(maximal)-343.2(in)]TJ
/F12 1 Tf
7.9397 0 TD
(T)Tj
/F2 1 Tf
0.7127 0 TD
[(.)-480.8(Then,)]TJ
-33.0234 -1.335 TD
[(there)-423.8(exists)-423.3(some)-422.6(prop)-30.1(er)-417.7(sp)-30.1(ecialization)]TJ
/F12 1 Tf
17.3448 0 TD
(P)Tj
/F13 1 Tf
7.9701 0 0 7.9701 297.27 226.86 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 305.07 222.54 Tm
(of)Tj
/F12 1 Tf
1.2045 0 TD
(P)Tj
/F2 1 Tf
0.7729 0 TD
[(,)-440.7(i.e.,)]TJ
/F12 1 Tf
2.69 0 TD
(P)Tj
/F21 1 Tf
1.2045 0 TD
()Tj
/F12 1 Tf
1.2246 0 TD
(P)Tj
/F13 1 Tf
7.9701 0 0 7.9701 399.15 226.86 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 401.79 222.54 Tm
[(,)-450.7(suc)30.1(h)-429.4(that)]TJ
/F12 1 Tf
5.2496 0 TD
(LO)Tj
/F2 1 Tf
1.4454 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
(P)Tj
/F2 1 Tf
0.7735 0 TD
0.4323 Tc
(\)=)Tj
/F12 1 Tf
-34.7097 -1.335 TD
0 Tc
(LO)Tj
/F2 1 Tf
1.4454 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
(P)Tj
/F13 1 Tf
7.9701 0 0 7.9701 111.75 210.9 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 114.51 206.58 Tm
(\).)Tj
-1.5558 -1.7064 TD
(If)Tj
/F12 1 Tf
0.9937 0 TD
(P)Tj
/F21 1 Tf
1.064 0 TD
()Tj
/F12 1 Tf
1.084 0 TD
(P)Tj
/F13 1 Tf
7.9701 0 0 7.9701 142.71 190.5 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 149.43 186.18 Tm
(then)Tj
/F12 1 Tf
2.2484 0 TD
(P)Tj
/F2 1 Tf
1.1041 0 TD
[(o)-30.1(ccurs)-330.3(in)]TJ
/F12 1 Tf
4.1957 0 TD
(P)Tj
/F13 1 Tf
7.9701 0 0 7.9701 248.91 190.5 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 255.63 186.18 Tm
[(at)-344.2(some)-332.3(p)-30.1(osition,)-344.9(sa)30.1(y)87.7(,)-350.3(1)]TJ
/F7 1 Tf
10.3888 0 TD
()Tj
/F12 1 Tf
1.074 0 TD
(p)Tj
/F7 1 Tf
0.793 0 TD
0.2962 Tc
(|)Tj
/F12 1 Tf
1.3518 0 TD
0 Tc
(P)Tj
/F13 1 Tf
7.9701 0 0 7.9701 427.59 190.5 Tm
()Tj
/F7 1 Tf
11.9552 0 0 11.9552 430.35 186.18 Tm
(|)Tj
/F2 1 Tf
0.2778 0 TD
[(.)-474.1(There)-338.4(are)-340.6(t)30.1(w)30.1(o)]TJ
-29.5271 -1.335 TD
[(cases)-326.8(b)-30.1(elo)30.1(w.)]TJ
1.2748 -1.7064 TD
[(\(i\))-321.5(The)-321.2(case)-321.6(where)]TJ
/F12 1 Tf
8.2508 0 TD
(p)Tj
/F7 1 Tf
0.7628 0 TD
()Tj
/F2 1 Tf
T*
[(=)-282.3(1)0(:)-432.9(Then,)]TJ
/F12 1 Tf
5.0589 0 TD
(P)Tj
/F13 1 Tf
7.9701 0 0 7.9701 273.39 154.14 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 279.99 149.82 Tm
[(is)-315.4(a)-323.5(non-prex)-312.8(sp)-30.1(ecialization)-325.5(of)]TJ
/F12 1 Tf
13.6008 0 TD
(P)Tj
/F2 1 Tf
0.7729 0 TD
[(.)-430.6(I)0(t)-319.9(immedi-)]TJ
-31.046 -1.335 TD
[(ately)-324.8(follo)30.1(ws)-336.5(from)-323.1(Lemma)-324.6(9)-323.5(that)]TJ
/F12 1 Tf
14.3837 0 TD
(LO)Tj
/F2 1 Tf
1.4454 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
(P)Tj
/F2 1 Tf
0.7735 0 TD
(\))Tj
/F7 1 Tf
0.6424 0 TD
()Tj
/F2 1 Tf
T*
(=)Tj
/F12 1 Tf
1.0439 0 TD
(LO)Tj
/F2 1 Tf
1.4454 0 TD
(\()Tj
/F12 1 Tf
0.3808 0 TD
(P)Tj
/F13 1 Tf
7.9701 0 0 7.9701 334.95 138.18 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 337.59 133.86 Tm
[(\).)-441.3(This)-319(is)-335.5(a)-323.5(con)30.1(tradiction.)]TJ
-20.2155 -1.7064 TD
[(\(ii\))-240.3(The)-240.9(case)-241.3(where)]TJ
/F12 1 Tf
8.2107 0 TD
(p)Tj
/F2 1 Tf
0.7628 0 TD
[(=)-282.3(1)0(:)-392.8(Then,)]TJ
/F12 1 Tf
4.9585 0 TD
(P)Tj
/F13 1 Tf
7.9701 0 0 7.9701 271.71 117.78 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 277.35 113.46 Tm
[(is)-235.1(a)-243.2(prex)-239.3(sp)-30.1(ecialization)-245.2(of)]TJ
/F12 1 Tf
11.3022 0 TD
(P)Tj
/F2 1 Tf
0.7729 0 TD
[(.)-400.5(B)0(y)-245.3(the)-236(denition)]TJ
-28.5266 -1.335 TD
[(of)-325.4(the)-326.4(paren)30.1(t,)-336.4(there)-323.5(are)-330.6(the)-326.4(follo)30.1(wing)-320.6(cases)-326.8(for)]TJ
/F12 1 Tf
20.0549 0 TD
(P)Tj
/F2 1 Tf
1.0941 0 TD
(and)Tj
/F12 1 Tf
1.9071 0 TD
(Q)Tj
/F2 1 Tf
0.7755 0 TD
(.)Tj
-22.5569 -1.7064 TD
[(\(ii.a\))-341.9(The)-331.2(case)-341.7(where)]TJ
/F12 1 Tf
9.3449 0 TD
(Q)Tj
/F2 1 Tf
1.074 0 TD
(=)Tj
/F12 1 Tf
1.064 0 TD
(aP)Tj
/F2 1 Tf
1.616 0 TD
[(for)-336.1(some)-332.3(letter)]TJ
/F12 1 Tf
6.5846 0 TD
(a)Tj
/F7 1 Tf
0.813 0 TD
()Tj
/F2 1 Tf
0.9636 0 TD
[(:)-456.2(Let)]TJ
/F12 1 Tf
3.202 0 TD
(Q)Tj
/F13 1 Tf
7.9701 0 0 7.9701 399.99 81.42 Tm
()Tj
/F2 1 Tf
11.9552 0 0 11.9552 406.35 77.1 Tm
(=)Tj
/F12 1 Tf
1.0539 0 TD
(aP)Tj
/F13 1 Tf
7.9701 0 0 7.9701 434.31 81.42 Tm
()Tj
/F7 1 Tf
11.9552 0 0 11.9552 440.55 77.1 Tm
0.2969 Tc
[(E)202.4(R)296.9(P)]TJ
/F2 1 Tf
3.212 0 TD
0 Tc
[(.)-460.8(Since)]TJ
/F12 1 Tf
-33.3145 -1.335 TD
(P)Tj
/F13 1 Tf
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