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b(is)g(a)h(con)m(tradiction)e(to)h(the)h (de\014nition)f(of)g(\011)3207 1720 y Fx(l)q(r)3266 1705 y FD(\()p FA(P)s(;)17 b(T)d FD([)p FA(q)3555 1720 y Fx(h)3616 1705 y FA(:)j(:)g(:)f(n)p FD(]\).)244 1825 y(Hence)47 b FA(q)590 1840 y Fx(h)666 1825 y FD(+)31 b FG(j)p FA(w)871 1840 y Fy(1)910 1825 y FA(c)952 1840 y Fy(1)1008 1825 y FA(:)17 b(:)g(:)f(w)1209 1840 y Fx(i)1237 1825 y FA(c)1279 1840 y Fx(i)1307 1825 y FA(w)1377 1840 y Fx(i)p Fy(+1)1495 1825 y FG(j)31 b(\000)h FD(1)50 b FG(\025)h FA(b)31 b FD(+)g FG(j)p FA(x)2152 1840 y Fy(1)2192 1825 y FA(a)2243 1840 y Fy(1)2299 1825 y FA(:)17 b(:)g(:)f(x)2485 1840 y Fx(i)2514 1825 y FA(a)2565 1840 y Fx(i)2593 1825 y FA(x)2648 1840 y Fx(i)p Fy(+1)2767 1825 y FG(j)31 b(\000)g FD(1.)83 b(Therefore)47 b(w)m(e)g(ha)m(v)m(e)244 1945 y FA(q)287 1960 y Fx(h)354 1945 y FD(+)22 b FG(j)p FA(s)526 1960 y Fx(h;k)629 1945 y FG(j)f(\000)i FD(1)28 b FG(\025)g FA(q)1003 1960 y Fx(h)1070 1945 y FD(+)22 b FG(j)p FA(w)1266 1960 y Fy(1)1305 1945 y FA(c)1347 1960 y Fy(1)1403 1945 y FA(:)17 b(:)g(:)f(w)1604 1960 y Fx(k)1643 1941 y Fs(0)1669 1945 y FA(c)1711 1960 y Fx(k)1750 1941 y Fs(0)1776 1945 y FA(w)1846 1960 y Fx(k)1885 1941 y Fs(0)1906 1960 y Fy(+1)2001 1945 y FG(j)21 b(\000)i FD(1)k FG(\025)i FA(b)22 b FD(+)g FG(j)p FA(v)t FG(j)g(\000)g FD(1.)0 2190 y(Finally)-8 b(,)29 b(w)m(e)k(ha)m(v)m(e)g FA(b)21 b FD(+)g FG(j)p FA(v)t FG(j)e(\000)i FD(1)28 b FG(\025)g FA(q)1325 2205 y Fx(h)1390 2190 y FD(+)21 b FA(l)i FG(\000)e FD(1,)31 b(since)i FA(R)2057 2205 y Fx(h)2133 2190 y FD(is)f(a)f(sub)m(w)m(ord)j (of)d FA(v)t FD(.)43 b(Hence)33 b FA(q)3259 2205 y Fx(h)3325 2190 y FD(+)20 b FG(j)p FA(s)3495 2205 y Fx(h;k)3598 2190 y FG(j)g(\000)h FD(1)27 b FG(\025)0 2311 y FA(b)c FD(+)f FG(j)p FA(v)t FG(j)f(\000)h FD(1)28 b FG(\025)g FA(q)614 2326 y Fx(h)681 2311 y FD(+)22 b FA(l)j FG(\000)d FD(1,)33 b(whic)m(h)g(implies)d FG(j)p FA(s)1725 2326 y Fx(h;k)1827 2311 y FG(j)e(\025)g FA(l)r FD(,)33 b(i.e.,)f(\()p FA(iii)p FD(\))h(holds.)44 b Ft(2)0 2641 y FB(3.4)135 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Fx(h)1488 3515 y FD(=)28 b(max)o FG(f)p FD(1)p FA(;)17 b(q)1959 3530 y Fx(h)2026 3515 y FG(\000)2125 3419 y Fv(l)2179 3476 y Fx(m)p Fy(+3)p Fx(k)p 2179 3492 192 4 v 2257 3549 a Fy(2)2380 3419 y Fv(m)2424 3515 y FG(g)p Black 0 3770 a Fu(Lemma)37 b(4)p Black 49 w FD(F)-8 b(or)32 b FA(k)e FG(\024)931 3723 y Fx(m)p Fw(\000)p Fy(12)12 b(log)1221 3740 y Fr(b)1265 3723 y Fx(m)p Fy(+3)p 874 3747 601 4 v 874 3804 a(2)g(log)1012 3822 y Fr(b)1055 3804 y Fx(m)p Fy(+3+4)g(log)1401 3822 y Fr(b)1444 3804 y Fx(c)1507 3770 y FG(\000)22 b FD(1,)33 b(where)g FA(c)28 b FD(=)2170 3704 y Fv(P)2257 3719 y Fx(m)p Fw(\000)p Fy(log)2466 3736 y Fr(b)2509 3719 y Fx(m)2257 3795 y(d)p Fy(=0)2575 3770 y FD(\()2650 3731 y Fy(1)p 2623 3747 89 4 v 2623 3756 a Fw(p)p 2682 3756 31 3 v 57 x Fx(b)2722 3770 y FD(\))2760 3734 y Fx(d)2801 3770 y FD(,)k(w)m(e)i(ha)m(v)m(e)1298 4013 y FA(P)14 b(r)s FD([)p FG(j)p FA(s)1523 4028 y Fx(h;k)1624 4013 y FG(j)28 b(\025)g FD(\()p FA(m)22 b FG(\000)h FA(k)s FD(\))p FA(=)p FD(2])k FA(<)h 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4965 y Fr(b)2561 4948 y Fx(m)2406 4982 y Fv(X)2403 5167 y Fx(d)p Fy(=1)2640 5065 y FA(d)22 b FG(\003)g FA(b)2826 5024 y Fw(\000)p Fx(d)0 5394 y Fu(Pro)s(of:)43 b FD(Recall)31 b(that)547 5614 y FA(P)14 b(r)s FD([)p FA(X)34 b FD(=)28 b(log)1043 5637 y Fx(b)1094 5614 y FA(m)22 b FD(+)g FA(d)p FD(])28 b FA(<)f(b)1549 5573 y Fw(\000)p Fx(d)1678 5614 y FD(for)32 b(all)e(in)m(tegers)j(1)28 b FG(\024)g FA(d)f FG(\024)h FA(m)23 b FG(\000)f FD(log)3021 5637 y Fx(b)3072 5614 y FA(m:)p Black 1900 5871 a FD(13)p Black eop %%Page: 14 16 14 15 bop Black Black 0 52 a FD(Let)29 b FA(Y)49 b FD(b)s(e)29 b(the)g(random)f(v)-5 b(ariable)26 b(whic)m(h)j(tak)m(es)h(v)-5 b(alues)28 b(log)2218 76 y Fx(b)2269 52 y FA(m)14 b FD(+)g FA(d)28 b FD(with)g(probabilit)m(y)f FA(b)3291 16 y Fw(\000)p Fx(d)3415 52 y FD(for)h(1)g FG(\024)g FA(d)f FG(\024)0 182 y FA(m)22 b FG(\000)h FD(log)333 205 y Fx(b)384 182 y FA(m)33 b FD(and)g(v)-5 b(alue)32 b(log)1070 205 y Fx(b)1121 182 y FA(m)g FD(with)h(probabilit)m(y)d(1)22 b FG(\000)2130 115 y Fv(P)2217 130 y Fx(m)p Fw(\000)p Fy(log)2426 147 y Fr(b)2469 130 y Fx(m)2217 207 y(d)p Fy(=1)2552 182 y FA(b)2593 146 y Fw(\000)p Fx(d)2689 182 y FD(.)43 b(Let)33 b(us)g(compute)g FA(E)6 b FD([)p FA(Y)21 b FD(].)647 487 y FA(E)6 b FD([)p FA(Y)22 b FD(])83 b(=)g(log)1226 510 y Fx(b)1277 487 y FA(m)p FD(\(1)22 b FG(\000)1571 369 y Fx(m)p Fw(\000)p Fy(log)1779 386 y Fr(b)1822 369 y Fx(m)1668 404 y Fv(X)1664 588 y Fx(d)p Fy(=1)1901 487 y FA(b)1942 446 y Fw(\000)p Fx(d)2038 487 y FD(\))g(+)2196 369 y Fx(m)p Fw(\000)p Fy(log)2405 386 y Fr(b)2448 369 y Fx(m)2293 404 y Fv(X)2290 588 y Fx(d)p Fy(=1)2510 487 y FD(\(log)2674 510 y Fx(b)2725 487 y FA(m)h FD(+)f FA(d)p FD(\))f FG(\003)h FA(b)3154 446 y Fw(\000)p Fx(d)941 802 y FD(=)83 b(log)1226 825 y Fx(b)1277 802 y FA(m)22 b FD(+)1482 684 y Fx(m)p Fw(\000)p Fy(log)1691 701 y Fr(b)1734 684 y Fx(m)1579 719 y Fv(X)1576 903 y Fx(d)p Fy(=1)1813 802 y FA(d)g FG(\003)g FA(b)1999 761 y Fw(\000)p Fx(d)941 1014 y FD(=)83 b(log)1226 1038 y Fx(b)1277 1014 y FA(m)22 b FD(+)g FA(c)1524 1029 y Fx(b)1559 1014 y FA(:)0 1234 y FD(Since)33 b FA(Y)54 b FD(has)32 b(the)h(largest)f(exp)s(ectation)h(among)e(all)g(random)h (v)-5 b(ariables)31 b FA(Y)2838 1198 y Fw(0)2894 1234 y FD(satisfying)540 1454 y FA(P)14 b(r)s FD([)p FA(Y)768 1413 y Fw(0)819 1454 y FD(=)28 b(log)1049 1478 y Fx(b)1100 1454 y FA(m)22 b FD(+)g FA(d)p FD(])27 b FG(\024)i FA(b)1557 1413 y Fw(\000)p Fx(d)1685 1454 y FD(for)j(all)e(in)m(tegers)j(1)28 b FG(\024)g FA(d)f FG(\024)i FA(m)22 b FG(\000)h FD(log)3029 1478 y Fx(b)3080 1454 y FA(m;)0 1674 y FD(w)m(e)34 b(obtain)1289 1794 y FA(E)6 b FD([)p FA(X)i FD(])28 b FA(<)f(E)6 b FD([)p FA(Y)22 b FD(])28 b(=)f(log)2109 1818 y Fx(b)2160 1794 y FA(m)c FD(+)f FA(c)2408 1809 y Fx(b)2442 1794 y FA(:)65 b Ft(2)p Black 0 2025 a Fu(Theorem)37 b(5)p Black 49 w FD(If)j FA(k)j FG(\024)964 1978 y Fx(m)p Fw(\000)p Fy(12)12 b(log)1255 1995 y Fr(b)1298 1978 y Fx(m)p Fy(+3)p 907 2002 601 4 v 907 2059 a(2)g(log)1045 2076 y Fr(b)1088 2059 y Fx(m)p Fy(+3+4)h(log)1434 2076 y Fr(b)1477 2059 y Fx(c)1545 2025 y FG(\000)27 b FD(1,)42 b(where)f FA(c)f FD(=)2254 1958 y Fv(P)2342 1973 y Fx(m)p Fw(\000)p Fy(log)2550 1990 y Fr(b)2593 1973 y Fx(m)2342 2050 y(d)p Fy(=0)2660 2025 y FD(\()2735 1986 y Fy(1)p 2708 2002 89 4 v 2708 2011 a Fw(p)p 2767 2011 31 3 v 57 x Fx(b)2807 2025 y FD(\))2845 1989 y Fx(d)2885 2025 y FD(,)i(and)e FA(T)54 b FD(is)39 b(stored)i(as)f(an)0 2154 y(arra)m(y)-8 b(,)33 b(then)g(the)g(exp)s(ected)h(running)e(time)g(of)g(Algorithm)d(SET)34 b(is)e(b)s(ounded)h(b)m(y)919 2374 y(\(2)p FA(n=)p FD(\()p FA(m)22 b FG(\000)h FA(k)s FD(\)\))f FG(\003)g FD(\(\()p FA(k)j FD(+)d(1\))g FG(\003)g FD(\(log)2177 2397 y Fx(b)2228 2374 y FA(m)g FD(+)g FA(c)2475 2389 y Fx(b)2532 2374 y FD(+)g(1\))g(+)g FA(C)7 b FD(\))p FA(:)0 2635 y Fu(Pro)s(of:)42 b FD(There)32 b(are)f(2)p FA(n=)p FD(\()p FA(m)18 b FG(\000)g FA(k)s FD(\))31 b(regions)f(in)g FA(T)14 b FD(.)42 b(F)-8 b(rom)29 b(eac)m(h)j(region)d FA(R)2718 2650 y Fx(h)2763 2635 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FA(v)j FD(is)32 b(a)h(sub)m(w)m(ord)i(of)e(the)h(in)m(teresting)244 2568 y(sub)m(w)m(ord)g FA(s)675 2583 y Fx(h)p Fw(\000)p Fy(1)p Fx(;k)r Fy(+1)958 2568 y FD([1)17 b FA(:)g(:)g(:)f(e)1227 2583 y Fx(h)p Fw(\000)p Fy(1)1362 2568 y FD(].)p Black 119 2771 a(2.)p Black 49 w(If)29 b FA(h)16 b FD(+)g FA(k)j FD(+)d(1)27 b FG(\024)i FA(r)18 b FD(+)e(1,)30 b(then)g FA(s)1371 2786 y Fx(h;k)r Fy(+1)1592 2771 y FD(=)d FA(y)1743 2786 y Fx(h)1788 2771 y FA(c)1830 2786 y Fx(h)1891 2771 y FA(:)17 b(:)g(:)f(w)2092 2786 y Fx(h)p Fy(+)p Fx(k)2230 2771 y FA(c)2272 2786 y Fx(h)p Fy(+)p Fx(k)2410 2771 y FA(z)2455 2786 y Fx(h)p Fy(+)p Fx(k)r Fy(+1)2684 2771 y FD(.)43 b(Let)29 b FA(q)j FD(=)27 b FA(mp)3237 2786 y Fx(h)3298 2771 y FG(\000)16 b(j)p FA(y)3467 2786 y Fx(h)3512 2771 y FG(j)p FD(,)30 b(that)f(is,)244 2892 y FA(q)36 b FD(p)s(oin)m(ts)c(to)h(the)g(\014rst)g(c)m(haracter)g(of)f (the)h(sub)m(w)m(ord)i FA(y)2243 2907 y Fx(h)2287 2892 y FA(c)2329 2907 y Fx(h)2374 2892 y FD(.)43 b(The)34 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Fs(0)2539 4042 y FA(a)2590 4057 y Fx(k)2629 4038 y Fs(0)2655 4042 y FG(j)p FD(,)40 b(there)f(is)f(a)g (pre\014x)i FA(u)e FD(of)g FA(x)3742 4057 y Fx(k)3781 4038 y Fs(0)3803 4057 y Fy(+1)458 4162 y FD(suc)m(h)k(that)e FA(mp)1039 4177 y Fx(h)p Fy(+)p Fx(k)1174 4158 y Fs(0)1227 4162 y FD(+)27 b(1)40 b(=)h FA(b)27 b FD(+)g FG(j)p FA(x)1790 4177 y Fy(1)1830 4162 y FA(a)1881 4177 y Fy(1)1937 4162 y FA(:)17 b(:)g(:)f(x)2123 4177 y Fx(k)2162 4158 y Fs(0)2189 4162 y FA(a)2240 4177 y Fx(k)2279 4158 y Fs(0)2305 4162 y FA(u)p FG(j)p FD(.)65 b(Let)40 b FA(u)2719 4126 y Fw(0)2782 4162 y FD(b)s(e)g(suc)m(h)i(that)e FA(x)3424 4177 y Fx(k)3463 4158 y Fs(0)3485 4177 y Fy(+1)3620 4162 y FD(=)g FA(uu)3848 4126 y Fw(0)3871 4162 y FD(.)458 4282 y(Ob)m(viously)-8 b(,)32 b FA(u)991 4246 y Fw(0)1045 4282 y FD(is)f(a)h(su\016x)g(of)g FA(P)44 b FD(since)32 b FA(x)1990 4297 y Fx(k)2029 4278 y Fs(0)2052 4297 y Fy(+1)2177 4282 y FD(is.)43 b(F)-8 b(urthermore,)31 b FA(u)2960 4246 y Fw(0)3015 4282 y 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FA(m)p 1666 3976 V 1666 4068 a FD(log)1792 4091 y Fx(b)1842 4068 y FA(m)23 b FD(+)f(5)p FA(:)p FD(56)2253 3999 y FG(\000)h FD(4)p FA(:)0 4257 y FD(In)33 b(case)g FA(b)c FD(=)e(4,)32 b(w)m(e)i(ha)m(v)m (e)815 4510 y FA(k)c FG(\024)1011 4443 y FA(m)23 b FG(\000)g FD(4)17 b(log)1410 4467 y Fx(b)1461 4443 y FA(m)22 b FG(\000)h FD(5)p 1011 4487 706 4 v 1064 4579 a(log)1190 4602 y Fx(b)1241 4579 y FA(m)f FD(+)g(2)g(+)g(1)1754 4510 y(=)2041 4443 y FA(m)p 1868 4487 432 4 v 1868 4579 a FD(log)1994 4602 y Fx(b)2045 4579 y FA(m)g FD(+)g(3)2331 4510 y FG(\000)h FD(4)2517 4443 y(log)2643 4467 y Fx(b)2694 4443 y FA(m)g FD(+)f(1)p FA(:)p FD(25)p 2517 4487 556 4 v 2580 4579 a(log)2706 4602 y Fx(b)2757 4579 y FA(m)g FD(+)g(3)0 4769 y(As)33 b(ab)s(o)m(v)m(e,)h(this)e(inequalit)m(y)f (certainly)h(holds)g(for)1531 4995 y FA(k)f FG(\024)1901 4928 y FA(m)p 1728 4972 432 4 v 1728 5064 a FD(log)1854 5087 y Fx(b)1905 5064 y FA(m)22 b FD(+)g(3)2191 4995 y FG(\000)h FD(4)p FA(:)0 5253 y FD(So,)43 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FA(p)899 3658 y Fy(1)938 3643 y FA(p)987 3658 y Fy(2)1026 3643 y FD(,)f FA(t)h FD(=)g FA(t)1282 3658 y Fy(1)1321 3643 y FA(t)1356 3658 y Fy(2)1396 3643 y FD(,)f(and)f(\000)f(b)s(e)g(an)h(alignmen)m(t)d(of)i FA(p)2571 3658 y Fy(2)2636 3643 y FD(and)g FA(t)2853 3658 y Fy(1)2918 3643 y FD(with)g FA(\016)t FD(\(\000\))j FG(\024)g FA(d)p FD(.)41 b(Assume)0 3764 y FG(j)p FD(\011)104 3779 y Fx(l)q(r)163 3764 y FD(\()p FA(p;)17 b(t)p FD(\))p FG(j)32 b(\025)g FA(d)j FD(and)g(let)g FA(w)1028 3779 y Fy(1)1067 3764 y FA(;)17 b(c)1153 3779 y Fy(1)1192 3764 y FA(;)g(:)g(:)g(:)f(;)h(w)1481 3779 y Fx(d)1521 3764 y FA(;)g(c)1607 3779 y Fx(d)1647 3764 y FA(;)g(w)1761 3779 y Fx(d)p Fy(+1)1926 3764 y FD(b)s(e)36 b(the)f(\014rst)h(2)p FA(d)23 b FD(+)h(1)35 b(elemen)m(ts)g(of)g(\011)3334 3779 y Fx(l)q(r)3394 3764 y FD(\()p FA(p;)17 b(t)p FD(\).)50 b(Then)0 3884 y FG(j)p FA(t)63 3899 y Fy(1)102 3884 y FG(j)28 b(\024)g(j)p FA(w)361 3899 y Fy(1)400 3884 y FA(c)442 3899 y Fy(1)498 3884 y FA(:)17 b(:)g(:)f(w)699 3899 y Fx(d)739 3884 y FA(c)781 3899 y Fx(d)821 3884 y FA(w)891 3899 y Fx(d)p Fy(+1)1022 3884 y FG(j)p FD(.)0 4071 y Fu(Pro)s(of:)42 b FD(By)32 b(Lemma)d(1,)i(there)h(is)e(a)g FA(d)1424 4035 y Fw(0)1447 4071 y FA(;)17 b FD(0)28 b FG(\024)g FA(d)1724 4035 y Fw(0)1774 4071 y FG(\024)h FA(d)p FD(,)h(and)h(a)g(partition)d(\011)g(=)g([)p FA(x)2950 4086 y Fy(1)2990 4071 y FA(;)17 b(a)3085 4086 y Fy(1)3124 4071 y FA(;)g(:)g(:)g(:)f(;)h(x)3398 4086 y Fx(d)3434 4067 y Fs(0)3461 4071 y FA(;)g(a)3556 4086 y Fx(d)3592 4067 y Fs(0)3618 4071 y FA(;)g(x)3717 4086 y Fx(d)3753 4067 y Fs(0)3776 4086 y Fy(+1)3871 4071 y FD(])0 4191 y(of)32 b FA(t)146 4206 y Fy(1)218 4191 y FD(w.r.t.)h FA(p)527 4206 y Fy(2)567 4191 y FD(,)f(hence)i(w.r.t.)g FA(p)p FD(.)43 b(W)-8 b(e)33 b(use)g(induction)f(on)g FA(i)h FD(to)g(sho)m(w)895 4411 y FG(j)p FA(x)978 4426 y Fy(1)1017 4411 y FA(a)1068 4426 y Fy(1)1124 4411 y FA(:)17 b(:)g(:)g(x)1311 4426 y Fx(i)1339 4411 y FG(j)28 b(\024)g(j)p FA(w)1598 4426 y Fy(1)1637 4411 y FA(c)1679 4426 y Fy(1)1735 4411 y FA(:)17 b(:)g(:)f(w)1936 4426 y Fx(i)1964 4411 y FG(j)32 b FD(for)g(all)e FA(i;)17 b FD(0)28 b FG(\024)g FA(i)g FG(\024)g FA(d)2784 4370 y Fw(0)2829 4411 y FD(+)22 b(1)p FA(:)p Black 770 w FD(\(5\))p Black 0 4631 a(If)g FA(i)28 b FD(=)f(0,)d(then)e(claim)e(\(5\))h(holds) g(trivially)-8 b(.)37 b(F)-8 b(or)21 b(a)g(pro)s(of)g(b)m(y)i(con)m (tradiction,)g(w)m(e)g(assume)f FG(j)p FA(x)3354 4646 y Fy(1)3393 4631 y FA(a)3444 4646 y Fy(1)3501 4631 y FA(:)17 b(:)g(:)f(x)3687 4646 y Fx(i)3715 4631 y FA(a)3766 4646 y Fx(i)3795 4631 y FA(x)3850 4646 y Fx(i)p Fy(+1)3968 4631 y FG(j)28 b FA(>)0 4752 y FG(j)p FA(w)98 4767 y Fy(1)137 4752 y FA(c)179 4767 y Fy(1)235 4752 y FA(:)17 b(:)g(:)f(w)436 4767 y Fx(i)464 4752 y FA(c)506 4767 y Fx(i)534 4752 y FA(w)604 4767 y Fx(i)p Fy(+1)722 4752 y FG(j)p FD(,)28 b(i.e.,)g FA(x)1040 4767 y Fx(i)p Fy(+1)1186 4752 y FD(ends)g(at)f FA(c)1559 4767 y Fx(i)p Fy(+1)1704 4752 y FD(or)g(righ)m(t)f(to)h(it.)41 b(By)27 b(the)h(induction)e(h)m (yp)s(othesis,)j(w)m(e)f(kno)m(w)0 4872 y(that)f FA(x)261 4887 y Fx(i)p Fy(+1)407 4872 y FD(starts)g(at)g(the)g(b)s(eginning)f (of)g FA(w)1566 4887 y Fx(i)p Fy(+1)1711 4872 y FD(or)h(left)f(to)h (it.)40 b(W)-8 b(e)28 b(conclude)f(that)g FA(w)3071 4887 y Fx(i)p Fy(+1)3189 4872 y FA(c)3231 4887 y Fx(i)p Fy(+1)3377 4872 y FD(is)f(a)h(sub)m(w)m(ord)0 4992 y(of)37 b FA(x)171 5007 y Fx(i)p Fy(+1)289 4992 y FD(,)i(hence)f(of)f FA(p)p FD(.)56 b(This,)39 b(ho)m(w)m(ev)m(er,)i(con)m(tradicts)c(the)g(fact)g (that)g FA(w)2702 5007 y Fx(i)p Fy(+1)2857 4992 y FD(and)g FA(c)3093 5007 y Fx(i)p Fy(+1)3249 4992 y FD(are)g(elemen)m(ts)g(of)0 5113 y(\011)76 5128 y Fx(l)q(r)136 5113 y FD(\()p FA(p;)17 b(t)p FD(\).)43 b(Th)m(us)34 b(\(5\))e(holds.)43 b(Consequen)m(tly)-8 b(,)35 b(w)m(e)f(infer)383 5333 y FG(j)p FA(t)446 5348 y Fy(1)486 5333 y FG(j)27 b FD(=)h FG(j)p FA(x)728 5348 y Fy(1)767 5333 y FA(a)818 5348 y Fy(1)875 5333 y FA(:)17 b(:)g(:)f(x)1061 5348 y Fx(d)1097 5329 y Fs(0)1124 5333 y FA(a)1175 5348 y Fx(d)1211 5329 y Fs(0)1238 5333 y FA(x)1293 5348 y Fx(d)1329 5329 y Fs(0)1352 5348 y Fy(+1)1446 5333 y FG(j)28 b(\024)g(j)p FA(w)1705 5348 y Fy(1)1744 5333 y FA(c)1786 5348 y Fy(1)1842 5333 y FA(:)17 b(:)g(:)f(w)2043 5348 y Fx(d)2079 5329 y Fs(0)2105 5333 y FA(c)2147 5348 y Fx(d)2183 5329 y Fs(0)2210 5333 y FA(w)2280 5348 y Fx(d)2316 5329 y Fs(0)2339 5348 y Fy(+1)2433 5333 y FG(j)27 b(\024)h(j)p FA(w)2691 5348 y Fy(1)2730 5333 y FA(c)2772 5348 y Fy(1)2828 5333 y FA(:)17 b(:)g(:)f(w)3029 5348 y Fx(d)3070 5333 y FA(c)3112 5348 y Fx(d)3152 5333 y FA(w)3222 5348 y Fx(d)p Fy(+1)3352 5333 y FG(j)p FA(:)33 b Ft(2)p Black 1900 5871 a FD(23)p Black eop %%Page: 24 26 24 25 bop Black Black Black 0 52 a Fu(De\014nition)36 b(3)p Black 49 w FD(F)-8 b(or)26 b(ev)m(ery)i FA(i)f FD(and)f FA(j)33 b FD(where)27 b(0)h FG(\024)g FA(i)g FG(\024)g FA(m)f FD(and)f(0)i FG(\024)g FA(j)34 b FG(\024)28 b FA(n)p FD(,)g(let)e FA(A)p FD(\()p FA(i;)17 b(j)6 b FD(\))26 b(b)s(e)h(the)g(alignmen)m(t)0 173 y(de\014ned)34 b(b)m(y)f(the)g(follo)m(wing)d(recurrences:)307 363 y FA(A)p FD(\(0)p FA(;)17 b(j)6 b FD(\))27 b(=)h FA(A)p FD(\()p FA(i;)17 b FD(0\))27 b(=)h([)17 b(])307 508 y FA(A)p FD(\()p FA(i)22 b FD(+)g(1)p FA(;)17 b(j)28 b FD(+)22 b(1\))432 772 y(=)536 573 y Fv(8)536 647 y(>)536 672 y(<)536 822 y(>)536 847 y(:)651 651 y FA(A)p FD(\()p FA(i;)17 b(j)28 b FD(+)22 b(1\)+)-33 b(+)o([)p FA(P)14 b FD([)p FA(i)22 b FD(+)g(1])11 b FG(!)g FA(")p FD(])208 b(if)32 b FA(D)s FD(\()p FA(i)22 b FD(+)g(1)p FA(;)17 b(j)27 b FD(+)22 b(1\))28 b(=)f FA(D)s FD(\()p FA(i;)17 b(j)28 b FD(+)22 b(1\))g(+)g(1)651 771 y FA(A)p FD(\()p FA(i;)17 b(j)6 b FD(\)+)-33 b(+)o([)p FA(P)14 b FD([)p FA(i)23 b FD(+)f(1])11 b FG(!)g FA(T)j FD([)p FA(j)27 b FD(+)22 b(1]])83 b(else)33 b(if)e FA(D)s FD(\()p FA(i)22 b FD(+)g(1)p FA(;)17 b(j)28 b FD(+)22 b(1\))27 b(=)h FA(D)s FD(\()p FA(i;)17 b(j)6 b FD(\))22 b(+)g FA(\016)3445 786 y Fx(i)p Fy(+1)p Fx(;j)t Fy(+1)651 891 y FA(A)p FD(\()p FA(i)g FD(+)h(1)p FA(;)17 b(j)6 b FD(\)+)-33 b(+)n([)p FA(")11 b FG(!)g FA(T)j FD([)p FA(j)28 b FD(+)22 b(1]])201 b(else)33 b(if)e FA(D)s FD(\()p FA(i)22 b FD(+)g(1)p FA(;)17 b(j)28 b FD(+)22 b(1\))27 b(=)h FA(D)s FD(\()p FA(i)22 b FD(+)g(1)p FA(;)17 b(j)6 b FD(\))21 b(+)i(1)0 1080 y(Here)33 b(the)g(sym)m(b)s(ol)f([)17 b(])32 b(denotes)i(the)f (empt)m(y)g(list)e(and)i(+)-33 b(+)32 b(denotes)i(list)d(app)s(end.)44 b Ft(2)0 1302 y FD(Note)22 b(that)g FA(A)p FD(\()p FA(i;)17 b(j)6 b FD(\))22 b(is)f(de\014ned)j(in)d(the)h(same)g(sc)m(heme)i(as)e FA(W)14 b FD(\()p FA(i;)j(j)6 b FD(\).)39 b(It)22 b(is)g(clear,)h(that) f FA(A)p FD(\()p FA(i;)17 b(j)6 b FD(\))22 b(is)g(an)f(optimal)0 1423 y(alignmen)m(t)31 b(of)h FA(P)14 b FD([1)j FA(:)g(:)g(:)e(i)p FD(])33 b(and)f FA(W)14 b FD(\()p FA(i;)j(j)6 b FD(\),)32 b(i.e.,)h FA(\016)t FD(\()p FA(A)p FD(\()p FA(i;)17 b(j)6 b FD(\)\))27 b(=)h FA(D)s FD(\()p FA(i;)17 b(j)6 b FD(\))27 b(=)g FA(edist)p FD(\()p FA(P)14 b FD([1)j FA(:)g(:)g(:)f(i)p FD(])p FA(;)h(W)d FD(\()p FA(i;)j(j)6 b FD(\)\).)p Black 0 1630 a Fu(Theorem)37 b(10)p Black 49 w FD(Algorithm)30 b(DLET)i(correctly)h(solv)m(es)g(the)g(appro)m(ximate)f(string)f(matc)m (hing)h(problem.)0 1812 y Fu(Pro)s(of:)42 b FD(Supp)s(ose)31 b(there)g(is)f(an)g(appro)m(ximate)f(matc)m(h)h(ending)g(at)f(p)s (osition)g FA(e)h FD(in)f FA(T)14 b FD(,)31 b(i.e.,)f FA(D)s FD(\()p FA(m;)17 b(e)p FD(\))27 b FG(\024)h FA(k)s FD(.)0 1933 y(Let)47 b FA(v)j FD(b)s(e)c(the)h(shortest)h(suc)m(h)g (appro)m(ximate)d(matc)m(h)h(and)h(assume)g(that)f FA(v)k FD(b)s(egins)c(at)g(p)s(osition)f FA(b)p FD(.)0 2053 y(Ob)m(viously)-8 b(,)34 b(there)g(are)f(in)m(tegers)h FA(h)1310 2068 y Fx(b)1378 2053 y FD(and)g FA(h)1625 2068 y Fx(e)1662 2053 y FD(,)g(1)29 b FG(\024)h FA(h)1964 2068 y Fx(b)2027 2053 y FG(\024)g FA(h)2190 2068 y Fx(e)2257 2053 y FG(\024)f FA(r)d FD(+)c(1,)34 b(suc)m(h)h(that)e FA(mp)3208 2068 y Fx(h)3249 2080 y Fr(b)3281 2068 y Fw(\000)p Fy(1)3404 2053 y FA(<)d(b)f FG(\024)h FA(mp)3821 2068 y Fx(h)3862 2080 y Fr(b)0 2174 y FD(and)44 b FA(mp)335 2189 y Fx(h)376 2197 y Fr(e)410 2189 y Fw(\000)p Fy(1)551 2174 y FA(<)j(e)g FG(\024)h FA(mp)1025 2189 y Fx(h)1066 2197 y Fr(e)1104 2174 y FD(.)77 b FA(v)48 b FD(will)42 b(b)s(e)i(detected)i(if)c(DLET)j(computes)f FA(D)s FD(\()p FA(i;)17 b(j)6 b FD(\))44 b(for)f(all)f FA(i;)17 b(j)50 b FD(with)0 2294 y(0)27 b FG(\024)i FA(i)f FG(\024)g FA(m)p FD(,)33 b FA(b)28 b FG(\024)g FA(j)34 b FG(\024)28 b FA(e)p FD(.)43 b(Therefore,)34 b(w)m(e)g(ha)m(v)m(e)g(to)e(sho)m(w:)p Black 70 2530 a(\(1\))p Black 49 w Fz(BPM)p FD(\()p FA(h)560 2545 y Fx(b)594 2530 y FD(\))h(is)f(true,)p Black 70 2725 a(\(2\))p Black 49 w Fz(CPM)p FD(\()p FA(h)p FD(\))h(is)f(true,)h (for)f(all)e FA(h;)17 b(h)1403 2740 y Fx(b)1465 2725 y FA(<)28 b(h)g FG(\024)g FA(h)1814 2740 y Fx(e)1851 2725 y FD(.)0 2960 y(Clearly)-8 b(,)40 b(\(1\))e(holds,)j(as)e(already) f(sho)m(wn)j(in)d(the)h(correctness)j(pro)s(of)c(of)g(Algorithm)e(LET.) k(T)-8 b(o)39 b(pro)m(v)m(e)0 3081 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b(Impro)l(v)l(ed)f(Linear)g(Exp)t (ected)h(Time)f(Algorithm)0 4843 y FD(In)g(this)f(section)g(w)m(e)i (sho)m(w)f(ho)m(w)g(to)f(incorp)s(orate)g(the)g(former)g(impro)m(v)m (emen)m(ts)g(of)g(Algorithm)e(LET)0 4964 y(\(see)33 b(Section)e(4.1\))g (in)m(to)g(Algorithm)e(DLET,)j(yielding)e(the)i Fz(dynamic)h(impr)-5 b(ove)g(d)33 b(line)-5 b(ar)34 b(exp)-5 b(e)g(cte)g(d)33 b(time)0 5084 y(algorithm)45 b FD(\(DILET)39 b(for)e(short\).)61 b(First)38 b(note,)i(that)e(table)g FA(W)51 b FD(con)m(tains)39 b(all)d(information)f(necessary)0 5204 y(to)45 b(determine)h(the)g (length)f(of)g(the)h(pre\014x)h(of)e(a)g(p)s(oten)m(tial)f(appro)m (ximate)h(matc)m(h.)82 b(Therefore,)50 b(the)0 5325 y(pre\014xes)38 b FA(y)407 5340 y Fx(h)487 5325 y FD(of)d FA(P)50 b FD(\(see)37 b(De\014nition)d(2\))h(are)h(not)g(needed)h(for)f(c)m(hec)m(king)h(the) f(dynamic)f(condition.)52 b(F)-8 b(or)0 5445 y(simplicit)m(y)g(,)31 b(w)m(e)k(also)e(do)g(not)h(use)g(them)g(in)e(the)i(static)g 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w(100.0)140 b(100.0)170 b(100.0)p 2597 2646 V 143 w(62.9)177 b(85.5)206 b(94.1)p 3571 2646 V 324 2745 V 418 2715 a(9)p 510 2745 V 253 w(61.7)190 b(95.7)214 b(99.9)p 1612 2745 V 102 w(100.0)140 b(100.0)170 b(100.0)p 2597 2745 V 184 w(2.5)178 b(28.5)206 b(41.3)p 3571 2745 V 324 2845 V 377 2815 a(10)p 510 2845 V 252 w(47.7)190 b(94.1)214 b(99.7)p 1612 2845 V 102 w(100.0)140 b(100.0)170 b(100.0)p 2597 2845 V 184 w(0.1)219 b(7.6)207 b(12.4)p 3571 2845 V 324 2944 V 377 2915 a(11)p 510 2944 V 252 w(49.7)190 b(88.1)214 b(99.0)p 1612 2944 V 102 w(100.0)140 b(100.0)170 b(100.0)p 2597 2944 V 184 w(0.0)219 b(0.3)249 b(0.5)p 3571 2944 V 324 3044 V 377 3014 a(12)p 510 3044 V 252 w(45.5)232 b(0.0)256 b(0.0)p 1612 3044 V 143 w(98.5)182 b(99.7)171 b(100.0)p 2597 3044 V 184 w(0.0)219 b(0.1)249 b(0.1)p 3571 3044 V 324 3144 V 377 3114 a(13)p 510 3144 V 1612 3144 V 1246 w(70.1)182 b(89.7)213 b(94.9)p 2597 3144 V 184 w(0.0)219 b(0.0)249 b(0.0)p 3571 3144 V 324 3243 V 377 3213 a(14)p 510 3243 V 1612 3243 V 1246 w(18.0)182 b(49.7)213 b(61.1)p 2597 3243 V 184 w(0.0)219 b(0.0)249 b(0.0)p 3571 3243 V 324 3343 V 377 3313 a(15)p 510 3343 V 1612 3343 V 1288 w(1.1)182 b(12.5)213 b(18.1)p 2597 3343 V 184 w(0.0)219 b(0.0)249 b(0.0)p 3571 3343 V 324 3443 V 377 3413 a(16)p 510 3443 V 1612 3443 V 1288 w(0.0)224 b(1.3)254 b(2.1)p 2597 3443 V 185 w(0.0)219 b(0.0)249 b(0.0)p 3571 3443 V 324 3446 3251 4 v Black Black 1900 5871 a FD(34)p Black eop %%Page: 35 37 35 36 bop Black Black 0 52 a FC(References)p Black 0 313 a FD([BM77])p Black 94 w(R.S.)33 b(Bo)m(y)m(er)h(and)g(J.S.)g(Mo)s (ore.)45 b(A)33 b(F)-8 b(ast)33 b(String)f(Searc)m(hing)i(Algorithm.)42 b Fz(Communic)-5 b(ations)404 433 y(of)34 b(the)h(A)n(CM)p FD(,)e Fu(20)p FD(\(10\):762{772,)d(1977.)p Black 0 637 a([Che52])p Black 85 w(H.)i(Cherno\013.)45 b(A)32 b(Measure)i(of)f (Asymptotic)f(E\016ciency)i(for)e(T)-8 b(ests)34 b(of)e(a)g(Hyp)s (othesis)i(Based)404 757 y(on)c(the)h(Sum)f(of)g(Observ)-5 b(ations.)40 b Fz(A)n(nnals)32 b(of)h(Mathematic)-5 b(al)32 b(Statistics)p FD(,)f Fu(23)p FD(:493{507,)e(1952.)p Black 0 961 a([CL92])p Black 121 w(W.I.)24 b(Chang)g(and)f(J.)h(Lamp)s (e.)k(Theoretical)23 b(and)g(Empirical)e(Comparisons)i(of)g(Appro)m (ximate)404 1081 y(String)28 b(Matc)m(hing)h(Algorithms.)36 b(In)30 b Fz(Pr)-5 b(o)g(c.)32 b(of)f(the)h(A)n(nnual)g(Symp)-5 b(osium)31 b(on)g(Combinatorial)404 1201 y(Pattern)k(Matching)g FD(\(CPM'92\))p Fz(,)g FD(LNCS)e(644,)f(pages)h(175{184,)e(1992.)p Black 0 1405 a([CL94])p Black 121 w(W.I.)e(Chang)h(and)f(E.L.)h(La)m (wler.)37 b(Sublinear)28 b(Appro)m(ximate)h(String)f(Matc)m(hing)h(and) h(Biolog-)404 1525 y(ical)h(Applications.)41 b Fz(A)n(lgorithmic)-5 b(a)p FD(,)32 b Fu(12)p FD(\(4/5\):327{344,)e(1994.)p Black 0 1729 a([EH88])p Black 113 w(A.)35 b(Ehrenfeuc)m(h)m(t)j(and)d (D.)g(Haussler.)52 b(A)36 b(New)g(Distance)f(Metric)g(on)g(Strings)g (Computable)404 1849 y(in)d(Linear)f(Time.)43 b Fz(Discr)-5 b(ete)34 b(Applie)-5 b(d)34 b(Mathematics)p FD(,)f Fu(20)p FD(:191{203,)e(1988.)p Black 0 2052 a([GP90])p Black 109 w(Z.)i(Galil)e(and)j(K.)g(P)m(ark.)48 b(An)35 b(Impro)m(v)m(ed)f (Algorithm)e(for)h(Appro)m(ximate)g(String)h(Matc)m(hing.)404 2173 y Fz(SIAM)h(Journal)f(on)h(Computing)p FD(,)d Fu(19)p FD(\(6\):989{999,)f(1990.)p Black 0 2376 a([JTU91])p Black 59 w(P)-8 b(.)45 b(Jokinen,)k(J.)c(T)-8 b(arhio,)48 b(and)e(E.)f(Ukk)m(onen.)83 b(A)46 b(Comparison)e(of)h(Appro)m(ximate)g (String)404 2497 y(Matc)m(hing)36 b(Algorithms.)52 b(T)-8 b(ec)m(hnical)37 b(Rep)s(ort)f(A-1991-7,)f(Departmen)m(t)h(of)g (Computer)g(Sci-)404 2617 y(ence,)e(Univ)m(ersit)m(y)f(of)f(Helsinki,)g (1991.)p Black 0 2820 a([Kur95])p Black 84 w(S.)44 b(Kurtz.)79 b Fz(F)-7 b(undamental)44 b(A)n(lgorithms)h(for)g(a)h(De)-5 b(clar)g(ative)44 b(Pattern)i(Matching)g(System)p FD(.)404 2941 y(Dissertation,)28 b(T)-8 b(ec)m(hnisc)m(he)31 b(F)-8 b(akult\177)-49 b(at,)28 b(Univ)m(ersit\177)-49 b(at)29 b(Bielefeld,)e(a)m(v)-5 b(ailable)27 b(as)i(Rep)s(ort)f(95-03,)404 3061 y(July)k(1995.)p Black 0 3264 a([LP85])p Black 125 w(D.J.)39 b(Lipman)e(and)j(W.R.)f(P)m(earson.)65 b(Rapid)38 b(and)h(Sensitiv)m(e)h(Protein)f(Similarit)m(y)c(Searc)m(h.)404 3385 y Fz(Scienc)-5 b(e)p FD(,)31 b Fu(227)p FD(:1435{1441,)g(1985.)p Black 0 3588 a([L)-11 b(V88])p Black 129 w(G.M.)37 b(Landau)g(and)g(U.) g(Vishkin.)55 b(F)-8 b(ast)37 b(String)f(Matc)m(hing)h(with)f FA(k)k FD(Di\013erences.)57 b Fz(Journal)404 3709 y(of)34 b(Computer)h(and)f(Systems)h(Scienc)-5 b(es)p FD(,)31 b Fu(37)p FD(:63{78,)h(1988.)p Black 0 3912 a([McC76])p Black 50 w(E.M.)24 b(McCreigh)m(t.)30 b(A)24 b(Space-Economical)e (Su\016x)j(T)-8 b(ree)25 b(Construction)f(Algorithm.)i Fz(Journal)404 4032 y(of)34 b(the)h(A)n(CM)p FD(,)e Fu(23)p FD(\(2\):262{272,)e(1976.)p Black 0 4236 a([My)m(e94])p Black 72 w(E.W.)37 b(My)m(ers.)57 b(A)37 b(Sublinear)e(Algorithm)f(for) i(Appro)m(ximate)g(Keyw)m(ord)i(Searc)m(hing.)55 b Fz(A)n(lgo-)404 4356 y(rithmic)-5 b(a)p FD(,)32 b Fu(12)p FD(\(4/5\):345{374,)e(1994.)p Black 0 4560 a([P)m(ea95])p Black 97 w(W.R.)e(P)m(earson.)37 b(Comparison)27 b(of)g(Metho)s(ds)i(for)e(Searc)m(hing)h(Protein)g (Sequence)i(Databases.)404 4680 y Fz(Pr)-5 b(otein)35 b(Scienc)-5 b(e)p FD(,)31 b Fu(4)p FD(:1145{1160,)g(1995.)p Black 0 4883 a([Sel80])p Black 127 w(P)-8 b(.H.)42 b(Sellers.)71 b(The)43 b(Theory)h(and)e(Computation)e(of)i(Ev)m(olutionary)f (Distances:)63 b(P)m(attern)404 5004 y(Recognition.)41 b Fz(Journal)35 b(of)g(A)n(lgorithms)p FD(,)d Fu(1)p FD(:359{373,)f(1980.)p Black 0 5207 a([ST95])p Black 128 w(E.)45 b(Sutinen)g(and)h(J.)f(T)-8 b(arhio.)80 b(On)45 b(Using)g FA(q)t FD(-Gram)e(Lo)s(cations)h(in)g(Appro)m(ximate)h (String)404 5328 y(Matc)m(hing.)53 b(In)36 b Fz(Pr)-5 b(o)g(c)g(e)g(e)g(dings)37 b(of)h(the)g(Eur)-5 b(op)g(e)g(an)37 b(Symp)-5 b(osium)37 b(on)h(A)n(lgorithms)p FD(,)e(pages)h(327{)404 5448 y(340.)32 b(Lecture)h(Notes)g(in)f(Computer)h(Science)g Fu(979)p FD(,)g(Springer)f(V)-8 b(erlag,)32 b(1995.)p Black 1900 5871 a(35)p Black eop %%Page: 36 38 36 37 bop Black Black Black 0 52 a FD([T)-8 b(ak94])p Black 90 w(T.)49 b(T)-8 b(ak)j(aok)g(a.)90 b(Appro)m(ximate)48 b(P)m(attern)h(Matc)m(hing)f(with)g(Samples.)90 b(In)49 b Fz(Pr)-5 b(o)g(c)g(e)g(e)g(dings)48 b(of)404 173 y(ISAA)n(C)43 b(1994)p FD(,)h(pages)e(234{242.)f(Lecture)i(Notes)g(in)e(Computer)h (Science)h Fu(834)p FD(,)i(Springer)404 293 y(V)-8 b(erlag,)31 b(1994.)p Black 0 497 a([TU93])p Black 109 w(J.)42 b(T)-8 b(arhio)42 b(and)g(E.)h(Ukk)m(onen.)75 b(Appro)m(ximate)41 b(Bo)m(y)m(er-Mo)s(ore)i(String)f(Matc)m(hing.)73 b Fz(SIAM)404 617 y(Journal)34 b(on)h(Computing)p FD(,)d Fu(22)p FD(\(2\):243{260,)f (1993.)p Black 0 820 a([Ukk85])p Black 77 w(E.)43 b(Ukk)m(onen.)77 b(Finding)41 b(Appro)m(ximate)h(P)m(atterns)i(in)f(Strings.)74 b Fz(Journal)44 b(of)g(A)n(lgorithms)p FD(,)404 941 y Fu(6)p FD(:132{137,)31 b(1985.)p Black 0 1144 a([Ukk92])p Black 77 w(E.)j(Ukk)m(onen.)51 b(Appro)m(ximate)33 b(String-Matc)m (hing)g(with)h FA(q)t FD(-Grams)e(and)j(Maximal)d(Matc)m(hes.)404 1265 y Fz(The)-5 b(or)g(etic)g(al)34 b(Computer)g(Scienc)-5 b(e)p FD(,)32 b Fu(92)p FD(\(1\):191{211,)e(1992.)p Black 0 1468 a([Ukk93])p Black 77 w(E.)48 b(Ukk)m(onen.)91 b(Appro)m(ximate)47 b(String-Matc)m(hing)g(o)m(v)m(er)i(Su\016x)g(T)-8 b(rees.)90 b(In)48 b Fz(Pr)-5 b(o)g(c.)49 b(of)g(the)404 1588 y(A)n(nnual)f(Symp)-5 b(osium)48 b(on)h(Combinatorial)e(Pattern)j (Matching)f FD(\(CPM'93\))p Fz(,)k FD(LNCS)48 b(684,)404 1709 y(pages)33 b(229{242,)e(1993.)p Black 0 1912 a([Ukk95])p Black 77 w(E.)i(Ukk)m(onen.)45 b(On-line)31 b(Construction)i(of)f (Su\016x-T)-8 b(rees.)45 b Fz(A)n(lgorithmic)-5 b(a)p FD(,)32 b Fu(14)p FD(\(3\),)g(1995.)p Black 0 2116 a([W)-8 b(ei73])p Black 89 w(P)g(.)32 b(W)-8 b(einer.)43 b(Linear)32 b(P)m(attern)h(Matc)m(hing)f(Algorithms.)40 b(In)33 b Fz(Pr)-5 b(o)g(c)g(e)g(e)g(dings)33 b(of)i(the)f(14th)h(IEEE)404 2236 y(A)n(nnual)24 b(Symp)-5 b(osium)25 b(on)f(Switching)h(and)f(A)n (utomata)h(The)-5 b(ory)p FD(,)24 b(pages)f(1{11,)g(The)g(Univ)m(ersit) m(y)404 2356 y(of)32 b(Io)m(w)m(a,)h(1973.)p Black 1900 5871 a(36)p Black eop %%Page: 37 39 37 38 bop Black Black 0 52 a Fu(Bisher)37 b(ersc)m(hienene)g(Rep)s (orts)g(an)h(der)g(T)-9 b(ec)m(hnisc)m(hen)37 b(F)-9 b(akult)o(\177)-55 b(at)0 173 y(Stand:)51 b(16.)37 b(F)-9 b(ebruar)39 b(2000)p Black 24 517 a(94{01)p Black 168 w FD(Mo)s(dular)32 b(Prop)s(erties)h(of)f(Comp)s(osable)f(T)-8 b(erm)33 b(Rewriting)e(Systems)472 638 y(\(Enno)i(Ohlebusc)m(h\))p Black 24 799 a Fu(94{02)p Black 168 w FD(Analysis)g(and)f(Applications) f(of)h(the)h(Direct)f(Cascade)i(Arc)m(hitecture)472 920 y(\(Enno)f(Littmann)e(und)i(Helge)g(Ritter\))p Black 24 1082 a Fu(94{03)p Black 168 w FD(F)-8 b(rom)31 b(Ukk)m(onen)k(to)d (McCreigh)m(t)h(and)g(W)-8 b(einer:)44 b(A)32 b(Unifying)g(View)472 1202 y(of)h(Linear-Time)d(Su\016x)k(T)-8 b(ree)33 b(Construction)472 1323 y(\(Rob)s(ert)g(Giegeric)m(h)e(und)i(Stefan)g(Kurtz\))p Black 24 1484 a Fu(94{04)p Black 168 w FD(Die)f(V)-8 b(erw)m(endung)34 b(unsc)m(harfer)g(Ma\031e)f(zur)g(Korresp)s (ondenzanalyse)472 1605 y(in)f(Stereo)h(F)-8 b(arbbildern)472 1725 y(\(Andr)m(\022)-46 b(e)34 b(W)-8 b(olfram)30 b(und)j(Alois)e (Knoll\))p Black 24 1887 a Fu(94{05)p Black 168 w FD(Searc)m(hing)i (Corresp)s(ondences)i(in)d(Colour)g(Stereo)h(Images)472 2007 y(|)g(Recen)m(t)g(Results)g(Using)f(the)h(F)-8 b(uzzy)34 b(In)m(tegral)472 2128 y(\(Andr)m(\022)-46 b(e)34 b(W)-8 b(olfram)30 b(und)j(Alois)e(Knoll\))p Black 24 2290 a Fu(94{06)p Black 168 w FD(A)i(Basic)f(Seman)m(tics)h(for)f(Computer)h (Arithmetic)472 2410 y(\(Markus)h(F)-8 b(reeric)m(ks,)34 b(A.)e(F)-8 b(auth)32 b(und)i(Alois)d(Knoll\))p Black 24 2572 a Fu(94{07)p Black 168 w FD(Rev)m(erse)k(Restructuring:)44 b(Another)33 b(Metho)s(d)g(of)f(Solving)472 2692 y(Algebraic)g (Equations)472 2813 y(\(Bernd)i(B)s(\177)-51 b(uto)m(w)32 b(und)h(Stephan)g(Thesing\))p Black 24 2975 a Fu(95{01)p Black 168 w FD(P)m(aNaMa)g(User)h(Man)m(ual)e(V1.3)472 3095 y(\(Bernd)i(B)s(\177)-51 b(uto)m(w)32 b(und)h(Stephan)g(Thesing\)) p Black 24 3257 a Fu(95{02)p Black 168 w FD(Computer)g(Based)g(T)-8 b(raining-Soft)m(w)m(are:)43 b(ein)32 b(in)m(teraktiv)m(er)h (Sequenzierkurs)472 3377 y(\(F)-8 b(rank)33 b(Meier,)g(Garrit)d(Skro)s (c)m(k)k(und)f(Rob)s(ert)f(Giegeric)m(h\))p Black 24 3539 a Fu(95{03)p Black 168 w FD(F)-8 b(undamen)m(tal)32 b(Algorithms)e(for)i(a)g(Declarativ)m(e)g(P)m(attern)h(Matc)m(hing)g (System)472 3660 y(\(Stefan)g(Kurtz\))p Black 24 3821 a Fu(95{04)p Black 168 w FD(On)g(the)g(Equiv)-5 b(alence)33 b(of)f(E-P)m(attern)h(Languages)472 3942 y(\(Enno)g(Ohlebusc)m(h)h(und) f(Esk)m(o)h(Ukk)m(onen\))p Black 24 4104 a Fu(96{01)p Black 168 w FD(Static)e(and)h(Dynamic)e(Filtering)f(Metho)s(ds)j(for)f (Appro)m(ximate)g(String)g(Matc)m(hing)472 4224 y(\(Rob)s(ert)h (Giegeric)m(h,)e(F)-8 b(rank)33 b(Hisc)m(hk)m(e,)h(Stefan)f(Kurtz)f (und)i(Enno)f(Ohlebusc)m(h\))p Black 24 4386 a Fu(96{02)p Black 168 w FD(Instructing)g(Co)s(op)s(erating)e(Assem)m(bly)j(Rob)s (ots)e(through)g(Situated)h(Dialogues)472 4506 y(in)f(Natural)g (Language)472 4627 y(\(Alois)f(Knoll,)g(Bernd)i(Hildebrandt)f(und)h (Jian)m(w)m(ei)f(Zhang\))p Black 24 4789 a Fu(96{03)p Black 168 w FD(Correctness)j(in)d(System)h(Engineering)472 4909 y(\(P)m(eter)h(Ladkin\))p Black 24 5071 a Fu(96{04)p Black 168 w FD(An)f(Algebraic)f(Approac)m(h)h(to)f(General)g(Bo)s (olean)f(Constrain)m(t)i(Problems)472 5191 y(\(Hans-W)-8 b(erner)34 b(G)s(\177)-51 b(usgen)32 b(und)h(P)m(eter)h(Ladkin\))p Black 24 5353 a Fu(96{05)p Black 168 w FD(F)-8 b(uture)33 b(Univ)m(ersit)m(y)g(Computing)f(Resources)472 5474 y(\(P)m(eter)i (Ladkin\))p Black Black eop %%Page: 38 40 38 39 bop Black Black Black 24 52 a Fu(96{06)p Black 168 w FD(Lazy)34 b(Cac)m(he)f(Implemen)m(ts)g(Complete)f(Cac)m(he)472 173 y(\(P)m(eter)i(Ladkin\))p Black 24 328 a Fu(96{07)p Black 168 w FD(F)-8 b(ormal)30 b(but)j(Liv)m(ely)f(Bu\013ers)i(in)e (TLA+)472 449 y(\(P)m(eter)i(Ladkin\))p Black 24 604 a Fu(96{08)p Black 168 w FD(The)g(X-31)e(and)g(A320)g(W)-8 b(arsa)m(w)34 b(Crashes:)45 b(Who)s(dunnit?)472 725 y(\(P)m(eter)34 b(Ladkin\))p Black 24 880 a Fu(96{09)p Black 168 w FD(Reasons)g(and)e (Causes)472 1001 y(\(P)m(eter)i(Ladkin\))p Black 24 1156 a Fu(96{10)p Black 168 w FD(Commen)m(ts)f(on)g(Confusing)f(Con)m(v)m (ersation)i(at)e(Cali)472 1277 y(\(Dafydd)h(Gibb)s(on)e(und)i(P)m(eter) h(Ladkin\))p Black 24 1432 a Fu(96{11)p Black 168 w FD(On)f(Needing)g (Mo)s(dels)472 1553 y(\(P)m(eter)h(Ladkin\))p Black 24 1708 a Fu(96{12)p Black 168 w FD(F)-8 b(ormalism)29 b(Helps)k(in)f (Describing)f(Acciden)m(ts)472 1829 y(\(P)m(eter)j(Ladkin\))p Black 24 1984 a Fu(96{13)p Black 168 w FD(Explaining)d(F)-8 b(ailure)31 b(with)h(T)-8 b(ense)34 b(Logic)472 2105 y(\(P)m(eter)g(Ladkin\))p Black 24 2260 a Fu(96{14)p Black 168 w FD(Some)f(Dubious)f(Theses)i(in)e(the)h(T)-8 b(ense)34 b(Logic)e(of)g(Acciden)m(ts)472 2381 y(\(P)m(eter)i(Ladkin\)) p Black 24 2536 a Fu(96{15)p Black 168 w FD(A)f(Note)g(on)f(a)h(Note)f (on)h(a)f(Lemma)f(of)h(Ladkin)472 2657 y(\(P)m(eter)i(Ladkin\))p Black 24 2812 a Fu(96{16)p Black 168 w FD(News)g(and)f(Commen)m(t)f(on) h(the)g(AeroP)m(eru)h(B757)d(Acciden)m(t)472 2933 y(\(P)m(eter)j (Ladkin\))p Black 24 3088 a Fu(97{01)p Black 168 w FD(Analysing)e(the)h (Cali)e(Acciden)m(t)j(With)e(a)g(WB-Graph)472 3209 y(\(P)m(eter)i (Ladkin\))p Black 24 3364 a Fu(97{02)p Black 168 w FD (Divide-and-Conquer)e(Multiple)f(Sequence)k(Alignmen)m(t)472 3485 y(\(Jens)f(Sto)m(y)m(e\))p Black 24 3640 a Fu(97{03)p Black 168 w FD(A)44 b(System)g(for)e(the)i(Con)m(ten)m(t-Based)h (Retriev)-5 b(al)41 b(of)i(T)-8 b(extual)43 b(and)h(Non-T)-8 b(extual)43 b(Do)s(cu-)472 3761 y(men)m(ts)34 b(Based)f(on)f(Natural)g (Language)g(Queries)472 3881 y(\(Alois)f(Knoll,)g(Ingo)i(Gl\177)-49 b(oc)m(kner,)32 b(Hermann)h(Helbig)e(und)i(Sv)m(en)h(Hartrumpf)7 b(\))p Black 24 4037 a Fu(97{04)p Black 168 w FD(Rose:)44 b(Generating)32 b(Sequence)j(F)-8 b(amilies)472 4157 y(\(Jens)34 b(Sto)m(y)m(e,)g(Dirk)e(Ev)m(ers)i(und)f(F)-8 b(olk)m(er)32 b(Mey)m(er\))p Black 24 4313 a Fu(97{05)p Black 168 w FD(F)-8 b(uzzy)48 b(Quan)m(ti\014ers)g(for)f(Pro)s(cessing) g(Natural)f(Language)h(Queries)g(in)g(Con)m(ten)m(t-Based)472 4433 y(Multimedia)30 b(Retriev)-5 b(al)31 b(Systems)472 4553 y(\(Ingo)i(Gl\177)-49 b(oc)m(kner)32 b(und)h(Alois)f(Knoll\))p Black 24 4709 a Fu(97{06)p Black 168 w FD(DFS)g(|)g(An)h(Axiomatic)e (Approac)m(h)i(to)f(F)-8 b(uzzy)34 b(Quan)m(ti\014cation)472 4829 y(\(Ingo)f(Gl\177)-49 b(oc)m(kner\))p Black 24 4985 a Fu(98{01)p Black 168 w FD(Kognitiv)m(e)33 b(Asp)s(ekte)i(b)s(ei)d (der)i(Realisierung)e(eines)h(robusten)i(Rob)s(otersystems)f(f)s(\177) -51 b(ur)32 b(Kon-)472 5105 y(struktionsaufgab)s(en)472 5226 y(\(Alois)f(Knoll)g(und)i(Bernd)g(Hildebrandt\))p Black 24 5381 a Fu(98{02)p Black 168 w FD(A)46 b(Declarativ)m(e)f (Approac)m(h)i(to)e(the)i(Dev)m(elopmen)m(t)e(of)h(Dynamic)e (Programming)f(Algo-)472 5502 y(rithms,)32 b(applied)f(to)i(RNA)f(F)-8 b(olding)472 5622 y(\(Rob)s(ert)33 b(Giegeric)m(h\))p Black Black eop %%Page: 39 41 39 40 bop Black Black Black 24 52 a Fu(98{03)p Black 168 w FD(Reducing)33 b(the)g(Space)g(Requiremen)m(t)g(of)f(Su\016x)i(T) -8 b(rees)472 173 y(\(Stefan)33 b(Kurtz\))p Black 24 335 a Fu(99{01)p Black 168 w FD(En)m(tsc)m(heidungsk)-5 b(alk)s(\177)-51 b(ule)472 455 y(\(Axel)50 b(Saalbac)m(h,)k(Christian) 49 b(Lange,)55 b(Sasc)m(ha)c(W)-8 b(endt,)55 b(Mathias)49 b(Katzer,)55 b(Guillaume)472 576 y(Dub)s(ois,)32 b(Mic)m(hael)g(H\177) -49 b(ohl,)32 b(Oliv)m(er)g(Kuhn,)h(Sv)m(en)h(W)-8 b(ac)m(hsm)m(uth)33 b(und)g(Gerhard)g(Sagerer\))p Black 24 737 a Fu(99{02)p Black 168 w FD(T)-8 b(ransforming)37 b(Conditional)f(Rewrite)i(Systems) h(with)f(Extra)g(V)-8 b(ariables)37 b(in)m(to)g(Uncondi-)472 858 y(tional)31 b(Systems)472 978 y(\(Enno)i(Ohlebusc)m(h\))p Black 24 1140 a Fu(99{03)p Black 168 w FD(A)g(F)-8 b(ramew)m(ork)32 b(for)g(Ev)-5 b(aluating)31 b(Approac)m(hes)j(to)f(F)-8 b(uzzy)33 b(Quan)m(ti\014cation)472 1260 y(\(Ingo)g(Gl\177)-49 b(oc)m(kner\))p Black 24 1422 a Fu(99{04)p Black 168 w FD(T)-8 b(o)m(w)m(ards)34 b(Ev)-5 b(aluation)31 b(of)h(Do)s(c)m(king) g(Hyp)s(otheses)j(using)d(elastic)g(Matc)m(hing)472 1543 y(\(Ste\013en)i(Neumann,)f(Stefan)f(P)m(osc)m(h)j(und)e(Gerhard)f (Sagerer\))p Black 24 1705 a Fu(99{05)p Black 168 w FD(A)e(Systematic)f (Approac)m(h)h(to)f(Dynamic)f(Programming)e(in)j(Bioinformatics.)39 b(P)m(art)29 b(1)g(and)472 1825 y(2:)44 b(Sequence)35 b(Comparison)c(and)i(RNA)f(F)-8 b(olding)472 1945 y(\(Rob)s(ert)33 b(Giegeric)m(h\))p Black 24 2107 a Fu(99{06)p Black 168 w FD(Autonomie)f(f)s(\177)-51 b(ur)31 b(situierte)h(Rob)s(otersysteme)h ({)g(Stand)g(und)g(En)m(t)m(wic)m(klungslinien)472 2228 y(\(Alois)e(Knoll\))p Black 24 2390 a Fu(2000{01)p Black 94 w FD(Adv)-5 b(ances)34 b(in)e(DFS)g(Theory)472 2510 y(\(Ingo)h(Gl\177)-49 b(oc)m(kner\))p Black 24 2672 a Fu(2000{02)p Black 94 w FD(A)33 b(Broad)f(Class)h(of)f(DFS)g(Mo)s(dels) 472 2792 y(\(Ingo)h(Gl\177)-49 b(oc)m(kner\))p Black 24 2954 a Fu(2000{03)p Black 94 w FD(An)33 b(Axiomatic)e(Theory)i(of)f (F)-8 b(uzzy)34 b(Quan)m(ti\014ers)f(in)f(Natural)f(Languages)472 3074 y(\(Ingo)i(Gl\177)-49 b(oc)m(kner\))p Black Black eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF