Subject: evolutionary_tree.ps Date: Sat, 2 Jan 1999 22:13:31 -0500 (EST) From: Ming-Yang Kao To: "Mark B. Gerstein" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: paper.dvi %%Pages: 35 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -f -pp 1-35 paper.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.01.01:1304 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N 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(the)g(random)g(v)m(ariables)-1 393 y Fm(X)39 400 y Fg(i)71 393 y Fq(for)f(di\013eren)o(t)f Fm(i)g Fq(are)h(not)g(necessarily)e (iden)o(tically)g(distributed.)24 b(F)l(or)18 b(brevit)o(y)l(,)e(w)o(e) h(often)h(omit)e(the)-1 454 y(index)f Fm(i)h Fq(of)h Fm(X)259 461 y Fg(i)290 454 y Fq(in)f(a)g(statemen)o(t)f(if)g(the)h (statemen)o(t)f(is)h(indep)q(enden)o(t)f(of)i Fm(i)p Fq(.)72 514 y(F)l(or)g(t)o(w)o(o)f(no)q(des)h Fm(X)j Fq(and)d Fm(Y)25 b Fi(2)14 b Fm(T)7 b Fq(,)15 b(let)753 624 y Fm(p)777 631 y Fg(X)s(Y)853 624 y Fq(=)f(Pr)p Fi(f)8 b Fm(X)18 b Fi(6)p Fq(=)c Fm(Y)19 b Fi(g)9 b Fm(:)-1 734 y Fq(Also,)16 b(for)g(brevit)o(y)l(,)e(let)849 803 y Fm(\013)g Fq(=)994 769 y Fm(m)p 951 791 128 2 v 951 837 a(m)d Fi(\000)g Fq(1)1084 803 y Fm(:)-1 907 y Fq(Then,)16 b(the)g Fj(closeness)i Fq(of)e Fm(X)21 b Fq(and)c Fm(Y)27 b Fq(is)388 1036 y Fm(\033)416 1043 y Fg(X)s(Y)491 1036 y Fq(=)14 b(Pr)p Fi(f)8 b Fm(X)18 b Fq(=)c Fm(Y)20 b Fi(g)11 b(\000)928 1003 y Fq(1)p 876 1025 V 876 1070 a Fm(m)g Fi(\000)g Fq(1)1009 1036 y(Pr)q Fi(f)d Fm(X)18 b Fi(6)p Fq(=)c Fm(Y)19 b Fi(g)14 b Fq(=)g(1)d Fi(\000)g Fm(\013p)1483 1043 y Fg(X)s(Y)1546 1036 y Fm(:)326 b Fq(\(2\))-1 1163 y(Notice)13 b(that)h Fm(\013)g Fq(equals)g(2)g(for)h Fm(m)e Fq(=)h(2)g(and)h(tends)f(to)g(1)g(as)h(the)f(alphab)q(et)g(size) f(gro)o(ws.)22 b(The)14 b(next)f(lemma)-1 1223 y(sho)o(ws)k(ho)o(w)g (to)f(calculate)g(the)g(closeness.)-1 1338 y Fb(Lemma)g(2.1)57 1452 y Fj(1.)24 b(If)17 b Fm(X)22 b Fj(and)c Fm(Y)28 b Fj(ar)n(e)17 b(the)h(endp)n(oints)g(of)f(an)h(e)n(dge)g Fm(e)f Fj(in)h Fm(T)7 b Fj(,)16 b(then)j Fm(\033)1305 1459 y Fg(X)s(Y)1380 1452 y Fq(=)14 b(\(det)8 b Fi(M)1587 1459 y Fg(e)1605 1452 y Fq(\))1655 1414 y Fc(1)p 1629 1420 67 2 v 1629 1440 a Ff(m)p Fe(\000)p Fc(1)1703 1452 y Fj(.)57 1553 y(2.)24 b(If)17 b(no)n(de)h Fm(Y)28 b Fj(is)18 b(on)f(the)h(p)n(ath)f(b)n(etwe)n(en)i(two)f(no)n(des)g Fm(X)j Fj(and)d Fm(Z)j Fj(in)d Fm(T)7 b Fj(,)17 b(then)h Fm(\033)1530 1560 y Fg(X)s(Z)1604 1553 y Fq(=)c Fm(\033)1684 1560 y Fg(X)s(Y)1745 1553 y Fm(\033)1773 1560 y Fg(Y)8 b(Z)1830 1553 y Fj(.)-1 1668 y(Pr)n(o)n(of.)42 b Fq(Statemen)o(t)14 b(1)i(follo)o(ws)g(from)e(the)i(fact)f(det)8 b Fi(M)1034 1675 y Fg(e)1066 1668 y Fq(=)14 b(\(1)d Fi(\000)e Fm(\013p)1275 1675 y Fg(e)1294 1668 y Fq(\))1313 1649 y Fg(m)p Fk(\000)p Fl(1)1392 1668 y Fq(.)21 b(Statemen)o(t)14 b(2)i(is)f(pro)o(v)o(ed)g (in)-1 1728 y(four)i(cases)f(as)h(follo)o(ws.)72 1788 y Fj(Case)24 b Fq(1:)38 b Fm(X)28 b Fq(is)c(an)h(ancestor)f(of)h Fm(Z)t Fq(.)44 b(W)l(e)24 b(pro)o(v)o(e)f(this)h(case)g(using)h (Statemen)o(t)d(1.)45 b(F)l(or)24 b(ev)o(ery)-1 1848 y(no)q(de)e(pair)g Fm(Q)267 1855 y Fl(1)309 1848 y Fq(and)g Fm(Q)448 1855 y Fl(2)489 1848 y Fq(where)g Fm(Q)675 1855 y Fl(1)716 1848 y Fq(is)g(an)g(ancestor)g(of)g Fm(Q)1142 1855 y Fl(2)1184 1848 y Fq(in)f Fm(T)7 b Fq(,)22 b(w)o(e)g(de\014ne)f (the)h Fj(p)n(ath)g(tr)n(ansition)-1 1908 y(matrix)c Fi(M)216 1915 y Fg(Q)244 1920 y Fc(1)261 1915 y Fg(Q)289 1920 y Fc(2)325 1908 y Fq(of)h(size)e Fm(m)11 b Fi(\002)h Fm(m)18 b Fq(from)e Fm(Q)798 1915 y Fl(1)836 1908 y Fq(to)i Fm(Q)936 1915 y Fl(2)973 1908 y Fq(b)o(y)f Fi(M)1102 1915 y Fg(Q)1130 1920 y Fc(1)1147 1915 y Fg(Q)1175 1920 y Fc(2)1194 1908 y Fq([)p Fm(k)r(;)8 b(k)1284 1890 y Fk(0)1295 1908 y Fq(])16 b(=)h(Pr)p Fi(f)8 b Fm(Q)1504 1915 y Fl(2)1537 1908 y Fq(=)14 b Fm(a)1615 1915 y Fg(k)1634 1906 y Fe(0)1656 1908 y Fi(j)f Fm(Q)1722 1915 y Fl(1)1755 1908 y Fq(=)h Fm(a)1833 1915 y Fg(k)1862 1908 y Fi(g)k Fq(to)-1 1968 y(generalize)e(Equation)i(\(1\).)25 b(By)17 b(the)g(indep)q(endence)f(of)i(m)o(utations)e(along)j(distinct)d (edges,)i Fi(M)1806 1975 y Fg(Q)1834 1980 y Fc(1)1851 1975 y Fg(Q)1879 1980 y Fc(2)1915 1968 y Fq(is)-1 2029 y(the)e(pro)q(duct)h(of)f(the)g(edge)h(transition)f(matrices)e(on)j (the)f(path)h(from)e Fm(Q)1354 2036 y Fl(1)1390 2029 y Fq(to)h Fm(Q)1488 2036 y Fl(2)1524 2029 y Fq(in)g Fm(T)7 b Fq(.)20 b(Th)o(us,)551 2167 y Fi(M)611 2174 y Fg(Q)639 2179 y Fc(1)656 2174 y Fg(Q)684 2179 y Fc(2)703 2167 y Fq([)p Fm(k)r(;)8 b(k)793 2146 y Fk(0)804 2167 y Fq(])13 b(=)883 2094 y Fd(\()937 2135 y Fq(1)f Fi(\000)f Fm(p)1047 2142 y Fg(Q)1075 2147 y Fc(1)1092 2142 y Fg(Q)1120 2147 y Fc(2)1181 2135 y Fq(if)k Fm(k)h Fq(=)e Fm(k)1345 2117 y Fk(0)1357 2135 y Fq(;)942 2168 y Fg(p)960 2174 y Ff(Q)984 2181 y Fc(1)1001 2174 y Ff(Q)1025 2181 y Fc(2)p 942 2184 103 2 v 955 2213 a Fg(m)p Fk(\000)p Fl(1)1181 2196 y Fq(if)h Fm(k)h Fi(6)p Fq(=)e Fm(k)1345 2177 y Fk(0)1357 2196 y Fq(.)-1 2325 y(Therefore,)i(w)o(e)h(can)g(generalize)f(Statemen) o(t)f(1)j(as)f Fm(\033)988 2332 y Fg(Q)1016 2337 y Fc(1)1033 2332 y Fg(Q)1061 2337 y Fc(2)1095 2325 y Fq(=)e(\(det)8 b Fi(M)1303 2332 y Fg(Q)1331 2337 y Fc(1)1348 2332 y Fg(Q)1376 2337 y Fc(2)1395 2325 y Fq(\))1445 2287 y Fc(1)p 1419 2293 67 2 v 1419 2313 a Ff(m)p Fe(\000)p Fc(1)1501 2325 y Fm(:)17 b Fq(Consequen)o(tly)l(,)f(since)-1 2385 y Fi(M)59 2392 y Fg(X)s(Z)133 2385 y Fq(=)d Fi(M)244 2392 y Fg(X)s(Y)306 2385 y Fi(M)366 2392 y Fg(Y)8 b(Z)423 2385 y Fq(,)15 b Fm(\033)480 2392 y Fg(X)s(Z)554 2385 y Fq(=)f Fm(\033)634 2392 y Fg(X)s(Y)695 2385 y Fm(\033)723 2392 y Fg(Y)8 b(Z)780 2385 y Fq(.)72 2445 y Fj(Case)17 b Fq(2:)k Fm(Z)f Fq(is)c(an)h(ancestor)g(of)f Fm(X)t Fq(.)22 b(This)16 b(case)h(is)f(symmetri)o(c)d(to)k(Case)f(1.)961 2682 y(5)p eop %%Page: 6 6 6 5 bop 72 120 a Fj(Case)12 b Fq(3:)20 b Fm(Y)j Fq(is)12 b(the)f(lo)o(w)o(est)h(common)e(ancestor)i(of)g Fm(X)17 b Fq(and)12 b Fm(Z)t Fq(.)20 b(Then)12 b(b)o(y)g(utilizing)e(the)i (indep)q(endence)-1 181 y(of)f(edge)g(m)o(utations)f(along)i(distinct)e (edges)h(and)h(b)o(y)e(conditioning)h(on)h(the)e(v)m(alue)h(of)g Fm(Y)23 b Fq(for)11 b(an)g(arbitrary)g Fm(y)-1 241 y Fq(with)16 b(Pr)p Fi(f)8 b Fm(Y)25 b Fq(=)14 b Fm(y)c Fi(g)k Fm(>)g Fq(0,)i(w)o(e)g(ha)o(v)o(e)372 351 y(Pr)p Fi(f)8 b Fm(X)19 b Fq(=)13 b Fm(Z)g Fi(j)g Fm(Y)25 b Fq(=)14 b Fm(y)9 b Fi(g)293 424 y Fq(=)384 382 y Fd(X)372 474 y Fg(a)391 479 y Ff(i)404 474 y Fk(2A)464 424 y Fq(Pr)p Fi(f)f Fm(X)19 b Fq(=)13 b Fm(Z)18 b Fq(=)c Fm(a)788 431 y Fg(i)810 424 y Fi(j)f Fm(Y)25 b Fq(=)14 b Fm(y)c Fi(g)293 545 y Fq(=)41 b(Pr)p Fi(f)8 b Fm(X)19 b Fq(=)13 b Fm(Z)18 b Fq(=)c Fm(y)c Fi(j)j Fm(Y)25 b Fq(=)14 b Fm(y)c Fi(g)h Fq(+)1007 504 y Fd(X)955 598 y Fg(a)974 603 y Ff(i)987 598 y Fk(2A\000f)p Fg(y)q Fk(g)1128 545 y Fq(Pr)p Fi(f)d Fm(X)18 b Fq(=)c Fm(Z)k Fq(=)c Fm(a)1452 552 y Fg(i)1474 545 y Fi(j)f Fm(Y)25 b Fq(=)14 b Fm(y)c Fi(g)293 698 y Fq(=)41 b(Pr)p Fi(f)8 b Fm(X)19 b Fq(=)13 b Fm(Y)20 b Fi(g)8 b Fq(Pr)p Fi(f)g Fm(Z)18 b Fq(=)c Fm(Y)19 b Fi(g)11 b Fq(+)g(\()p Fm(m)g Fi(\000)g Fq(1\))1139 664 y(Pr)p Fi(f)e Fm(X)18 b Fi(6)p Fq(=)c Fm(Y)19 b Fi(g)p 1139 686 268 2 v 1209 732 a Fm(m)11 b Fi(\000)g Fq(1)1417 664 y(Pr)p Fi(f)d Fm(Y)25 b Fi(6)p Fq(=)14 b Fm(Z)e Fi(g)p 1417 686 261 2 v 1483 732 a Fm(m)f Fi(\000)g Fq(1)1682 698 y Fm(:)-1 828 y Fq(Since)k(Pr)p Fi(f)9 b Fm(X)18 b Fq(=)c Fm(Z)e Fi(j)h Fm(Y)25 b Fq(=)14 b Fm(y)c Fi(g)16 b Fq(is)g(constan)o(t)h(for)f(all)g Fm(y)r Fq(,)451 938 y Fm(\033)479 945 y Fg(X)s(Z)580 938 y Fq(=)41 b(1)12 b Fi(\000)e Fm(\013)p Fq(Pr)q Fi(f)e Fm(X)18 b Fi(6)p Fq(=)c Fm(Z)e Fi(g)580 1010 y Fq(=)41 b(1)12 b Fi(\000)e Fm(\013)f Fq(\(1)j Fi(\000)e Fq(Pr)q Fi(f)e Fm(X)18 b Fq(=)c Fm(Z)e Fi(j)h Fm(Y)25 b Fq(=)14 b Fm(y)c Fi(g)p Fq(\))580 1083 y(=)41 b(\(1)12 b Fi(\000)e Fm(\013)p Fq(Pr)q Fi(f)e Fm(X)18 b Fi(6)p Fq(=)c Fm(Y)20 b Fi(g)o Fq(\)\(1)12 b Fi(\000)f Fm(\013)p Fq(Pr)p Fi(f)d Fm(Z)18 b Fi(6)p Fq(=)c Fm(Y)19 b Fi(g)p Fq(\))580 1156 y(=)41 b Fm(\033)687 1163 y Fg(X)s(Y)749 1156 y Fm(\033)777 1163 y Fg(Z)r(Y)833 1156 y Fm(:)72 1266 y Fj(Case)15 b Fq(4:)21 b(Otherwise.)f(Let)14 b Fm(W)22 b Fq(b)q(e)14 b(the)h(lo)o(w)o(est)f(common)e(ancestor)j(of)g Fm(X)k Fq(and)c Fm(Z)t Fq(.)21 b(Assume)13 b(without)-1 1326 y(loss)18 b(of)g(generalit)o(y)f(that)h Fm(Y)29 b Fq(is)18 b(on)g(the)g(path)g(b)q(et)o(w)o(een)f Fm(X)23 b Fq(and)18 b Fm(W)25 b Fq(in)17 b Fm(T)7 b Fq(.)26 b(Then)18 b(b)o(y)f(the)h(ab)q (o)o(v)o(e)g(three)-1 1386 y(cases,)e Fm(\033)163 1393 y Fg(X)s(Z)236 1386 y Fq(=)e Fm(\033)316 1393 y Fg(X)s(W)388 1386 y Fm(\033)416 1393 y Fg(W)5 b(Z)496 1386 y Fq(=)14 b Fm(\033)576 1393 y Fg(X)s(Y)637 1386 y Fm(\033)665 1393 y Fg(Y)8 b(W)734 1386 y Fm(\033)762 1393 y Fg(W)d(Z)842 1386 y Fq(=)14 b Fm(\033)922 1393 y Fg(X)s(Y)983 1386 y Fm(\033)1011 1393 y Fg(Y)8 b(Z)1068 1386 y Fm(:)p 1114 1356 21 2 v 1114 1384 2 28 v 1132 1384 V 1114 1386 21 2 v 72 1446 a Fq(The)16 b(next)g(corollary)g(is)g(used)g(b)o(y)g(the)g (HGT)g(algorithm)g(to)g(lo)q(cate)h(in)o(ternal)e(no)q(des)i(of)f Fm(T)7 b Fq(.)-1 1548 y Fb(Corollary)18 b(2.2)24 b Fj(F)l(or)16 b(any)g Fm(\033)h Fj(and)g(any)f(two)g(no)n(des)g Fm(X)q(;)8 b(Y)26 b Fi(2)14 b Fm(T)22 b Fj(with)17 b Fm(\033)1327 1555 y Fg(X)s(Y)1402 1548 y Fi(\024)d Fm(\033)r Fj(,)i(ther)n(e)g(is)g (a)g(no)n(de)g Fm(P)24 b Fj(on)-1 1608 y(the)18 b(p)n(ath)f(b)n(etwe)n (en)i Fm(X)j Fj(and)17 b Fm(Y)29 b Fj(in)18 b Fm(T)23 b Fj(such)18 b(that)g Fm(\033)r Fq(\(1)11 b Fi(\000)g Fm(\013g)r Fq(\))1109 1590 y Fl(1)p Fg(=)p Fl(2)1178 1608 y Fi(\024)j Fm(\033)1259 1615 y Fg(X)s(P)1333 1608 y Fi(\024)g Fm(\033)r Fq(\(1)d Fi(\000)f Fm(\013g)r Fq(\))1594 1590 y Fk(\000)p Fl(1)p Fg(=)p Fl(2)1677 1608 y Fm(:)-1 1710 y Fj(Pr)n(o)n(of.)53 b Fq(This)18 b(corollary)g(follo)o(ws)g(from) f(Lemma)e(2.1\(2\))k(and)g(the)f(fact)g(that)g(the)g(closeness)g(b)q (et)o(w)o(een)-1 1770 y(the)e(endp)q(oin)o(ts)g(of)h(an)g(edge)f(in)g Fm(T)22 b Fq(is)16 b(at)h(least)f(1)c Fi(\000)e Fm(\013g)r Fq(.)p 1064 1740 V 1064 1768 2 28 v 1083 1768 V 1064 1770 21 2 v 72 1830 a(If)16 b Fm(X)21 b Fq(and)16 b Fm(Y)28 b Fq(are)16 b(lea)o(v)o(es,)e(their)i(closeness)g(can)g(b)q(e)h (estimated)d(using)j(sample)e(sequences)g(as)786 1975 y(^)-26 b Fm(\033)812 1982 y Fg(X)s(Y)887 1975 y Fq(=)944 1941 y(1)p 944 1963 25 2 v 946 2009 a Fm(`)1005 1921 y Fg(`)982 1934 y Fd(X)983 2025 y Fg(i)p Fl(=1)1050 1975 y Fm(I)1082 1981 y Fl(^)1072 1989 y Fg(X)1101 1994 y Ff(i)1119 1981 y Fl(^)1114 1989 y Fg(Y)1134 1994 y Ff(i)1149 1975 y Fm(;)723 b Fq(\(3\))-1 2130 y(where)152 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Fm(\033)388 760 y Fg(X)s(Y)468 753 y Fq(and)j(\001)606 760 y Fg(X)s(Y)668 753 y Fq(.)27 b(This)19 b(in)o(tuition)e(leads)i(to)f(a)h(natural)g(greedy)f (strategy)g(outlined)-1 813 y(b)q(elo)o(w)e(that)h(fa)o(v)o(ors)f(leaf) g(pairs)g(with)g(small)f Fm(d)858 820 y Fg(X)s(Y)937 813 y Fq(and)h(large)h Fm(\033)1179 820 y Fg(X)s(Y)1240 813 y Fq(.)72 873 y(The)e Fj(g-depth)g Fq(of)g(a)g(no)q(de)g(in)f(a)h (ro)q(oted)g(tree)f(is)g(the)g(smallest)f(n)o(um)o(b)q(er)g(of)h(edges) h(in)f(a)h(path)g(from)e(the)-1 933 y(no)q(de)18 b(to)h(a)f(leaf.)26 b(Let)18 b Fm(e)f Fq(b)q(e)h(an)h(edge)e(b)q(et)o(w)o(een)g(no)q(des)i Fm(u)1073 940 y Fl(1)1110 933 y Fq(and)g Fm(u)1235 940 y Fl(2)1272 933 y Fq(in)f(a)g(ro)q(oted)h(tree)e Fm(T)1663 915 y Fk(0)1674 933 y Fq(.)26 b(Let)18 b Fm(T)1839 915 y Fk(0)1832 945 y Fl(1)1869 933 y Fq(and)-1 993 y Fm(T)35 975 y Fk(0)28 1006 y Fl(2)66 993 y Fq(b)q(e)g(the)g(subtrees)g(of)h Fm(T)506 975 y Fk(0)535 993 y Fq(obtained)g(b)o(y)f(cutting)g Fm(e)g Fq(whic)o(h)f(con)o(tain)h Fm(u)1362 1000 y Fl(1)1400 993 y Fq(and)h Fm(u)1525 1000 y Fl(2)1545 993 y Fq(,)f(resp)q(ectiv)o (ely)l(.)25 b(The)-1 1053 y Fj(g-depth)18 b Fq(of)g Fm(e)e Fq(in)h Fm(T)359 1035 y Fk(0)387 1053 y Fq(is)g(the)g(larger)g(of)h (the)e(g-depth)i(of)g Fm(u)1067 1060 y Fl(1)1103 1053 y Fq(in)f Fm(T)1197 1035 y Fk(0)1190 1066 y Fl(1)1226 1053 y Fq(and)h(that)g(of)f Fm(u)1513 1060 y Fl(2)1550 1053 y Fq(in)f Fm(T)1643 1035 y Fk(0)1636 1066 y Fl(2)1656 1053 y Fq(.)24 b(The)17 b Fj(g-depth)-1 1114 y Fq(of)f(a)f(ro)q(oted)i (tree)d(is)i(the)f(largest)g(p)q(ossible)h(g-depth)g(of)g(an)g(edge)f (in)g(the)g(tree.)20 b(\(The)c(pre\014x)f(g)g(is)h(added)-1 1174 y(to)h(the)f(term)e(depth)i(b)q(ecause)h(this)f(usage)h(of)f (depth)h(is)f(nonstandard)i(in)e(graph)h(theory)l(.\))72 1234 y(Let)22 b Fm(d)g Fq(b)q(e)f(the)g Fm(g)r Fq(-depth)h(of)g Fm(T)7 b Fq(.)36 b(V)l(arian)o(ts)21 b(of)h(the)f(next)g(lemma)e(ha)o (v)o(e)h(pro)o(v)o(en)h(v)o(ery)f(useful)h(and)-1 1294 y(insigh)o(tful)15 b([10)q(,)g(11)q(,)h(15)q(,)g(27].)-1 1395 y Fb(Lemma)g(2.7)57 1508 y Fj(1.)24 b Fm(d)14 b Fi(\024)g Fq(1)d(+)g Fi(b)p Fq(log)382 1520 y Fl(2)402 1508 y Fq(\()p Fm(n)g Fi(\000)g Fq(1\))p Fi(c)p Fm(:)57 1610 y Fj(2.)24 b(Every)h(internal)h(no)n(de)e Fm(P)32 b Fj(of)25 b Fm(T)31 b Fj(exc)n(ept)26 b(the)f(r)n(o)n(ot)e(has)h(a)h (de\014ning)h(triplet)g Fm(X)t(Y)11 b(Z)29 b Fj(such)c(that)121 1670 y Fm(d)146 1677 y Fg(X)s(P)207 1670 y Fm(;)8 b(d)254 1677 y Fg(Y)h(P)312 1670 y Fj(,)18 b(and)h Fm(d)466 1677 y Fg(Z)r(P)540 1670 y Fj(ar)n(e)f(al)r(l)h(at)g(most)f Fm(d)12 b Fq(+)g(1)18 b Fj(and)h(thus,)f Fm(\033)1246 1677 y Fg(X)s(Y)8 b(Z)1350 1670 y Fi(\025)15 b Fq(\(1)d Fi(\000)g Fm(\013g)r Fq(\))1585 1652 y Fl(2\()p Fg(d)p Fl(+1\))1695 1670 y Fj(.)25 b(Every)19 b(le)n(af)121 1730 y(of)e Fm(T)24 b Fj(is)17 b(in)h(such)g(a)f(triplet.)-1 1843 y(Pr)n(o)n(of.)52 b Fq(The)18 b(pro)q(of)h(is)e(straigh)o(tforw)o (ard.)27 b(Note)18 b(that)g(the)g(more)e(un)o(balanced)i Fm(T)24 b Fq(is,)18 b(the)f(smaller)f(its)-1 1904 y(g-depth)h(is.)p 262 1874 V 262 1902 2 28 v 281 1902 V 262 1904 21 2 v 72 1964 a(In)h Fm(T)7 b Fq(,)17 b(the)h(star)g(formed)f(b)o(y)g(a)h (de\014ning)g(triplet)f(of)h(an)h(in)o(ternal)d(no)q(de)j(con)o(tains)f (the)g(three)f(edges)-1 2024 y(inciden)o(t)g(to)i(the)g(in)o(ternal)f (no)q(de.)29 b(Th)o(us,)19 b(\011\()p Fm(T)7 b Fq(\))18 b(can)h(b)q(e)g(reconstructed)g(from)e(triplets)h(describ)q(ed)g(in)-1 2084 y(Lemma)13 b(2.7\(2\))j(or)g(those)f(with)h(similarly)c(large)j (closeness)h(v)m(alues.)k(This)c(observ)m(ation)g(motiv)m(ates)e(the)-1 2144 y(follo)o(wing)i(de\014nitions.)21 b(Let)571 2294 y Fm(\033)599 2301 y Fl(lg)670 2294 y Fq(=)754 2260 y(3)778 2219 y Fi(p)p 820 2219 25 2 v 41 x Fq(2)p 754 2282 91 2 v 787 2328 a(2)858 2221 y Fd( )896 2219 y Fi(p)p 937 2219 25 2 v 937 2260 a Fq(2)12 b Fi(\000)f Fq(1)p 896 2282 152 2 v 896 2291 a Fi(p)p 938 2291 25 2 v 41 x Fq(2)g(+)g(1)1052 2221 y Fd(!)1085 2229 y Fl(2)1113 2294 y Fq(\(1)g Fi(\000)g Fm(\013g)r Fq(\))1292 2273 y Fl(2)p Fg(d)p Fl(+4)1375 2294 y Fq(;)555 2416 y Fm(\033)583 2423 y Fl(sm)670 2416 y Fq(=)758 2382 y Fm(\033)786 2389 y Fl(lg)p 754 2404 66 2 v 754 2413 a Fi(p)p 796 2413 25 2 v 41 x Fq(2)825 2416 y(;)550 2540 y Fm(\033)578 2547 y Fl(md)670 2540 y Fq(=)754 2506 y Fm(\033)782 2513 y Fl(lg)823 2506 y Fq(+)g Fm(\033)900 2513 y Fl(sm)p 754 2528 191 2 v 838 2574 a Fq(2)950 2540 y Fm(:)961 2682 y Fq(9)p eop %%Page: 10 10 10 9 bop -1 120 a Fj(R)n(emark.)21 b Fq(The)16 b(c)o(hoice)f(of)i Fm(\033)528 127 y Fl(lg)574 120 y Fq(is)f(obtained)g(b)o(y)g(solving)g (Equations)h(\(17\),)f(\(18\))h(and)g(\(19\).)72 181 y(A)k(triplet)e Fm(X)t(Y)12 b(Z)24 b Fq(is)d Fj(lar)n(ge)g Fq(if)f Fm(\033)676 188 y Fg(X)s(Y)8 b(Z)786 181 y Fi(\025)21 b Fm(\033)874 188 y Fl(lg)904 181 y Fq(;)i(it)d(is)h Fj(smal)r(l)h Fq(if)e Fm(\033)1258 188 y Fg(X)s(Y)8 b(Z)1368 181 y Fi(\024)21 b Fm(\033)1456 188 y Fl(sm)1501 181 y Fq(.)35 b(By)20 b(Lemma)f(2.7\(2\),)-1 241 y(eac)o(h)h(nonro)q(ot)j (in)o(ternal)d(no)q(de)h(of)g Fm(T)28 b Fq(has)21 b(at)g(least)g(one)g (large)g(de\014ning)g(triplet.)34 b(The)21 b(next)f(lemma)-1 301 y(and)c(corollary)e(sho)o(w)i(that)f(the)g(large)g(triplets)f(all)g (ha)o(v)o(e)h(larger)g(estimated)e(closeness)i(v)m(alues)g(than)g(the) -1 361 y(small)g(triplets)g(with)h(high)g(probabilit)o(y)l(.)-1 446 y Fb(Lemma)g(2.8)24 b Fj(The)18 b(\014rst)f(ine)n(quality)i(b)n (elow)g(holds)f(for)f(al)r(l)i(lar)n(ge)g(triplets)f Fm(X)t(Y)11 b(Z)t Fj(,)18 b(and)g(the)h(se)n(c)n(ond)e(for)-1 507 y(al)r(l)i(smal)r(l)f(triplets.)456 652 y Fq(Pr)p Fi(f)11 b Fq(^)-26 b Fm(\033)570 659 y Fg(X)s(Y)7 b(Z)671 652 y Fi(\024)14 b Fm(\033)752 659 y Fl(md)811 652 y Fi(g)41 b(\024)h Fq(exp)1041 554 y Fd(0)1041 627 y(B)1041 654 y(@)1077 652 y Fi(\000)1121 561 y Fd(\020)1145 568 y Fi(p)p 1187 568 25 2 v 41 x Fq(2)11 b Fi(\000)g Fq(1)1296 561 y Fd(\021)1321 572 y Fl(2)p 1121 640 221 2 v 1181 686 a Fq(36)p Fm(\013)1260 672 y Fl(2)1346 652 y Fm(`\033)1396 631 y Fl(2)1394 664 y(lg)1424 554 y Fd(1)1424 627 y(C)1424 654 y(A)1469 652 y Fq(;)378 b(\(11\))456 845 y(Pr)p Fi(f)11 b Fq(^)-26 b Fm(\033)570 852 y Fg(X)s(Y)7 b(Z)671 845 y Fi(\025)14 b Fm(\033)752 852 y Fl(md)811 845 y Fi(g)41 b(\024)h Fq(exp)1041 747 y Fd(0)1041 820 y(B)1041 846 y(@)1077 845 y Fi(\000)1121 753 y Fd(\020)1145 760 y Fi(p)p 1187 760 25 2 v 41 x Fq(2)11 b Fi(\000)g Fq(1)1296 753 y Fd(\021)1321 765 y Fl(2)p 1121 833 221 2 v 1181 879 a Fq(36)p Fm(\013)1260 864 y Fl(2)1346 845 y Fm(`\033)1396 824 y Fl(2)1394 857 y(lg)1424 747 y Fd(1)1424 820 y(C)1424 846 y(A)1469 845 y Fm(:)378 b Fq(\(12\))-1 988 y Fj(Pr)n(o)n(of.)42 b Fq(See)16 b Fi(x)p Fq(A.2.)p 406 959 21 2 v 406 987 2 28 v 425 987 V 406 989 21 2 v -1 1080 a Fb(Corollary)i(2.9)24 b Fj(L)n(et)c Fm(E)467 1087 y Fl(g)508 1080 y Fj(denote)h(the)g(event)h (that)h Fq(^)-26 b Fm(\033)1012 1087 y Fg(X)s(Y)7 b(Z)1119 1080 y Fm(>)22 b Fq(^)-27 b Fm(\033)1204 1087 y Fg(X)1236 1078 y Fe(0)1247 1087 y Fg(Y)1275 1078 y Fe(0)1286 1087 y Fg(Z)1312 1078 y Fe(0)1346 1080 y Fj(for)20 b(every)h(lar)n(ge)f (triplet)h Fm(X)t(Y)12 b(Z)-1 1141 y Fj(and)18 b(every)g(smal)r(l)g (triplet)g Fm(X)541 1123 y Fk(0)553 1141 y Fm(Y)592 1123 y Fk(0)604 1141 y Fm(Z)641 1123 y Fk(0)653 1141 y Fj(.)k(Then)535 1286 y Fq(Pr)587 1238 y Fd(n)p 623 1247 40 2 v 623 1286 a Fm(E)662 1293 y Fl(g)690 1238 y Fd(o)732 1286 y Fi(\024)784 1213 y Fd( )817 1252 y Fm(n)820 1320 y Fq(3)846 1213 y Fd(!)888 1286 y Fq(exp)970 1188 y Fd(0)970 1261 y(B)970 1288 y(@)1007 1286 y Fi(\000)1051 1195 y Fd(\020)1075 1202 y Fi(p)p 1117 1202 25 2 v 41 x Fq(2)11 b Fi(\000)g Fq(1)1226 1195 y Fd(\021)1251 1206 y Fl(2)p 1051 1275 221 2 v 1111 1320 a Fq(36)p Fm(\013)1190 1306 y Fl(2)1276 1286 y Fm(`\033)1326 1266 y Fl(2)1324 1298 y(lg)1354 1188 y Fd(1)1354 1261 y(C)1354 1288 y(A)1399 1286 y Fm(:)-1 1439 y Fj(Pr)n(o)n(of.)p 180 1400 40 2 v 51 w Fm(E)219 1446 y Fl(g)257 1439 y Fq(implies)k(that)j(there)g(is)f(a)h(large)g (triplet)f Fm(X)t(Y)11 b(Z)22 b Fq(with)e(^)-27 b Fm(\033)1300 1446 y Fg(X)s(Y)8 b(Z)1405 1439 y Fi(\024)16 b Fm(\033)1488 1446 y Fl(md)1557 1439 y Fq(or)i(there)f(is)g(a)i(small)-1 1499 y(triplet)c Fm(X)192 1481 y Fk(0)204 1499 y Fm(Y)243 1481 y Fk(0)255 1499 y Fm(Z)292 1481 y Fk(0)320 1499 y Fq(with)j(^)-26 b Fm(\033)459 1506 y Fg(X)491 1497 y Fe(0)501 1506 y Fg(Y)530 1497 y Fe(0)541 1506 y Fg(Z)567 1497 y Fe(0)594 1499 y Fi(\025)14 b Fm(\033)675 1506 y Fl(md)726 1499 y Fq(.)21 b(Th)o(us,)16 b(b)o(y)g(Lemma)e(2.8,)155 1590 y(Pr)208 1542 y Fd(n)p 244 1551 V 244 1590 a Fm(E)283 1597 y Fl(g)311 1542 y Fd(o)380 1590 y Fi(\024)533 1548 y Fd(X)461 1652 y Fq(large)i Fg(X)s(Y)8 b(Z)674 1590 y Fq(Pr)p Fi(f)j Fq(^)-26 b Fm(\033)788 1597 y Fg(X)s(Y)7 b(Z)889 1590 y Fi(\024)14 b Fm(\033)970 1597 y Fl(md)1029 1590 y Fi(g)d Fq(+)1208 1548 y Fd(X)1114 1652 y Fq(small)k Fg(X)1274 1643 y Fe(0)1285 1652 y Fg(Y)1313 1643 y Fe(0)1324 1652 y Fg(Z)1350 1643 y Fe(0)1370 1590 y Fq(Pr)p Fi(f)c Fq(^)-27 b Fm(\033)1483 1597 y Fg(X)1515 1588 y Fe(0)1526 1597 y Fg(Y)1554 1588 y Fe(0)1565 1597 y Fg(Z)1591 1588 y Fe(0)1618 1590 y Fi(\025)14 b Fm(\033)1699 1597 y Fl(md)1758 1590 y Fi(g)380 1784 y(\024)461 1711 y Fd( )493 1750 y Fm(n)496 1818 y Fq(3)523 1711 y Fd(!)564 1784 y Fq(exp)647 1686 y Fd(0)647 1759 y(B)647 1786 y(@)683 1784 y Fi(\000)727 1693 y Fd(\020)751 1700 y Fi(p)p 793 1700 25 2 v 41 x Fq(2)d Fi(\000)g Fq(1)902 1693 y Fd(\021)927 1704 y Fl(2)p 727 1772 221 2 v 787 1818 a Fq(36)p Fm(\013)866 1804 y Fl(2)952 1784 y Fm(`\033)1002 1763 y Fl(2)1000 1796 y(lg)1030 1686 y Fd(1)1030 1759 y(C)1030 1786 y(A)1075 1784 y Fm(:)p 15 1895 21 2 v 15 1923 2 28 v 34 1923 V 15 1925 21 2 v 72 1985 a Fq(A)23 b(nonro)q(ot)i(in)o(ternal)e(no)q(de)h (of)g Fm(T)30 b Fq(ma)o(y)22 b(ha)o(v)o(e)h(more)f(than)i(one)g(large)f (de\014ning)h(triplet.)42 b(Con-)-1 2045 y(sequen)o(tly)l(,)23 b(since)f(distance)h(estimates)f(con)o(tain)h(errors,)h(w)o(e)f(ma)o(y) f(obtain)h(an)h(erroneous)g(estimate)-1 2105 y(of)c(\011\()p Fm(T)7 b Fq(\))19 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Fl(min)1147 2557 y Fm(;)700 b Fq(\(13\))949 2682 y(10)p eop %%Page: 11 11 11 10 bop -1 120 a Fq(where)240 168 y(^)232 181 y(\001)273 188 y Fg(X)s(P)348 181 y Fq(=)13 b(\()426 168 y(^)418 181 y(\001)459 188 y Fg(X)s(Y)532 181 y Fq(+)589 168 y(^)581 181 y(\001)622 188 y Fg(X)s(Z)693 181 y Fi(\000)751 168 y Fq(^)743 181 y(\001)784 188 y Fg(Y)7 b(Z)840 181 y Fq(\))p Fm(=)p Fq(2)17 b(and)1027 168 y(^)1019 181 y(\001)1060 188 y Fg(X)s(Q)1135 181 y Fq(=)c(\()1213 168 y(^)1205 181 y(\001)1246 188 y Fg(X)s(U)1318 181 y Fq(+)1375 168 y(^)1367 181 y(\001)1408 188 y Fg(X)s(V)1481 181 y Fi(\000)1539 168 y Fq(^)1531 181 y(\001)1572 188 y Fg(X)s(V)1633 181 y Fq(\))p Fm(=)p Fq(2)p Fm(:)-1 268 y Fq(Notice)i(that)i(t)o(w)o(o)g(triplet)e(cen)o(ters)g(can)i(b)q(e)g (compared)e(in)h(this)g(manner)g(only)g(if)g(the)g(triplets)g(share)h (at)-1 328 y(least)f(one)g(leaf.)21 b(Let)464 457 y Fm(c)14 b Fq(=)g(min)641 384 y Fd(\()764 423 y Fq(\001)805 430 y Fl(min)p 679 445 272 2 v 679 491 a Fi(\000)8 b Fq(ln\(1)j 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2457 y Fq(\))j Fi(6)p Fq(=)g Fi(;)p Fj(.)57 2557 y(2.)24 b Fi(S)d Fj(c)n(onsists)d(of)f(the)h(splitting)h(tuples)f (for)f(the)h(e)n(dges)g(in)g Fm(T)1199 2539 y Fk(\003)1192 2570 y Fg(k)1218 2557 y Fj(.)949 2682 y Fq(12)p eop %%Page: 13 13 13 12 bop -1 120 a Fj(Pr)n(o)n(of.)42 b Fq(The)17 b(statemen)o(ts)d (are)j(pro)o(v)o(ed)e(separately)l(.)72 181 y(Statemen)o(t)h(1.)27 b(The)18 b(pro)q(of)h(is)f(b)o(y)f(induction)g(on)i Fm(k)r Fq(.)26 b(The)18 b(base)g(case)g(follo)o(ws)g(from)f(the)h(fact)f(that) -1 241 y(the)e(statemen)o(t)f(holds)i(for)g Fm(T)545 223 y Fk(\003)538 253 y Fl(3)580 241 y Fq(at)g(line)e(H4.)21 b(The)16 b(induction)f(step)g(follo)o(ws)h(from)e(the)h(use)h(of)g(a)g (relev)m(an)o(t)-1 301 y(triplet)f(at)i(line)e(H10.)72 361 y(Statemen)o(t)c(2.)20 b(The)13 b(pro)q(of)h(is)e(b)o(y)g (induction)g(on)i Fm(k)g Fq(based)f(on)h(the)e(follo)o(wing)g(facts.)21 b(Let)12 b Fm(Q)1743 368 y Fl(1)1763 361 y Fm(Q)1802 368 y Fl(2)1834 361 y Fq(b)q(e)g(an)-1 421 y(edge)i(in)f Fm(T)198 403 y Fk(\003)191 434 y Fg(k)231 421 y Fq(that)i(also)f (exists)g(in)f Fm(T)653 403 y Fk(\003)646 434 y Fg(k)q Fl(+1)712 421 y Fq(.)21 b(Let)14 b Fm(M)5 b(U)g(V)26 b Fq(b)q(e)14 b(a)g(triplet)f(with)g(cen)o(ter)g Fm(Q)h Fq(suc)o(h)f(that)i Fm(M)k Fi(62)14 b Fm(T)1889 403 y Fk(\003)1882 434 y Fg(k)q Fl(+1)-1 482 y Fq(and)19 b Fm(U;)8 b(V)28 b Fi(2)17 b Fm(T)293 464 y Fk(\003)286 494 y Fg(k)312 482 y Fq(.)27 b(Then,)18 b(at)h(line)e(H14,)h Fi(h)p Fm(Q)827 489 y Fl(1)847 482 y Fm(Q)886 489 y Fl(2)905 482 y Fm(;)8 b(M)d(U)g(V)s(;)j(Q;)g Fq(\001)1172 464 y Fk(\003)1172 494 y Fg(Q)1200 499 y Fc(1)1217 494 y Fg(Q)1247 482 y Fm(;)g Fq(\001)1310 464 y Fk(\003)1310 494 y Fg(Q)1338 499 y Fc(2)1354 494 y Fg(Q)1384 482 y Fm(;)g Fq(\001)1447 464 y Fk(\003)1447 494 y Fg(M)t(Q)1514 482 y Fi(i)19 b Fq(is)e(a)i(splitting)f(tuple)-1 542 y(for)e Fm(T)109 524 y Fk(\003)102 554 y Fg(k)144 542 y Fq(if)f(and)i(only)e(if)h(it)f(is)h(also)g(one)g(in)f Fm(T)809 524 y Fk(\003)802 554 y Fg(k)q Fl(+1)868 542 y Fq(.)22 b(Also,)15 b(after)h(a)g(new)g(leaf)f Fm(N)21 b Fq(and)16 b(a)h(new)f(in)o(ternal)e(no)q(de)-1 602 y Fm(P)27 b Fq(are)19 b(inserted)g(to)h Fm(T)428 584 y Fk(\003)421 614 y Fg(k)466 602 y Fq(at)g(lines)e(H11)i(and)g(H12,)g (eac)o(h)f(edge)h Fm(Q)1232 609 y Fl(1)1251 602 y Fm(Q)1290 609 y Fl(2)1329 602 y Fi(2)f Fm(T)1417 584 y Fk(\003)1410 614 y Fg(k)q Fl(+1)1489 602 y Fi(\000)13 b(f)p Fm(P)1597 609 y Fl(1)1617 602 y Fm(P)q(;)8 b(P)1702 609 y Fl(2)1723 602 y Fm(P)q(;)g(N)d(P)i Fi(g)21 b Fq(at)-1 662 y(line)15 b(H16)i(ma)o(y)d(ha)o(v)o(e)i(new)g(relev)m(an)o(t)f(triplets,)g(whic)o (h)g(m)o(ust)g(b)q(e)i(of)f(the)g(form)f Fm(N)1502 644 y Fk(0)1514 662 y Fm(N)5 b(Z)20 b Fq(with)d Fm(N)1767 644 y Fk(0)1792 662 y Fi(62)d Fm(T)1875 644 y Fk(\003)1868 675 y Fg(k)q Fl(+1)1934 662 y Fq(.)-1 722 y(Suc)o(h)i Fm(N)161 704 y Fk(0)173 722 y Fm(N)5 b(Z)20 b Fq(is)c(irrelev)m(an)o(t) f(in)h Fm(T)629 704 y Fk(\003)622 735 y Fg(k)664 722 y Fq(b)q(ecause)h Fm(N)889 704 y Fk(0)901 722 y Fm(;)8 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b(based)h(on)g(the)g(input)f(sequences.)25 b(Also,)-1 2398 y(in)14 b(practice,)g(if)g Fm(f)20 b Fq(is)15 b(unkno)o(wn)g(and)g (th)o(us)g(\001)833 2405 y Fl(min)908 2398 y Fq(is)f(also)i(unkno)o (wn,)f(then)f(the)h(HGT)f(algorithm)g(can)h(b)q(e)-1 2458 y(rep)q(eatedly)i(run)h(b)o(y)g(iterativ)o(ely)d(halving)j(an)g (initial)f(guess)h(of)h Fm(f)k Fq(un)o(til)17 b(w)o(e)g(build)h(the)f (whole)h(\011)1833 2465 y Fl(w)1861 2458 y Fq(\()p Fm(T)7 b Fq(\).)-1 2518 y(Ho)o(w)o(ev)o(er,)20 b(the)h(ending)f Fm(f)27 b Fq(should)21 b(nev)o(er)f(b)q(e)h(smaller)e(than)1178 2498 y Fl(1)p 1178 2506 18 2 v 1180 2535 a Fg(`)1200 2518 y Fq(,)j(whic)o(h)e(is)h(the)g(smallest)e(probabilit)o(y)949 2682 y(13)p eop %%Page: 14 14 14 13 bop -1 120 a Fq(observ)m(able)19 b(from)g(the)g(input)g (sequences.)30 b(This)19 b(iterativ)o(e)f(approac)o(h)i(results)f(in)g (an)h(extra)f Fi(O)9 b Fq(\(log)g Fm(`)p Fq(\))-1 181 y(factor)17 b(in)e(the)h(running)h(time.)-1 325 y Fh(3.3)66 b(Lemm)o(as)19 b(for)j(b)r(ounding)i(the)f(sample)d(size)72 478 y Fq(Let)e Fm(L)194 485 y Fg(k)234 478 y Fq(b)q(e)h(the)f(set)g(of) g(the)g(lea)o(v)o(es)f(of)i(\011\()p Fm(T)7 b Fq(\))17 b(that)i(are)f(in)g Fm(T)1225 460 y Fk(\003)1218 490 y Fg(k)1244 478 y Fq(.)27 b(Let)18 b(\011)1412 485 y Fg(k)1452 478 y Fq(b)q(e)g(the)g(subtree)g(of)h(\011\()p Fm(T)7 b Fq(\))-1 538 y(formed)17 b(b)o(y)g(the)h(edges)g(on)g(paths)h (b)q(et)o(w)o(een)e(lea)o(v)o(es)f(in)i Fm(L)1079 545 y Fg(k)1100 538 y Fq(.)26 b(A)18 b Fj(br)n(anchless)g Fq(path)h(in)e(\011)1639 545 y Fg(k)1679 538 y Fq(is)g(one)h(whose)-1 598 y(in)o(ternal)e(no)q(des)i(are)g(all)f(of)g(degree)g(2)h(in)f(\011) 817 605 y Fg(k)838 598 y Fq(.)25 b(W)l(e)17 b(sa)o(y)g(that)h Fm(T)1190 580 y Fk(\003)1183 610 y Fg(k)1226 598 y Fj(matches)g Fm(T)24 b Fq(if)17 b Fm(T)1549 580 y Fk(\003)1542 610 y Fg(k)1585 598 y Fq(without)h(the)f(edge)-1 658 y(lengths)g(can)g(b)q (e)g(obtained)g(from)e(\011)679 665 y Fg(k)717 658 y Fq(b)o(y)h(replacing)h(ev)o(ery)e(maximal)e(branc)o(hless)k(path)g (with)g(an)g(edge)-1 718 y(b)q(et)o(w)o(een)e(its)h(t)o(w)o(o)g(endp)q (oin)o(ts.)72 779 y(F)l(or)h Fm(k)e Fq(=)f(3)p Fm(;)8 b(:)g(:)g(:)g(;)g(n)p Fq(,)16 b(w)o(e)g(de\014ne)f(the)h(follo)o(wing)g (conditions:)72 880 y Fi(\017)24 b(A)161 887 y Fg(k)182 880 y Fq(:)d Fm(T)253 862 y Fk(\003)246 893 y Fg(k)289 880 y Fq(matc)o(hes)14 b Fm(T)7 b Fq(.)72 982 y Fi(\017)24 b(B)154 989 y Fg(k)175 982 y Fq(:)d(F)l(or)c(ev)o(ery)d(in)o(ternal)h (no)q(de)i Fm(Q)d Fi(2)g Fm(T)860 964 y Fk(\003)853 994 y Fg(k)879 982 y Fq(,)i(the)g(triplet)f(formed)g(b)o(y)g(def)t(\()p Fm(Q)p Fq(\))g(is)h(not)h(small.)72 1084 y Fi(\017)24 b(C)147 1091 y Fg(k)168 1084 y Fq(:)e(F)l(or)16 b(ev)o(ery)f(edge)h Fm(Q)570 1091 y Fl(1)589 1084 y Fm(Q)628 1091 y Fl(2)661 1084 y Fi(2)e Fm(T)744 1066 y Fk(\003)737 1096 y Fg(k)764 1084 y Fq(,)h Fi(j)p Fq(\001)848 1066 y Fk(\003)848 1096 y Fg(Q)876 1101 y Fc(1)893 1096 y Fg(Q)921 1101 y Fc(2)951 1084 y Fi(\000)c Fq(\001)1042 1091 y Fg(Q)1070 1096 y Fc(1)1087 1091 y Fg(Q)1115 1096 y Fc(2)1134 1084 y Fi(j)i Fm(<)h Fq(2\001)1278 1091 y Fl(min)1339 1084 y Fq(.)72 1185 y(In)j(this)g(section,)f(Lemmas)g(3.3,)h(3.4,)g(and)h(3.5)f (analyze)g(under)g(what)h(conditions)f(Split-Edge)g(can)-1 1246 y(help)g(correctly)f(insert)h(a)h(new)f(leaf)g(and)h(a)g(new)f(in) o(ternal)g(no)q(de)h(to)f Fm(T)1317 1228 y Fk(\003)1310 1258 y Fg(k)1336 1246 y Fq(.)25 b(Later)18 b(in)f Fi(x)p Fq(3.4,)h(w)o(e)f(use)g(these)-1 1306 y(lemm)o(as)12 b(to)j(sho)o(w)g(b)o(y)e(induction)h(in)g(Lemma)e(3.6)i(that)h(the)f (ev)o(en)o(ts)f Fm(E)1301 1313 y Fl(g)1335 1306 y Fq(and)i Fm(E)1464 1313 y Fl(c)1496 1306 y Fq(in)f(Corollaries)g(2.9)g(and)-1 1366 y(2.11)k(imply)e(that)i Fi(A)389 1373 y Fg(k)410 1366 y Fq(,)f Fi(B)474 1373 y Fg(k)495 1366 y Fq(,)h(and)g Fi(C)649 1373 y Fg(k)689 1366 y Fq(hold)f(for)h(all)f Fm(k)r Fq(.)26 b(This)18 b(leads)f(to)h(Theorem)f(3.7,)h(stating)g (that)g(the)-1 1426 y(HGT)e(algorithm)f(solv)o(es)g(the)h(w)o(eigh)o (ted)f(ev)o(olutionary)g(top)q(ology)i(problem)e(with)g(a)i(p)q (olynomial-sized)-1 1486 y(sample.)72 1547 y(Lemmas)d(3.3,)j(3.4,)f (and)h(3.5)f(mak)o(e)e(the)j(follo)o(wing)e(assumptions)i(for)f(some)f Fm(k)h(<)e(n)p Fq(:)72 1648 y Fi(\017)24 b Fq(The)16 b(\()p Fm(k)d Fi(\000)e Fq(3\)-th)17 b(iteration)f(of)g(the)g(rep)q (eat)h(at)f(line)f(H8)i(has)g(b)q(een)f(completed.)72 1750 y Fi(\017)24 b Fm(T)157 1732 y Fk(\003)150 1762 y Fg(k)192 1750 y Fq(has)17 b(b)q(een)f(constructed,)g(and)h Fi(A)806 1757 y Fg(k)827 1750 y Fq(,)f Fi(B)890 1757 y Fg(k)911 1750 y Fq(,)g(and)h Fi(C)1062 1757 y Fg(k)1099 1750 y Fq(hold.)72 1852 y Fi(\017)24 b Fq(The)16 b(HGT)g(algorithm)g (is)g(curren)o(tly)e(in)i(the)g(\()p Fm(k)d Fi(\000)e Fq(2\)-th)17 b(iteration)f(of)g(the)g(rep)q(eat.)-1 1966 y Fb(Lemma)g(3.3)24 b Fj(Assume)19 b(that)h Fm(E)615 1973 y Fl(c)653 1966 y Fj(holds)f(and)h(the)g(triplet)g Fm(N)5 b(X)t(Y)31 b Fj(input)20 b(to)g(Split-Edge)i(is)d(not)h(smal)r (l.)-1 2026 y(Then,)e(the)g(test)g(of)f(line)i(S7)f(fails)f(if)h(and)f (only)h(if)f Fm(P)22 b Fi(6)p Fq(=)13 b Fm(P)1070 2033 y Fl(1)1108 2026 y Fj(and)18 b Fm(P)j Fi(6)p Fq(=)13 b Fm(P)1337 2033 y Fl(2)1375 2026 y Fj(in)18 b Fm(T)7 b Fj(.)-1 2140 y(Pr)n(o)n(of.)42 b Fq(There)16 b(are)h(t)o(w)o(o)f (directions,)f(b)q(oth)i(using)g(the)f(follo)o(wing)f(equation.)22 b(F)l(rom)15 b(line)g(S6,)286 2250 y(\001)327 2257 y Fl(1)360 2250 y Fq(=)f(\()439 2238 y(^)431 2250 y(\001)472 2257 y Fg(X)501 2262 y Fc(1)518 2257 y Fg(P)558 2250 y Fi(\000)d Fq(\001)649 2257 y Fg(X)678 2262 y Fc(1)695 2257 y Fg(P)724 2250 y Fq(\))g Fi(\000)g Fq(\()831 2238 y(^)823 2250 y(\001)864 2257 y Fg(X)893 2262 y Fc(1)910 2257 y Fg(P)932 2262 y Fc(1)963 2250 y Fi(\000)g Fq(\001)1054 2257 y Fg(X)1083 2262 y Fc(1)1100 2257 y Fg(P)1122 2262 y Fc(1)1141 2250 y Fq(\))h(+)f(\(\001)1281 2257 y Fg(X)1310 2262 y Fc(1)1326 2257 y Fg(P)1367 2250 y Fi(\000)g Fq(\001)1458 2257 y Fg(X)1487 2262 y Fc(1)1504 2257 y Fg(P)1526 2262 y Fc(1)1545 2250 y Fq(\))p Fm(:)283 b Fq(\(16\))72 2360 y(\(=)-8 b Fi(\))p Fq(\))21 b(T)l(o)h(pro)o(v)o(e)f(b)o(y)g(con)o (tradiction,)g(assume)g Fm(P)30 b Fq(=)23 b Fm(P)1141 2367 y Fl(1)1182 2360 y Fq(or)f Fm(P)30 b Fq(=)23 b Fm(P)1400 2367 y Fl(2)1442 2360 y Fq(in)e Fm(T)7 b Fq(.)36 b(If)21 b Fm(P)30 b Fq(=)23 b Fm(P)1797 2367 y Fl(1)1817 2360 y Fq(,)f(then)-1 2420 y(\001)40 2427 y Fg(X)69 2432 y Fc(1)86 2427 y Fg(P)138 2420 y Fq(=)h(\001)240 2427 y Fg(X)269 2432 y Fc(1)286 2427 y Fg(P)308 2432 y Fc(1)328 2420 y Fq(,)f(and)g(b)o(y)f Fi(A)577 2427 y Fg(k)598 2420 y Fq(,)i Fm(P)666 2427 y Fl(1)707 2420 y Fq(is)f(an)g(in)o(ternal) e(no)q(de)i(in)f Fm(T)1241 2402 y Fk(\003)1234 2433 y Fg(k)1260 2420 y Fq(.)37 b(By)21 b Fi(B)1426 2427 y Fg(k)1447 2420 y Fq(,)i(the)e(triplet)f(formed)g(b)o(y)-1 2481 y(def)s(\()p Fm(P)116 2488 y Fl(1)136 2481 y Fq(\))15 b(is)f(not)h(small.)k(Th)o(us,)c(b)o(y)f Fm(E)686 2488 y Fl(c)718 2481 y Fq(and)i(Equation)f(\(16\),)g Fi(j)p Fq(\001)1191 2488 y Fl(1)1210 2481 y Fi(j)f Fm(<)f Fq(\001)1330 2488 y Fl(min)1391 2481 y Fq(.)21 b(By)13 b(symmetry)l(,)e(if)j Fm(P)21 b Fq(=)14 b Fm(P)1914 2488 y Fl(2)1934 2481 y Fq(,)-1 2541 y(then)i Fi(j)p Fq(\001)165 2548 y Fl(2)184 2541 y Fi(j)e Fm(<)g Fq(\001)305 2548 y Fl(min)365 2541 y Fq(.)21 b(In)16 b(either)g(case,)f(the)h(test)h(of)f(line)f(S7)i (passes.)949 2682 y(14)p eop %%Page: 15 15 15 14 bop 72 120 a Fq(\()p Fi(\()-8 b Fq(=\))14 b(Since)f Fm(P)21 b Fi(6)p Fq(=)14 b Fm(P)464 127 y Fl(1)484 120 y Fq(,)g(\001)553 127 y Fg(X)582 132 y Fc(1)599 127 y Fg(P)635 120 y Fi(\000)7 b Fq(\001)722 127 y Fg(X)751 132 y Fc(1)767 127 y Fg(P)789 132 y Fc(1)822 120 y Fi(\025)14 b(\000)8 b Fq(ln\(1)f Fi(\000)g Fm(\013f)e Fq(\))13 b Fi(\025)h Fq(2\001)1269 127 y Fl(min)1330 120 y Fq(.)20 b(If)14 b Fm(P)1442 127 y Fl(1)1476 120 y Fq(is)g(a)g(leaf)g(in)f Fm(T)1740 102 y Fk(\003)1733 133 y Fg(k)1760 120 y Fq(,)h(then)f(b)o(y) -1 181 y Fi(A)39 188 y Fg(k)60 181 y Fq(,)j Fm(P)121 188 y Fl(1)158 181 y Fq(is)g(leaf)h Fm(X)339 188 y Fl(1)375 181 y Fq(in)g Fm(T)7 b Fq(,)15 b(and)602 168 y(^)593 181 y(\001)634 188 y Fg(X)663 193 y Fc(1)680 188 y Fg(P)702 193 y Fc(1)737 181 y Fq(=)f(\001)830 188 y Fg(X)859 193 y Fc(1)876 188 y Fg(P)898 193 y Fc(1)932 181 y Fq(=)g(0.)23 b(By)16 b Fm(E)1158 188 y Fl(c)1192 181 y Fq(and)i(Equation)f(\(16\),)g Fi(j)p Fq(\001)1671 188 y Fl(1)1690 181 y Fi(j)d Fm(>)g Fq(1)p Fm(:)p Fq(5\001)1873 188 y Fl(min)1934 181 y Fq(.)-1 241 y(If)h Fm(P)78 248 y Fl(1)114 241 y Fq(is)g(an)h(in)o(ternal)f(no)q (de)h(in)f Fm(T)618 223 y Fk(\003)611 253 y Fg(k)637 241 y Fq(,)g(then)h(b)o(y)f Fi(B)877 248 y Fg(k)898 241 y Fq(,)g Fm(E)963 248 y Fl(c)981 241 y Fq(,)g(and)h(Equation)g(\(16\),) g(w)o(e)f(ha)o(v)o(e)g Fi(j)p Fq(\001)1668 248 y Fl(1)1687 241 y Fi(j)f Fm(>)g Fq(\001)1808 248 y Fl(min)1868 241 y Fq(.)21 b(In)-1 301 y(either)13 b(case,)i Fi(j)p Fq(\001)306 308 y Fl(1)325 301 y Fi(j)e Fm(>)h Fq(\001)445 308 y Fl(min)506 301 y Fq(.)20 b(By)14 b(symmetry)l(,)d(since)i Fm(P)22 b Fi(6)p Fq(=)13 b Fm(P)1103 308 y Fl(2)1123 301 y Fq(,)i Fi(j)p Fq(\001)1207 308 y Fl(2)1226 301 y Fi(j)f Fm(>)f Fq(\001)1346 308 y Fl(min)1407 301 y Fq(.)21 b(Th)o(us,)14 b(the)g(test)g(of)h(line)e(S7)-1 361 y(fails.)p 136 332 21 2 v 136 360 2 28 v 155 360 V 136 362 21 2 v -1 466 a Fb(Lemma)j(3.4)24 b Fj(In)d(addition)g(to)g (the)g(assumption)g(in)g(L)n(emma)f(3.3,)i(also)f(assume)g(that)g Fm(P)27 b Fi(6)p Fq(=)21 b Fm(P)1830 473 y Fl(1)1871 466 y Fj(and)-1 526 y Fm(P)i Fi(6)p Fq(=)15 b Fm(P)137 533 y Fl(2)175 526 y Fj(in)k Fm(T)7 b Fj(,)18 b(i.e.,)g(the)h(test)g (of)f(line)h(S7)g(has)f(faile)n(d.)25 b(Then,)19 b(the)g(test)f(of)h (line)g(S14)f(fails)h(if)f(and)g(only)-1 586 y(if)f Fm(P)25 b Fj(is)17 b(on)h(the)g(p)n(ath)f(b)n(etwe)n(en)i Fm(P)620 593 y Fl(1)657 586 y Fj(and)f Fm(P)783 593 y Fl(2)821 586 y Fj(in)f Fm(T)7 b Fj(.)-1 691 y(Pr)n(o)n(of.)42 b Fq(There)16 b(are)h(t)o(w)o(o)f(directions.)72 751 y(\()p Fi(\()-8 b Fq(=\))16 b(F)l(rom)f(lines)g(S6,)h(S10)i(and)e (Corollary)h(2.4\(2\),)180 859 y(\(\001)240 838 y Fk(0)240 871 y Fl(1)271 859 y Fi(\000)10 b Fq(\001)361 838 y Fk(0)361 871 y Fl(2)381 859 y Fq(\))h Fi(\000)g Fq(\(\001)521 866 y Fg(P)543 871 y Fc(1)560 866 y Fg(P)600 859 y Fi(\000)g Fq(\001)691 866 y Fg(P)713 871 y Fc(2)730 866 y Fg(P)760 859 y Fq(\))41 b(=)h Fi(\006)947 811 y Fd(\020)972 859 y Fq(\()999 846 y(^)991 859 y(\001)1032 866 y Fg(X)1061 871 y Fc(1)1077 866 y Fg(P)1118 859 y Fi(\000)11 b Fq(\001)1209 866 y Fg(X)1238 871 y Fc(1)1255 866 y Fg(P)1284 859 y Fq(\))g Fi(\000)g Fq(\()1391 846 y(^)1383 859 y(\001)1424 866 y Fg(X)1453 871 y Fc(1)1470 866 y Fg(P)1492 871 y Fc(1)1523 859 y Fi(\000)f Fq(\001)1613 866 y Fg(X)1642 871 y Fc(1)1659 866 y Fg(P)1681 871 y Fc(1)1701 859 y Fq(\))1720 811 y Fd(\021)900 950 y Fi(\006)947 902 y Fd(\020)972 950 y Fq(\()999 938 y(^)991 950 y(\001)1032 957 y Fg(X)1061 962 y Fc(2)1077 957 y Fg(P)1118 950 y Fi(\000)h Fq(\001)1209 957 y Fg(X)1238 962 y Fc(2)1255 957 y Fg(P)1284 950 y Fq(\))g Fi(\000)g Fq(\()1391 938 y(^)1383 950 y(\001)1424 957 y Fg(X)1453 962 y Fc(2)1470 957 y Fg(P)1492 962 y Fc(2)1523 950 y Fi(\000)f Fq(\001)1613 957 y Fg(X)1642 962 y Fc(2)1659 957 y Fg(P)1681 962 y Fc(2)1701 950 y Fq(\))1720 902 y Fd(\021)1753 950 y Fm(:)-1 1052 y Fq(Th)o(us,)16 b(whether)g Fm(P)354 1059 y Fl(1)390 1052 y Fq(and)h Fm(P)516 1059 y Fl(2)553 1052 y Fq(are)f(lea)o(v)o(es)f (or)h(in)o(ternal)g(no)q(des)h(in)f Fm(T)1244 1034 y Fk(\003)1237 1064 y Fg(k)1263 1052 y Fq(,)g(b)o(y)f Fi(A)1400 1059 y Fg(k)1421 1052 y Fq(,)h Fi(B)1484 1059 y Fg(k)1505 1052 y Fq(,)g(and)h Fm(E)1666 1059 y Fl(c)1684 1052 y Fq(,)558 1154 y Fi(j)p Fq(\(\001)632 1133 y Fk(0)632 1166 y Fl(1)662 1154 y Fi(\000)11 b Fq(\001)753 1133 y Fk(0)753 1166 y Fl(2)772 1154 y Fq(\))g Fi(\000)g Fq(\(\001)912 1161 y Fg(P)934 1166 y Fc(1)951 1161 y Fg(P)992 1154 y Fi(\000)g Fq(\001)1083 1161 y Fg(P)1105 1166 y Fc(2)1122 1161 y Fg(P)1151 1154 y Fq(\))p Fi(j)j Fm(<)g Fq(2\001)1315 1161 y Fl(min)1376 1154 y Fm(:)-1 1255 y Fq(By)i(line)f(S13)i(and)g (Corollary)f(2.4\(2\),)384 1386 y(\001)425 1366 y Fk(00)425 1398 y Fl(1)488 1386 y Fm(<)572 1351 y Fq(2\001)637 1358 y Fl(min)709 1351 y Fq(+)11 b(\(\001)818 1358 y Fg(P)840 1363 y Fc(1)857 1358 y Fg(P)898 1351 y Fi(\000)f Fq(\001)988 1358 y Fg(P)1010 1363 y Fc(2)1028 1358 y Fg(P)1057 1351 y Fq(\))h(+)g(\001)1177 1333 y Fk(\003)1177 1363 y Fg(P)1199 1368 y Fc(1)1217 1363 y Fg(P)1239 1368 y Fc(2)p 572 1374 687 2 v 903 1420 a Fq(2)488 1509 y(=)572 1474 y(2\(2\001)680 1481 y Fl(min)752 1474 y Fi(\000)g Fq(\001)843 1481 y Fg(P)865 1486 y Fc(2)882 1481 y Fg(P)912 1474 y Fq(\))g(+)g(\()p Fi(\000)p Fq(2\001)1114 1481 y Fl(min)1186 1474 y Fq(+)g(\001)1276 1481 y Fg(P)1298 1486 y Fc(1)1315 1481 y Fg(P)1337 1486 y Fc(2)1357 1474 y Fq(\))g(+)g(\001)1477 1456 y Fk(\003)1477 1487 y Fg(P)1499 1492 y Fc(1)1516 1487 y Fg(P)1538 1492 y Fc(2)p 572 1498 986 2 v 1053 1544 a Fq(2)-1 1626 y(Then,)20 b(since)f Fm(P)27 b Fi(6)p Fq(=)19 b Fm(P)413 1633 y Fl(2)453 1626 y Fq(and)h(th)o(us)g(\001)703 1633 y Fg(P)725 1638 y Fc(2)742 1633 y Fg(P)791 1626 y Fi(\025)f Fq(2\001)914 1633 y Fl(min)975 1626 y Fq(,)h(b)o(y)f Fi(C)1106 1633 y Fg(k)1128 1626 y Fq(,)h(w)o(e)f(ha)o(v)o(e)g(\001)1394 1608 y Fk(00)1394 1638 y Fl(1)1434 1626 y Fm(<)h Fq(\001)1533 1608 y Fk(\003)1533 1638 y Fg(P)1555 1643 y Fc(1)1572 1638 y Fg(P)1594 1643 y Fc(2)1614 1626 y Fm(:)f Fq(By)g(symmetry)-5 b(,)-1 1686 y(\001)40 1668 y Fk(00)40 1698 y Fl(2)75 1686 y Fm(<)13 b Fq(\001)167 1668 y Fk(\003)167 1698 y Fg(P)189 1703 y Fc(1)207 1698 y Fg(P)229 1703 y Fc(2)248 1686 y Fq(.)22 b(Th)o(us,)16 b(the)g(test)g(of)g(line)g(S14)h(fails.)72 1746 y(\(=)-8 b Fi(\))p Fq(\))17 b(T)l(o)g(pro)o(v)o(e)g(b)o(y)g(con)o (tradiction,)f(assume)h(that)g Fm(P)25 b Fq(is)17 b(not)g(on)h(the)f (path)h(b)q(et)o(w)o(een)e Fm(P)1750 1753 y Fl(1)1787 1746 y Fq(and)i Fm(P)1914 1753 y Fl(2)1934 1746 y Fq(.)-1 1806 y(By)j(similar)e(argumen)o(ts,)j(if)f(\001)595 1813 y Fg(P)617 1818 y Fc(1)634 1813 y Fg(P)687 1806 y Fm(>)i Fq(\001)789 1813 y Fg(P)811 1818 y Fc(1)828 1813 y Fg(P)850 1818 y Fc(2)891 1806 y Fq(\(resp)q(ectiv)o(ely)l(,)d(\001)1233 1813 y Fg(P)1255 1818 y Fc(2)1273 1813 y Fg(P)1325 1806 y Fm(>)j Fq(\001)1427 1813 y Fg(P)1449 1818 y Fc(1)1466 1813 y Fg(P)1488 1818 y Fc(2)1508 1806 y Fq(\),)g(then)e(\001)1721 1788 y Fk(00)1721 1819 y Fl(1)1765 1806 y Fm(>)i Fq(\001)1867 1788 y Fk(\003)1867 1819 y Fg(P)1889 1824 y Fc(1)1906 1819 y Fg(P)1928 1824 y Fc(2)-1 1867 y Fq(\(resp)q(ectiv)o(ely)l(,)13 b(\001)334 1848 y Fk(00)334 1879 y Fl(2)369 1867 y Fm(>)h Fq(\001)462 1848 y Fk(\003)462 1879 y Fg(P)484 1884 y Fc(1)501 1879 y Fg(P)523 1884 y Fc(2)543 1867 y Fq(\).)21 b(Th)o(us,)16 b(the)g(test)g(of)h(line)e(S14)i(passes.)p 1335 1837 21 2 v 1335 1865 2 28 v 1354 1865 V 1335 1867 21 2 v -1 1971 a Fb(Lemma)f(3.5)24 b Fj(Assume)19 b(that)f Fm(P)608 1978 y Fl(1)629 1971 y Fm(P)660 1978 y Fl(2)698 1971 y Fj(is)h(an)g(e)n(dge)g(in)g Fm(T)1028 1953 y Fk(\003)1021 1983 y Fg(k)1066 1971 y Fj(and)g(some)f(no)n(de)h(is)g(strictly)g(b)n (etwe)n(en)h Fm(P)1832 1978 y Fl(1)1871 1971 y Fj(and)-1 2031 y Fm(P)30 2038 y Fl(2)65 2031 y Fj(in)15 b Fm(T)7 b Fj(.)21 b(Then)16 b(ther)n(e)f(is)g(a)f(lar)n(ge)i(triplet)f Fm(N)5 b(Q)868 2038 y Fl(1)888 2031 y Fm(Q)927 2038 y Fl(2)961 2031 y Fj(with)15 b(c)n(enter)h Fm(P)22 b Fj(such)16 b(that)f Fm(N)k Fi(62)14 b Fm(T)1608 2013 y Fk(\003)1601 2044 y Fg(k)1627 2031 y Fj(,)h Fm(Q)1696 2038 y Fl(1)1730 2031 y Fi(2)f Fq(def)s(\()p Fm(P)1894 2038 y Fl(1)1914 2031 y Fq(\))p Fj(,)-1 2092 y Fm(Q)38 2099 y Fl(2)71 2092 y Fi(2)g Fq(def)s(\()p Fm(P)235 2099 y Fl(2)255 2092 y Fq(\))p Fj(,)k(and)f Fm(P)25 b Fj(is)17 b(strictly)h(b)n(etwe)n (en)h Fm(P)882 2099 y Fl(1)920 2092 y Fj(and)f Fm(P)1046 2099 y Fl(2)1083 2092 y Fj(in)g Fm(T)7 b Fj(.)-1 2196 y(Pr)n(o)n(of.)44 b Fq(By)15 b(Lemma)g(2.7\(2\),)i(for)f(ev)o(ery)f(no) q(de)i Fm(P)24 b Fq(strictly)15 b(b)q(et)o(w)o(een)h Fm(P)1342 2203 y Fl(1)1378 2196 y Fq(and)h Fm(P)1504 2203 y Fl(2)1541 2196 y Fq(in)f Fm(T)7 b Fq(,)16 b(there)g(exists)g(a) -1 2256 y(leaf)f Fm(N)20 b Fi(62)14 b Fm(T)231 2238 y Fk(\003)224 2269 y Fg(k)266 2256 y Fq(with)h Fm(\033)404 2263 y Fg(P)5 b(N)479 2256 y Fi(\025)13 b Fq(\(1)e Fi(\000)f Fm(\013g)r Fq(\))709 2238 y Fg(d)p Fl(+1)775 2256 y Fq(.)21 b(T)l(o)c(c)o(ho)q(ose)f Fm(P)7 b Fq(,)16 b(there)f(are)h(t)o(w)o(o)g (cases:)21 b(\(1\))16 b(b)q(oth)h Fm(P)1766 2263 y Fl(1)1802 2256 y Fq(and)g Fm(P)1928 2263 y Fl(2)-1 2317 y Fq(are)f(in)o(ternal)f (no)q(des)j(in)d Fm(T)490 2299 y Fk(\003)483 2329 y Fg(k)510 2317 y Fq(,)g(and)i(\(2\))g Fm(P)744 2324 y Fl(1)780 2317 y Fq(or)g Fm(P)871 2324 y Fl(2)907 2317 y Fq(is)f(a)h(leaf)f(in)g Fm(T)1181 2299 y Fk(\003)1174 2329 y Fg(k)1200 2317 y Fq(.)72 2377 y(Case)i(1.)24 b(By)16 b(Lemma)f(3.1\(1\),)i(let)g(def)s (\()p Fm(P)848 2384 y Fl(1)868 2377 y Fq(\))e(=)g Fi(f)p Fm(X)q(;)8 b(Y)1071 2384 y Fl(1)1092 2377 y Fm(;)g(Z)1147 2384 y Fl(1)1167 2377 y Fi(g)17 b Fq(and)g(def)t(\()p Fm(P)1422 2384 y Fl(2)1442 2377 y Fq(\))e(=)g Fi(f)p Fm(X)q(;)8 b(Y)1645 2384 y Fl(2)1665 2377 y Fm(;)g(Z)1720 2384 y Fl(2)1740 2377 y Fi(g)p Fq(.)24 b(By)16 b Fi(B)1913 2384 y Fg(k)1934 2377 y Fq(,)-1 2437 y(neither)e Fm(X)t(Y)235 2444 y Fl(2)255 2437 y Fm(Z)288 2444 y Fl(2)323 2437 y Fq(nor)i Fm(X)t(Y)481 2444 y Fl(1)501 2437 y Fm(Z)534 2444 y Fl(1)569 2437 y Fq(is)f(small.)k(T)l(o)d(\014x)f(the)f(notation) i(for)f(def)t(\()p Fm(P)1369 2444 y Fl(1)1389 2437 y Fq(\))g(and)g(def)s(\()p Fm(P)1633 2444 y Fl(2)1653 2437 y Fq(\))g(with)g(resp)q(ect)-1 2497 y(to)e(their)f(top)q(ological)i(la) o(y)o(out,)f(w)o(e)f(assume)g(without)i(loss)f(of)g(generalit)o(y)f (that)h(Figure)g(5)g(or)g(equiv)m(alen)o(tly)-1 2557 y(the)j(follo)o(wing)g(statemen)o(ts)f(hold:)949 2682 y(15)p eop %%Page: 16 16 16 15 bop 72 120 a Fi(\017)24 b Fq(In)15 b Fm(T)217 102 y Fk(\003)210 133 y Fg(k)251 120 y Fq(and)h(th)o(us)f(in)g Fm(T)21 b Fq(b)o(y)15 b Fi(A)664 127 y Fg(k)685 120 y Fq(,)g Fm(P)745 127 y Fl(2)780 120 y Fq(is)g(on)h(the)f(paths)h(b)q(et) o(w)o(een)e Fm(P)1328 127 y Fl(1)1363 120 y Fq(and)i Fm(Y)1485 127 y Fl(2)1505 120 y Fq(,)f(b)q(et)o(w)o(een)f Fm(P)1752 127 y Fl(1)1788 120 y Fq(and)h Fm(Z)1914 127 y Fl(2)1934 120 y Fq(,)121 181 y(and)i(b)q(et)o(w)o(een)e Fm(P)435 188 y Fl(1)471 181 y Fq(and)i Fm(Y)594 188 y Fl(1)614 181 y Fq(,)f(resp)q(ectiv)o(ely)l(.)72 280 y Fi(\017)24 b Fq(Similarly)l(,)13 b Fm(P)369 287 y Fl(1)405 280 y Fq(is)j(on)h(the)f(paths)h(b)q(et)o(w)o(een)e Fm(P)958 287 y Fl(2)995 280 y Fq(and)i Fm(Z)1123 287 y Fl(1)1159 280 y Fq(and)g(b)q(et)o(w)o(een)e Fm(P)1473 287 y Fl(2)1509 280 y Fq(and)i Fm(X)t Fq(.)72 380 y Fi(\017)24 b Fq(\001)162 387 y Fg(P)184 392 y Fc(2)201 387 y Fg(Y)221 392 y Fc(2)255 380 y Fi(\024)13 b Fq(\001)348 387 y Fg(P)370 392 y Fc(2)388 387 y Fg(Z)412 392 y Fc(2)431 380 y Fq(.)-1 475 y(Both)h Fm(N)5 b(X)t(Y)234 482 y Fl(2)269 475 y Fq(and)15 b Fm(N)5 b(Z)439 482 y Fl(1)459 475 y Fm(Y)487 482 y Fl(2)521 475 y Fq(de\014ne)14 b Fm(P)7 b Fq(,)14 b(and)h(the)f(target)g(triplet) f(is)h(one)h(of)f(these)g(t)o(w)o(o)g(for)g(some)f(suitable)-1 535 y Fm(P)7 b Fq(.)21 b(T)l(o)c(c)o(ho)q(ose)g Fm(P)7 b Fq(,)16 b(w)o(e)g(further)g(divide)f(Case)h(1)h(in)o(to)f(three)g (sub)q(cases.)72 595 y Fj(Case)26 b Fq(1a:)41 b Fm(\033)331 602 y Fg(X)s(P)385 607 y Fc(2)434 595 y Fm(<)30 b(\033)530 602 y Fg(Y)550 607 y Fc(2)567 602 y Fg(P)589 607 y Fc(2)609 595 y Fq(\(1)18 b Fi(\000)f Fm(\013g)r Fq(\))26 b(and)h Fm(\033)960 602 y Fg(Y)980 607 y Fc(2)997 602 y Fg(P)1019 607 y Fc(1)1069 595 y Fm(<)j(\033)1165 602 y Fg(X)s(P)1219 607 y Fc(1)1238 595 y Fq(\(1)18 b Fi(\000)f Fm(\013g)r Fq(\).)50 b(The)26 b(target)g(triplet)e(is)-1 655 y Fm(N)5 b(X)t(Y)115 662 y Fl(2)136 655 y Fq(.)34 b(Since)19 b Fm(\033)343 662 y Fg(X)s(Y)395 667 y Fc(2)436 655 y Fi(\024)496 626 y(p)p 537 626 100 2 v 537 655 a Fm(\033)565 662 y Fg(X)s(Y)617 667 y Fc(2)636 655 y Fq(,)j(b)o(y)e(Corollary)g(2.2,)i (let)e Fm(P)28 b Fq(b)q(e)20 b(a)h(no)q(de)g(on)g(the)g(path)g(b)q(et)o (w)o(een)f Fm(X)-1 731 y Fq(and)h Fm(Y)126 738 y Fl(2)167 731 y Fq(in)g Fm(T)27 b Fq(with)400 681 y Fd(q)p 442 681 279 2 v 50 x Fm(\033)470 738 y Fg(X)s(Y)522 743 y Fc(2)541 731 y Fq(\(1)11 b Fi(\000)g Fm(\013g)r Fq(\))22 b Fi(\024)f Fm(\033)830 738 y Fg(X)s(P)913 731 y Fi(\024)973 681 y Fd(q)p 1014 681 327 2 v 1014 731 a Fm(\033)1042 738 y Fg(X)s(Y)1094 743 y Fc(2)1114 731 y Fq(\(1)11 b Fi(\000)g Fm(\013g)r Fq(\))1293 717 y Fk(\000)p Fl(1)1361 731 y Fq(and)22 b(th)o(us)e(b)o(y)h(Lemma)d(2.1\(2\))-1 761 y Fd(q)p 40 761 279 2 v 40 811 a Fm(\033)68 818 y Fg(X)s(Y)120 823 y Fc(2)140 811 y Fq(\(1)11 b Fi(\000)g Fm(\013g)r Fq(\))j Fi(\024)g Fm(\033)414 818 y Fg(Y)434 823 y Fc(2)451 818 y Fg(P)494 811 y Fi(\024)547 761 y Fd(q)p 588 761 327 2 v 588 811 a Fm(\033)616 818 y Fg(X)s(Y)668 823 y Fc(2)688 811 y Fq(\(1)d Fi(\000)g Fm(\013g)r Fq(\))867 797 y Fk(\000)p Fl(1)914 811 y Fq(.)21 b(By)13 b(the)g(condition)h(of)g (Case)g(1a)h(and)f(Lemma)e(2.1\(2\),)-1 880 y Fm(P)27 b Fq(is)19 b(strictly)f(b)q(et)o(w)o(een)h Fm(P)502 887 y Fl(1)542 880 y Fq(and)h Fm(P)671 887 y Fl(2)711 880 y Fq(in)f Fm(T)7 b Fq(.)30 b(Also,)20 b(b)o(y)f(Corollary)h(2.5,)g Fm(\033)1393 887 y Fg(X)s(Y)1445 892 y Fc(2)1484 880 y Fi(\025)1547 860 y Fl(2)p 1547 868 18 2 v 1547 897 a(3)1570 880 y Fm(\033)1598 887 y Fg(X)s(Y)1650 892 y Fc(2)1667 887 y Fg(Z)1691 892 y Fc(2)1710 880 y Fq(.)31 b(Th)o(us,)20 b(b)o(y)-1 940 y(Lemma)14 b(2.1\(2\),)j(since)e Fm(X)t(Y)519 947 y Fl(2)539 940 y Fm(Z)572 947 y Fl(2)609 940 y Fq(is)h(not)h(small,)470 1064 y Fm(\033)498 1071 y Fg(N)t(X)s(Y)582 1076 y Fc(2)642 1064 y Fq(=)981 1031 y(3)p 727 1053 534 2 v 796 1079 a Fl(1)p 732 1087 146 2 v 732 1116 a Fg(\033)752 1122 y Ff(X)r(P)805 1116 y Fg(\033)825 1122 y Ff(P)t(N)893 1099 y Fq(+)1015 1079 y Fl(1)p 947 1087 154 2 v 947 1116 a Fg(\033)967 1122 y Ff(Y)985 1129 y Fc(2)1003 1122 y Ff(P)1028 1116 y Fg(\033)1048 1122 y Ff(P)t(N)1117 1099 y Fq(+)1204 1079 y Fl(1)p 1171 1087 85 2 v 1171 1116 a Fg(\033)1191 1122 y Ff(X)r(Y)1236 1129 y Fc(2)642 1211 y Fi(\025)1087 1178 y Fq(1)p 727 1200 746 2 v 727 1208 a Fd(q)p 769 1208 28 2 v 774 1238 a Fl(2)p 774 1246 18 2 v 774 1275 a(3)796 1257 y Fm(\033)826 1232 y Fk(\000)p Fl(1)p Fg(=)p Fl(2)824 1270 y Fg(X)s(Y)876 1275 y Fc(2)893 1270 y Fg(Z)917 1275 y Fc(2)937 1257 y Fq(\(1)11 b Fi(\000)g Fm(\013g)r Fq(\))1116 1243 y Fk(\000)p Fg(d)p Fk(\000)p Fl(3)p Fg(=)p Fl(2)1255 1257 y Fq(+)1309 1238 y Fl(1)p 1309 1246 V 1309 1275 a(2)1332 1257 y Fm(\033)1362 1237 y Fk(\000)p Fl(1)1360 1270 y Fg(X)s(Y)1412 1275 y Fc(2)1429 1270 y Fg(Z)1453 1275 y Fc(2)642 1335 y Fm(>)42 b(\033)750 1342 y Fl(lg)780 1335 y Fm(:)1067 b Fq(\(17\))-1 1437 y(So)17 b Fm(N)5 b(X)t(Y)183 1444 y Fl(2)220 1437 y Fq(is)16 b(as)h(desired)e(for)i (Case)f(1a.)72 1497 y Fj(Case)k Fq(1b:)28 b Fm(\033)315 1504 y Fg(X)s(P)369 1509 y Fc(2)407 1497 y Fi(\025)19 b Fm(\033)493 1504 y Fg(Y)513 1509 y Fc(2)531 1504 y Fg(P)553 1509 y Fc(2)572 1497 y Fq(\(1)14 b Fi(\000)f Fm(\013g)r Fq(\).)31 b(The)20 b(target)g(triplet)e(is)h Fm(N)5 b(X)t(Y)1372 1504 y Fl(2)1393 1497 y Fq(.)31 b(Let)19 b Fm(P)27 b Fq(b)q(e)19 b(the)h(\014rst)f(no)q(de)-1 1558 y(after)14 b Fm(P)143 1565 y Fl(2)178 1558 y Fq(on)h(the)f(path)h (from)f Fm(P)583 1565 y Fl(2)617 1558 y Fq(to)o(w)o(ard)h Fm(P)809 1565 y Fl(1)844 1558 y Fq(in)f Fm(T)7 b Fq(.)20 b(Then,)14 b Fm(\033)1136 1565 y Fg(Y)1156 1570 y Fc(2)1174 1565 y Fg(P)1217 1558 y Fi(\025)f Fm(\033)1297 1565 y Fg(Y)1317 1570 y Fc(2)1335 1565 y Fg(P)1357 1570 y Fc(2)1377 1558 y Fq(\(1)7 b Fi(\000)g Fm(\013g)r Fq(\).)22 b(By)13 b(Corollary)i(2.5,)-1 1618 y Fm(\033)29 1600 y Fl(2)27 1630 y Fg(Y)47 1635 y Fc(2)64 1630 y Fg(P)108 1618 y Fi(\025)e Fm(\033)188 1625 y Fg(X)s(Y)240 1630 y Fc(2)257 1625 y Fg(Z)281 1630 y Fc(2)301 1618 y Fq(\(1)e Fi(\000)g Fm(\013g)r Fq(\))480 1600 y Fl(2)500 1618 y Fm(=)p Fq(3.)22 b(Next,)15 b(since)g Fm(\033)864 1625 y Fg(X)s(Y)916 1630 y Fc(2)949 1618 y Fi(\025)f Fm(\033)1030 1625 y Fg(X)s(Z)1086 1630 y Fc(2)1121 1618 y Fq(and)j Fm(\033)1244 1625 y Fg(P)1266 1630 y Fc(2)1283 1625 y Fg(Y)1303 1630 y Fc(2)1337 1618 y Fi(\025)c Fm(\033)1417 1625 y Fg(P)1439 1630 y Fc(2)1457 1625 y Fg(Z)1481 1630 y Fc(2)1500 1618 y Fq(,)599 1747 y Fm(\033)627 1754 y Fg(X)s(Y)679 1759 y Fc(2)696 1754 y Fg(Z)720 1759 y Fc(2)781 1747 y Fi(\024)1054 1713 y Fq(3)p 866 1735 402 2 v 866 1782 a(2)p Fm(\033)920 1762 y Fk(\000)p Fl(1)918 1794 y Fg(X)s(Y)970 1799 y Fc(2)1001 1782 y Fq(+)e Fm(\033)1080 1762 y Fk(\000)p Fl(1)1078 1794 y Fg(Y)1098 1799 y Fc(2)1115 1794 y Fg(P)1137 1799 y Fc(2)1157 1782 y Fm(\033)1187 1762 y Fk(\000)p Fl(1)1185 1794 y Fg(P)1207 1799 y Fc(2)1224 1794 y Fg(Z)1248 1799 y Fc(2)781 1881 y Fi(\024)1054 1848 y Fq(3)p 866 1870 400 2 v 866 1917 a(2)p Fm(\033)920 1896 y Fk(\000)p Fl(1)918 1929 y Fg(X)s(P)972 1934 y Fc(2)991 1917 y Fm(\033)1021 1896 y Fk(\000)p Fl(1)1019 1929 y Fg(Y)1039 1934 y Fc(2)1057 1929 y Fg(P)1079 1934 y Fc(2)1110 1917 y Fq(+)g Fm(\033)1189 1896 y Fk(\000)p Fl(2)1187 1929 y Fg(Y)1207 1934 y Fc(2)1224 1929 y Fg(P)1246 1934 y Fc(2)781 2025 y Fi(\024)1035 1990 y Fq(3)p Fm(\033)1089 1971 y Fl(2)1087 2002 y Fg(X)s(P)1141 2007 y Fc(2)p 866 2013 464 2 v 866 2059 a Fq(2\(1)h Fi(\000)f Fm(\013g)r Fq(\))g(+)g(\(1)h Fi(\000)e Fm(\013g)r Fq(\))1309 2045 y Fl(2)1334 2025 y Fm(:)-1 2159 y Fq(Th)o(us,)16 b Fm(\033)166 2141 y Fl(2)164 2171 y Fg(X)s(P)239 2159 y Fm(>)d(\033)320 2141 y Fl(2)318 2171 y Fg(X)s(P)372 2176 y Fc(2)405 2159 y Fm(>)h(\033)485 2166 y Fg(X)s(Y)537 2171 y Fc(2)554 2166 y Fg(Z)578 2171 y Fc(2)597 2159 y Fq(\(1)e Fi(\000)f Fm(\013g)r Fq(\))777 2141 y Fl(2)797 2159 y Fq(.)21 b(Since)15 b Fm(\033)987 2166 y Fg(X)s(Y)1039 2171 y Fc(2)1072 2159 y Fi(\025)1130 2139 y Fl(2)p 1130 2147 18 2 v 1130 2176 a(3)1153 2159 y Fm(\033)1181 2166 y Fg(X)s(Y)1233 2171 y Fc(2)1250 2166 y Fg(Z)1274 2171 y Fc(2)1309 2159 y Fq(and)i Fm(X)t(Y)1476 2166 y Fl(2)1496 2159 y Fm(Z)1529 2166 y Fl(2)1566 2159 y Fq(is)f(not)h(small,)442 2288 y Fm(\033)470 2295 y Fg(N)t(X)s(Y)554 2300 y Fc(2)614 2288 y Fq(=)953 2254 y(3)p 699 2276 534 2 v 768 2303 a Fl(1)p 704 2311 146 2 v 704 2339 a Fg(\033)724 2345 y Ff(X)r(P)776 2339 y Fg(\033)796 2345 y Ff(P)t(N)865 2322 y Fq(+)987 2303 y Fl(1)p 919 2311 154 2 v 919 2339 a Fg(\033)939 2345 y Ff(Y)957 2352 y Fc(2)974 2345 y Ff(P)1000 2339 y Fg(\033)1020 2345 y Ff(P)t(N)1089 2322 y Fq(+)1176 2303 y Fl(1)p 1143 2311 85 2 v 1143 2339 a Fg(\033)1163 2345 y Ff(X)r(Y)1208 2352 y Fc(2)614 2435 y Fm(>)1087 2401 y Fq(1)p 699 2423 801 2 v 699 2430 a Fd(\020)729 2459 y Fl(1+)774 2431 y Fk(p)p 803 2431 18 2 v 28 x Fl(3)p 729 2467 93 2 v 766 2496 a(3)826 2430 y Fd(\021)859 2478 y Fm(\033)889 2453 y Fk(\000)p Fl(1)p Fg(=)p Fl(2)887 2491 y Fg(X)s(Y)939 2496 y Fc(2)956 2491 y Fg(Z)980 2496 y Fc(2)999 2478 y Fq(\(1)12 b Fi(\000)e Fm(\013g)r Fq(\))1178 2464 y Fk(\000)p Fg(d)p Fk(\000)p Fl(2)1283 2478 y Fq(+)1337 2459 y Fl(1)p 1337 2467 18 2 v 1337 2496 a(2)1359 2478 y Fm(\033)1389 2458 y Fk(\000)p Fl(1)1387 2491 y Fg(X)s(Y)1439 2496 y Fc(2)1456 2491 y Fg(Z)1480 2496 y Fc(2)614 2557 y Fm(>)42 b(\033)722 2564 y Fl(lg)752 2557 y Fm(:)1095 b Fq(\(18\))949 2682 y(16)p eop %%Page: 17 17 17 16 bop -1 120 a Fq(So)17 b Fm(N)5 b(X)t(Y)183 127 y Fl(2)220 120 y Fq(is)16 b(as)h(desired)e(for)i(Case)f(1b.)72 181 y Fj(Case)h Fq(1c:)23 b Fm(\033)302 188 y Fg(Y)322 193 y Fc(2)339 188 y Fg(P)361 193 y Fc(1)396 181 y Fi(\025)15 b Fm(\033)478 188 y Fg(X)s(P)532 193 y Fc(1)551 181 y Fq(\(1)d Fi(\000)f Fm(\013g)r Fq(\).)24 b(If)16 b Fm(\033)846 188 y Fg(Z)870 193 y Fc(1)887 188 y Fg(P)909 193 y Fc(1)944 181 y Fm(>)f(\033)1025 188 y Fg(X)s(P)1079 193 y Fc(1)1098 181 y Fq(,)i(the)g(target)g(triplet)f(is)g Fm(N)5 b(Z)1634 188 y Fl(1)1655 181 y Fm(Y)1683 188 y Fl(2)1703 181 y Fq(;)17 b(otherwise,)-1 241 y(it)j(is)h Fm(N)5 b(X)t(Y)222 248 y Fl(2)243 241 y Fq(.)35 b(The)21 b(t)o(w)o(o)g(cases)h(are)f (symmetr)o(ic,)d(and)k(w)o(e)f(assume)f Fm(\033)1342 248 y Fg(X)s(P)1396 253 y Fc(1)1437 241 y Fi(\025)i Fm(\033)1526 248 y Fg(Z)1550 253 y Fc(1)1566 248 y Fg(P)1588 253 y Fc(1)1608 241 y Fq(.)36 b(Let)21 b Fm(P)28 b Fq(b)q(e)21 b(the)-1 301 y(\014rst)e(no)q(de)h(after)f Fm(P)372 308 y Fl(1)412 301 y Fq(on)g(the)g(path)h(from)e Fm(P)835 308 y Fl(1)874 301 y Fq(to)o(w)o(ard)i Fm(P)1071 308 y Fl(2)1110 301 y Fq(in)f Fm(T)7 b Fq(.)29 b(Then,)20 b Fm(\033)1422 308 y Fg(X)s(P)1501 301 y Fi(\025)f Fm(\033)1587 308 y Fg(X)s(P)1641 313 y Fc(1)1660 301 y Fq(\(1)13 b Fi(\000)g Fm(\013g)r Fq(\).)31 b(By)-1 361 y(Corollary)15 b(2.5,)f Fm(\033)333 343 y Fl(2)331 374 y Fg(X)s(P)406 361 y Fi(\025)f Fm(\033)488 343 y Fl(2)486 374 y Fg(X)s(P)540 379 y Fc(1)560 361 y Fq(\(1)7 b Fi(\000)g Fm(\013g)r Fq(\))731 343 y Fl(2)766 361 y Fi(\025)13 b Fm(\033)846 368 y Fg(X)s(Y)898 373 y Fc(1)915 368 y Fg(Z)939 373 y Fc(1)959 361 y Fq(\(1)7 b Fi(\000)g Fm(\013g)r Fq(\))1130 343 y Fl(2)1151 361 y Fm(=)p Fq(3.)21 b(Since)14 b Fm(\033)1388 368 y Fg(X)s(Y)1440 373 y Fc(2)1473 361 y Fi(\025)f Fm(\033)1553 368 y Fg(X)s(Z)1609 373 y Fc(2)1643 361 y Fq(and)i Fm(\033)1764 368 y Fg(Y)1784 373 y Fc(2)1801 368 y Fg(Z)1825 373 y Fc(2)1858 361 y Fm(>)f Fq(0,)453 507 y Fm(\033)481 514 y Fg(X)s(Y)533 519 y Fc(2)550 514 y Fg(Z)574 519 y Fc(2)624 507 y Fm(<)746 473 y Fq(3)p 697 495 124 2 v 697 542 a(2)p Fm(\033)751 522 y Fk(\000)p Fl(1)749 554 y Fg(X)s(Y)801 559 y Fc(2)855 507 y Fi(\024)1033 473 y Fq(3)p 929 495 233 2 v 929 542 a(2)p Fm(\033)983 522 y Fk(\000)p Fl(1)981 554 y Fg(Y)1001 559 y Fc(2)1019 554 y Fg(P)1041 559 y Fc(1)1061 542 y Fm(\033)1091 522 y Fk(\000)p Fl(1)1089 554 y Fg(X)s(P)1143 559 y Fc(1)1197 507 y Fi(\024)1307 472 y Fq(3)p Fm(\033)1361 454 y Fl(2)1359 484 y Fg(Y)1379 489 y Fc(2)1397 484 y Fg(P)1419 489 y Fc(1)p 1271 495 205 2 v 1271 541 a Fq(2\(1)e Fi(\000)e Fm(\013g)r Fq(\))1480 507 y Fm(:)-1 652 y Fq(Hence)17 b Fm(\033)176 634 y Fl(2)174 665 y Fg(Y)194 670 y Fc(2)212 665 y Fg(P)259 652 y Fm(>)h(\033)345 634 y Fl(2)343 665 y Fg(Y)363 670 y Fc(2)380 665 y Fg(P)402 670 y Fc(1)440 652 y Fm(>)g Fq(2)p Fm(\033)548 659 y Fg(X)s(Y)600 664 y Fc(2)617 659 y Fg(Z)641 664 y Fc(2)660 652 y Fq(\(1)13 b Fi(\000)g Fm(\013g)r Fq(\))p Fm(=)p Fq(3.)29 b(Then,)19 b(since)f(neither)g Fm(X)t(Y)1440 659 y Fl(2)1460 652 y Fm(Z)1493 659 y Fl(2)1532 652 y Fq(nor)h Fm(X)t(Y)1693 659 y Fl(1)1713 652 y Fm(Z)1746 659 y Fl(1)1785 652 y Fq(is)g(small)-1 713 y(and)e Fm(\033)122 720 y Fg(X)s(Y)174 725 y Fc(2)207 713 y Fi(\025)264 693 y Fl(2)p 264 701 18 2 v 264 730 a(3)287 713 y Fm(\033)315 720 y Fg(X)s(Y)367 725 y Fc(2)384 720 y Fg(Z)408 725 y Fc(2)427 713 y Fq(,)211 850 y Fm(\033)239 857 y Fg(N)t(X)s(Y)323 862 y Fc(2)383 850 y Fq(=)722 816 y(3)p 468 838 534 2 v 536 864 a Fl(1)p 473 872 146 2 v 473 901 a Fg(\033)493 907 y Ff(X)r(P)545 901 y Fg(\033)565 907 y Ff(P)t(N)634 884 y Fq(+)756 864 y Fl(1)p 688 872 154 2 v 688 901 a Fg(\033)708 907 y Ff(Y)726 914 y Fc(2)743 907 y Ff(P)769 901 y Fg(\033)789 907 y Ff(P)t(N)857 884 y Fq(+)945 864 y Fl(1)p 911 872 85 2 v 911 901 a Fg(\033)931 907 y Ff(X)r(Y)976 914 y Fc(2)383 997 y Fm(>)1087 963 y Fq(1)p 468 985 1264 2 v 487 1021 a Fl(1)p 473 1029 48 2 v 473 1033 a Fk(p)p 502 1033 18 2 v 28 x Fl(3)525 1040 y Fm(\033)555 1015 y Fk(\000)p Fl(1)p Fg(=)p Fl(2)553 1052 y Fg(X)s(Y)605 1057 y Fc(1)622 1052 y Fg(Z)646 1057 y Fc(1)665 1040 y Fq(\(1)12 b Fi(\000)e Fm(\013g)r Fq(\))844 1026 y Fk(\000)p Fg(d)p Fk(\000)p Fl(2)949 1040 y Fq(+)1017 1021 y Fl(1)p 1003 1029 48 2 v 1003 1033 a Fk(p)p 1032 1033 18 2 v 28 x Fl(6)1055 1040 y Fm(\033)1085 1015 y Fk(\000)p Fl(1)p Fg(=)p Fl(2)1083 1052 y Fg(X)s(Y)1135 1057 y Fc(2)1152 1052 y Fg(Z)1176 1057 y Fc(2)1195 1040 y Fq(\(1)h Fi(\000)g Fm(\013g)r Fq(\))1374 1026 y Fk(\000)p Fg(d)p Fk(\000)p Fl(3)p Fg(=)p Fl(2)1514 1040 y Fq(+)1568 1021 y Fl(1)p 1568 1029 V 1568 1057 a(2)1590 1040 y Fm(\033)1620 1020 y Fk(\000)p Fl(1)1618 1052 y Fg(X)s(Y)1670 1057 y Fc(2)1688 1052 y Fg(Z)1712 1057 y Fc(2)383 1119 y Fm(>)42 b(\033)491 1126 y Fl(lg)520 1119 y Fm(:)1327 b Fq(\(19\))-1 1229 y(So)17 b Fm(N)5 b(X)t(Y)183 1236 y Fl(2)220 1229 y Fq(is)16 b(as)h(desired)e(for)i(Case)f(1c)h(with)f Fm(\033)887 1236 y Fg(X)s(P)941 1241 y Fc(1)974 1229 y Fi(\025)d Fm(\033)1054 1236 y Fg(Z)1078 1241 y Fc(1)1095 1236 y Fg(P)1117 1241 y Fc(1)1137 1229 y Fq(.)72 1289 y Fj(Case)18 b Fq(2.)24 b(By)17 b(symmetry)-5 b(,)14 b(assume)j(that)h Fm(P)878 1296 y Fl(2)913 1289 y Fq(=)e Fm(X)21 b Fq(is)c(a)h(leaf)f(in) g Fm(T)1306 1271 y Fk(\003)1299 1301 y Fg(k)1325 1289 y Fq(.)24 b(Then,)17 b(since)g Fm(k)g Fi(\025)f Fq(3,)h Fm(P)1809 1296 y Fl(1)1846 1289 y Fq(is)g(an)-1 1349 y(in)o(ternal)g(no)q(de)h(in)g Fm(T)395 1331 y Fk(\003)388 1362 y Fg(k)414 1349 y Fq(.)27 b(Let)18 b(def)s(\()p Fm(P)661 1356 y Fl(1)681 1349 y Fq(\))f(=)f Fi(f)p Fm(X)q(;)8 b(Y)s(;)g(Z)t Fi(g)p Fq(.)27 b(By)17 b(symmetry)l(,)e(further)i(assume) h Fm(\033)1697 1356 y Fg(Y)8 b(P)1747 1361 y Fc(1)1783 1349 y Fi(\025)17 b Fm(\033)1867 1356 y Fg(Z)r(P)1915 1361 y Fc(1)1934 1349 y Fq(.)-1 1410 y(There)e(are)g(t)o(w)o(o)g(sub)q (cases.)21 b(If)15 b Fm(\033)601 1417 y Fg(X)s(P)655 1422 y Fc(1)688 1410 y Fm(<)f(\033)768 1417 y Fg(Y)8 b(P)818 1422 y Fc(1)837 1410 y Fq(\(1)h Fi(\000)g Fm(\013g)r Fq(\),)15 b(then)g(the)g(pro)q(of)h(is)e(similar)f(to)j(that)f(of)h (Case)f(1a)-1 1470 y(and)h(the)f(desired)f Fm(P)23 b Fq(is)15 b(in)f(the)h(middle)e(of)j(the)f(path)g(b)q(et)o(w)o(een)g Fm(X)k Fq(and)d Fm(Y)26 b Fq(in)15 b Fm(T)7 b Fq(.)21 b(Otherwise,)14 b(the)h(pro)q(of)-1 1530 y(is)h(similar)e(that)j(of)f (Case)h(1b)g(and)f Fm(P)24 b Fq(is)16 b(the)g(\014rst)g(no)q(de)h (after)f Fm(P)1200 1537 y Fl(1)1236 1530 y Fq(on)h(the)f(path)h(from)e Fm(P)1648 1537 y Fl(1)1684 1530 y Fq(to)o(w)o(ard)i Fm(X)j Fq(in)-1 1590 y Fm(T)7 b Fq(.)21 b(In)16 b(b)q(oth)h(cases,)f(the)g (desired)f(triplet)g(is)h Fm(N)5 b(X)t(Y)12 b Fq(.)p 1009 1561 21 2 v 1009 1588 2 28 v 1027 1588 V 1009 1590 21 2 v -1 1735 a Fh(3.4)66 b(The)22 b(sample)f(size)h(required)j(b)n(y) e(the)g Fv(HGT)f Fh(algorithm)-1 1941 y Fb(Lemma)16 b(3.6)24 b Fm(E)330 1948 y Fl(g)367 1941 y Fj(and)18 b Fm(E)498 1948 y Fl(c)533 1941 y Fj(imply)f(that)h Fi(A)806 1948 y Fg(k)827 1941 y Fj(,)g Fi(B)893 1948 y Fg(k)914 1941 y Fj(,)f(and)h Fi(C)1067 1948 y Fg(k)1106 1941 y Fj(hold)f(for)g(al)r (l)i Fm(k)d Fq(=)d(3)p Fm(;)8 b(:)g(:)g(:)g(;)g(n)p Fj(.)-1 2055 y(Pr)n(o)n(of.)42 b Fq(The)17 b(pro)q(of)g(is)f(b)o(y)g(induction) g(on)g Fm(k)r Fq(.)72 2115 y Fj(Base)21 b(c)n(ase)p Fq(:)28 b Fm(k)21 b Fq(=)e(3.)31 b(By)19 b(Lemma)e(2.7\(2\),)j Fm(E)946 2122 y Fl(c)964 2115 y Fq(,)g(and)g(the)f(greedy)g(selection)f (of)i(line)e(H2,)i(line)e(H4)-1 2176 y(constructs)g Fm(T)270 2158 y Fk(\003)263 2188 y Fl(3)307 2176 y Fq(without)g(edge)g(lengths.) 25 b(Then,)18 b Fi(A)977 2183 y Fl(3)1014 2176 y Fq(holds)h(trivially)l (.)k Fi(B)1380 2183 y Fl(3)1417 2176 y Fq(follo)o(ws)18 b(from)f Fm(E)1732 2183 y Fl(c)1750 2176 y Fq(,)g Fm(E)1817 2183 y Fl(g)1838 2176 y Fq(,)g(and)-1 2236 y(line)e(H2.)21 b Fi(C)213 2243 y Fl(3)249 2236 y Fq(follo)o(ws)c(from)e Fi(B)558 2243 y Fl(3)577 2236 y Fq(,)h Fm(E)643 2243 y Fl(c)677 2236 y Fq(and)h(the)f(use)g(of)h(Equation)f(\(8\))h(at)g (line)e(H5.)72 2296 y Fj(Induction)k(hyp)n(othesis)p Fq(:)h Fi(A)582 2303 y Fg(k)603 2296 y Fq(,)c Fi(B)666 2303 y Fg(k)687 2296 y Fq(,)g(and)h Fi(C)838 2303 y Fg(k)876 2296 y Fq(hold)f(for)g(some)g Fm(k)f(<)f(n)p Fq(.)72 2356 y Fj(Induction)24 b(step)p Fq(.)41 b(The)22 b(induction)g(step)g (is)h(concerned)e(with)i(the)f(\()p Fm(k)17 b Fi(\000)e Fq(2\)-th)23 b(iteration)f(of)h(the)-1 2416 y(rep)q(eat)e(at)g(line)f (H8.)35 b(Righ)o(t)20 b(b)q(efore)h(this)f(iteration,)h(b)o(y)g(the)f (induction)h(h)o(yp)q(othesis,)g(since)f Fm(k)k(<)d(n)p Fq(,)-1 2477 y(some)14 b Fm(N)164 2458 y Fk(0)176 2477 y Fm(Q)215 2484 y Fl(1)235 2477 y Fm(Q)274 2484 y Fl(2)308 2477 y Fq(satis\014es)i(Lemma)d(3.5.)21 b(Therefore,)15 b(during)h(this)f(iteration,)g(b)o(y)f Fm(E)1552 2484 y Fl(c)1586 2477 y Fq(and)i(Lemmas)d(3.3,)-1 2537 y(3.4,)k(and)g (3.1\(2\),)h Fi(S)i Fq(at)e(line)e(H9)g(has)i(a)f(splitting)g(tuple)f (for)h Fm(T)1183 2519 y Fk(\003)1176 2549 y Fg(k)1219 2537 y Fq(that)g(con)o(tains)g(a)h(triplet)d Fm(N)5 b(X)t(Y)29 b Fq(with)949 2682 y(17)p eop %%Page: 18 18 18 17 bop 2 120 a Fq(^)-27 b Fm(\033)27 127 y Fg(N)t(X)s(Y)134 120 y Fi(\025)16 b Fq(^)-26 b Fm(\033)215 127 y Fg(N)247 119 y Fe(0)257 127 y Fg(Q)285 132 y Fc(1)302 127 y Fg(Q)330 132 y Fc(2)349 120 y Fq(.)20 b(F)l(urthermore,)12 b(line)g(H10)h (\014nds)g(suc)o(h)g(a)g(tuple.)20 b(By)12 b Fm(E)1373 127 y Fl(g)1393 120 y Fq(,)i Fm(N)5 b(X)t(Y)24 b Fq(is)13 b(not)g(small.)19 b(Lines)-1 181 y(H11)g(and)h(H12)g(create)e Fm(T)488 163 y Fk(\003)481 193 y Fg(k)q Fl(+1)566 181 y Fq(using)i(this)f(triplet.)29 b(Th)o(us,)19 b Fi(B)1145 188 y Fg(k)q Fl(+1)1231 181 y Fq(follo)o(ws)g(from)f Fi(B)1545 188 y Fg(k)1566 181 y Fq(.)30 b(By)18 b(Lemmas)g(3.3)-1 241 y(and)g(3.4,)h Fi(A)230 248 y Fg(k)q Fl(+1)314 241 y Fq(follo)o(ws)e(from)g Fi(A)632 248 y Fg(k)653 241 y Fq(.)26 b Fi(C)719 248 y Fg(k)q Fl(+1)804 241 y Fq(follo)o(ws)18 b(from)e Fi(C)1108 248 y Fg(k)1148 241 y 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Fq(This)14 b(section)g(presen)o(ts)f(the)h(F)l (ast-HGT)g(algorithm)f(and)i(its)e(subroutine)i(Up)q(date-)p Fi(S)j Fq(in)13 b(Figures)h(6)g(and)-1 273 y(7,)22 b(resp)q(ectiv)o (ely)l(.)34 b(The)21 b(algorithm)f(is)h(parallel)g(to)g(the)g(HGT)h (algorithm.)35 b(Ev)o(ery)20 b(line)g(in)h(Figure)g(6)-1 333 y(p)q(erforms)15 b(the)g(same)g(function)g(as)h(the)g(line)e(with)i (the)f(same)g(n)o(um)o(b)q(ering)f(in)h(Figure)g(3.)21 b(Also,)16 b(for)f(eac)o(h)-1 393 y(leaf)21 b Fm(N)27 b Fi(2)22 b Fm(T)7 b Fq(,)21 b Fi(S)t Fq([)p Fm(N)5 b Fq(])21 b(pla)o(ys)f(the)h(same)f(role)h(as)g(the)g(column)f(of)h Fi(S)k Fq(indexed)20 b(b)o(y)h Fm(N)26 b Fq(in)21 b(the)f(pro)q(of)j (of)-1 454 y(Theorem)15 b(3.2.)72 514 y(The)j(analysis)h(of)f(the)g(F)l (ast-HGT)h(algorithm)e(is)h(also)g(parallel)g(to)g(that)h(of)f(the)g (HGT)g(algorithm.)-1 574 y(Hence,)h(w)o(e)h(adapt)h(for)f(the)g(F)l (ast-HGT)h(algorithm)e(in)g(a)i(straigh)o(tforw)o(ard)f(manner)f(the)h (de\014nitions)-1 634 y(of)f Fm(T)93 616 y Fk(\003)112 634 y Fq(,)h Fm(T)182 616 y Fk(\003)175 646 y Fg(k)201 634 y Fq(,)f(def)s(\()p Fi(\001)p Fq(\),)g(relev)m(an)o(t)f(triplets,)h (edge)g(lengths,)g(splitting)f(triplets,)h(and)g(splitting)g(tuples,)g Fm(T)1929 616 y Fk(\003)1922 646 y Fg(k)-1 694 y Fq(matc)o(hing)c Fm(T)7 b Fq(,)15 b Fi(A)317 701 y Fg(k)338 694 y Fq(,)h Fi(B)401 701 y Fg(k)422 694 y Fq(,)g Fi(C)478 701 y Fg(k)515 694 y Fq(from)g Fi(x)p Fq(3.1,)g Fi(x)p Fq(3.2,)g(and)h Fi(x)p Fq(3.3.)72 755 y(A)h(triplet)e Fm(N)5 b(X)t(Y)30 b Fq(is)18 b Fj(str)n(ongly)g Fq(relev)m(an)o(t)f(for)h(an)h(edge)f Fm(P)1133 762 y Fl(1)1153 755 y Fm(P)1184 762 y Fl(2)1221 755 y Fi(2)f Fm(T)1307 736 y Fk(\003)1300 767 y Fg(k)1344 755 y Fq(if)g(not)i(only)e(it)h(is)g(relev)m(an)o(t)f(but)-1 815 y(also)j Fm(X)j Fi(2)c Fq(def)s(\()p Fm(P)332 822 y Fl(1)352 815 y Fq(\))h(and)f Fm(Y)30 b Fi(2)19 b Fq(def)t(\()p Fm(P)716 822 y Fl(2)736 815 y Fq(\).)30 b Fm(N)5 b(X)t(Y)31 b Fq(is)19 b(a)g Fj(str)n(ongly)h Fq(splitting)e(triplet)g(for)i 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y(T)l(o)20 b(sa)o(v)o(e)e (running)h(time,)e(the)i(F)l(ast-HGT)h(algorithm)e(di\013ers)h(from)f (the)g(HGT)i(algorithm)d(in)i(the)-1 1356 y(follo)o(wing)d(four)h(k)o (ey)e(asp)q(ects.)72 1417 y(A)o(t)h(line)f(F2,)h Fm(AB)s(C)i Fq(is)e(selected)f(for)i(a)f(\014xed)g(arbitrary)g Fm(A)p Fq(.)21 b(This)16 b(reduces)g(the)g(n)o(um)o(b)q(er)e(of)j(triplets)-1 1477 y(considered)c(at)h(line)f(H2)g(from)g Fi(O)c Fq(\()p Fm(n)666 1459 y Fl(3)686 1477 y Fq(\))14 b(to)g Fi(O)9 b Fq(\()p Fm(n)873 1459 y Fl(2)893 1477 y Fq(\).)20 b(This)14 b(impro)o(v)o(em)o(en)n(t)d(is)j(supp)q(orted)g(b)o(y)f(the)h(fact)f (that)-1 1537 y(eac)o(h)j(leaf)f(in)h Fm(T)23 b Fq(is)16 b(con)o(tained)g(in)g(a)g(large)h(triplet.)72 1597 y(A)o(t)i(lines)f (F7)i(and)g(F15,)g Fi(S)k Fq(k)o(eeps)18 b(only)h(strongly)h(splitting) f(ones.)31 b(This)19 b(decreases)h(the)f(n)o(um)o(b)q(er)-1 1657 y(of)k(triplets)f(considered)g(at)h(lines)f(H7)h(and)g(H15)h(for)f (eac)o(h)f(in)o(v)o(olv)o(ed)f(edge)h(from)g Fi(O)9 b Fq(\()p 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Fb(Theorem)17 b(3.9)23 b Fj(The)18 b Fq(F)l(ast-HGT)g Fj(algorithm)g(runs)f(in)h Fi(O)9 b Fq(\()p Fm(n)1153 102 y Fl(2)1184 120 y Fq(+)i Fm(`n)1282 102 y Fl(2)1303 120 y Fq(\))17 b Fj(time.)-1 230 y(Pr)n(o)n(of.)43 b Fq(As)16 b(in)g(Theorem)f(3.2,)h(the)g Fi(O)10 b Fq(\()p Fm(`n)801 212 y Fl(2)821 230 y Fq(\))16 b(term)f(accoun)o(ts)i(for)f (estimating)f(the)h(pairwise)g(closeness)-1 290 y(v)m(alues)h(and)g (distances)g(b)q(et)o(w)o(een)e(terminal)g(taxa.)23 b(It)16 b(is)h(straigh)o(tforw)o(ard)h(to)f(sho)o(w)g(that)g(the)g(running)-1 350 y(time)h(afterw)o(ards)j(is)g Fi(O)9 b Fq(\()p Fm(n)508 332 y Fl(2)528 350 y Fq(\))20 b(based)h(on)h(the)e(ab)q(o)o(v)o(e)g (four)h(k)o(ey)f(di\013erences)f(b)q(et)o(w)o(een)h(the)g(F)l(ast-HGT) -1 410 y(algorithm)15 b(and)i(the)f(HGT)g(algorithm.)p 784 381 21 2 v 784 409 2 28 v 803 409 V 784 411 21 2 v 72 471 a Fj(R)n(emark.)49 b Fq(The)25 b(F)l(ast-HGT)i(algorithm)d(sa) o(v)o(es)h(time)f(b)o(y)h(examining)e(only)j(strongly)f(splitting)-1 531 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1198 2 28 v 906 1198 V 887 1200 21 2 v 72 1260 a(W)l(e)i(need)g(the)g(follo)o(wing) h(v)o(ersion)e(of)i(Lemma)d(3.1\(2\))j(for)g(the)f(F)l(ast-HGT)h (algorithm.)30 b(F)l(or)20 b Fm(k)h Fq(=)-1 1320 y(3)p Fm(;)8 b(:)g(:)g(:)g(;)g(n)14 b Fi(\000)h Fq(1)21 b(and)h(eac)o(h)f (leaf)g Fm(M)28 b Fi(2)23 b Fm(T)7 b Fq(,)22 b(let)e Fi(S)894 1327 y Fg(k)916 1320 y Fq([)p Fm(M)5 b Fq(])21 b(b)q(e)g(the)g(v)o(ersion)g(of)h Fi(S)t Fq([)p Fm(M)5 b Fq(])21 b(at)g(the)h(start)f(of)h(the)-1 1380 y(\()p Fm(k)13 b Fi(\000)e Fq(2\)-th)17 b(iteration)f(of)g(the)g(rep)q(eat)h (at)f(line)f(F8.)-1 1478 y Fb(Lemma)h(3.11)24 b Fj(Assume)c(that)g(for) g(a)g(given)i Fm(k)f Fi(\024)d Fm(n)13 b Fi(\000)g Fq(1)p Fj(,)41 b Fm(E)1177 1485 y Fl(g)1198 1478 y Fj(,)21 b Fm(E)1270 1485 y Fl(c)1287 1478 y Fj(,)g Fi(A)1363 1485 y Fg(k)1382 1476 y Fe(0)1395 1478 y Fj(,)g Fi(B)1464 1485 y Fg(k)1483 1476 y Fe(0)1496 1478 y Fj(,)g(and)f Fi(C)1655 1485 y Fg(k)1674 1476 y Fe(0)1708 1478 y Fj(hold)g(for)g(al)r (l)-1 1539 y Fm(k)26 1520 y Fk(0)51 1539 y Fi(\024)14 b Fm(k)r Fj(.)57 1637 y(1.)24 b(If)17 b Fi(S)202 1644 y Fg(k)224 1637 y Fq([)p Fm(M)5 b Fq(])16 b Fj(is)i(not)g(nul)r(l,)h (then)f(it)g(is)f(a)h(str)n(ongly)f(splitting)i(tuple)f(for)f(some)h(e) n(dge)g(in)g Fm(T)1718 1618 y Fk(\003)1711 1649 y Fg(k)1737 1637 y Fj(.)57 1737 y(2.)24 b(If)16 b(an)h(e)n(dge)h Fm(Q)386 1744 y Fl(1)405 1737 y Fm(Q)444 1744 y Fl(2)477 1737 y Fi(2)c Fm(T)560 1719 y Fk(\003)553 1749 y Fg(k)596 1737 y Fj(and)j(a)g(triplet)g Fm(M)5 b(R)965 1744 y Fl(1)986 1737 y Fm(R)1023 1744 y Fl(2)1059 1737 y Fj(with)17 b Fm(M)j Fi(62)14 b Fm(T)1314 1719 y Fk(\003)1307 1749 y Fg(k)1350 1737 y Fj(satisfy)i(L)n(emma)g(3.10)p Fq(\()p Fj(2)p Fq(\))p Fj(,)g(then)121 1797 y Fi(S)151 1804 y Fg(k)172 1797 y Fq([)p Fm(M)5 b Fq(])17 b Fj(is)g(a)h(str)n(ongly)f (splitting)i(tuple)f(for)f Fm(Q)969 1804 y Fl(1)988 1797 y Fm(Q)1027 1804 y Fl(2)1064 1797 y Fj(in)h Fm(T)1160 1779 y Fk(\003)1153 1810 y Fg(k)1196 1797 y Fj(that)g(c)n(ontains)g(a)f (triplet)h Fm(M)5 b(R)1765 1779 y Fk(0)1765 1810 y Fl(1)1785 1797 y Fm(R)1822 1779 y Fk(0)1822 1810 y Fl(2)1859 1797 y Fj(with)123 1857 y Fq(^)-26 b Fm(\033)149 1866 y Fg(M)t(R)214 1854 y Fe(0)214 1877 y Fc(1)230 1866 y Fg(R)257 1854 y Fe(0)257 1877 y Fc(2)290 1857 y Fi(\025)16 b Fq(^)-27 b Fm(\033)370 1864 y Fg(M)t(R)435 1869 y Fc(1)451 1864 y Fg(R)478 1869 y Fc(2)497 1857 y Fj(.)-1 1955 y(Pr)n(o)n(of.)42 b Fq(The)17 b(t)o(w)o(o)f(statemen)o(ts)f(are)h(pro)o(v)o(ed)f(as)i (follo)o(ws.)72 2016 y(Statemen)o(t)e(1.)24 b(This)17 b(statemen)o(t)e(follo)o(ws)i(directly)e(from)h(the)h(initialization)e (of)i Fi(S)k Fq(at)c(line)f(F7,)h(the)-1 2076 y(deletions)e(from)h Fi(S)k Fq(at)c(line)g(F14,)g(and)h(the)f(insertions)g(in)o(to)g Fi(S)k Fq(at)c(lines)g(F7)g(and)h(F15.)72 2136 y(Statemen)o(t)e(2.)21 b(The)16 b(pro)q(of)i(is)e(b)o(y)g(induction)f(on)i Fm(k)r Fq(.)72 2196 y Fj(Base)g(c)n(ase)p Fq(:)k Fm(k)16 b Fq(=)e(3.)21 b(By)15 b Fm(E)576 2203 y Fl(c)594 2196 y Fq(,)g Fi(A)663 2203 y Fl(3)683 2196 y 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b(a)h(general)e(transition)h(matrix.)18 b(It)10 b(w)o(ould)h(also)h(b)q(e)f(of)g(imp)q(ortance)e(to)j(broaden)f (the)g(exp)q(erimen)o(tal)-1 1253 y(analysis)18 b(b)o(y)g(sim)o (ulating)e(sequence)h(ev)o(olution)g(on)i(a)f(wide)g(range)g(of)h(ev)o (olutionary)e(trees,)g(esp)q(ecially)-1 1313 y(biologically)e (realistic)g(ones.)-1 1480 y Fr(Ac)n(kno)n(wledgmen)m(ts)-1 1589 y Fq(W)l(e)h(w)o(ould)g(lik)o(e)f(to)h(thank)h(Dana)g(Angluin)f (and)h(Stanley)e(Eisenstat)i(for)f(v)o(ery)g(useful)f(discussions.)-1 1756 y Fr(A)81 b(Pro)r(ofs)27 b(of)g(some)e(lemm)o(as)-1 1880 y Fh(A.1)66 b(Pro)r(of)22 b(of)f(Lemma)e(2.6)-1 1972 y Fq(Let)d Fm(h)114 1979 y Fg(X)s(Y)190 1972 y Fq(=)249 1952 y Fl(^)-20 b Fg(\033)267 1958 y Ff(X)r(Y)p 247 1961 74 2 v 247 1989 a Fg(\033)267 1995 y Ff(X)r(Y)325 1972 y Fq(;)16 b Fm(h)383 1979 y Fg(X)s(Z)457 1972 y Fq(=)515 1952 y Fl(^)-19 b Fg(\033)534 1958 y Ff(X)r(Z)p 514 1961 72 2 v 514 1989 a Fg(\033)534 1995 y Ff(X)r(Z)590 1972 y Fq(;)16 b Fm(h)648 1979 y Fg(Y)8 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Fm(\033)696 1843 y Fg(X)s(Y)758 1836 y Fm(\033)788 1816 y Fl(2)786 1849 y Fg(Y)8 b(Z)842 1836 y Fm(\033)870 1843 y Fg(X)s(Z)930 1836 y Fm(\017:)-1 1932 y Fq(Using)16 b(T)l(a)o(ylor's)g(expansion,)g(for)h Fm(u)c(>)h Fq(0,)746 2053 y(\()p Fm(t)d Fi(\000)844 2010 y(p)p 885 2010 126 2 v 885 2053 a Fm(t)903 2039 y Fl(2)934 2053 y Fi(\000)f Fm(u)p Fq(\))1030 2033 y Fl(2)1064 2053 y Fm(>)1128 2020 y(u)1156 2002 y Fl(2)p 1120 2042 62 2 v 1120 2088 a Fq(4)p Fm(t)1162 2073 y Fl(2)1187 2053 y Fm(:)-1 2161 y Fq(Th)o(us,)527 2248 y Fm(r)550 2228 y Fl(2)584 2248 y Fm(>)886 2215 y(\017)906 2197 y Fl(2)p 641 2237 531 2 v 641 2252 a Fd(\020)697 2280 y Fl(1)p 670 2288 72 2 v 670 2317 a Fg(\033)690 2323 y Ff(X)r(Z)758 2300 y Fq(+)817 2280 y Fl(1)p Fk(\000)p Fg(\017)p 812 2288 70 2 v 812 2317 a(\033)832 2323 y Ff(Y)d(Z)897 2300 y Fq(+)979 2280 y Fl(1)p 951 2288 74 2 v 951 2317 a Fg(\033)971 2323 y Ff(X)r(Y)1030 2252 y Fd(\021)1055 2263 y Fl(2)1083 2300 y Fm(\033)1113 2283 y Fl(2)1111 2312 y Fg(X)s(Z)1189 2248 y Fm(>)1246 2215 y(\017)1266 2197 y Fl(2)1286 2215 y Fm(\033)1316 2197 y Fl(2)1314 2227 y Fg(X)s(Y)g(Z)p 1246 2237 156 2 v 1268 2283 a Fq(9)p Fm(\033)1322 2265 y Fl(2)1320 2295 y Fg(X)s(Z)1406 2248 y Fm(:)-1 2384 y Fq(Therefore,)91 2508 y(Pr)143 2434 y Fd(\()193 2495 y Fq(^)185 2508 y(\001)226 2515 y Fg(X)s(P)298 2508 y Fi(\000)j Fq(\001)388 2515 y Fg(X)s(P)463 2508 y Fi(\025)520 2474 y(\000)e Fq(ln\(1)k Fi(\000)e Fm(\017)p Fq(\))p 520 2496 231 2 v 624 2542 a(2)764 2434 y Fd(\))812 2508 y Fi(\024)j Fq(3)8 b(exp)980 2447 y Fd(\022)1010 2508 y Fi(\000)1067 2474 y Fq(2)p 1054 2496 52 2 v 1054 2542 a Fm(\013)1085 2527 y Fl(2)1110 2508 y Fm(`\033)1160 2487 y Fl(2)1158 2520 y Fg(X)s(Z)1218 2508 y Fm(r)1241 2487 y Fl(2)1261 2447 y Fd(\023)1306 2508 y Fm(<)13 b Fq(3)8 b(exp)1473 2447 y Fd(\022)1503 2508 y Fi(\000)1573 2474 y Fq(2)p 1547 2496 76 2 v 1547 2542 a(9)p Fm(\013)1602 2527 y Fl(2)1628 2508 y Fm(`\033)1678 2487 y Fl(2)1676 2520 y Fg(X)s(Y)g(Z)1764 2508 y Fm(\017)1784 2487 y Fl(2)1803 2447 y Fd(\023)1842 2508 y Fm(:)949 2682 y Fq(23)p eop %%Page: 24 24 24 23 bop -1 120 a Fh(A.2)66 b(Pro)r(of)22 b(of)f(Lemma)e(2.8)-1 213 y Fq(W)l(e)d(use)g(the)g(follo)o(wing)g(basic)g(inequalities.)337 347 y(min)426 274 y Fd(\()467 313 y Fq(^)-26 b Fm(\033)493 320 y Fg(X)s(Y)p 465 335 90 2 v 465 381 a Fm(\033)493 388 y Fg(X)s(Y)559 347 y Fm(;)589 313 y Fq(^)f Fm(\033)614 320 y Fg(X)s(Z)p 586 335 88 2 v 586 381 a Fm(\033)614 388 y Fg(X)s(Z)679 347 y Fm(;)708 313 y Fq(^)h Fm(\033)734 320 y Fg(Y)8 b(Z)p 706 335 85 2 v 706 381 a Fm(\033)734 388 y Fg(Y)g(Z)795 274 y Fd(\))842 347 y Fi(\024)902 313 y Fq(^)-26 b Fm(\033)928 320 y Fg(X)s(Y)7 b(Z)p 900 335 116 2 v 900 381 a Fm(\033)928 388 y Fg(X)s(Y)g(Z)1035 347 y Fi(\024)13 b Fq(max)1186 274 y Fd(\()1227 313 y Fq(^)-26 b Fm(\033)1253 320 y Fg(X)s(Y)p 1225 335 90 2 v 1225 381 a Fm(\033)1253 388 y Fg(X)s(Y)1319 347 y Fm(;)1349 313 y Fq(^)f Fm(\033)1374 320 y Fg(X)s(Z)p 1346 335 88 2 v 1346 381 a Fm(\033)1374 388 y Fg(X)s(Z)1439 347 y Fm(;)1468 313 y Fq(^)h Fm(\033)1494 320 y Fg(Y)8 b(Z)p 1466 335 85 2 v 1466 381 a Fm(\033)1494 388 y Fg(Y)g(Z)1555 274 y Fd(\))1597 347 y Fq(;)250 b(\(21\))649 468 y Fm(\033)677 475 y Fg(X)s(Y)8 b(Z)p 649 490 116 2 v 695 536 a Fq(3)784 502 y Fi(\024)13 b Fq(min)7 b Fi(f)p Fm(\033)979 509 y Fg(X)s(Y)1040 502 y Fm(;)h(\033)1090 509 y Fg(X)s(Z)1150 502 y Fm(;)g(\033)1200 509 y Fg(Y)g(Z)1256 502 y Fi(g)g Fm(:)558 b Fq(\(22\))72 602 y(W)l(e)22 b(\014rst)h(pro)o(v)o(e)e (Equation)i(\(11\).)39 b(Pic)o(k)21 b Fm(\025)k Fi(\025)e Fq(1)g(suc)o(h)f(that)h Fm(\033)1295 609 y Fg(X)s(Y)7 b(Z)1406 602 y Fq(=)24 b Fm(\033)1496 609 y Fl(lg)1526 602 y Fm(\025)p Fq(.)40 b(Without)22 b(loss)h(of)-1 663 y(generalit)o(y)l(,)14 b(w)o(e)i(ma)o(y)f(supp)q(ose)638 797 y(min)728 724 y Fd(\()769 763 y Fq(^)-27 b Fm(\033)794 770 y Fg(X)s(Y)p 766 785 90 2 v 766 831 a Fm(\033)794 838 y Fg(X)s(Y)861 797 y Fm(;)890 763 y Fq(^)h Fm(\033)916 770 y Fg(X)s(Z)p 888 785 88 2 v 888 831 a Fm(\033)916 838 y Fg(X)s(Z)980 797 y Fm(;)1010 763 y Fq(^)f Fm(\033)1035 770 y Fg(Y)8 b(Z)p 1007 785 85 2 v 1007 831 a Fm(\033)1035 838 y Fg(Y)g(Z)1096 724 y Fd(\))1144 797 y Fq(=)1203 763 y(^)-26 b Fm(\033)1229 770 y Fg(X)s(Y)p 1201 785 90 2 v 1201 831 a Fm(\033)1229 838 y Fg(X)s(Y)1295 797 y Fm(:)-1 934 y Fq(Then)16 b(b)o(y)g(Equations)h(\(4\),)f(\(21\),)h (and)f(\(22\),)394 1070 y(Pr)p Fi(f)11 b Fq(^)-27 b Fm(\033)507 1077 y Fg(X)s(Y)8 b(Z)609 1070 y Fi(\024)14 b Fm(\033)690 1077 y Fl(md)749 1070 y Fi(g)42 b Fq(=)g(Pr)948 997 y Fd(\()997 1037 y Fq(^)-27 b Fm(\033)1022 1044 y Fg(X)s(Y)8 b(Z)p 994 1059 116 2 v 994 1105 a Fm(\033)1022 1112 y Fg(X)s(Y)g(Z)1129 1070 y Fi(\024)1191 1037 y Fm(\033)1219 1044 y Fl(md)p 1187 1059 87 2 v 1187 1105 a Fm(\033)1215 1112 y Fl(lg)1245 1105 y Fm(\025)1286 997 y Fd(\))815 1212 y Fi(\024)42 b Fq(Pr)948 1138 y Fd(\()997 1178 y Fq(^)-27 b Fm(\033)1022 1185 y Fg(X)s(Y)p 994 1200 90 2 v 994 1246 a Fm(\033)1022 1253 y Fg(X)s(Y)1103 1212 y Fi(\024)1164 1178 y Fm(\033)1192 1185 y Fl(md)p 1161 1200 87 2 v 1161 1246 a Fm(\033)1189 1253 y Fl(lg)1218 1246 y Fm(\025)1260 1138 y Fd(\))815 1365 y Fi(\024)42 b Fq(exp)978 1280 y Fd(0)978 1354 y(@)1015 1365 y Fi(\000)1072 1331 y Fq(2)p 1059 1354 52 2 v 1059 1399 a Fm(\013)1090 1385 y Fl(2)1115 1365 y Fm(`)1143 1292 y Fd( )1176 1365 y Fq(1)12 b Fi(\000)1270 1331 y Fm(\033)1298 1338 y Fl(md)p 1266 1354 87 2 v 1266 1399 a Fm(\033)1294 1406 y Fl(lg)1324 1399 y Fm(\025)1358 1292 y Fd(!)1390 1304 y Fl(2)1418 1365 y Fm(\033)1448 1345 y Fl(2)1446 1377 y Fg(X)s(Y)1508 1280 y Fd(1)1508 1354 y(A)815 1548 y Fi(\024)42 b Fq(exp)978 1450 y Fd(0)978 1523 y(B)978 1550 y(@)1015 1548 y Fi(\000)1059 1502 y Fq(2)1091 1454 y Fd(\020)1116 1502 y Fq(1)11 b Fi(\000)1206 1481 y Fg(\033)1226 1487 y Fc(md)p 1206 1490 64 2 v 1215 1519 a Fg(\033)1235 1525 y Fc(lg)1275 1454 y Fd(\021)1300 1465 y Fl(2)p 1059 1537 262 2 v 1151 1582 a Fq(9)p Fm(\013)1206 1568 y Fl(2)1325 1548 y Fm(`\033)1375 1528 y Fl(2)1373 1561 y(lg)1403 1450 y Fd(1)1403 1523 y(C)1403 1550 y(A)1447 1548 y Fm(:)-1 1707 y Fq(By)16 b(the)g(c)o(hoice)e(of)j Fm(\033)388 1714 y Fl(md)439 1707 y Fq(,)652 1825 y(2)684 1776 y Fd(\020)710 1825 y Fq(1)11 b Fi(\000)800 1804 y Fg(\033)820 1810 y Fc(md)p 800 1813 64 2 v 809 1842 a Fg(\033)829 1848 y Fc(lg)869 1776 y Fd(\021)894 1788 y Fl(2)p 652 1859 262 2 v 745 1905 a Fq(9)p Fm(\013)800 1891 y Fl(2)932 1871 y Fq(=)989 1837 y(\()1008 1796 y Fi(p)p 1050 1796 25 2 v 41 x Fq(2)g Fi(\000)g Fq(1\))1178 1819 y Fl(2)p 989 1859 209 2 v 1044 1905 a Fq(36)p Fm(\013)1123 1891 y Fl(2)1203 1871 y Fm(;)644 b Fq(\(23\))-1 1992 y(and)17 b(th)o(us)f(Equation)h(\(11\))g (holds.)72 2052 y(W)l(e)i(next)g(pro)o(v)o(e)f(Equation)i(\(12\))g (similarly)l(.)27 b(Pic)o(k)18 b Fm(\025)h Fi(\024)f Fq(1)i(suc)o(h)f(that)h Fm(\033)1459 2059 y Fg(X)s(Y)7 b(Z)1565 2052 y Fq(=)19 b Fm(\033)1650 2059 y Fl(sm)1695 2052 y Fm(\025)p Fq(.)31 b(Without)-1 2112 y(loss)17 b(of)f(generalit)o(y)l(,)f(w)o(e)g(ma)o(y)g(assume)633 2246 y(max)732 2173 y Fd(\()773 2213 y Fq(^)-26 b Fm(\033)799 2220 y Fg(X)s(Y)p 771 2235 90 2 v 771 2281 a Fm(\033)799 2288 y Fg(X)s(Y)866 2246 y Fm(;)895 2213 y Fq(^)f Fm(\033)920 2220 y Fg(X)s(Z)p 892 2235 88 2 v 892 2281 a Fm(\033)920 2288 y Fg(X)s(Z)985 2246 y Fm(;)1014 2213 y Fq(^)h Fm(\033)1040 2220 y Fg(Y)8 b(Z)p 1012 2235 85 2 v 1012 2281 a Fm(\033)1040 2288 y Fg(Y)g(Z)1101 2173 y Fd(\))1148 2246 y Fq(=)1208 2213 y(^)-27 b Fm(\033)1233 2220 y Fg(X)s(Y)p 1205 2235 90 2 v 1205 2281 a Fm(\033)1233 2288 y Fg(X)s(Y)1300 2246 y Fm(:)-1 2383 y Fq(Then)16 b(b)o(y)g(Equations)h(\(5\),)f (\(21\),)h(and)f(\(22\),)365 2524 y(Pr)p Fi(f)11 b Fq(^)-27 b Fm(\033)478 2531 y Fg(X)s(Y)8 b(Z)580 2524 y Fi(\025)14 b Fm(\033)661 2531 y Fl(md)720 2524 y Fi(g)41 b Fq(=)h(Pr)919 2451 y Fd(\()968 2490 y Fq(^)-27 b Fm(\033)993 2497 y Fg(X)s(Y)8 b(Z)p 965 2513 116 2 v 965 2558 a Fm(\033)993 2565 y Fg(X)s(Y)g(Z)1100 2524 y Fi(\025)1158 2490 y Fm(\033)1186 2497 y Fl(md)1236 2449 y Fi(p)p 1278 2449 25 2 v 41 x Fq(2)p 1158 2513 145 2 v 1187 2558 a Fm(\033)1215 2565 y Fl(lg)1245 2558 y Fm(\025)1316 2451 y Fd(\))949 2682 y Fq(24)p eop %%Page: 25 25 25 24 bop 786 149 a Fi(\024)41 b Fq(Pr)919 76 y Fd(\()968 116 y Fq(^)-27 b Fm(\033)993 123 y Fg(X)s(Y)p 965 138 90 2 v 965 184 a Fm(\033)993 191 y Fg(X)s(Y)1074 149 y Fi(\025)1131 116 y Fm(\033)1159 123 y Fl(md)1210 75 y Fi(p)p 1252 75 25 2 v 41 x Fq(2)p 1131 138 145 2 v 1161 184 a Fm(\033)1189 191 y Fl(lg)1218 184 y Fm(\025)1289 76 y Fd(\))786 303 y Fi(\024)41 b Fq(exp)949 217 y Fd(0)949 292 y(@)985 303 y Fi(\000)1043 269 y Fq(2)p 1029 291 52 2 v 1029 337 a Fm(\013)1060 323 y Fl(2)1085 303 y Fm(`)1113 230 y Fd( )1152 269 y Fm(\033)1180 276 y Fl(md)1231 228 y Fi(p)p 1272 228 25 2 v 1272 269 a Fq(2)p 1152 291 145 2 v 1181 337 a Fm(\033)1209 344 y Fl(lg)1239 337 y Fm(\025)1313 303 y Fi(\000)10 b Fq(1)1386 230 y Fd(!)1420 237 y Fl(2)1448 303 y Fm(\033)1478 282 y Fl(2)1476 315 y Fg(X)s(Y)1537 217 y Fd(1)1537 292 y(A)786 488 y Fi(\024)41 b Fq(exp)949 390 y Fd(0)949 463 y(B)949 489 y(@)985 488 y Fi(\000)1029 393 y Fd(\020)1059 421 y Fg(\033)1079 427 y Fc(md)1123 393 y Fk(p)p 1152 393 18 2 v 28 x Fl(2)p 1059 430 112 2 v 1091 459 a Fg(\033)1111 465 y Fc(lg)1186 441 y Fi(\000)11 b Fq(1)1260 393 y Fd(\021)1285 403 y Fl(2)p 1029 476 276 2 v 1129 522 a Fq(9)p Fm(\013)1184 507 y Fl(2)1310 488 y Fm(`\033)1358 495 y Fl(lg)1388 390 y Fd(1)1388 463 y(C)1388 489 y(A)1433 488 y Fm(:)-1 642 y Fq(Th)o(us,)16 b(b)o(y)g(Equation)g(\(23\),)h(Equation)g(\(12\))g (holds.)-1 785 y Fh(A.3)66 b(Pro)r(of)22 b(of)f(Lemma)e(2.10)-1 877 y Fq(Since)i(Lemma)f(2.6)j(can)f(help)g(establish)g(only)g(one)g (half)h(of)f(the)g(desired)g(inequalit)o(y)l(,)f(w)o(e)h(split)f(the)-1 937 y(probabilit)o(y)15 b(on)i(the)f(left-hand)g(side)g(of)h(Equation)f (\(15\).)176 1059 y(Pr)228 999 y Fd(\032)267 1010 y(\014)267 1035 y(\014)267 1059 y(\014)289 1047 y Fq(^)281 1059 y(\001)322 1066 y Fg(X)s(P)394 1059 y Fi(\000)11 b Fq(\001)485 1066 y Fg(X)s(P)545 1010 y Fd(\014)545 1035 y(\014)545 1059 y(\014)j Fi(\025)630 1026 y Fq(\001)671 1033 y Fl(min)p 630 1048 102 2 v 669 1094 a Fq(2)745 999 y Fd(\033)96 1181 y Fq(=)42 b(Pr)228 1121 y Fd(\032)275 1169 y Fq(^)267 1181 y(\001)308 1188 y Fg(X)s(P)380 1181 y Fi(\000)11 b Fq(\001)471 1188 y Fg(X)s(P)545 1181 y Fi(\025)603 1147 y Fq(\001)644 1154 y Fl(min)p 603 1170 V 641 1215 a Fq(2)718 1121 y Fd(\033)760 1181 y Fq(+)g(Pr)861 1121 y Fd(\032)908 1169 y Fq(^)900 1181 y(\001)941 1188 y Fg(X)s(P)1013 1181 y Fi(\000)g Fq(\001)1104 1188 y Fg(X)s(P)1178 1181 y Fi(\024)j(\000)1275 1147 y Fq(\001)1316 1154 y Fl(min)p 1275 1170 V 1313 1215 a Fq(2)1389 1121 y Fd(\033)95 1303 y Fi(\024)42 b Fq(Pr)228 1242 y Fd(\032)275 1290 y Fq(^)267 1303 y(\001)308 1310 y Fg(X)s(P)380 1303 y Fi(\000)11 b Fq(\001)471 1310 y Fg(X)s(P)545 1303 y Fi(\025)603 1269 y Fq(\001)644 1276 y Fl(min)p 603 1291 V 641 1337 a Fq(6)718 1242 y Fd(\033)760 1303 y Fq(+)176 1425 y(Pr)228 1364 y Fd(\032)275 1412 y Fq(^)267 1425 y(\001)308 1432 y Fg(X)s(P)380 1425 y Fi(\000)g Fq(\001)471 1432 y Fg(X)s(P)545 1425 y Fi(\024)j(\000)642 1391 y Fq(\001)683 1398 y Fl(min)p 642 1413 V 680 1459 a Fq(2)756 1362 y Fd(\014)756 1387 y(\014)756 1412 y(\014)756 1437 y(\014)792 1412 y Fq(^)783 1425 y(\001)824 1432 y Fg(Y)8 b(P)893 1425 y Fi(\000)j Fq(\001)984 1432 y Fg(Y)d(P)1055 1425 y Fi(\025)1112 1391 y Fq(\001)1153 1398 y Fl(min)p 1112 1413 V 1151 1459 a Fq(6)1227 1364 y Fd(\033)1267 1425 y Fq(Pr)1319 1364 y Fd(\032)1366 1412 y Fq(^)1358 1425 y(\001)1399 1432 y Fg(Y)g(P)1467 1425 y Fi(\000)j Fq(\001)1558 1432 y Fg(Y)d(P)1629 1425 y Fi(\025)1687 1391 y Fq(\001)1728 1398 y Fl(min)p 1687 1413 V 1726 1459 a Fq(6)1802 1364 y Fd(\033)1844 1425 y Fq(+)176 1546 y(Pr)228 1486 y Fd(\032)275 1534 y Fq(^)267 1546 y(\001)308 1553 y Fg(X)s(P)380 1546 y Fi(\000)j Fq(\001)471 1553 y Fg(X)s(P)545 1546 y Fi(\024)j(\000)642 1513 y Fq(\001)683 1520 y Fl(min)p 642 1535 V 680 1580 a Fq(2)756 1484 y Fd(\014)756 1509 y(\014)756 1534 y(\014)756 1559 y(\014)792 1534 y Fq(^)783 1546 y(\001)824 1553 y Fg(Y)8 b(P)893 1546 y Fi(\000)j Fq(\001)984 1553 y Fg(Y)d(P)1055 1546 y Fm(<)1111 1513 y Fq(\001)1152 1520 y Fl(min)p 1111 1535 V 1150 1580 a Fq(6)1226 1486 y Fd(\033)1266 1546 y Fq(Pr)1318 1486 y Fd(\032)1365 1534 y Fq(^)1357 1546 y(\001)1398 1553 y Fg(Y)g(P)1467 1546 y Fi(\000)i Fq(\001)1557 1553 y Fg(Y)e(P)1629 1546 y Fm(<)1685 1513 y Fq(\001)1726 1520 y Fl(min)p 1685 1535 V 1724 1580 a Fq(6)1800 1486 y Fd(\033)1840 1546 y Fm(:)-1 1673 y Fq(Since)134 1660 y(^)126 1673 y(\001)167 1680 y Fg(X)s(Y)240 1673 y Fi(\000)j Fq(\001)331 1680 y Fg(X)s(Y)406 1673 y Fq(=)j(\()485 1660 y(^)477 1673 y(\001)518 1680 y Fg(X)s(P)590 1673 y Fi(\000)d Fq(\001)681 1680 y Fg(X)s(P)741 1673 y Fq(\))g(+)g(\()847 1660 y(^)839 1673 y(\001)880 1680 y Fg(Y)d(P)948 1673 y Fi(\000)j Fq(\001)1039 1680 y Fg(Y)d(P)1097 1673 y Fq(\))p Fm(;)294 1795 y Fq(Pr)347 1735 y Fd(\032)386 1746 y(\014)386 1770 y(\014)386 1795 y(\014)408 1783 y Fq(^)400 1795 y(\001)441 1802 y Fg(X)s(P)512 1795 y Fi(\000)j Fq(\001)603 1802 y Fg(X)s(P)664 1746 y Fd(\014)664 1770 y(\014)664 1795 y(\014)j Fi(\025)749 1762 y Fq(\001)790 1769 y Fl(min)p 749 1784 V 788 1830 a Fq(2)864 1735 y Fd(\033)937 1795 y Fi(\024)41 b Fq(Pr)1069 1735 y Fd(\032)1117 1783 y Fq(^)1108 1795 y(\001)1149 1802 y Fg(X)s(P)1221 1795 y Fi(\000)11 b Fq(\001)1312 1802 y Fg(X)s(P)1386 1795 y Fi(\025)1444 1762 y Fq(\001)1485 1769 y Fl(min)p 1444 1784 V 1482 1830 a Fq(6)1559 1735 y Fd(\033)1601 1795 y Fq(+)1017 1917 y(Pr)1069 1856 y Fd(\032)1117 1904 y Fq(^)1108 1917 y(\001)1149 1924 y Fg(Y)d(P)1218 1917 y Fi(\000)j Fq(\001)1309 1924 y Fg(Y)d(P)1380 1917 y Fi(\025)1437 1883 y Fq(\001)1478 1890 y Fl(min)p 1437 1905 V 1476 1951 a Fq(6)1552 1856 y Fd(\033)1594 1917 y Fq(+)1017 2039 y(Pr)1069 1978 y Fd(\032)1117 2026 y Fq(^)1108 2039 y(\001)1149 2046 y Fg(X)s(Y)1222 2039 y Fi(\000)j Fq(\001)1313 2046 y Fg(X)s(Y)1388 2039 y Fi(\024)j(\000)1485 2005 y Fq(\001)1526 2012 y Fl(min)p 1485 2027 V 1523 2073 a Fq(3)1600 1978 y Fd(\033)1639 2039 y Fm(:)-1 2156 y Fq(F)l(rom)h(Equation)i(\(10\),)265 2285 y(Pr)317 2224 y Fd(\032)365 2272 y Fq(^)356 2285 y(\001)397 2292 y Fg(X)s(P)469 2285 y Fi(\000)11 b Fq(\001)560 2292 y Fg(X)s(P)634 2285 y Fi(\025)692 2251 y Fq(\001)733 2258 y Fl(min)p 692 2273 V 731 2319 a Fq(6)807 2224 y Fd(\033)852 2285 y Fi(\024)i Fq(3)8 b(exp)1020 2212 y Fd( )1053 2285 y Fi(\000)1122 2251 y Fq(2)p 1097 2273 76 2 v 1097 2319 a(9)p Fm(\013)1152 2305 y Fl(2)1177 2285 y Fm(`\033)1227 2264 y Fl(2)1225 2297 y Fg(X)s(Y)g(Z)1322 2224 y Fd(\022)1352 2285 y Fq(1)k Fi(\000)e Fm(e)1460 2264 y Fk(\000)1492 2245 y Fc(\001)1517 2254 y(min)p 1492 2257 78 2 v 1524 2277 a(3)1577 2224 y Fd(\023)1608 2236 y Fl(2)1627 2212 y Fd(!)1668 2285 y Fm(:)-1 2412 y Fq(By)16 b(T)l(a)o(ylor's)f(expansion,)490 2476 y Fd(\022)520 2537 y Fq(1)d Fi(\000)e Fm(e)628 2516 y Fk(\000)660 2497 y Fc(\001)685 2505 y(min)p 660 2509 V 692 2529 a(3)745 2476 y Fd(\023)775 2487 y Fl(2)809 2537 y Fi(\025)862 2488 y Fd(\020)886 2537 y Fq(1)i Fi(\000)f Fq(\(1)g Fi(\000)g Fm(\013f)5 b Fq(\))1161 2502 y Ff(c)p 1161 2508 16 2 v 1161 2529 a Fc(3)1183 2488 y Fd(\021)1208 2500 y Fl(2)1241 2537 y Fm(>)1298 2503 y(c)1319 2485 y Fl(2)p 1298 2525 41 2 v 1306 2571 a Fq(9)1344 2537 y Fm(\013)1375 2516 y Fl(2)1395 2537 y Fm(f)1424 2516 y Fl(2)1444 2537 y Fm(;)949 2682 y Fq(25)p eop %%Page: 26 26 26 25 bop -1 120 a Fq(and)17 b(th)o(us)440 199 y(Pr)492 139 y Fd(\032)540 187 y Fq(^)531 199 y(\001)572 206 y Fg(X)s(P)644 199 y Fi(\000)11 b Fq(\001)735 206 y Fg(X)s(P)809 199 y Fi(\025)867 166 y Fq(\001)908 173 y Fl(min)p 867 188 102 2 v 905 234 a Fq(6)982 139 y Fd(\033)1027 199 y Fi(\024)i Fq(3)8 b(exp)1195 126 y Fd( )1228 199 y Fi(\000)1275 166 y Fm(c)1296 148 y Fl(2)p 1272 188 49 2 v 1272 234 a Fq(81)1325 199 y Fm(`\033)1375 179 y Fl(2)1373 212 y(lg)1403 199 y Fm(f)1432 179 y Fl(2)1452 126 y Fd(!)1494 199 y Fm(:)353 b Fq(\(24\))-1 313 y(By)16 b(symme)o(try)l(,)440 402 y(Pr)492 341 y Fd(\032)540 389 y Fq(^)531 402 y(\001)572 409 y Fg(X)s(P)644 402 y Fi(\000)11 b Fq(\001)735 409 y Fg(X)s(P)809 402 y Fi(\025)867 368 y Fq(\001)908 375 y Fl(min)p 867 390 102 2 v 905 436 a Fq(6)982 341 y Fd(\033)1027 402 y Fi(\024)i Fq(3)8 b(exp)1195 328 y Fd( )1228 402 y Fi(\000)1275 368 y Fm(c)1296 350 y Fl(2)p 1272 390 49 2 v 1272 436 a Fq(81)1325 402 y Fm(`\033)1375 381 y Fl(2)1373 414 y(lg)1403 402 y Fm(f)1432 381 y Fl(2)1452 328 y Fd(!)1494 402 y Fm(:)353 b Fq(\(25\))-1 518 y(F)l(rom)15 b(Equation)i(\(5\),)300 656 y(Pr)352 596 y Fd(\032)400 644 y Fq(^)391 656 y(\001)432 663 y Fg(X)s(Y)505 656 y Fi(\000)11 b Fq(\001)596 663 y Fg(X)s(Y)671 656 y Fi(\024)j(\000)768 623 y Fq(\001)809 630 y Fl(min)p 768 645 102 2 v 806 691 a Fq(3)883 596 y Fd(\033)928 656 y Fi(\024)f Fq(exp)1063 583 y Fd( )1096 656 y Fi(\000)1153 623 y Fq(2)p 1140 645 52 2 v 1140 691 a Fm(\013)1171 676 y Fl(2)1196 656 y Fm(`\033)1246 636 y Fl(2)1244 669 y Fg(X)s(Y)1314 596 y Fd(\022)1345 656 y Fm(e)1373 617 y Fc(\001)1398 625 y(min)p 1372 628 78 2 v 1403 649 a(3)1468 656 y Fi(\000)e Fq(1)1542 596 y Fd(\023)1573 607 y Fl(2)1592 583 y Fd(!)1633 656 y Fm(:)-1 794 y Fq(F)l(rom)k(Equation)i(\(22\),)752 892 y Fm(\033)780 899 y Fg(X)s(Y)856 892 y Fi(\025)913 858 y Fq(\(1)12 b Fi(\000)f Fm(\013g)r Fq(\))1093 834 y Fl(2)p Fg(d)p Fl(+2)p 913 880 263 2 v 1000 930 a Fq(3)1024 889 y Fi(p)p 1065 889 25 2 v 1065 930 a Fq(2)1181 892 y Fm(:)-1 1001 y Fq(By)16 b(T)l(a)o(ylor's)f(expansion,)490 1074 y Fd(\022)520 1135 y Fm(e)548 1095 y Fc(\001)573 1104 y(min)p 548 1107 78 2 v 579 1127 a(3)643 1135 y Fi(\000)c Fq(1)717 1074 y Fd(\023)748 1086 y Fl(2)782 1135 y Fi(\025)834 1087 y Fd(\020)859 1135 y Fq(\(1)g Fi(\000)g Fm(\013f)5 b Fq(\))1042 1114 y Fk(\000)1076 1101 y Ff(c)p 1075 1107 16 2 v 1075 1127 a Fc(3)1109 1135 y Fi(\000)10 b Fq(1)1182 1087 y Fd(\021)1208 1098 y Fl(2)1241 1135 y Fm(>)1298 1101 y(c)1319 1083 y Fl(2)p 1298 1123 41 2 v 1306 1169 a Fq(9)1344 1135 y Fm(\013)1375 1114 y Fl(2)1395 1135 y Fm(f)1424 1114 y Fl(2)1444 1135 y Fm(:)-1 1258 y Fq(Therefore,)436 1346 y(Pr)488 1286 y Fd(\032)535 1334 y Fq(^)527 1346 y(\001)568 1353 y Fg(X)s(Y)641 1346 y Fi(\000)h Fq(\001)732 1353 y Fg(X)s(Y)807 1346 y Fi(\024)j(\000)904 1313 y Fq(\001)945 1320 y Fl(min)p 904 1335 102 2 v 942 1380 a Fq(3)1018 1286 y Fd(\033)1063 1346 y Fi(\024)g Fq(exp)1199 1273 y Fd( )1232 1346 y Fi(\000)1279 1313 y Fm(c)1300 1294 y Fl(2)p 1276 1335 49 2 v 1276 1380 a Fq(81)1329 1346 y Fm(`\033)1379 1326 y Fl(2)1377 1359 y(lg)1407 1346 y Fm(f)1436 1326 y Fl(2)1456 1273 y Fd(!)1498 1346 y Fm(:)349 b Fq(\(26\))-1 1463 y(Putting)16 b(Equations)h(\(24\),)g(\(25\),)f(and)h(\(26\))g (together,)295 1594 y(Pr)347 1534 y Fd(\032)386 1544 y(\014)386 1569 y(\014)386 1594 y(\014)408 1582 y Fq(^)400 1594 y(\001)441 1601 y Fg(X)s(Y)514 1594 y Fi(\000)11 b Fq(\001)605 1601 y Fg(X)s(Y)666 1544 y Fd(\014)666 1569 y(\014)666 1594 y(\014)j Fi(\025)751 1560 y Fq(\001)792 1567 y Fl(min)p 751 1582 102 2 v 790 1628 a Fq(2)866 1534 y Fd(\033)939 1594 y Fi(\024)41 b Fq(Pr)1071 1534 y Fd(\032)1119 1582 y Fq(^)1111 1594 y(\001)1152 1601 y Fg(X)s(P)1223 1594 y Fi(\000)11 b Fq(\001)1314 1601 y Fg(X)s(P)1389 1594 y Fi(\025)1446 1560 y Fq(\001)1487 1567 y Fl(min)p 1446 1582 V 1485 1628 a Fq(6)1561 1534 y Fd(\033)1603 1594 y Fq(+)1019 1716 y(Pr)1071 1655 y Fd(\032)1119 1703 y Fq(^)1111 1716 y(\001)1152 1723 y Fg(Y)d(P)1220 1716 y Fi(\000)j Fq(\001)1311 1723 y Fg(Y)d(P)1382 1716 y Fi(\025)1440 1682 y Fq(\001)1481 1689 y Fl(min)p 1440 1704 V 1478 1750 a Fq(6)1554 1655 y Fd(\033)1597 1716 y Fq(+)1019 1838 y(Pr)1071 1777 y Fd(\032)1119 1825 y Fq(^)1111 1838 y(\001)1152 1845 y Fg(X)s(Y)1224 1838 y Fi(\000)j Fq(\001)1315 1845 y Fg(X)s(Y)1391 1838 y Fi(\024)i(\000)1487 1804 y Fq(\001)1528 1811 y Fl(min)p 1487 1826 V 1526 1872 a Fq(3)1602 1777 y Fd(\033)939 1966 y Fi(\024)41 b Fq(7)8 b(exp)1135 1893 y Fd( )1168 1966 y Fi(\000)1215 1932 y Fm(c)1236 1914 y Fl(2)p 1212 1955 49 2 v 1212 2000 a Fq(81)1265 1966 y Fm(`\033)1315 1946 y Fl(2)1313 1979 y(lg)1343 1966 y Fm(f)1372 1946 y Fl(2)1392 1893 y Fd(!)1433 1966 y Fm(;)-1 2101 y Fq(concluding)16 b(the)g(pro)q(of)h(of)g(Lemma)d(2.10.)-1 2267 y Fr(References)23 2377 y Fq([1])24 b(R.)13 b(Agarw)o(ala,)g(V.)f(Bafna,)i(M.)e(F)l(arac)o (h,)h(B.)f(Nara)o(y)o(anan,)i(M.)e(P)o(aterson,)i(and)f(M.)f(Thorup.)17 b(On)c(the)99 2437 y(appro)o(ximabilit)o(y)h(of)j(n)o(umerical)d (taxonom)o(y)i(\(\014tting)h(distances)f(b)o(y)g(tree)h(metrics\).)j (In)c Fj(Pr)n(o)n(c)n(e)n(e)n(d-)99 2497 y(ings)h(of)f(the)g(7th)g(A)o (nnual)i(A)o(CM-SIAM)e(Symp)n(osium)f(on)h(Discr)n(ete)g(A)o(lgorithms) p Fq(,)f(pages)h(365{372,)99 2557 y(1996.)949 2682 y(26)p eop %%Page: 27 27 27 26 bop 23 120 a Fq([2])24 b(A.)15 b(Am)o(bainis,)e(R.)i(Desp)q(er,)g (M.)g(F)l(arac)o(h,)g(and)h(S.)f(Kannan.)21 b(Nearly)14 b(tigh)o(t)h(b)q(ounds)i(on)f(the)f(learn-)99 181 y(abilit)o(y)c(of)h (ev)o(olution.)i(In)e Fj(Pr)n(o)n(c)n(e)n(e)n(dings)g(of)i(the)g(38th)g (A)o(nnual)h(IEEE)e(Symp)n(osium)g(on)h(F)l(oundations)99 241 y(of)k(Computer)f(Scienc)n(e)p Fq(,)h(pages)f(524{533,)h(1997.)23 340 y([3])24 b(K.)18 b(A)o(tteson.)26 b(The)18 b(p)q(erformance)f(of)h (neigh)o(b)q(or-joining)h(algorithms)e(of)i(ph)o(ylogen)o(y)e (reconstruc-)99 401 y(tion.)37 b(In)21 b(T.)h(Jiang)g(and)g(D.)f(T.)g (Lee,)i(editors,)f Fj(L)n(e)n(ctur)n(e)f(Notes)i(in)g(Computer)f (Scienc)n(e)i(1276:)99 461 y(Pr)n(o)n(c)n(e)n(e)n(dings)14 b(of)i(the)f(3r)n(d)f(A)o(nnual)j(International)g(Computing)f(and)f (Combinatorics)h(Confer)n(enc)n(e)p Fq(,)99 521 y(pages)h(101{110.)i (Springer-V)l(erlag,)c(New)h(Y)l(ork,)f(NY,)g(1997.)23 620 y([4])24 b(J.)16 b(Ca)o(v)o(ender.)k(T)l(axonom)o(y)c(with)g (con\014dence.)21 b Fj(Mathematic)n(al)c(Bioscienc)n(es)p Fq(,)h(40:271{280,)h(1978.)23 720 y([5])24 b(J.)14 b(T.)f(Chang.)19 b(F)l(ull)12 b(reconstruction)i(of)g(Mark)o(o)o(v)f(mo)q(dels)g(on)h (ev)o(olutionary)f(trees:)20 b(iden)o(ti\014abilit)o(y)99 780 y(and)d(consistency)l(.)k Fj(Mathematic)n(al)c(Bioscienc)n(es)p Fq(,)h(137:51{73,)g(1996.)23 880 y([6])24 b(M.)d(Cry)o(an,)h(L.)f(A.)f (Goldb)q(erg,)j(and)f(P)l(.)f(W.)f(Goldb)q(erg.)37 b(Ev)o(olutionary)21 b(trees)g(can)g(b)q(e)g(learned)99 940 y(in)c(p)q(olynomial)g(time)e (in)i(the)g(t)o(w)o(o-state)h(general)f(Mark)o(o)o(v-mo)q(del.)23 b(In)17 b Fj(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)g(the)h(39th)99 1000 y(A)o(nnual)g(IEEE)e(Symp)n(osium)g(on)h(F)l(oundations)g(of)f (Computer)g(Scienc)n(e)p Fq(,)h(pages)f(436{445,)i(1998.)23 1099 y([7])24 b(M.)d(Cs)q(})-26 b(ur\177)i(os)24 b(and)f(M.)e(Y.)g (Kao.)38 b(Reco)o(v)o(ering)20 b(ev)o(olutionary)h(trees)h(through)h (harmonic)d(greedy)99 1160 y(triplets.)25 b(In)18 b Fj(Pr)n(o)n(c)n(e)n (e)n(dings)f(of)i(the)h(10th)e(A)o(nnual)j(A)o(CM-SIAM)d(Symp)n(osium)g (on)i(Discr)n(ete)f(A)o(lgo-)99 1220 y(rithms)p Fq(,)c(1999.)23 b(T)l(o)17 b(app)q(ear.)23 1319 y([8])24 b(W.)c(H.)g(E.)g(Da)o(y)l(.)34 b(Computational)20 b(complexit)o(y)d(of)k(inferring)e(ph)o(ylogenies)h (from)f(dissimilarit)o(y)99 1379 y(matrices.)h Fj(Bul)r(letin)g(of)d (Mathematic)n(al)h(Biolo)n(gy)p Fq(,)e(49:461{467,)i(1987.)23 1479 y([9])24 b(W.)12 b(H.)f(E.)h(Da)o(y)l(,)g(D.)g(S.)g(Johnson,)i (and)e(D.)g(Sank)o(o\013.)j(The)d(computational)g(complexit)o(y)c(of)13 b(inferring)99 1539 y(ro)q(oted)k(ph)o(ylogenies)f(b)o(y)f(parsimon)o (y)l(.)20 b Fj(Mathematic)n(al)e(Bioscienc)n(es)p Fq(,)f(81:33{42,)h (1986.)-1 1639 y([10])24 b(D.-Z.)12 b(Du,)i(Y.-J.)d(Zhang,)j(and)g(Q.)e (F)l(eng.)j(On)e(b)q(etter)g(heuristic)e(for)i(Euclidean)f(Steiner)g (minim)n(um)99 1699 y(trees)23 b(\(extended)f(abstract\).)42 b(In)22 b Fj(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)g(the)i(32nd)e(A)o(nnual)i (IEEE)f(Symp)n(osium)f(on)99 1759 y(F)l(oundations)18 b(of)g(Computer)f(Scienc)n(e)p Fq(,)h(pages)f(431{439,)h(1991.)-1 1858 y([11])24 b(P)l(.)d(Erd})-24 b(os,)24 b(K.)d(Rice,)g(M.)f(A.)h (Steel,)g(L.)h(A.)e(Sz)o(\023)-23 b(ek)o(ely)l(,)21 b(and)h(T.)f(W)l (arno)o(w.)37 b(The)22 b(Short)g(Quartet)99 1919 y(Metho)q(d.)g Fj(Mathematic)n(al)17 b(Mo)n(deling)h(and)g(Scienti\014c)i(Computing)p Fq(,)c(1998.)23 b(to)17 b(app)q(ear.)-1 2018 y([12])24 b(M.)17 b(F)l(arac)o(h)g(and)h(S.)e(Kannan.)25 b(E\016cien)o(t)16 b(algorithms)g(for)i(in)o(v)o(erting)d(ev)o(olution.)23 b(In)17 b Fj(Pr)n(o)n(c)n(e)n(e)n(dings)99 2078 y(of)h(the)f(28th)h(A)o (nnual)h(A)o(CM)d(Symp)n(osium)h(on)h(The)n(ory)e(of)h(Computing)p Fq(,)g(pages)g(230{236,)h(1996.)-1 2178 y([13])24 b(J.)19 b(F)l(elsenstein.)30 b(Cases)20 b(in)f(whic)o(h)g(parsimon)o(y)f(or)i (compatibilit)o(y)c(metho)q(ds)j(will)g(b)q(e)g(p)q(ositiv)o(ely)99 2238 y(misleading.)h Fj(Systematic)e(Zo)n(olo)n(gy)p Fq(,)d(22:240{249,)k(1978.)-1 2337 y([14])24 b(J.)11 b(F)l(elsenstein.)g(Numerical)d(metho)q(ds)i(for)h(inferring)f(ev)o (olutionary)g(trees.)i Fj(The)h(Quarterly)f(R)n(eview)99 2398 y(of)18 b(Biolo)n(gy)p Fq(,)d(57\(4\):379{404)q(,)k(Decem)o(b)q (er)14 b(1982.)-1 2497 y([15])24 b(G.)14 b(N.)f(F)l(rederic)o(kson.)j (A)d(data)i(structure)f(for)g(dynamically)d(main)o(taining)h(ro)q(oted) j(trees.)i Fj(Journal)99 2557 y(of)h(A)o(lgorithms)p Fq(,)d(24:37{65,)j(1997.)949 2682 y(27)p eop %%Page: 28 28 28 27 bop -1 120 a Fq([16])24 b(D.)14 b(Gus\014eld.)j Fj(A)o(lgorithms)e(on)h(Strings,)g(T)l(r)n(e)n(es,)g(and)f(Se)n(quenc)n (es:)23 b(Computer)15 b(Scienc)n(e)i(and)f(Com-)99 181 y(putational)j(Biolo)n(gy)p Fq(.)i(Cam)o(bridge)15 b(Univ)o(ersit)o(y)e (Press,)k(New)e(Y)l(ork,)h(NY,)f(1997.)-1 282 y([17])24 b(D.)13 b(M.)g(Hillis.)h(Molecular)f(v)o(ersus)g(morphological)f (approac)o(hes)i(to)g(systematics.)g Fj(A)o(nnual)i(R)n(eview)99 343 y(of)i(Ec)n(olo)n(gy)f(and)g(Systematics)p Fq(,)g(18:23{42,)h (1987.)-1 444 y([18])24 b(D.)e(M.)f(Hillis,)g(C.)h(Moritz,)g(and)h(B.)e (K.)g(Mable,)i(editors.)38 b Fj(Mole)n(cular)23 b(Systematics)p Fq(.)39 b(Sinauer)99 504 y(Asso)q(ciates,)16 b(Sunderland,)g(Ma,)g(2nd) h(edition,)e(1996.)-1 606 y([19])24 b(W.)16 b(Ho)q(e\013ding.)23 b(Probabilit)o(y)16 b(inequalities)e(for)j(sums)f(of)h(b)q(ounded)g (random)f(v)m(ariables.)22 b Fj(Journal)99 666 y(of)c(the)f(A)o(meric)n (an)h(Statistic)n(al)g(Asso)n(ciation)p Fq(,)e(58:13{30,)i(1963.)-1 768 y([20])24 b(T.)15 b(H.)g(Juk)o(es)g(and)h(C.)f(R.)g(Can)o(tor.)20 b(Ev)o(olution)15 b(of)h(protein)f(molecules.)i(In)e(H.)g(N.)f(Munro,)i (editor,)99 828 y Fj(Mammalian)f(Pr)n(otein)f(Metab)n(olism)p Fq(,)f(v)o(olume)e(I)q(I)q(I,)h(c)o(hapter)h(24,)h(pages)g(21{132.)h (Academic)10 b(Press,)99 888 y(New)16 b(Y)l(ork,)f(NY,)g(1969.)-1 990 y([21])24 b(J.)19 b(Kim.)27 b(Large)20 b(scale)e(ph)o(ylogenies)g (and)i(measuring)e(the)h(p)q(erformance)e(of)j(ph)o(ylogenetic)d(esti-) 99 1050 y(mators.)k Fj(Systematic)d(Biolo)n(gy)p Fq(,)e(47:43{60,)i (1998.)-1 1152 y([22])24 b(J.)18 b(Neyman.)27 b(Molecular)18 b(studies)g(of)h(ev)o(olution:)25 b(a)19 b(source)g(of)g(no)o(v)o(el)e (statistical)h(problems.)27 b(In)99 1212 y(S.)12 b(S.)g(Gupta)h(and)g (J.)f(Y)l(ac)o(k)o(el,)e(editors,)j Fj(Statistic)n(al)h(De)n(cision)g (The)n(ory)f(and)h(R)n(elate)n(d)f(T)l(opics)p Fq(,)g(pages)99 1272 y(1{27.)18 b(Academic)13 b(Press,)j(New)g(Y)l(ork,)f(NY,)g(1971.) -1 1374 y([23])24 b(M.)18 b(Noro,)i(R.)e(Masuda,)i(I.)e(A.)g(Dubro)o(v) o(o,)i(M.)e(C.)g(Y)l(oshida,)i(and)f(M.)f(Kato.)30 b(Molecular)18 b(ph)o(ylo-)99 1434 y(genetic)c(inference)f(of)i(the)g(Wo)q(olly)g (Mammoth)e Fj(mammuthus)i(primigenius)p Fq(,)h(based)f(on)g(complete)99 1494 y(sequences)20 b(of)h(mito)q(c)o(hondrial)e(cyto)q(c)o(hrome)f Fj(b)j Fq(and)h(12S)f(rib)q(osomal)f(RNA)g(genes.)34 b Fj(Journal)21 b(of)99 1555 y(Mole)n(cular)d(Evolution)p Fq(,)f(46:314{326,)i(1998.)-1 1656 y([24])24 b(C.-Y.)18 b(Ou,)g(C.)g(A.)g(Cieselski,)e(G.)i(My)o(ers,)g(C.)g(I.)f(Bandea,)i (C.-C.)f(Luo,)h(B.)f(T.)g(M.)f(Korb)q(er,)i(J.)f(I.)99 1717 y(Mullins,)12 b(G.)h(Sc)o(ho)q(c)o(hetman,)e(R.)i(L.)g(Berk)o(elm) o(an,)e(A.)h(N.)g(Economou,)h(J.)f(J.)h(Witte,)f(L.)h(J.)f(F)l(urman,) 99 1777 y(G.)j(A.)f(Satten,)h(K.)g(A.)f(MacInnes,)g(J.)h(W.)f(Curran,)i (and)g(H.)e(W.)g(Ja\013e.)20 b(Molecular)14 b(epidemiology)99 1837 y(of)j(HIV)e(transmission)g(in)h(a)h(den)o(tal)f(practice.)k Fj(Scienc)n(e)p Fq(,)e(256:1165{1171)q(,)h(1992.)-1 1939 y([25])24 b(N.)16 b(Saitou)g(and)h(M.)e(Nei.)20 b(The)c(neigh)o(b)q (or-joining)h(metho)q(d:)k(a)16 b(new)g(metho)q(d)g(for)g (reconstructing)99 1999 y(ph)o(ylogenetic)f(trees.)21 b Fj(Mole)n(cular)d(Biolo)n(gy)f(and)h(Evolution)p Fq(,)f (4\(4\):406{425,)i(1987.)-1 2101 y([26])24 b(J.)15 b(C.)g(Setubal)g (and)g(J.)g(Meidanis.)j Fj(Intr)n(o)n(duction)e(to)h(Computational)g (Mole)n(cular)f(Biolo)n(gy)p Fq(.)k(PWS)99 2161 y(Publishing)c(Compan)o (y)l(,)f(Boston,)i(MA,)e(1997.)-1 2262 y([27])24 b(D.)12 b(D.)f(Sleator)g(and)i(R.)d(E.)i(T)l(arjan.)i(A)d(data)h(structure)g (for)f(dynamic)f(trees.)j Fj(Journal)g(of)g(Computer)99 2323 y(and)18 b(System)g(Scienc)n(es)p Fq(,)f(26:362{391,)i(1983.)-1 2424 y([28])24 b(M.)17 b(Steel.)25 b(The)18 b(complexit)o(y)d(of)j (reconstructing)g(trees)f(from)g(qualitativ)o(e)f(c)o(haracters)i(and)g (sub-)99 2485 y(trees.)j Fj(Journal)c(of)h(Classi\014c)n(ation)p Fq(,)e(9:91{116,)i(1992.)949 2682 y(28)p eop %%Page: 29 29 29 28 bop -1 120 a Fq([29])24 b(M.)d(Steel.)38 b(Reco)o(v)o(ering)20 b(a)j(tree)e(from)g(the)h(leaf)f(colourations)i(it)e(generates)h(under) g(a)h(mark)o(o)o(v)99 181 y(mo)q(del.)d Fj(Applie)n(d)e(Mathematics)g (L)n(etters)p Fq(,)d(7\(2\):19{23,)j(1994.)-1 282 y([30])24 b(D.)15 b(L.)g(Sw)o(o\013ord,)h(G.)f(J.)g(Olsen,)f(P)l(.)h(J.)f(W)l (addell,)g(and)i(D.)f(M.)f(Hillis.)j(Ph)o(ylogenetic)d(inference.)j(In) 99 343 y(D.)h(M.)g(Hillis,)e(C.)i(Moritz,)g(and)h(B.)e(K.)h(Mable,)g (editors,)g Fj(Mole)n(cular)i(Systematics)p Fq(,)f(c)o(hapter)f(11,)99 403 y(pages)f(407{514.)i(Sinauer)d(Asso)q(ciates,)g(Sunderland,)g(Ma,)g (2nd)h(edition,)e(1996.)-1 504 y([31])24 b(C.)16 b(T)l(u\017ey)g(and)h (M.)e(Steel.)21 b(Links)16 b(b)q(et)o(w)o(een)g(maxim)n(um)c(lik)o (eliho)q(o)q(d)k(and)g(maxim)o(um)c(parsimon)o(y)99 565 y(under)j(a)g(simple)e(mo)q(del)g(of)i(site)g(substitution.)k Fj(Bul)r(letin)g(of)d(Mathematic)n(al)g(Biolo)n(gy)p Fq(,)f(59\(3\):581{)99 625 y(607,)i(1997.)-1 727 y([32])24 b(M.)29 b(S.)g(W)l(aterman.)60 b Fj(Intr)n(o)n(duction)29 b(to)g(Computational)i(Biolo)n(gy:)46 b(Maps,)32 b(Se)n(quenc)n(es)g (and)99 787 y(Genomes)p Fq(.)22 b(Chapman)16 b(&)g(Hall,)f(New)h(Y)l (ork,)f(NY,)g(1995.)949 2682 y(29)p eop %%Page: 30 30 30 29 bop 109 218 a Fb(Algorithm)14 b Fq(Harmonic)g(Greedy)i(T)l (riplets)109 320 y Fb(Input:)21 b Fq(\001)323 327 y Fl(min)400 320 y Fq(and)c Fm(n)f Fq(m)o(utated)f(length-)p Fm(`)i Fq(sequences)e(generated)h(from)f Fm(T)7 b Fq(.)109 422 y Fb(Output:)21 b Fq(\011)363 429 y Fl(w)391 422 y Fq(\()p Fm(T)7 b Fq(\).)23 523 y(H1)25 b(F)l(or)16 b(all)g(lea)o(v)o(es)f Fm(X)20 b Fq(and)d Fm(Y)28 b Fq(of)16 b Fm(T)7 b Fq(,)15 b(use)i(Equations)f(\(3\))h(and)g(\(7\))f(to)h(compute)g(^)-26 b Fm(\033)1587 530 y Fg(X)s(Y)1665 523 y Fq(and)1768 511 y(^)1760 523 y(\001)1801 530 y Fg(X)s(Y)1862 523 y Fq(.)23 625 y(H2)25 b(Find)16 b(a)g(triplet)f Fm(AB)s(C)k Fq(with)d(the)g(maxim)o(um)e(^)-26 b Fm(\033)996 632 y Fg(AB)r(C)1080 625 y Fq(.)23 727 y(H3)25 b Fb(if)16 b Fm(AB)s(C)j Fq(is)d(not)g(p)q(ositiv)o(e)g Fb(then)g Fq(let)f Fm(T)837 709 y Fk(\003)872 727 y Fq(b)q(e)i(the)f(empt)o(y)e (tree,)h Fb(fail)p Fq(,)g(and)i Fb(stop)p Fq(.)23 828 y(H4)25 b(Let)16 b Fm(T)232 810 y Fk(\003)267 828 y Fq(b)q(e)h(the)f (star)h(with)f(three)f(edges)i(formed)e(b)o(y)g Fm(AB)s(C)k Fq(and)e(its)f(cen)o(ter)f Fm(D)q Fq(.)23 930 y(H5)25 b(Use)15 b(Equation)i(\(8\))g(to)f(set)g(\001)668 912 y Fk(\003)668 942 y Fg(AD)740 930 y Fi( )812 917 y Fq(^)804 930 y(\001)845 937 y Fg(AD)903 930 y Fq(,)g(\001)974 912 y Fk(\003)974 942 y Fg(B)r(D)1048 930 y Fi( )1119 917 y Fq(^)1111 930 y(\001)1152 937 y Fg(B)r(D)1212 930 y Fq(,)g(\001)1283 912 y Fk(\003)1283 942 y Fg(C)r(D)1356 930 y Fi( )1428 917 y Fq(^)1420 930 y(\001)1461 937 y Fg(C)r(D)1520 930 y Fq(.)23 1032 y(H6)25 b(Set)16 b(def)s(\()p Fm(D)q Fq(\))e Fi( )g(f)p Fm(A;)8 b(B)s(;)g(C)t Fi(g)p Fq(.)23 1133 y(H7)25 b(Let)16 b Fi(S)k Fq(b)q(e)d(the)f(set)g(of)g (splitting)g(tuples)g(for)g Fm(AD)q Fq(,)h Fm(B)s(D)g Fq(and)g Fm(C)t(D)q Fq(.)23 1235 y(H8)25 b Fb(rep)r(eat)23 1337 y Fq(H9)106 b Fb(if)16 b Fi(S)i Fq(=)13 b Fi(;)k Fb(then)e(fail)h Fq(and)h Fb(stop)p Fq(.)-1 1439 y(H10)106 b(Find)16 b Fi(h)p Fm(P)356 1446 y Fl(1)376 1439 y Fm(P)407 1446 y Fl(2)428 1439 y Fm(;)8 b(N)d(X)t(Y)s(;)j(P)q(;)g Fq(\001)686 1420 y Fk(\003)686 1451 y Fg(P)708 1456 y Fc(1)726 1451 y Fg(P)755 1439 y Fm(;)g Fq(\001)818 1420 y Fk(\003)818 1451 y Fg(P)840 1456 y Fc(2)857 1451 y Fg(P)887 1439 y Fm(;)g Fq(\001)950 1420 y Fk(\003)950 1451 y Fg(N)t(P)1010 1439 y Fi(i)14 b(2)g(S)20 b Fq(with)d(the)f(maxim) n(um)f(^)-27 b Fm(\033)1591 1446 y Fg(N)t(X)s(Y)1684 1439 y Fq(.)-1 1540 y(H11)106 b(Split)16 b Fm(P)338 1547 y Fl(1)358 1540 y Fm(P)389 1547 y Fl(2)425 1540 y Fq(in)o(to)g(t)o(w)o (o)g(edges)g Fm(P)777 1547 y Fl(1)798 1540 y Fm(P)23 b Fq(and)17 b Fm(P)978 1547 y Fl(2)998 1540 y Fm(P)24 b Fq(in)15 b Fm(T)1145 1522 y Fk(\003)1181 1540 y Fq(with)h(lengths)g (\001)1501 1522 y Fk(\003)1501 1553 y Fg(P)1523 1558 y Fc(1)1540 1553 y Fg(P)1586 1540 y Fq(and)h(\001)1722 1522 y Fk(\003)1722 1553 y Fg(P)1744 1558 y Fc(2)1761 1553 y Fg(P)1791 1540 y Fq(.)-1 1642 y(H12)106 b(Add)16 b(to)h Fm(T)393 1624 y Fk(\003)428 1642 y Fq(a)g(leaf)e Fm(N)22 b Fq(and)17 b(an)f(edge)h Fm(N)5 b(P)23 b Fq(with)16 b(length)g(\001)1293 1624 y Fk(\003)1293 1654 y Fg(N)t(P)1354 1642 y Fq(.)-1 1744 y(H13)106 b(Set)16 b(def)s(\()p Fm(P)7 b Fq(\))14 b Fi( )g(f)p Fm(N)r(;)8 b(X)q(;)g(Y)k Fi(g)p Fq(.)-1 1845 y(H14)106 b(Remo)o(v)o(e)14 b(from)h Fi(S)20 b Fq(all)c(tuples)g(con)o(taining)g Fm(P)1019 1852 y Fl(1)1039 1845 y Fm(P)1070 1852 y Fl(2)1106 1845 y Fq(or)h Fm(N)5 b Fq(.)-1 1947 y(H15)106 b(Add)16 b(to)h Fi(S)j Fq(the)c(splitting)f(tuples)h(for)h Fm(P)933 1954 y Fl(1)953 1947 y Fm(P)7 b Fq(,)16 b Fm(P)1052 1954 y Fl(2)1072 1947 y Fm(P)7 b Fq(,)16 b Fm(N)5 b(P)i Fq(.)-1 2049 y(H16)98 b(F)l(or)12 b(eac)o(h)g(edge)g Fm(Q)517 2056 y Fl(1)537 2049 y Fm(Q)576 2056 y Fl(2)609 2049 y Fi(2)i Fm(T)692 2031 y Fk(\003)714 2049 y Fi(\000)s(f)p Fm(P)812 2056 y Fl(1)831 2049 y Fm(P)q(;)8 b(P)916 2056 y Fl(2)937 2049 y Fm(P)q(;)g(N)d(P)i Fi(g)p Fq(,)14 b(add)f(to)f Fi(S)17 b Fq(the)12 b(splitting)f(tuples)h(con)o(taining)190 2130 y(relev)m(an)o(t)j Fm(N)418 2112 y Fk(0)430 2130 y Fm(N)5 b(Z)21 b Fq(with)16 b Fm(N)683 2112 y Fk(0)708 2130 y Fi(62)f Fm(T)792 2112 y Fk(\003)811 2130 y Fq(.)-1 2231 y(H17)25 b Fb(un)n(til)16 b Fq(all)f(lea)o(v)o(es)g(of)i Fm(T)22 b Fq(are)17 b(inserted)e(to)i Fm(T)915 2213 y Fk(\003)934 2231 y Fq(;)f(i.e.,)e(this)i(lo)q(op)h(has)g(iterated)e Fm(n)d Fi(\000)e Fq(3)17 b(times.)-1 2333 y(H18)25 b(Output)16 b Fm(T)318 2315 y Fk(\003)337 2333 y Fq(.)644 2489 y(Figure)f(3:)22 b(The)16 b(HGT)h(algorithm.)949 2682 y(30)p eop %%Page: 31 31 31 30 bop 99 159 a Fb(Algorithm)14 b Fq(Split-Edge)99 261 y Fb(Input:)22 b Fq(an)16 b(edge)g Fm(P)482 268 y Fl(1)503 261 y Fm(P)534 268 y Fl(2)570 261 y Fq(in)g Fm(T)663 243 y Fk(\003)698 261 y Fq(and)h(a)g(relev)m(an)o(t)e(triplet) g Fm(N)5 b(X)t(Y)28 b Fq(with)16 b(cen)o(ter)f Fm(P)7 b Fq(.)99 363 y Fb(Output:)20 b Fq(If)13 b Fm(P)20 b Fq(is)14 b(strictly)e(b)q(et)o(w)o(een)g Fm(P)838 370 y Fl(1)871 363 y Fq(and)i Fm(P)994 370 y Fl(2)1028 363 y Fq(in)f Fm(T)20 b Fq(and)13 b(th)o(us)h(can)f(b)q(e)h(inserted)e(on)i Fm(P)1754 370 y Fl(1)1774 363 y Fm(P)1805 370 y Fl(2)1826 363 y Fq(,)f(then)99 423 y(w)o(e)j(return)g(the)h(message)e(\\split")i (and)g(the)f(edge)g(lengths)h(\001)1250 405 y Fk(\003)1250 435 y Fg(P)1272 440 y Fc(1)1289 435 y Fg(P)1318 423 y Fq(,)f(\001)1389 405 y Fk(\003)1389 435 y Fg(P)1411 440 y Fc(2)1428 435 y Fg(P)1458 423 y Fq(,)g(and)h(\001)1624 405 y Fk(\003)1624 435 y Fg(N)t(P)1684 423 y Fq(.)22 b(Otherwise,)99 483 y(w)o(e)16 b(return)g(a)h(reason)g(wh)o(y)f Fm(P)23 b Fq(cannot)17 b(b)q(e)f(inserted.)23 585 y(S1)25 b(Use)16 b(Equation)h(\(8\))f(to)h(compute)748 572 y(^)740 585 y(\001)781 592 y Fg(X)s(P)841 585 y Fq(,)879 572 y(^)871 585 y(\001)912 592 y Fg(Y)8 b(P)969 585 y Fq(,)1007 572 y(^)999 585 y(\001)1040 592 y Fg(N)t(P)1117 585 y Fq(for)16 b Fm(N)5 b(X)t(Y)12 b Fq(.)23 686 y(S2)25 b(Let)17 b Fm(X)227 693 y Fl(1)261 686 y Fi(2)d(f)p Fm(X)q(;)8 b(Y)j Fi(g)g(\\)h Fq(def)s(\()p Fm(P)633 693 y Fl(1)653 686 y Fq(\))k(and)h Fm(X)823 693 y Fl(2)857 686 y Fi(2)d(f)p Fm(X)q(;)8 b(Y)k Fi(g)f(\\)g Fq(def)s(\()p Fm(P)1229 693 y Fl(2)1249 686 y Fq(\).)23 788 y(S3)25 b(F)l(or)17 b(eac)o(h)e Fm(i)f Fq(=)g(1)i(or)h(2,)f Fb(if)g Fm(P)613 795 y Fg(i)644 788 y Fq(is)g(an)h(in)o(ternal)e(no)q(de)i(of)f Fm(T)1150 770 y Fk(\003)23 890 y Fq(S4)107 b Fb(then)15 b Fq(use)i(Equation)f(\(8\))h(to)g(compute)945 877 y(^)937 890 y(\001)978 897 y Fg(X)1007 902 y Ff(i)1020 897 y Fg(P)1042 902 y Ff(i)1074 890 y Fq(for)g(the)f(triplet)f(formed)g(b)o (y)g(def)s(\()p Fm(P)1730 897 y Fg(i)1745 890 y Fq(\))23 992 y(S5)107 b Fb(else)15 b Fq(set)370 979 y(^)362 992 y(\001)403 999 y Fg(X)432 1004 y Ff(i)445 999 y Fg(P)467 1004 y Ff(i)496 992 y Fi( 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