IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, VOL. TK, NO. TK, MONTHTK YEARTK 1. Hamiltonian Walks of Phylogenetic Treespaces

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1 IEEE TRNSTIONS ON OMPUTTIONL IOLOGY ND IOINFORMTIS, VOL. TK, NO. TK, MONTHTK YERTK Hmiltonin Wlks of Phylognti Trsps Kvughn Goron, Eri For, n Kthrin St. John strt W nswr rynt s omintoril hllng on miniml wlks of phylognti trsp unr th nrst nighor intrhng (NNI) mtri. W show tht th shortst pth through th NNI-trsp of n-lf trs is Hmiltonin for ll n. Tht is, thr is miniml pth tht visits ll inry trs xtly on, unr NNI movs. Inx Trms Phylognti Tr onstrution, Grph Thory, nlysis of lgorithms. INTRODUTION PHYLOGENETI trs pit th volutionry rltionships within st of tx, rprsnt s lf lls []. Th trs my root in whih s thy illustrt th nstry of th tx or unroot. In this ppr, w look t unroot phylognis. Fining th tr tht st fits th t, whr th t is st of tx n orr hrtrs, is ntrl gol of volutionry iology. Howvr, th numr of possil trs grows s n xponntil funtion of th numr of tx, n fining th optiml tr unr th ritri most us y iologists is NP-hr [], []. Du to th siz of th srh sp, xhustiv srh is oftn not possil, so huristi srh is oftn us to isovr th st tr. To systmtilly trvrs th sp, it is nssry tht it rrng in som mnnr. ommon rrngmnt is to link trs tht r singl mov prt unr som tr rrrngmnt oprtion; th rsulting grph is oftn ll trsp. Fousing on trs tht iffr y singl nrst nighor intrhng (NNI) mov, Dvi rynt sk for th lngth of th shortst wlk tht visits ll trs in NNI trsp []. It is known tht using mtris tht yil mor nighors thn NNI (nmly sutr prun n rgrft (SPR) n tr istion n ronntion (TR)) hv th shortst wlk possil: Hmiltonin pth [5]. Prvious to our urrnt work, th st known NNI-wlk of ll inry trs visit vry tr t most twi [5]. W show tht, for ll n, Hmiltonin pth xists on th sp of ll inry trs on n lvs unr th NNI mtri, sttling rynt s hllng. W follow th strtgy of prvious work in xpning Hmiltonin K. Goron is with th Dprtmnt of Mth & omputr Sin, Lhmn ollg, ity Univrsity of Nw York, ronx, Nw York, 068; Support y n unrgrut rsrh fllowship from th Louis Stoks llin for Minority Prtiiption in Rsrh, NSF # E. For is with th Dprtmnt of omputr Sin, th Grut ntr, ity Univrsity of Nw York, Nw York, Nw York 006. K. St. John is with th Dprtmnt of Mth & omputr Sin, Lhmn ollg n th Dprtmnt of omputr Sin, th Grut ntr, ity Univrsity of Nw York; orrsponing uthor: stjohn@lhmn.uny.u; Support provi y NSF Grnt # pths on n-lf trs to th sp of ll inry trs on (n + ) lvs [5]. This i os not work irtly for NNI-wlks ut n mploy with sutl twist. Inst of vloping wlks on th xpnsion of singl n-lf tr, w look t ll trs tht n rt from susqunt tripls of n-lf trs on th Hmiltonin pth for th smllr sp. Using th Hmiltonin pth of th smllr sp s kon, w n thn glu togthr th unions of th xpnsions to form Hmiltonin pth on th (n + )-lf trs. Sin vry NNI mov n simult y n SPR or TR mov, this ppr provis n ltrntiv proof for th xistn of Hmiltonin pths for th SPR n TR trsps. KGROUND W rifly fin inry phylognti trs n th ssoit trms us in this ppr. For mor til trtmnt, s Smpl n Stl []. Phylognti trs pit volutionry rltionships twn tx pl t th lvs. Trs n root, in whih s thy illustrt th nstry of th tx, or unroot. W look t unroot inry phylognti trs (hrftr rfrr to s trs). Formlly, s fin y Roinson: Dfinition : [6]: (inry) phylognti tr is grph G on olltion of lll nos L (th tx) n unlll intrior vrtis. Th lll nos form th lvs of th tr n, thrfor, hv vlny on, n h intrior vrtx hs vlny thr. W will us wll-known ft out th numr of unroot inry trs: Lmm : [6]: For n tx, thr r (n 5)!! = (n 5)(n ) 5 possil unroot trs. Not tht for ll n, (n 5)!! is ivisil y thr. W will us this hrtristi of trsp to prtition pths of n-lf trs into tripls. W will lso xmin pirs of lvs: Dfinition : siling pir, or hrry, is pir of lvs whos inint gs shr ommon vrtx. Diffrn msurs on trs inu mtri sps on th st of n-lf trs. W will fous on mtris tht msur shp iffrns twn trs (n ignor

2 IEEE TRNSTIONS ON OMPUTTIONL IOLOGY ND IOINFORMTIS, VOL. TK, NO. TK, MONTHTK YERTK T l n+ l n+ D D D l n+ Fig.. Th lft si shows NNI trnsformtions. To trnsform into T or vi-vrs, intrhng th sutr with th sutr ontining th lf l n+. To trnsform T into or vi-vrs, ithr intrhng th sutr ontining th lf l n+ with th sutr ontining n D or intrhng th sutr ontining with th sutr ontining., T n r ll nighors in NNI trsp. Th right si illustrts how pth in NNI trsp will rprsnt in this ppr. Not tht th top sris of movs is quivlnt to moving from to T to, s in th lft si of th figur. Th ottom right rprsnts th sm movs with th urv g rprsnting th pth of th (n + ) st lf, l n+. iffrns in th lngths of gs or rnhs). Ths mtris inu isrt sp tht n rprsnt y grph: Dfinition : Givn st of trs T = {, T,..., T k } with n lvs lll y S, G = (T, E), or trsp, is th grph G with vrtis lll y T n th gs E onnting vrtis tht r nighors istn on prt unr givn mtri. rynt s hllng fouss on nrst nighor intrhng (NNI). Othr populr mtris inlu sutr prun rgrft (SPR) n tr istion n ronntion (TR) [7]. Dfinition : [7]: nrst nighor intrhng (NNI) swps ny two sutrs onnt to opposit ns of n g (s Figur ). Th NNI istn ( NNI ) twn two trs is th minimum numr of NNI oprtions tht trnsforms on of th trs into th othr. W not tht th NNI oprtion is symmtri in tht ny NNI tr rrrngmnt oprtion n rvrs. Ths movs fin nighorhoos on th sp: Dfinition 5: Lt isrt tr mtri. Th st of ll trs T m whr (T, T m ) is th -nighorhoo (or simply nighorhoo) of T. n n-lf tr hs n intrnl gs. Using n NNI mov, nw tr n form y swpping on of th four sutrs on opposit sis of th intrnl g (s Figur ). Only two of ths swps will prou nw trs, n thus vry n-lf tr hs (n ) nighors in NNI trsp. Th hllng on whih w fous is phrs in trms of wlks, w will us this trm intrhngly with th ommon trm from grph thory: pths. Dfinition 6: []: n NNI-wlk is squn, T,..., T k of unroot inry phylognti trs whr h onsutiv pir of trs iffr y singl NNI mov. Dfinition 7: [8]: Hmiltonin pth in grph is simpl pth tht visits vry no xtly on. This pth n rprsnt s n orr st of nos, v, v,..., v n, whr v i is onnt to v i+ y n g. Dtrmining whthr n ritrry grph hs Hmiltonin pth is NP-hr [8]. Howvr, for mny lsss of grphs (for xmpl, omplt grphs), Hmiltoniity n trmin sily in polynomil tim. MIN RESULTS W prov tht Hmiltonin pth xists through th st of n-lf trs unr th NNI mtri, for ll n. Th proof onstruts Hmiltonin pth of th (n + )-lf trsp from Hmiltonin pth of th n-lf trsp (s Figur ). This is on y tking susqunt tripls from th pth of th n-lf trsp n ontruting pth through ll (n + )-lf trs tht rt from thos thr trs (formlly fin s th xpnsion of trs, low). Sin vry (n + )-lf tr longs to xtly on suh xpnsion of tripl, linking th pths of th xpnsions yils pth tht visits vry (n + )- lf tr xtly on. Dfinition 8: Lt T n n-lf tr n n g of T. Th xpnsion of n g,, is th (n+)-lf tr, T (), gnrt whn nw lf is to tht g. Lt th xpnsion of n n-lf tr, T, th st of (n + )-lf trs tht n gnrt from xpning ll gs of T (s Figur ). If two trs iffr y only singl NNI mov, thn th gs of th trs r intil, xpt for singl g, tht w ll th g of iffrn. Formlly: Dfinition 9: onsir two trs, n T, tht iffr y on NNI mov. Lt, th g of iffrn, th singl g in th symmtri iffrn twn th st of gs of n th st of gs of T.

3 IEEE TRNSTIONS ON OMPUTTIONL IOLOGY ND IOINFORMTIS, VOL. TK, NO. TK, MONTHTK YERTK Fig.. Th proof of th min rsult onstruts Hmiltonin pths of th (n + )-lf trsp from Hmiltonin pths of th n-lf trsp. This is on y using Hmiltonin pth of th smllr sp (ol pth) s kon for pth of th lrgr sp. For vry tripl of trs in th pth of th smllr sp, pth is rt through its xpnsion. n xmpl of pth through tripl is ox. Th vrious trs in th xpnsions of th tripl r visit in ll orr. Ths pths r link to form th Hmiltonin pth of th (n + )-lf trsp. W not tht th siz of th xpnsion of n n-lf tr is inpnnt of th givn tr topology n pns only on th numr of intrnl gs. Likwis, in inry tr, th numr of intrnl gs is funtion of th numr of lvs. For givn tr T with n lvs, thr r n trs with n + lvs ontin in th xpnsion of T. W first prov svrl usful lmms out xpnsions of gs: Lmm : Lt T n unroot inry tr, n lt n jnt gs on T. T ( ) n T ( ) iffr y on NNI mov. Proof: Lt n jnt gs in n n-lf tr. Lt S th sutr whos root g is inint with n. Th ition of nw lf, l n+, rts two nw gs:, whih onnts th nw lf no to th tr n, whih sprts S n l n+. In T ( ), l n+ is twn n n S is twn n. Th opposit ours in th T ( ). In tht s, S is twn n n l n+ is twn n. Tht is, T ( ) n T ( ) hv th sm tr topology sv for th rrngmnt roun. Sin th nw txon n th root sutr r on opposit sis of, n intrnl rnh, swpping thm osts only on NNI mov. Thrfor, T ( ) n T ( ) iffr y on NNI mov. Lmm : Lt n T two unroot inry trs whr n T iffr y on NNI mov. Lt n g tht is not th g of iffrn,. () n T () iffr y on NNI mov. Proof: Lt,,, n D th four sutrs whos root gs r inint to, twn n T. y finition, th rrngmnt of th four sutrs in iffr from thir rrngmnt in T. ssum, without loss of gnrlity, tht is n g of, n not y th sutr rt y th ition of th nw lf to in. Not tht is intil in () n T (). Th rrngmnt of,,, n D roun is th only iffrn twn th two trs. Thrfor, () n T () iffr y on NNI mov. Whn fousing on tripl of onsutiv trs on Hmiltonin pth of n-sp, thr r mny sutrs whih r intil ross ll thr trs. Th nxt lmm shows how th xpnsions of ths sutrs n trvrs suh tht h no in th xpnsions is visit only on (s Figur ): Lmm : Lt, T, n thr unroot inry n-lf trs whr n T r NNI nighors n whr T n r NNI nighors. Lt S i som root sutr on T i whr i =,, or. If S = S = S, th union of th xpnsions of th gs in S, S, n S hs Hmiltonin pth suh tht th wlk strts on T i (p i ), whr p i is th root g of S i, n ns on T j (p j ), whr p j is th root g of S j n i j. Proof: W pro y inution on th siz of th sutr. s s: Th sutr hs two lvs n thr gs: p i, whih onnts th root no to th intrnl no; l i, whih onnts th intrnl no to lf no; n r i, whih onnts th intrnl no to th othr lf no. ll thr gs r jnt. y Lmm, th xpnsions of ths gs r NNI nighors. Sin n T r on NNI mov prt, (p ) n T (p ) r NNI nighors y Lmm. Th rst of th gs follow suit. Tht is, T z (y z ) n T z+ (y z+ ) r NNI nighors whr y {p, l, r} n z {, }. W n numrt th possil wlks tht strt on T i (p i ) n n t T j (p j ) whr i j (s Figur ). W intify th pth through th (n + )-lf trs y th g tht is xpn: p l r r r l l p p, p l r r r p l l p, p r r p l l l r p. W not tht sin th gs r not irt, h of th ov thr pths oul trvrs in rvrs. Thus, w hv Hmiltonin pth of th xpnsions of th gs of th sutrs tht gins on T i (p i ) n ns on

4 IEEE TRNSTIONS ON OMPUTTIONL IOLOGY ND IOINFORMTIS, VOL. TK, NO. TK, MONTHTK YERTK Fig.. Expnsion of n unroot tr on n lvs to (n + )-lf trs. Whn n =, thr r fiv possil gs to whih to tth nw lf, rsulting in fiv 5-lf trs in th xpnsion of th initil tr. T j (p j ) whr i j. Inutiv Stp: ssum tht th sutr, S i, hs thr or mor lvs n t lst fiv gs: p i, whih onnts th root no to n intrnl no; n gs l i n r i whih r inint with p i. y Lmm, th xpnsions of ths gs r NNI nighors. Furthr, sin n T r on NNI mov prt, (p ) n T (p ) r NNI nighors y Lmm. Th rst of th gs follow suit. Tht is, T z (y z ) n T z+ (y z+ ) r NNI nighors whr y {p, l, r} n z {, }. W show tht Hmiltonin pth n strt on T i (p i ) n n on T j (p j ) whr i j. Without loss of gnrlity, ssum tht l i is th root g of th innr sutr, i. Lt T (,, ) th union of th xpnsions of ll th gs in,, n xpt for th xpnsions of two of th root gs, l i n l j, whos visit w xpliitly show. y th inutiv hypothsis, Hmiltonin pth n strt on T i (l i ), n n on T j (l j ) whr i j. s I: S i is omplx root sutr with lf tth to th root. r i onnts th first intrnl no to lf (s Figur ). Th following r pths of th union strting t T i (p i ) n ning t T j (p j ), i j: p r l T (,, ) l r r p p, p r l T (,, ) l r r p p, p p r r l T (,, ) l r p. s II: S i is omplx root sutr with nothr omplx root sutr tth to its root. r i onnts th first intrnl no to nothr omplx sutr, D i. Th following r pths of th union strting t T i (p i ) n ning t T j (p j ), i j: p r T (D, D, D ) r l T (,, ) l p p, p r T (D, D, D ) r l T (,, ) l p p, p r T (D, D, D ) r p l T (,, ) l p. This omplts th proof. Lmm fouss on th union of th xpnsions of gs in root sutrs tht r intil ross tripl of susqunt trs in pth of th n-lf trsp, giving multipl pths tht n trvrs tht union from iffrnt strting n stopping points. Th following lmm shows how ths pths n glu togthr to form pth for th unions of th xpnsions of th omplt trs. Th iffiulty in th proof is th lining up of th npoints of pths to rt singl longr pth. Sin Lmm provis pths from ny root g of intil sutrs, it suffis to show how to trvrs th gs of iffrn in th xpnsions. Lmm 5: Lt, T, n thr unroot inry trs whr n T r NNI nighors n whr T n r NNI nighors. For ny g of, thr xists Hmiltonin pth of th union of th xpnsions of, T, n strting t (). Proof: Lt, T, n thr unroot inry trs whr n T r NNI nighors n whr T n r NNI nighors. Lt th g of iffrn twn n T, n lt th g of iffrn twn T n. Lt th xpnsion of sutr, S, in n n-lf tr, T, th union of th xpnsions of th gs in S. Lt n g in. W will onstrut wlk tht trvrss, xtly on, vry tr in th union of th xpnsions of, T, n. Th lotion of, whos xpnsion, (), is th strting point, trmins our strtgy for uiling th Hmiltonin pth. Dnot y n th sutrs tht rsult from rmoving (ut not its npoints) from. If on of th sutrs, sy, is intil ross, T, n, thn nithr nor is in, n y Lmm, thr is Hmiltonin pth ross th union of th xpnsions of,, n, ginning t in n ning t in T i whr i = or. So, ssum tht nithr nor is intil ross ll trs. W pro y ss on th rltiv lotions of th gs of iffrn, or, to : s I: ssum tht is on th pth twn n (tht is, is th losr g of iffrn to ). Lt,..., m th pth twn n in,

5 IEEE TRNSTIONS ON OMPUTTIONL IOLOGY ND IOINFORMTIS, VOL. TK, NO. TK, MONTHTK YERTK 5 T T T T T T Fig.. Lft: s s I for Lmm. Th lls on th rrows n sutrs init th orr of th trvrsl. S x is root hrry tht is th sm ross, T, n. Not tht th pth through th xpnsions of th gs in S x n nott s sris of NNI movs, moving th lf in th xpnsions through th vrious gs in S x Right: Th thr wlks show tht Hmiltonin pth n strt t T i (p i ) n n on T j (p j ) whr i j. n lt S, S,..., S m th sutrs long th pth. y hypothsis, th sutrs, S, S,..., S m, our in ll thr trs. W pply Lmm to h of th sutrs n link th rsulting pths y visiting th th xpnsion of th sutr, S i, follow y th xpnsion of th pth g, i+, in, T, n, for h i, rting pth tht xtns to T ( ) or ( ). W my ssum tht our pth thus fr ns t ( ). Lt not th sutr, intil in, T n, tht w hv trvrs thus fr. Th root g of is inint with. Lt D th sutr whos root g is inint with in T n (tht is, n D r on th sm si of in T n ) n lt F n G th rmining sutrs whos root gs r inint with in T n (tht is, F n G r on th opposit si of from n D in T n ). Sin th root gs, r n r D, of n D r jnt in, w n mov from (r ) to (r D ) without hving to trvrs th xpnsion of (tht trvrsl is optionl). If D os not ontin, it is intil ross ll thr trs n w n pply Lmm to yil n xtnsion of our pth tht trvrss th union of th xpnsions of ll thr opis of D, ning in. iffrs from roun, so th root g of in is not inint with D, ut with on of th othr two sutrs, sy F. To simplify th rgumnt, w will ssum tht F os not ontin th g (if F os ontin, th proof follows y slightly mor omplit, ut similr, rgumnt). W n similrly xtn our pth to th union of th xpnsions of th sutr F in, T, n, ning in. In ition, us D n F r on opposit sis of in, our pth must first visit ( ). W now hv pth tht trvrss th xpnsion of thr out of four of th sutrs whos root gs r inint with n ( ). To rh th finl of th four sutrs, G, w ross ( ), thn T ( ). From thr, w visit th xpnsion of th root g of G, whih w trvrs y n rgumnt similr to th on ov. W not tht if D, ov, h ontin, thn w woul us th rgumnt for trvrsing ( ), T ( ), n ( ) to trvrs ( ), T ( ), n ( ). On w hv trvrs th xpnsion of th thr opis of G, w hv pth tht visits ll th gs of, T, n, n thus ll trs in th union of th xpnsions of thos trs. s II: W must lso onsir th s whr lis on th pth twn th gs of iffrns, n. Whil th rgumnt for this s is similr to tht ov, thr is th itionl iffiulty of strting in th mil. It is nssry to trvrs th pth from to n still hv unvisit gs upon whih to rturn so tht th othr si of n lso visit. This n omplish y trvrsing only th pth gs (n non of th tth sutrs) in th tr until th g of iffrn is rh. Th pth is thn uilt, s ov, ut on th rturn, th sutrs on th pth r link y visits to th pth gs in only T n. On tht stion of th xpnsions of th trs hs n visit, th rmining trs in th union of th xpnsions r visit (nmly th union of th xpnsion of whr is th sutr rsulting from rmoving n tht ontins ). Th rsult is Hmiltonin pth of th union of th xpnsions of, T, n. s III: Not th spil s whr =. In this s, n T r NNI nighors, T n r NNI nighors, n n r NNI nighors. Suh s is simplifi vrsion of th prvious ss, n thus is ovr ov. Thorm : For ll n, thr xists Hmiltonin pth through th n-lf NNI trsp. Proof: y inution on n, th numr of lvs. s s: Whn n =, (n 5)!! =. Lt th four lvs,,, n. Thn, without loss of gnrlity, in sprts, from, ; in T sprts, from, ; n in sprts, from,. n T r NNI nighors, n T n r NNI nighors. y

6 IEEE TRNSTIONS ON OMPUTTIONL IOLOGY ND IOINFORMTIS, VOL. TK, NO. TK, MONTHTK YERTK 6 th prvious lmm, thr is Hmiltonin pth through th -lf NNI trsp (s Figur ). Inutiv Stp: ssum thr is Hmiltonin pth through th n-lf NNI trsp. Th wlk visits th orr st of trs,, T,,... T (n 5)!!. y th finition of Hmiltonin pth, T x n T x+ r NNI nighors whr x < (n 5)!!. y th prvious lmm, th union of th xpnsions of th triplt T y, T y+, n T y+ hs Hmiltonin pth whr y = z n z (n 5)!!. us (n 5)!! is ivisil y whn n, thr is n orr st of sussiv triplts, R, R,..., R (n 5)!!, whr R is th triplt of trs, T,, n R (n 5)!! is th triplt T (n 5)!!, T (n 5)!!, T (n 5)!!. Th unions of xpnsions on h of ths triplts hs Hmiltonin pth. onsir T y, th thir tr in triplt R v, n T y+, th first tr in R v+. T y n T y+ r NNI nighors. Thn, y Lmm, T y (), whr is not, is n NNI nighor of T Y + (). Th n of th Hmiltonin pth through n xpnsion of triplt n thus onnt to th ginning of th Hmiltonin pth through th xpnsion of th suing triplt. s shown ov, thr is n orr st of triplts whih ovrs n-lf trsp with Hmiltonin pth. Th xpnsions of h of ths triplts hs Hmiltonin pth, n h of th wlks n link y singl NNI mov. Thrfor, Hmiltonin pth xists through th (n + )-lf NNI trsp. ONLUSION W hv shown tht th shortst wlk on th sp of inry phylognti trs with n lvs unr th NNI mtri is Hmiltonin pth. Sin visiting h no xtly on is th miniml pth lngth possil, this nswrs rynt s First omintoril hllng on th lngth of th shortst wlk of trs unr th NNI mtri. In ition, sin vry NNI mov n simult y n SPR or TR mov, this lso givs n ltrntiv proof to th Hmiltonin pths of SPR n TR trsp [5]. Our itrtiv pproh to uiling Hmiltonin pth for th sp of trs with n + lvs from pth of th smllr sp of trs with n lvs os not yil n lgorithm for prouing Hmiltonin pth irtly nor o w s n ovious wy to o this. 5 KNOWLEDGMENTS Th uthors woul lik to thnk th nonymous rviwrs for thir thoughtful n hlpful ommnts, whih grtly improv this ppr. For spirit isussions, th uthors thnk th mmrs of th Trsp Working Group t UNY: ln Josph rs, nn Mri lor, Kin rown, Jun stillo, Smnth Dly, John DJsus, Mihl Hintz, Dnil Ippolito, Jinni L, Jon Mr, Olivr Mnz, Diqun Moor, n Rhl Sprtt. This work ws support y grnts from th US Ntionl Sin Fountion progrms in mthmtil iology n omputtionl mthmtis (NSF #09-090) n th Nw York ity Louis Stoks llin for Minority Prtiiption in Rsrh (NSF #07-09). REFERENES []. Smpl n M. Stl, Phylogntis, sr. Oxfor Ltur Sris in Mthmtis n its pplitions. Oxfor: Oxfor Univrsity Prss, 00, vol.. [] L. Fouls n R. Grhm, Th Stinr prolm in phylogny is NP-omplt, vns in ppli Mthmtis, vol., no., pp. 9, 98. [] S. Roh, short proof tht phylognti tr ronstrution y mximum liklihoo is hr, IEEE/M Trnstions on omputtionl iology n ioinformtis, vol., no., pp. 9 9, 006. [] D. rynt, Pnny nt: mthmtil hllng, 008, [5]. J. J. rs, J. DJsus, M. Hintz, D. Moor, n K. St. John, Wlks in phylognti trsp, Informtion Prossing Lttrs, vol., pp , 0. [6] D. Roinson, omprison of ll trs with vlny thr, Journl of omintoril Thory, Sris, vol., no., pp. 05 9, 97. [7]. L. lln n M. Stl, Sutr trnsfr oprtions n thir inu mtris on volutionry trs, nnls of omintoris, vol. 5, no., pp. 5, 00. [8] T. H. ormn,. E. Lisrson, R. L. Rivst, n. Stin, Introution to lgorithms, n. mrig, M: MIT Prss, 00. rif uthor iogrphis Kvughn Goron is pursuing hlors gr in omputr sin t Lhmn ollg, ity Univrsity of Nw York (UNY). H riv hlors gr in iohmistry from inghmpton Univrsity. His rsrh intrsts inlu omputtionl iology n nturl lngug prossing. ontt him t kvughn.goron@l.uny.u. Eri For is otorl nit t th UNY Grut ntr n lturr t Lhmn ollg, UNY. For hols mstrs gr in linguistis from th Grut ntr. His rsrh intrsts inlu omputtionl linguistis n ioinformtis. ontt him t for@g.uny.u. Kthrin St. John is profssor of mthmtis n omputr sin t Lhmn ollg, UNY n hols ppointmnts to th otorl fultis of nthropology n omputr sin t th Grut ntr of UNY, s wll s th invrtrt zoology n plontology ivisions of th mrin Musum of Nturl History. St. John riv hr otorl gr from UL. Hr rsrh intrsts inlu omputtionl iology, rnom struturs, n lgorithms. Sh

7 IEEE TRNSTIONS ON OMPUTTIONL IOLOGY ND IOINFORMTIS, VOL. TK, NO. TK, MONTHTK YERTK 7 is mmr of M, MS, n SIM. ontt hr t stjohn@lhmn.uny.u.