Adebiyi, E. F.

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f *ru rnal of Computer Scien'ce & Its A pplications. l)eccmht'r 2{X)6. \"o1. l3 No.l()f lXl'r Dictionarv from\ Subquadratic Time GenerationString(s) using Suffix TreeF"zekiel F. Adebiyi. )cp;u't ment of Computer and Intllrmat ion Se iercr-Sil.-r)\ errant uni versityP\18 1023. Ota, Nigeria.: - mai I : eadebiyi @ sdsc.edu.lbstract.Given a set of alphaitbas y @hose size is lyl) and set of subject strings X = {x n x,..., t J bver t such that n = y,i=t.rxi, the problem consi^dereil in this paper, is thegeneration 0f items of the dbtionary l2la, where q is a positive htteger greaterthan I, from the strings in X. This stringology problem finds application in thebioinfoimatics problem of moffi findingf10] and recently in the selection ofoligonucleotides for Microarray and PCR[13, I4]. Microarray and PCR are bothacademic and industri"al tools of Bioinfornmtics used in Life Sciences.A naive algorithm that runs in O(lylt) time was given recently by Zheng et al.[13,141 to solve this problem. In this paper, we present a novel algorithm based onthe saffix tree tlmt solves this problem in subquadratic time.Keywords . Stringology, Suffix tree, dictionanry, seeds, motifs, oligonacleoti.des,microarray, PCR.1.0 IntroductionIn bioinformatics, the following stringology problem isan impofiant step in motif finclingl l0] and rccently inthe selecticn of oligonuclsotiCes for micro array andIl{iR[ 13, l4].'fypical setup include k subject strings X= {xl, x2, .. !, **}, such that n - f,=r..k xi, where stringsxl, x?, . .., xk are generated over a set of alphabets I,whose size is a = Itl. The problem is then to list out alllf lo substrings that appear in X, whel e q is a po$itiveinteger greater than l. Since atl possible of llltsubstrings, not known in advance may occLrr in X, anaive approach employed in [3, 14] is to create l;l"buckets (if q = 2 andltl = 4 for DI\A ,4t = 16 bucketsinclude AA, AC, AG, ^dl', . r., TT) and slicle a ruler oflength q through each sutlject stnngs to collect all pos-sible lflo substrings.To collect the occuffence statisticsof the lf lt buckets that occur in X, in the worst case, aO(lfl't) time is requircd. We show in this paper how anovel algorithm designed using the suffix tree is cleverlyengineered to do this in O(n ln n/ln a) time. For compactness, let us call the ltlt substrings seeds of length q(hencefofih, q-seeds). Our resulting algorithm has be :nused in Adebiyi[3] andAdebiyi and Olarenwaju[4] todesign efficient sequential and parallel algorithm foroli gon ucleoti des selecti on.1.1 Organization of the paperThis paper is structured as follows. In section 2, wediscuss the suffix tree andprcsent our novel algorithm insection 3. We discuss the implementation of our algo-rithm and oLlr expenmental experience using this algo-26 A Subquadratic Time Generation of lll', nictionary from String(s) using Suffix Treerithm in section 4. We conclude this paper in section 5.1.0 Suffix treeSome good materials on suffix tree can be found inAdebiyifl I and Gusfield[6].'Wegive abriefdescription below. The followittg informaldescription is taken from [1].A suffix tree is a lexicographically inter-connected datastructure, that provide efficient access to all substringsof a stringsz over which it is built. This datastructurecan be constructed and represented in linear time andspace. And this has enable the solution ofmany strin._qsproblem in lineartime. The construction ofa sufrr-x ffeein lineartime can be found in Weinerfl2l, McCreight[8]and [Jkkonenll1].Arecent paper by Kurtz[7] discusrdhow an economical construction of suffix tree uith re-spect to space can be done.We give here the definition of a suffix tree for an arbi-trary string x of length N over an alphabet t as presentin [6]. We follow this to show how a suffix tree can beEzekiel F. Adebiyibuilt for a set of strings.Definition 1 A suffix tree of a N-character string x is arooted directed tree with exactlyN leaves, numbered 1to N. Each internal node, other than the root., has atleast tr,r-o children and each edge is labelled rn'ith anonempt)- substring ofx. No two edges out a node canhave edge-labels beginning with the same characters.The ke)'feature ofthe suffix tree is that for any leafi, thelabel of the path from the root to leaf i exactly spells outthe suffix of x that start at position i. Note that the defi-nition abor-e does not guarantee the existence of a suflth tree tor any string x. The problem is that if apreffixof a sutJ-rx of r matches a suffix of x, the palh for thelater suftir u'ould not end ataleaf. Therefore, to guar-antee the eristence of a suffix tree for any string x, weplace at the end of r a special symbol that is not in thealphabet I.\\'e use in this paper. the symbol $, forthetermination character. Below is a suffix tree for x -GTAICTAGG The number at the leaves indicate thestarting po sition of the corresponding suffi xe s.fCt.{r;rt'r'drr;rf,i1'Fig" I", Th* ssJfix [rs* fur x: * #T.4'I'C7'A{IG\-\\_\ (:r,+c$sA{i{}5..'?dgrtii TATCTAOC3I3e.q:e[ F, Adebiyilh: suffix tree constructed for a set of strings is called aJenerahzedsuffix tree, can be easily achieved by con-iicutively building the suffix tree for each string in the:et.The resulting suffix tree is built in time proportional to:he sum of all the string lengths. The leafnumber oftheA Subquadratic Time Generation of lflo Dictionary from String(s) using Suffix Tree 27single string suffix tree can easily be converted to twonumbers, one identifying the string and the other identi-fying the startingposition inthat string. Belowis the gen-eralized suffix tree (henceforth GST) for the stringsxl -TACTAand x2: CACTCA.FiS. il" 'I'J:* -q*usrtrlirr:cl sr:f$ix trex {{:;'$tf}3.0 The Subquadratic running timeAlgorithmWb state first some important preliminaries. Note thattwo types ofexact repeats exist: maximal and non-maxi-mal. An exact repeat F is maximal if aBc and bBd occurin S for some a*b and cfd, &,b, c, d t I. Otherwisethe repeat is non-maximal.A maximal repeat is aproper substring ofanother mari-mal repeat, called the supermaximal repeat. Note thatthe q-seeds formally described above can be define asfollows.Definition 2 Aq-seed is a substring of length q thatoccur in at least one location in the set X.Let each bucket obtained via q-seed i be B . Elsewhere lzl,r-rsing the suffix ffee for solving a different prob-lem, we use among other definitions and observationthe following definition and corollary to characterize seedsl,fj?Tft:r ;n1 :x 3,'dil1'']',4 .-tnCI ;Itg x {.-,'.-{.i-"''J-t-)tto extractthem. We discovered that the characteriza-tion also finds application here partly.Observation 1 A substring that is completely non-maxi-mal in all strings in {*i,l i : I . . .k} cannot be a possibleseed, since it is contained in some supermaximal re-peats.Definition 3 Arepeat seed is a repeat (i.e., non-maxi-mal, maximal or supeffnaximal) in at least two ofthesubject strings"Noting the above, the seeds of interest in this paper asdefined in definition 2 canbe charactertzed using thefollor,ving coro llary :Corollary 1 In addition to the q-length constraint, ob-serue that definition 3 above will capture all the q-seedsof the dictionary 4q extractable from the strings in X,,q3,1."i128 A Subquadratic Time Generation of f ;lr Dictionary from String(s) using Suffix Treeexcept for the q-seeds that exist only on one sequerlce.And because mrximality as depicted in observation I isnot a constraint in definition 2,theq-seeds we desirecan also end between two nodes, in addition to thosethat exist at the nodes.Using the following definitions and notations, the algo-rithm encapsulating all the observations regarding thegeneration ofq-seeds is given in figures 3, 5, and 4 ofthe appendix.' seeded*iAboolean variable labelling all loca-tions (leafs (seed ending in between or at theend) and nodes (seed ending at the end or inbetween trvo edges)) where seeds are found.And seeded* is true if a leaf or a node spelled(after Sagot[9] ),a seed.e bucklen: Total# ofentries in a bucker. bricklen4- 0.e hitscount:' otal # of buckets. i.e. Total + of q-mers extracteC. hitscount +- 0"Ezekiel F. Adebiyilencount: Accumulated sum ofedge lengths aswe do the depth first traverse ofthe suffix free.succ: successive children of a node x.depth: accumulated length ofedges alongapathin a suffix tree.Lemma 1. Using corollary l, all the q-seeds of thedictionary lllo extractable from the strings in X can bedone in O(n ln n/ln a) worst case time.Proof. The suffix tree for X can be build in o(n) time.Note that for each q-seeds required for aparticular qlength: \\€ will not need to traverse all nodes and edgesof the suffix tree. But consiclering a worst case sce-nario, in uhich case, we neecl to traverse all edges andnodes. Blunrer[5] had shor,vn thatthe number ofnodesin a suffi\ ri'ee lor a random text x on an alphabet ofsize a : I-i. rincer a nlodel of equiprobabilit,v and in-ceprgndcnce ct rire characters in X, is F(n) - iFc +Fcin))n, r,r,here in this caseooY-rl'f*' :I;il{*rln{;'*lci* til*i* lil iI i;t r' "lanriF.trti 3;; !4 -Ji 5-it *- cjII # J*-**d"5id$f 'i i!-Jis a small oscillating factorwith exponentially increasingperiod and n accounts for the prefixes ofX. It is straight-forward now to extend that the number of edges of asuffix is bound by O(n ln n/ln a). hiote that the numberof edges on the path from the root to any leafis (ln n/lna) and the numer of leaf is O(n). Therefnre, the numberof edges on a suffix tree can not tre ffiot'* than (}(n ln #}n e). Sin*e t"he travorsing execllfecl in ourr aXgorithn: is nfuirctian ofthr Nrurnher ofnodes and edges, therefure.our algorithm wilrst case nm time (plus the tims l*e"iuirecfto build sutTix tree) is bound from above by O(n ln r:llna).4.0 Experimental ExperienceV/e have implemented our subquadratic algorithm in Cnnder Linux based on KufizlTl space efficient imple-rnentntion ofa modified iV{cCreight sufrr-x tree andtestedit als* under Linux on a PC n-ith Pentium IV Cpu and5l?hdn I{.AM. We bench-marked our algorithrn (AlgF3) wittrr th* neive algorithm in [13. l1] (AlgA).E:tid F. AdebiyfTo compare output, the dataset used is the barley ESTsfrom HARVEST 1.45 downloaded from http:llharvest.ucr.edu/. It contains k - 53;240 unigenes, withfr : 43,464, 144 bases. 'We downloaded from http:llharvest.ucr.edt/ anoth er 7.41 KB of EST unigenes tocompare their effi ciency.Algorithm A and algorithm B, for q - 11 and q- 12extracted the following same number of seeds in tablel(a,b), grouped according to the number oftimes, theyocculred. For q: l3,our algorithm (Alg B) producedA Subquadratic Time Generation of lllq Dictionary from String(s) using Suffix Tree 29the seeds oftable 1(c), for which the naive algorithm(AlgA) failed to run. This is because the exponentialspace requirement ofthe naive algorithm is astronomi-cal too large to be accofirmodated on the PC we usedto carry out our experiment.As expected theoretically, the run times (on the samesystem) ofthe two algorithms, that is,AlgAandAlg Bon the 7 .41 KB of EST unigenes dataset are depictedin table 1(d). For q : 13,because of the astronomicalincrease in 4t, algorithm A will not ruIt.Table 1" Disrrihrtior of recd* extrn*ted frrrm the berlsy E$Ts f*r a) {' * 33 {g * 1"1} 'b}I = 3S ig r 1?)1 c) I * 39 qE = 13) and di th* *rnning time* {x1$} of alguriih.r:r of ,4 *rrcl"S far different vdue* of g on th* *-axis# *f $ucuffen*ss # qrf s*scls # trf'*serlrrs$f*$ # *f se,eris{h}{ur}g${}t.?$ Ifl?Jt?l !}3$ri?$ilss?37Srlslrs!s$s"{1*$lfi-1Sts-?$}an-gg4$-49ss-$?lI^?31,?s2gs.s{]ll$s?.3{?1S{}{s*4{}{1.5$si*1{*}# *f $ssrJrr{;}I}r-:e$ # *f srrcrds{d}SG1-SlrJ- 1'$x{}-!$3{}-;lll,.1*-4$sil-$.llfitt);}SiiSI1iFrr.r?...+ Ff f .j{*$T.*?i3il$'tr'1.$gi^it{t}$30 A Subquadratic Time Generation of l;le Dictionary from String(s)5.0 ConclusionA subquadratic algorithm has been designed to gener-ate the lpla dictionary which finds applications in thebioffirmatics problems ofmotifs finding and selectionofoligonucleotides. The algorithm is designed using thesuffix fee, clever$ engineered viathe fine lexicographicinterconnected structure ofthe suffix tree to run in asubqradratic time instead ofan existing novel algorithmthat runs in exponential time.AcknowledgmentWe thank the anonymous referee of our paper acceptedat the Intl Workshop and Conf. on new trends in theMathematical and Computer Science with applicationto real world problems, June l9-23,2006,whose care,firl review work has necessitated ttre new technical ex-position inthis paper. This work is partially supportedby the Covenant University Senate Research Grant200412005. Partofthis workwas done, while theauthor was at LIRMM, France on a CNRS-NEPADspecial grant.References1. Adebiyi, E. F. (2002) Pattern Discovery inBiology and String Sortine: Theon- andExperimentation. Shaker Publisher. -\achen.Germany.2. Adebiyi, E. F., Kaufrnann,Iv{ (2002) ExtractinsCommon Motifs under the Levenshtein \Ieasure :Theory and Experimentation. Proc. of the 2ndWorkshop on Algorithm of Bioinformatics(V/ABI), Italy and also appeared in LectureNotes in Computer Science, 2452, I 40- 1 56.3. Adebiyi, E. F. (2006) Using Suffix Tree for Effi-cient Selection of Unique oligos for large ESTdatabarses, Submiuedto WABI 2006 andlnter-national Joumal of Bioinformaticr nnd Computa-tionalBioloryflJBCB)4. Adebiyi, E. Fand Oyelade, J. O. (2006)ACom-parative Analysis of existing Oligonucleotidesusing Suffix Tree Ezekiel F" Adebiyis election Algorithms and optimal parallel oligo sselection for large EST Databases (ExtendedAbstract). Accepted (peer reviewed) proc" oftheIntl workshop and conf. on new trends in theMathematical and computer science with appli-cation to real world probleffis, June 19-23,2006.5. A. Blumer A. Ehrenfeucht, and D. Haussler.Average size of Suffix trees and DAWGS.Discrete Applied Mathematics, 24, 37 -45, 1 9g9.6. D. Gusfield. Algorithms on Strings, Tiees and Se-quen ce s . CAmbridge ljniversity press, New york,1997 .7 . Kurtz, s. (1999) Reducing the space Require-ment of Suffix trees. Software-practice and Ex-perien ce. 29( I 3): I 149-ll7 I .8. Mccreight. E. M . (1976)A Space-EconornicalSuffix tree Construction Algorithm. Journal ofACM 23(2):262-272.9. sagot, N4-F. SpellingApproximate Repeated orcornmon ir{otifs using a Suffix tree, LhICS 1390,111-127,1999.fompa, M. (1 999) An Exact Method fbr FindingShort Motifs in Sequences withApplicationto theItibosome Bincting Site Problem, Tthlntl. Conf.Intelligent Systems for Molecular Biol*gy flsh{B}.,262-27 1 .'--kkonen. E. f 1995) on-line Construction ofS i:fr .r uees. { gorithmica- | I :Zl9 -26A.\\ einer. P. ( 1973) Linear Pattern MatchingAlgo-rirlun. Proc. 14th IEEE Sym. on Switching andAutonrata theorl', 1-1 1.Zheng, J., Close, T., Jiang, T., and Lonardi, S.(2003) Eftrcient Selection ofljnique and popularoligos for large EST Databases, cpM 2003,LNC S 267 6, 394-40 1 .Zheng, J., Close, T., Jiang, T., and Lonardi, S.(2001) Efficient Selection ofUnique and popularoligos for large EST Databases, Bioinformatics,2A(1 3), 2101-211,2.10.ii11.13.14.Ezekiel F. AdebiyiAppendixA Subquadratic Time Generation of lllq Dictionary from String(s) using Suffix Tree 31I"f;drs.4,lri),$"ttI,8.g.l$.LL.12.J"$.14"15"I$,17"18.FINnIfiT;${X, {),S:l' '.* Build #"$',F'{"{ }fiur r$trt, *hilelr*r: # {rn d*y,,**. fr,*t Jirnr*t}if{r ix ;* le;*f}{*rl fr # nf chara*t*rs ehet r$Erstitnt# t he lsaf bl"&ur:}r.iffterl H s)s**rJ*cJ* * 3ftrif.srourui : Jr.tige$u.n* + Ifiu*l*fgtUadt*c<rnnt * I .^.^sh1***ou11l. {-* ;Hetrse if{m ls a nclcl*}Ie'ncount a, d*ptft, &vc,tJsn : giftg ** Ien*ou,r# ll E < len*rxrnf)*eed*d* -*a IIri Lscotr p{.'x h i* scu unt * Il,IS'f0C*P$$ {t*, S?, h.f f *cr*r.rnf }ffTttf*n*rnt +* 3jelse DEPTIIFII€T{", Jeraeuu,nt, {, Sf}Fi$. S, 'fhe subelu:ulratie r*lg*ritlun for sxtreeting _q*se*ds\-

A Subquadratic Time Generation of |E|q Dictionary from

String(s) using Suffix Tree

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A Subquadratic Time Generation of |E|q Dictionary from String(s) using Suffix Tree

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