Introduction to Computational Molecular Biology. Suffix Trees

Transcription

1 Itroductio to Computtiol Moleculr Biology Lecture 11: October 14, 004 Scribe: Athich Muthitchroe Lecturer: Ross Lippert Editor: Toy Scelfo Suffix Trees This is oe of the most complicted dt structures we will cover i clss. We will strt with the history of suffix trees. I 1970, Kuth proposed tht the problem Give strigs S1 d S, fid the lrgest commo substrig is hrder th just redig i the two strigs (> O( S1 + S )). Bck the it ws hrd to defie lower boud. I 1973, Peter Wieer estblished tht the lower boud ws O( S1 + S ). I 1976, McCreight gve prcticl dt structure to solve the problem. Review: Keyword tree Give the ptters P 1...P k. C perform lookup P s i text T i O( A T ) time. To represet ll substrigs of S, we c build keyword tree o ll suffixes of S, sice every suffix is some prefix of suffix. Exmple of pplictio i biology: Lookig for ucleotide sequece i the hum geome is logous to fidig how my times substrig ppers i chrcter sequece. Wht is suffix tree? Figure 11.1 illustrtes exmple of suffix tree with suffix liks. If we build keyword tree of ll suffixes, it will tke O( A S ) time d O( S ) spce. Sice filure liks re prllel, We c exploit the fct tht ech strig is some other strig s suffix d compress the tree. 11-1

2 11- Lecture 11: October 14, b Figure 11.1: A suffix tree for the strig b Redig the tree: pth from the root to lef is lwys suffix. We crete ew brch oly if it is topologiclly iterestig. This results i reductio of the odes, d we ed up with S leves d less th S o-leves, ccordig to some theorem. Ech ode c be substrig, ot just chrcter. To coserve spce, we c represet ech ode by pir (idex, le) ito the strig, isted of the etire substrig. O( S ) spce. O(1) per elemet. The oly filure liks retied is betwee iterl odes d ot to the leves. I keyword trees, filure lik F () = m, where m is the ode correspodig to the lrgest prefix i the FSM, which is suffix of strig. I suffix trees, m is the ode correspodig to the first proper suffix of. Why do filure liks poit to brchig iterl odes? Becuse w hs to be brchig suffix. Filures i suffix trees re bit more complicted. 1. follow pret s lik d remtch to positio i the middle of the Filures = substrig or t ode.. follow your ode s lik. STFSM hs sc time O( A T ), oce the tree is built. T is the whole text.

3 Lecture 11: October 14, x w w y z b Figure 11.: Digrm of brchig odes d filure lik from ode xw to w Ptter P lookup tkes O( A P ) time. A is the whole lphbet. Applictios of Suffix trees Ex. Lookig for ptter. ptter Oce foud ptter, the umber of leves re the umber of times the ptter occurs. Ex. Fid ptter A[10-0 chrs]b[10-00 chrs]. Ex. MUM (Mximl Uique Mtch) G1 % G Figure 11.3: Coctetio of two geomes to i the MUM exmple

4 11-4 Lecture 11: October 14, 004 Coctete G 1 d G, with specil symbols t the ed of ech geome, d the build tree of the cocteted strig (which is esy), see Figure MUM is the mtch tht the rightmost chrcter does ot mtch the other strig s rightmost chrcter AND tht the leftmost chrcter does ot mtch the other strig s leftmost chrcter. Ex. Logest Commo Prefix x y i j depth their pret ode, if i d j re sibligs. lcp(i, j) = depth of the lest commo cestor (LCA), otherwise. It tkes O(1) to fid the LCA. Other hcks: order childre lexicogrphiclly. Mkes it esier to compre to other trees. C use more complicted suffix tree to expli MUMs. Buildig Suffix Trees Tkes O(). Actully O( A T ), but the size of the lphbet is costt. There re severl proposls, such s Uckoe d McCreight, but Uckoe is just reorderig of McCreight s loops. McCreight s method 1. Add S[0...] to tree, resultig i big lef.. The, dd S[1...], S[...],... Follow some existig brchig util mismtch. If so, the brch, s i Figure 11.4 At every step dd up to oe itertl ode dd exctly oe lef, S[i...]

5 Lecture 11: October 14, S[0..] S[..] S[1..] Figure 11.4: Spshot of suffix tree durig the McCreight s method. Code expltio (refer to lecture slide Buildig suffix tree ) Procedures fork: dds ew iterl ode. get-child: give ode N, d chrcter ch, returs the ode tht correspods to ch. get-brch: give suffix s, fid s. If cot fid, crete ew ode d brch. How to speed up get-brch to get O(1) time? Every time lookup up S[i+1,...], c do fst-mtch up to d 1 while ot mkig mistke. d is the mximum depth tht the previous substrig mtched for which we creted ew ode. If we did ot crete ew ode lst time, we c move up to the pret ode d fst-mtch dow. The crete ew ode if eeded.