Suffix trees, suffix arrays, BWT

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1 ALGORITHMES POUR LA BIO-INFORMATIQUE ET LA VISUALISATION COURS 3 Rluc Uricru Suffix trees, suffix rrys, BWT Bsed on: Suffix trees nd suffix rrys presenttion y Him Kpln Suffix trees course y Pco Gomez Liner-Time Construction of Suffix Trees y Dn Gusfield Introduction to the Burrows-Wheeler Trnsform nd FM Index, Ben Lngmed

2 Trie A tree representing set of strings. { eef d fe fg c } f e e d f e c g

3 Trie Assume no string is prefix of nother Ech edge is leled y letter, no two edges outgoing from the sme node re leled the sme. Ech string corresponds to lef. e e d f c f e g

4 Compressed Trie Compress unry nodes, lel edges y strings c è c e e d f eef d f f e g e g

5 Suffix tree Given string s suffix tree of s is compressed trie of ll suffixes of s. Oservtion: To mke suffixes prefix-free we dd specil chrcter, sy, t the end of s

6 Suffix tree (Exmple) Let s=. A suffix tree of s is compressed trie of ll suffixes of s= { }

7 Trivil lgorithm to uild Suffix tree Put the lrgest suffix in Put the suffix in

8 Put the suffix in

9 Put the suffix in

10 Put the suffix in

11 We will lso lel ech lef with the strting point of the corresponding suffix

12

13 Anlysis Tkes O(n 2 ) time to uild. But we cn do it in O(n) time with Ukkonen lgorithm It needs : implicit suffix trees, suffix links

14 Implicit suffix trees 1. Remove ll the terminl symols 2. From the resulting tree, remove edges without lel 3. Finlly, from the resulting tree, remove nodes tht do not hve t lest two children T={c} I(T) implicit suffix tree

15 Suffix links Let A e n ritrry sustring of T, including the possiility of eing the empty string. Let z e chrcter of T. Suppose there re two nodes v, w, the former with pth-lel za nd the ltter with A. A pointer from v to w is suffix link. Oservtion Every internl node hs one suffix link.

16 Suffix links Suffix links in the implicit suffix tree of string T =

17 Wht cn we do with it? Exct string mtching Given text T ( T = n), preprocess it such tht when pttern P ( P =m) rrives, you cn quickly decide whether it occurs in T. We my lso wnt to find ll occurrences of P in T.

18 Exct string mtching In preprocessing we just uild suffix tree in O(n) time Given pttern P = we trverse the tree ccording to the pttern. 2 5

19 If we did not get stuck trversing the tree then the pttern occurs in the text. Ech lef in the sutree elow the node we rech corresponds to n occurrence. By trversing this sutree we get ll k occurrences in O(n+k) time

20 Generlized suffix tree Given set of strings S, generlized suffix tree of S is compressed trie of ll suffixes of s S To mke these suffixes prefix-free we dd specil chr, sy, t the end of s To ssocite ech suffix with unique string in S dd different specil chr to ech s

21 Generlized suffix tree (Exmple) Let s 1 = nd s 2 =, nd generlized suffix tree for s 1 nd s 2 { } # # # # 3 # 2 # 1 2 # 4 #

22 So wht cn we do with it? Mtch pttern ginst dtse of strings

23 Longest common sustring (of two strings) E v e r y n o d e w i t h l e f descendnt from string s 1 nd lef descendnt from string s 2 represents mximl common sustring nd vice vers. 3 # 2 # 1 2 # 4 #

24 Lowest common ncestors A lot more cn e gined from the suffix tree if we preprocess it so tht we cn nswer LCA queries on it

25 Why? The LCA of two leves represents the longest common prefix (LCP) of these 2 suffixes # # 2 # 1 2 # 4 3 1

26 Finding mximl plindromes The mximl plindrome with center etween i-1 nd i in string s, is the LCP of the suffix t position i of s nd the suffix t position m-i of s r. Let s = c. Prepre generlized suffix tree for s = c nd s r = c# For every i find the LCA of suffix i of s nd suffix m-i+1 of s r.

27 Let s = c then s r = c# 6 c # c #

28 Drwcks Suffix trees consume lot of spce It is O(n) ut the constnt is quite ig

29 Suffix rry (SA) We loose some of the functionlity ut we sve spce. Let s = Sort the suffixes of s lexicogrphiclly:,,,. The suffix rry gives the indices of the suffixes in sorted order

30 How do we uild it? Build suffix tree Trverse the tree in DFS, lexicogrphiclly picking edges outgoing from ech node nd fill the suffix rry. O(n) time

31 How do we serch for pttern? If P occurs in T then ll its occurrences re consecutive in the suffix rry. Do inry serch on the suffix rry Tkes O(m logn) time

32 Exmple Let S = mississippi Let P = iss L M R i ippi issippi ississippi mississippi pi ppi sippi sisippi ssippi ssissippi

33 Burrows-Wheeler Trnsform (BWT) A wy of permuting the chrcters of string T into nother string BWT (T) Reversile permuttion 2 min pplictions: compression nd indexing

34 BWT vi BWM T = => 6 x 6 mtrix (BWM(T)) contining the rottions of T sort the rows lexicogrphiclly BWT(T)

35 BWT vi suffix rrys (SA) For BWM we sort T s rottions nd for SA we sort T s suffixes. So, for i from 0 to T -1 SA[i]>0? BWT[i]=T[SA[i]-1] : BWM(T) SA Suffixes for SA

36 LF Mpping BWM(T) with rnks on T F L LF Mpping the i th occurrence of chrcter c in the lst column hs the sme rnk s the i th occurrence of c in the first column. Exmple in the lst column hve rnks 3, 1, 2, 0 in the first column hve the sme rnks

37 Reversing the BWT with LF Mpping on the BWT(T) F L rnk L[1] = 0 is to the left of F[1]= in T 2. Find the chr to the left of 0 ó find the row strting with 0 Bsed on the LF Mpping, 0 hs rnk 0 thus it corresponds to the first in F L[2] = 0 is to the left of F[2]= 0 in T T = nd so on for rows 6, 4, 3, 7, 5

38 Applictions of the BWT Finding ll occurrences of P in T By pplying LF Mpping repetedly we find the rnge of rows prefixed y successively longer proper suffixes of P The size of the finl rnge gives the numer of times P occurs in T (if empty, P does not occur in T) Compression BWT( tomorrow nd tomorrow nd tomorrow ) = wwwdd nnoootttmmmrrrrrrooo ooo

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