Suffix Tries. Slides adapted from the course by Ben Langmead

Transcription

1 Suffix Tries Slides dpted from the course y Ben Lngmed en.lngmed@gmil.com

2 Indexing with suffixes Until now, our indexes hve een sed on extrcting sustrings from T A very different pproch is to extrct suffixes from T. This will led us to some interesting nd prcticl index dt structures: A ANA ANANA BANANA NA NANA B A N A N A A B A N A N A N A B A N A N A N A B B A N A N A N A B A N A N A N A B A Suffix Trie Suffix Tree Suffix Arry FM Index

3 Tries A trie (pronounced try ) is tree representing collection of strings with one node per common prefix Smllest tree such tht: Ech edge is leled with chrcter c Σ A node hs t most one outgoing edge leled c, for c Σ Ech key is spelled out long some pth strting t the root Nturl wy to represent either set or mp where keys re strings

4 Build trie contining ll suffixes of text T T: GTTATAGCTGATCGCGGCGTAGCGG GTTATAGCTGATCGCGGCGTAGCGG TTATAGCTGATCGCGGCGTAGCGG TATAGCTGATCGCGGCGTAGCGG ATAGCTGATCGCGGCGTAGCGG TAGCTGATCGCGGCGTAGCGG AGCTGATCGCGGCGTAGCGG GCTGATCGCGGCGTAGCGG CTGATCGCGGCGTAGCGG TGATCGCGGCGTAGCGG GATCGCGGCGTAGCGG ATCGCGGCGTAGCGG TCGCGGCGTAGCGG CGCGGCGTAGCGG GCGGCGTAGCGG CGGCGTAGCGG GGCGTAGCGG GCGTAGCGG CGTAGCGG GTAGCGG TAGCGG AGCGG GCGG CGG GG G m(m+1)/2 chrs

5 First dd specil terminl chrcter to the end of T is chrcter tht does not pper elsewhere in T, nd we define it to e less thn other chrcters (for DNA: < A < C < G < T) enforces rule we re ll used to using: e.g. s comes efore sh in the dictionry. lso gurntees no suffix is prefix of ny other suffix. T: GTTATAGCTGATCGCGGCGTAGCGG GTTATAGCTGATCGCGGCGTAGCGG TTATAGCTGATCGCGGCGTAGCGG TATAGCTGATCGCGGCGTAGCGG ATAGCTGATCGCGGCGTAGCGG TAGCTGATCGCGGCGTAGCGG AGCTGATCGCGGCGTAGCGG GCTGATCGCGGCGTAGCGG CTGATCGCGGCGTAGCGG TGATCGCGGCGTAGCGG GATCGCGGCGTAGCGG ATCGCGGCGTAGCGG TCGCGGCGTAGCGG CGCGGCGTAGCGG GCGGCGTAGCGG CGGCGTAGCGG GGCGTAGCGG

6 Tries Smllest tree such tht: Ech edge is leled with chrcter from Σ A node hs t most one outgoing edge leled with c, for ny c Σ Ech key is spelled out long some pth strting t the root

7 T: T: Shortest (non-empty) suffix Ech pth from root to lef represents suffix; ech suffix is represented y some pth from root to lef Would this still e the cse if we hdn t dded? Longest suffix

8 T: Ech pth from root to lef represents suffix; ech suffix is represented y some pth from root to lef Would this still e the cse if we hdn t dded? No

9 We cn think of nodes s hving lels, where the lel spells out chrcters on the pth from the root to the node

10 How do we check whether string S is sustring of T? Note: Ech of T s sustrings is spelled out long pth from the root. I.e., every sustring is prefix of some suffix of T. Strt t the root nd follow the edges leled with the chrcters of S S = Yes, it s sustring If we fll off the trie -- i.e. there is no outgoing edge for next chrcter of S, then S is not sustring of T If we exhust S without flling off, S is sustring of T

11 How do we check whether string S is sustring of T? Note: Ech of T s sustrings is spelled out long pth from the root. I.e., every sustring is prefix of some suffix of T. Strt t the root nd follow the edges leled with the chrcters of S If we fll off the trie -- i.e. there is no outgoing edge for next chrcter of S, then S is not sustring of T If we exhust S without flling off, S is sustring of T S = Yes, it s sustring

12 How do we check whether string S is sustring of T? Note: Ech of T s sustrings is spelled out long pth from the root. I.e., every sustring is prefix of some suffix of T. Strt t the root nd follow the edges leled with the chrcters of S If we fll off the trie -- i.e. there is no outgoing edge for next chrcter of S, then S is not sustring of T x S = No, not sustring If we exhust S without flling off, S is sustring of T

13 How do we check whether string S is suffix of T? Sme procedure s for sustring, ut dditionlly check whether the finl node in the wlk hs n outgoing edge leled S = Not suffix

14 How do we check whether string S is suffix of T? Sme procedure s for sustring, ut dditionlly check whether the finl node in the wlk hs n outgoing edge leled S = Is suffix

15 How do we count the numer of times string S occurs s sustring of T? Follow pth corresponding to S. Either we fll off, in which cse nswer is 0, or we end up t node n nd the nswer = # of lef nodes in the sutree rooted t n. n S = 2 occurrences Leves cn e counted with depth-first trversl.

16 How do we find the longest repeted sustring of T? Find the deepest node with more thn one child

17 Suffix Trie implementtion (derived from Ben Lngmed) clss SuffixTrie(oject): ''' uilding suffix Trie ''' def init (self, t): """ Mke suffix trie from t """ if t[-1]!='': t += '' # specil termintor symol self.root = {} for i in rnge(len(t)): # for ech suffix cur = self.root for c in t[i:]: # for ech chrcter in i'th suffix if c == '': cur[c] = i # dd outgoing edge nd suffix position elif c not in cur: cur[c] = {} # dd outgoing edge if necessry cur = cur[c]

18 Suffix Trie implementtion: followpth clss SuffixTrie(oject):. def followpth(self, s): """ Follow pth given y chrcters of s. Return node t end of pth, or None if we fll off. """ cur = self.root for c in s: if c not in cur: return None cur = cur[c] return cur

19 Suffix Trie implementtion: find ll positons clss SuffixTrie(oject):. def findleves(self,v): """ Return the leves from given vertex v""" leves=[] if v == None: return leves for c in v: if c == '': leves+=[v[c]] else: leves+=self.findleves(v[c]) return leves def findpositions(self,s): """ Return list of mtching positions of s """ return self.findleves(self.followpth(s))

20 Exmples if nme == ' min ': seq='' print "seq=",seq strie=suffixtrie(seq) for p in ['','','','']: print "find postion of ",p,"in seq",strie.findpositions(p) print "find the leves=",strie.findleves(strie.root) python../codes/st/strie.py seq= find postion of in seq [2, 0, 3, 5] find postion of in seq [1, 4] find postion of in seq [2] find postion of in seq [] find the leves= [2, 0, 3, 5, 1, 4, 6]

21 How mny nodes does the suffix trie hve? Is there clss of string where the numer of suffix trie nodes grows linerly with m? T = Yes: e.g. string of m s in row ( m ) 1 Root m nodes with incoming edge m + 1 nodes with incoming edge 2m + 2 nodes

22 Is there clss of string where the numer of suffix trie nodes grows with m 2? Yes: n n 1 root n nodes long chin, right n nodes long chin, middle n chins of n nodes hnging off ech chin node 2n + 1 leves (not shown) Figure & exmple y Crl Kingsford n 2 + 4n + 2 nodes, where m = 2n

23 : upper ound on size Could worst-cse # nodes e worse thn O(m 2 )? Root Suffix trie Mx # nodes from top to ottom = length of longest suffix + 1 = m + 1 Deepest lef Mx # nodes from left to right = mx # distinct sustrings of ny length m O(m 2 ) is worst cse

24 : ctul growth Built suffix tries for the first 500 prefixes of the lmd phge virus genome Blck curve shows how # nodes increses with prefix length # suffix trie nodes m^2 ctul m Length prefix over which suffix trie ws uilt