Suffix Trees. Philip Bille

Transcription

1 Suffix Trees Philip Bille

2 Outline String ictinr prlem Tries Suffix trees Applictins f suffix trees Cnstructing suffix trees Suffix srting

3 String Dictinries

4 String Dictinr Prlem The string ictinr prlem: Let S e string f chrcters frm lphet Σ. Preprcess S int t structure t supprt serch(p): Return the strting psitins f ll ccurrences f P in S. Exmple: S = serch() = {,6}

5 Tries n Suffix Trees

6 Tries [Frekin 96] Retrievl Stre set f strings in rte tree such tht Ech ege is lele chrcter. Eges t chilren f ne re srte their chrcters. Ech rt-t-lef pth represents string in the set. (tine cnctenting the lels f eges n the pth). Cmmn prefixes shre sme pth mximll. T mke ech string crrespn t unique lef we mke the strings prefixfree ppening specil chrcter Σ t ech string.

7 Trie f ll suffixes fr

8 Spce n Preprcessing Time Hw much spce? Spce: O(n ) Preprcessing: O(n )

9 Serching Prcess P frm left-t-right while ing tp-wn serch f trie: At ech ne ientif utging ege crrespning t next chrcter in P. If there is n such ege, P is nt sustring f S. Reprt lels f ll leves elw finl ne.

10 S= Serch fr P =

11 Serching Hw much time? Time: O(m) fr tp-wn serch + time fr reprting lels f leves. Time: O(m + cc) with extr infrmtin t reprt leves.

12 Suffix Tries Therem: We cn slve the string ictinr prlem in O(n ) spce n preprcessing time. O(m + cc) time fr queries.

13 Suffix Trees

14 Suffix Trees Suffix tree is the cmpct trie f ll suffixes f S. Chins f nes with single chil re cmpcte int single ege. => Ege lels ecme strings. Stre S n stre ege lels reference t S.

15 Suffix tree fr

16 Spce n Preprcessing Time Hw much spce? Spce: O(n) Preprcessing: O(srt(n, Σ )) srt(n, Σ ) = time t srt n chrcters frm n lphet Σ.

17 Serching As in tries.

18 S= Serch fr P =

19 Serching Hw much time? Time: O(m + cc).

20 Suffix Trees Therem: We cn slve the string ictinr prlem in O(n) spce n srt(n, Σ ) preprcessing time. O(m + cc) time fr queries.

21 Applictins f Suffix Trees

22 Applictins Apprximte string mtching prlems Cmpressin schemes (Lempel-Ziv fmil,...) Repetitive string prlems (plinrmes, tnem repets,...) Infrmtin retrievl prlems (cument retrievl, tp-k retrievl,...)...

23 Lngest Cmmn Extensins The lngest cmmn extensin prlem: Let S e string f chrcters frm lphet Σ. Preprcess S int t structure t supprt LCP(i,j): Return the length f the lngest cmmn prefix f S[i,n] n S[j,n]. Exmple: S = LCP(,6) =

24 S= LCP(,6) =

25 Lngest Cmmn Extensins Slutin: Suffix tree + string epth f ech ne + nerest cmmn ncestr t structure. Spce: O(n) Time: O()

26 Cnstructing Suffix Trees

27 Cnstructing Suffix Trees Therem [Ksi et l. ]: Given the srte rer f suffixes f S, we cn cnstruct the suffix tree fr S in O(n) time. Therem [Frch-Cltn et l. ]: We cn srt the suffixes f S in O(srt(n, Σ )) time. => Therem: We cn cnstruct suffix tree fr S in O(srt(n, Σ )) time.

28 Suffix Srting

29 Overview Srting smll universes Srting its Srting smll integers Srting plnmil universes Suffix srting Rix srting Prefix uling Difference cver smpling

30 Srting Smll Universes

31 Srting Smll Universes Let X e sequence f n integers frm universe U = {,,..., u-}. Hw fst cn we srt if the size f the universe is nt t ig? U = {,}? U = {,..., n-}? U = {,..., n - }?

32 Rix Srt [Hllerith 887] X is sequence f n integers frm U = {,..., n -}. Write ech x X s se n integer (x, x, x): x = x n + x n + x Exmple: in se 7 = (,, ) Srt X ccring t rightmst (lest significnt) igit Srt X ccring t mile igit Srt X ccring t leftmst (mst significnt) igit Ech srt shul e stle. Finl result is the srte sequence f X.

33 n =, U = {,..., n - = 999}

34 Rix Srt Therem: We cn srt n integers frm universe U = {,..., n - } in O(n) time. Therem: We cn srt n integers frm universe U = {,..., n k - } in O(kn) time. Lrger universes? Therem [Hn n Thrup ]: We cn srt n integers in O(n lg lg n) time r O(n (lg lg n) / ) expecte time.

35 Suffix Srting

36 Suffix Srting Given string S f length n ver lphet Σ, the suffix srting prlem is t cmpute the lexicgrphic rer f ll suffixes f S. Exmple: S =

37 Overview Alphet reuctin slutins: Rix srting Prefix uling Difference cver smpling

38 Alphet Reuctin Initil step in ll lgrithms. If Σ > n: Srt the chrcters in S. Replce them their rnk in the srte rer. => new lphet is {,..., n-} => Lemm: If we cn suffix srt string f length n ver lphet f size n in time t(n), we cn suffix srt string f length n ver lphet Σ in time O(t(n) + srt(n, Σ )).

39 Rix Srt Generte ll suffixes n rix srt them. Hw fst is this? Therem: We cn suffix srt string f length n ver lphet f size n in time O(n ).

40 Prefix Duling [Mner n Mers 99] Srt sustrings f lengths,,, 8,..., n. Ech step uses rix srt n pir frm previus step. Hw fst is this? Therem: We cn suffix srt string f length n ver lphet f size n in time O(n lg n).

41 Difference Cver Smpling

42 DC [Krkkinen et l. ] Srt suffixes in three steps: Step : Srt smple suffixes Smple ll suffixes strting t psitins i = m n i = m. Recursivel srt smple suffixes. Step : Srt nn-smple suffixes Srt the remining suffixes (strting t psitins i = m ). Step : Merge Merge smple n nn-smple suffixes.

43 Step : Srt Smple Suffixes

44 Step : Srt Smple Suffixes 7

45 Step : Srt Smple Suffixes 8

46 Step : Srt Smple Suffixes 7 6 8

47 Step : Srt Smple Suffixes

48 Step : Srt Nn-Smple Suffixes

49 Step : Merge 6 7 8

50 Step : Merge

51 Step : Merge

52 Step : Merge

53 Step : Merge

54 Step : Merge

55 Step : Merge

56 Step : Merge

57 Step : Merge

58 Step : Merge

59 Step : Merge

60 Step : Merge

61 Cmplexit T(n) = time t suffix srt string f length n ver lphet f size n Step : Srt smple suffixes Smple ll suffixes strting t psitins i = m n i = m. Recursivel srt smple suffixes. O(n) T(n/) Step : Srt nn-smple suffixes Srt the remining suffixes (strting t psitins i = m ). O(n) Step : Merge Merge smple n nn-smple suffixes. O(n) T(n) = T(n/) + O(n) = O(n)

62 Suffix Srting Therem: We cn suffix srt string f length n ver lphet f size n in time O(n). => Therem: We cn suffix srt string f length n ver lphet Σ in O(n + srt(n, Σ )) = O(srt(n, Σ )) time. => Therem: We cn cnstruct suffix tree fr string f length n ver lphet Σ in O(srt(n, Σ )) time. Bun is ptiml.

63 Summr String ictinr prlem Tries Suffix trees Applictins f suffix trees Cnstructing suffix trees Suffix srting

64 References Mrtin Frch-Cltn, Pl Ferrgin, S. Muthukrishnn: On the srtingcmplexit f suffix tree cnstructin, J. ACM,. Juh Kärkkäinen, Peter Sners, Stefn Burkhrt: Liner wrk suffix rr cnstructin. J. ACM, 6. Dn Gusfiel. Algrithms n Strings, Trees, n Sequences, Chp. -9. Scrie ntes frm MIT.