String comparison by transposition networks

Transcription

1 String omprison y trnsposition networks Alexnder Tiskin (Joint work with Peter Krushe) Deprtment of Computer Siene University of Wrwik (inludes n extended version of this presenttion) Alexnder Tiskin (Wrwik) String omprison y networks / 36

2 Semi-lol string omprison 2 Algorithmi tehniques 3 Algorithmi pplitions: the seweed lgorithm 4 Algorithmi pplitions: trnsposition networks 5 Algorithmi pplitions: sprse omprison 6 Conlusions nd future work Alexnder Tiskin (Wrwik) String omprison y networks 2 / 36

3 Semi-lol string omprison 2 Algorithmi tehniques 3 Algorithmi pplitions: the seweed lgorithm 4 Algorithmi pplitions: trnsposition networks 5 Algorithmi pplitions: sprse omprison 6 Conlusions nd future work Alexnder Tiskin (Wrwik) String omprison y networks 3 / 36

4 Semi-lol string omprison String mthing: finding n ext pttern in string String omprison: finding similr ptterns in two strings (Also known s pproximte string mthing, no reltion to pproximtion lgorithms!) Applitions: omputtionl iology, imge reognition,... Alexnder Tiskin (Wrwik) String omprison y networks 4 / 36

5 Semi-lol string omprison String mthing: finding n ext pttern in string String omprison: finding similr ptterns in two strings (Also known s pproximte string mthing, no reltion to pproximtion lgorithms!) Applitions: omputtionl iology, imge reognition,... Stndrd types of string omprison: glol: whole string ginst whole string lol: sustrings ginst sustrings Min interest of this work: semi-lol: whole string ginst sustrings; prefixes ginst suffixes Alexnder Tiskin (Wrwik) String omprison y networks 4 / 36

6 Semi-lol string omprison Consider strings (= sequenes) over n lphet of size σ Distinguish ontiguous sustrings nd not neessrily ontiguous susequenes Speil ses of sustring: prefix, suffix Stndrd nottion: strings, of length m, n respetively Assume when neessry: m n; m, n resonly lose Alexnder Tiskin (Wrwik) String omprison y networks 5 / 36

7 Semi-lol string omprison Consider strings (= sequenes) over n lphet of size σ Distinguish ontiguous sustrings nd not neessrily ontiguous susequenes Speil ses of sustring: prefix, suffix Stndrd nottion: strings, of length m, n respetively Assume when neessry: m n; m, n resonly lose The longest ommon susequene (LCS) sore: length of the longest string tht is susequene of oth nd The LCS prolem: determine the LCS sore for ginst Alexnder Tiskin (Wrwik) String omprison y networks 5 / 36

8 Semi-lol string omprison The LCS prolem is speil se of the edit distne prolem: minimum ost to trnsform into y hrter edits Chrter edits: insertions, deletions, sustitutions Levenshtein distne ED Lev (, ): ost in = ost del = ost su = LCS distne ED LCS (, ): ost in = ost del =, ost su 2 Equivlently, sustitutions not llowed ED LCS (, ) = m + n 2 LCS(, ) An edit distne is rtionl, if ost in+ost del ost su is rtionl numer All tehniques in this work extend to rtionl edit distnes Alexnder Tiskin (Wrwik) String omprison y networks 6 / 36

9 Semi-lol string omprison The lignment grph (direted, yli) lue = 0 red = LCS("", "") = "" LCS = highest-sore orner-to-orner pth Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

10 Semi-lol string omprison LCS prolem: omputtion time, ssuming σ = O() O(mn) [Wgner, Fisher: 974] O ( ) mn log n [Msek, Pterson: 980], [Crohemore+: 2003] Alexnder Tiskin (Wrwik) String omprison y networks 8 / 36

11 Semi-lol string omprison LCS prolem: omputtion time, ssuming σ = O() O(mn) [Wgner, Fisher: 974] O ( ) mn log n [Msek, Pterson: 980], [Crohemore+: 2003] Stndrd pproh: dynmi progrmming Speedup y exhustive preomputtion of smll loks Blok size: t = O(log n) Blok interfe: O(t) vlues, eh of size O() Totl work O ( mn log n), log-ost RAM Alexnder Tiskin (Wrwik) String omprison y networks 8 / 36

12 Semi-lol string omprison The semi-lol LCS prolem: determine the length of LCS string-sustring LCS: string ginst every sustring of prefix-suffix LCS: every prefix of ginst every suffix of symmetrilly, sustring-string nd suffix-prefix LCS Alexnder Tiskin (Wrwik) String omprison y networks 9 / 36

13 Semi-lol string omprison The semi-lol LCS prolem: determine the length of LCS string-sustring LCS: string ginst every sustring of prefix-suffix LCS: every prefix of ginst every suffix of symmetrilly, sustring-string nd suffix-prefix LCS Output: the highest-sore mtrix of O(n 2 ) LCS sores, llowed to e represented impliitly Cf.: stndrd dynmi progrmming gives prefix-prefix LCS Alexnder Tiskin (Wrwik) String omprison y networks 9 / 36

14 Semi-lol string omprison The lignment grph lue = 0 red = LCS("", "...") = "" Semi-lol LCS = ll highest-sore order-to-order pths (string-sustring = top-to-ottom, et.) Alexnder Tiskin (Wrwik) String omprison y networks 0 / 36

15 Semi-lol string omprison A: the highest-sore mtrix for semi-lol LCS of nd = "" = "" A(0, 3) = LCS(, ) = 8 = "..." A(4, ) = LCS(, ) = 5 A(i, j) = j i if i > j Alexnder Tiskin (Wrwik) String omprison y networks / 36

16 Semi-lol string omprison Semi-lol LCS prolem: output representtion size query time O(n 2 ) O() trivil O(m /2 n) O(log n) string-sustring, [Alves+: 2003] O(n) O(n) string-sustring, [Alves+: 2005] O(n log n) O(log 2 n) [T: 2006], using [Bentley: 980] Alexnder Tiskin (Wrwik) String omprison y networks 2 / 36

17 Semi-lol string omprison Semi-lol LCS prolem: output representtion size query time O(n 2 ) O() trivil O(m /2 n) O(log n) string-sustring, [Alves+: 2003] O(n) O(n) string-sustring, [Alves+: 2005] O(n log n) O(log 2 n) [T: 2006], using [Bentley: 980] Semi-lol LCS prolem: omputtion time, ny σ O(mn 2 ) nive O(mn) string-sustring, [Shmidt: 998], [Alves+: 2005] O ( ) mn [T: 2006] log 0.5 n O ( mn(log log n) 2 ) log n [T: 2007] Bsed on the seweed lgorithm, log-ost RAM Alexnder Tiskin (Wrwik) String omprison y networks 2 / 36

18 Semi-lol string omprison 2 Algorithmi tehniques 3 Algorithmi pplitions: the seweed lgorithm 4 Algorithmi pplitions: trnsposition networks 5 Algorithmi pplitions: sprse omprison 6 Conlusions nd future work Alexnder Tiskin (Wrwik) String omprison y networks 3 / 36

19 Algorithmi tehniques The lignment grph nd the seweeds = "" = "" = "..." A(4, ) = LCS(, ) = 4 P Σ (i, j) = 4 2 = 5 P gives n impliit representtion of A P(i, j) = orresponds to seweed (top, i) (ottom, j) Alexnder Tiskin (Wrwik) String omprison y networks 4 / 36

20 Algorithmi tehniques The lignment grph nd the seweeds = "" = "" = "..." A(4, ) = LCS(, ) = 4 P Σ (i, j) = 4 2 = 5 P gives n impliit representtion of A P(i, j) = orresponds to seweed (top, i) (ottom, j) Also define top right, left right, left ottom seweeds Gives omplete order-to-order grph-theoreti mthing Alexnder Tiskin (Wrwik) String omprison y networks 4 / 36

21 Algorithmi tehniques Gudí s seweeds (Cs Milà, Brelon) Alexnder Tiskin (Wrwik) String omprison y networks 5 / 36

22 Semi-lol string omprison 2 Algorithmi tehniques 3 Algorithmi pplitions: the seweed lgorithm 4 Algorithmi pplitions: trnsposition networks 5 Algorithmi pplitions: sprse omprison 6 Conlusions nd future work Alexnder Tiskin (Wrwik) String omprison y networks 6 / 36

23 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

24 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

25 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

26 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

27 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

28 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

29 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

30 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

31 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

32 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

33 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

34 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

35 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

36 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

37 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

38 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

39 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

40 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

41 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

42 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

43 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

44 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

45 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

46 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

47 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

48 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

49 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

50 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

51 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

52 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

53 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

54 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

55 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

56 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

57 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

58 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

59 Algorithmi pplitions: the seweed lgorithm The seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Eh pir of seweeds is llowed to ross t most one Time O(mn) Alexnder Tiskin (Wrwik) String omprison y networks 7 / 36

60 Algorithmi pplitions: the seweed lgorithm Alexnder Tiskin (Wrwik) String omprison y networks 8 / 36

61 Semi-lol string omprison 2 Algorithmi tehniques 3 Algorithmi pplitions: the seweed lgorithm 4 Algorithmi pplitions: trnsposition networks 5 Algorithmi pplitions: sprse omprison 6 Conlusions nd future work Alexnder Tiskin (Wrwik) String omprison y networks 9 / 36

62 Algorithmi pplitions: trnsposition networks Comprison network: iruit of omprtors A omprtor ompres two inputs, then outputs them in presried order Comprison networks trditionlly used to otin non-rnhing merging nd sorting lgorithms Clssil omprison networks: # omprtors merging O(n log n) [Bther: 968] sorting O(n log 2 n) [Bther: 968] O(n log n) [Ajti+: 983] Alexnder Tiskin (Wrwik) String omprison y networks 20 / 36

63 Algorithmi pplitions: trnsposition networks Comprison network: iruit of omprtors A omprtor ompres two inputs, then outputs them in presried order Comprison networks trditionlly used to otin non-rnhing merging nd sorting lgorithms Clssil omprison networks: # omprtors merging O(n log n) [Bther: 968] sorting O(n log 2 n) [Bther: 968] O(n log n) [Ajti+: 983] Comprison networks re desried y wire digrms Alexnder Tiskin (Wrwik) String omprison y networks 20 / 36

64 Algorithmi pplitions: trnsposition networks The dimond network [Munter: 993] k inputs rrive t top-left k outputs otined t ottom-right Θ(k 2 ) omprtors Eh omprtor swps min to ottom-left mx to top-right A trnsposition network: ll omprisons re etween djent wires A merging network: if oth hlves of input pre-sorted, then output sorted Alexnder Tiskin (Wrwik) String omprison y networks 2 / 36

65 Algorithmi pplitions: trnsposition networks The dimond network If oth hlves of input re sorted, the output is sorted nd n e omputed in time O(n) y merging We will onsider the dimond network s output for n unsorted input, for whih the sorting order is known (s seprte permuttion) In this se, the (unsorted) output n still e omputed in time O(n) Therefore, trnsposition network n e omputed effiiently, if it n e deomposed into smll numer of dimond sunetworks Alexnder Tiskin (Wrwik) String omprison y networks 22 / 36

66 Algorithmi pplitions: trnsposition networks The seweed lgorithm s trnsposition network = "" = "" Comprtors orrespond to hrter mismthes Inputs distint nd pre-sorted in reverse; outputs normlly unsorted Eh input-to-output vlue pth tres seweed Alexnder Tiskin (Wrwik) String omprison y networks 23 / 36

67 Algorithmi pplitions: trnsposition networks Prmeterised omprison: ompring highly similr or dissimilr strings Assume m = n Let ls = LCS(, ), ed = ED LCS (, ) = 2(n ls) High-similrity omprison: (some kind of) ed smll only need to look t O(n ed) hrter mthes High-dissimilrity omprison: ls smll my hve to look t ll hrter mthes ssume the mthes re given s prt of the input; otherwise extr hrge e.g. O(n log n) or O(n σ) (depending on the model) Flexile omprison: sensitive to oth similrity nd dissimilrity Alexnder Tiskin (Wrwik) String omprison y networks 24 / 36

68 Algorithmi pplitions: trnsposition networks Prmeterised omprison (ontd.) O(n ls + n log n) [Hirsherg: 977], [Apostolio, Guerr: 985] O(n ed Lev ) [Ukkonen: 985] O(n ed) [Myers: 986] O(ls ed log n) [Myers: 986], [Wu+: 990] O(n ls) [Apostolio+: 992] O(ls ed) [Rik: 995] Alexnder Tiskin (Wrwik) String omprison y networks 25 / 36

69 Algorithmi pplitions: trnsposition networks Prmeterised omprison (ontd.) O(n ls + n log n) [Hirsherg: 977], [Apostolio, Guerr: 985] O(n ed Lev ) [Ukkonen: 985] O(n ed) [Myers: 986] O(ls ed log n) [Myers: 986], [Wu+: 990] O(n ls) [Apostolio+: 992] O(ls ed) [Rik: 995] O(ls ed) = O(n ls) nd O(n ed) running together Both re mzingly simple using trnsposition networks The network tkes inputs (no longer distint) Vlue pths now depend on tion of 0 vs 0 nd vs omprisons. Define suh tion to e lwys non-swp. Alexnder Tiskin (Wrwik) String omprison y networks 25 / 36

70 Algorithmi pplitions: trnsposition networks Prmeterised omprison (ontd.) High-similrity Alexnder Tiskin (Wrwik) String omprison y networks 26 / 36

71 Algorithmi pplitions: trnsposition networks Prmeterised omprison (ontd.) High-similrity Alexnder Tiskin (Wrwik) String omprison y networks 26 / 36

72 Algorithmi pplitions: trnsposition networks Prmeterised omprison (ontd.) High-similrity High-dissimilrity Alexnder Tiskin (Wrwik) String omprison y networks 26 / 36

73 Algorithmi pplitions: trnsposition networks Prmeterised omprison (ontd.) High-similrity High-dissimilrity Alexnder Tiskin (Wrwik) String omprison y networks 26 / 36

74 Algorithmi pplitions: trnsposition networks Prmeterised omprison (ontd.) High-similrity High-dissimilrity Impliit minstrem of 0s/s ross the network, roken y stry 0s/s There re t most ed or ls stry 0s/s, we tre them expliitly O(n ed) O(n ls) Alexnder Tiskin (Wrwik) String omprison y networks 26 / 36

75 Algorithmi pplitions: trnsposition networks Bit-prllel omprison: using stndrd instrutions on words of size w O(mn/w) [Allison, Dix: 986], [Myers: 999], [Crohemore+: 200] Alexnder Tiskin (Wrwik) String omprison y networks 27 / 36

76 Algorithmi pplitions: trnsposition networks Bit-prllel omprison: using stndrd instrutions on words of size w O(mn/w) [Allison, Dix: 986], [Myers: 999], [Crohemore+: 200] A simple interprettion using trnsposition networks The network tkes inputs ; let it M = mth M S C S M S C + S C S C M S C C S C M S C Alexnder Tiskin (Wrwik) String omprison y networks 27 / 36

77 Algorithmi pplitions: trnsposition networks Bit-prllel omprison: using stndrd instrutions on words of size w O(mn/w) [Allison, Dix: 986], [Myers: 999], [Crohemore+: 200] A simple interprettion using trnsposition networks The network tkes inputs ; let it M = mth M S C S M S C + S C S C M S C C S C M S C Therefore S ( S + ( S M)) ( S M), where S, M re words Alexnder Tiskin (Wrwik) String omprison y networks 27 / 36

78 Algorithmi pplitions: trnsposition networks Alexnder Tiskin (Wrwik) String omprison y networks 28 / 36

79 Semi-lol string omprison 2 Algorithmi tehniques 3 Algorithmi pplitions: the seweed lgorithm 4 Algorithmi pplitions: trnsposition networks 5 Algorithmi pplitions: sprse omprison 6 Conlusions nd future work Alexnder Tiskin (Wrwik) String omprison y networks 29 / 36

80 Algorithmi pplitions: sprse omprison The sprse LCS prolem: ompring strings with few hrter mthes Speil se: the LCS prolem on permuttions, m = n mthes Equivlent to the longest inresing susequene (LIS) prolem O(n 2 ) nive O(n log n) impliit in [Erdös, Szekeres: 935] [Roinson: 938], [Knuth: 970], [Dijkstr: 980] O(n log log n) unit-ost RAM [Chng, Wng: 992] unit-ost RAM [Bespmytnikh, Segl: 2000] Alexnder Tiskin (Wrwik) String omprison y networks 30 / 36

81 Algorithmi pplitions: sprse omprison Generl se: r mthes. Suppose m = n, n r n 2. O(n 2 ) nive O(r log n) [Hunt, Szymnski: 977] Alexnder Tiskin (Wrwik) String omprison y networks 3 / 36

82 Algorithmi pplitions: sprse omprison Generl se: r mthes. Suppose m = n, n r n 2. O(n 2 ) nive O(r log n) [Hunt, Szymnski: 977] A simple interprettion using trnsposition networks Impliit minstrem of 0s ross the network, split into sustrems y s moving digonlly t mthes For every mth, O(log n) inry serh to find its sustrem Totl work O(r log n) Alexnder Tiskin (Wrwik) String omprison y networks 3 / 36

83 Algorithmi pplitions: sprse omprison The sprse semi-lol LCS prolem Speil se: the semi-lol LCS prolem on permuttions, m = n mthes O(n 2 log n) nive O(n 2 ) restrited, [Alert+: 2003], [Chen+: 2005] O(n.5 log n) rndomised restrited, [Alert+: 2007] O(n.5 ) [T: 2006] Alexnder Tiskin (Wrwik) String omprison y networks 32 / 36

84 Algorithmi pplitions: sprse omprison The sprse semi-lol LCS prolem Speil se: the semi-lol LCS prolem on permuttions, m = n mthes O(n 2 log n) nive O(n 2 ) restrited, [Alert+: 2003], [Chen+: 2005] O(n.5 log n) rndomised restrited, [Alert+: 2007] O(n.5 ) [T: 2006] Algorithm: divide the lignment grph reursively into strips Hlf of seweeds in eh strip is trivil Otin the non-trivil seweeds in eh strip y reursion Conquer the resulting two seweed sets y (min, +)-multiplition Top-level onquer domintes, totl work O(n.5 ) Alexnder Tiskin (Wrwik) String omprison y networks 32 / 36

85 Algorithmi pplitions: sprse omprison The sprse semi-lol LCS prolem Generl se: r mthes. Suppose m = n, n r n 2. O(nr 0.5 ) [Krushe, T: 2008] Alexnder Tiskin (Wrwik) String omprison y networks 33 / 36

86 Algorithmi pplitions: sprse omprison The sprse semi-lol LCS prolem Generl se: r mthes. Suppose m = n, n r n 2. O(nr 0.5 ) [Krushe, T: 2008] Algorithm: divide the lignment grph reursively into squre loks Reursion se: lok with no mthes or lok of size with mth Compute eh lok of size h > s dimond trnsposition network. Cn e done in time O(h) insted of the nive O(h 2 ). In the divide-nd-onquer tree: root: one lok of size n, work O(n) level log 4 r: r loks of size n/r 0.5, work O(r n/r 0.5 ) = O(nr 0.5 ) leves: r loks of size, work O(n) Level log 4 r domintes, totl work O(nr 0.5 ) Alexnder Tiskin (Wrwik) String omprison y networks 33 / 36

87 Algorithmi pplitions: sprse omprison Alexnder Tiskin (Wrwik) String omprison y networks 34 / 36

88 Semi-lol string omprison 2 Algorithmi tehniques 3 Algorithmi pplitions: the seweed lgorithm 4 Algorithmi pplitions: trnsposition networks 5 Algorithmi pplitions: sprse omprison 6 Conlusions nd future work Alexnder Tiskin (Wrwik) String omprison y networks 35 / 36

89 Conlusions nd future work String omprison y trnsposition networks: the seweed lgorithm s network simple interprettion of high-similrity nd dissimilrity LCS simple interprettion of it-prllel LCS sprse semi-lol LCS vi trnsposition networks Further pplitions of the method: ompring run-length ompressed strings online pproximte mthing...? Alexnder Tiskin (Wrwik) String omprison y networks 36 / 36