Text Searching. Thierry Lecroq Laboratoire d Informatique, du Traitement de l Information et des

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1 Text Searching Thierry Lecroq Laboratoire d Informatique, du Traitement de l Information et des Systèmes. International PhD School in Formal Languages and Applications Tarragona, November 20th and 21st, 2006

2 Plan 1 Introduction 2 Left-to-right scan shift 3 Right-to-left scan shift Thierry Lecroq Text Searching 2/77

3 Plan 1 Introduction 2 Left-to-right scan shift 3 Right-to-left scan shift Thierry Lecroq Text Searching 3/77

4 Pattern matching text position of occurrence pattern Problem Find all the occurrences of pattern x of length m inside the text y of length n Two types of solutions Fixed pattern Use of combinatorial properties Static text Solutions based on indexes preprocessing time O(m) searching time O(n) preprocessing time O(n) searching time O(m) Thierry Lecroq Text Searching 4/77

5 Searching for a fixed string String matching given a pattern x, find all its locations in any text y Pattern a string x of symbols, of length m t a t a Text a string y of symbols, of length n c a c g t a t a t a t g c g t t a t a a t Thierry Lecroq Text Searching 5/77

6 String matching Occurrences c a c g t a t a t a t g c g t t a t a a t t a t a t a t a t a t a Basic operation symbol comparison (=, ) Thierry Lecroq Text Searching 6/77

7 Interest Practical basic problem for search for various patterns lexical analysis approximate string matching comparisons of strings alignments... Theoretical design of algorithms analysis of algorithms complexity combinatorics on strings... Thierry Lecroq Text Searching 7/77

8 Sliding window strategy window text y shift scan scan scan pattern x shift Scan-and-shift mechanism put window at the beginning of text while window on text do scan: if window = pattern then report it shift: shift window to the right and memorize some information for use during next scans and shifts Thierry Lecroq Text Searching 8/77

9 Naive search Principles No memorization, shift by 1 position Complexity O(m n) time, O(1) extra space Number of symbol comparisons maximum m n expected 2 n on a two-letter alphabet, with equiprobablity and independence conditions Thierry Lecroq Text Searching 9/77

10 Naive string-searching algorithm text y 0 pos u b j n 1 Algorithm pattern x u a 0 i m 1 NaiveSearch(string x, y; integer m, n) pos 0 while pos n m do i 0 while i < m and x[i] = y[pos + i] do i i + 1 if i = m then output( x occurs in y at position, pos) pos pos + 1 Thierry Lecroq Text Searching 10/77

11 Complexity of the problem pattern x of length m text y of length n (n > m) Theorem The search can be done optimally in time O(n) and space O(1). [Galil and Seiferas, 1983] Theorem The search can be done in optimal expected time O( log m m n). [Yao, 1979] Theorem The maximal number of comparisons done during the search is n m (n m), and can be made n + 3(m+1) (n m). [Cole et alii, 1995] Thierry Lecroq Text Searching 11/77

12 Known bounds on symbol comparisons Lower bounds Upper bounds access to the whole text n [Folklore] 2n 1 [Morris and Pratt, 1970] n [Zwick and Paterson, 1991] 4 3 search with a sliding window of size m n n 1 3 n + 9 4m n [Apost. and Croch., 1991] n [Galil and Giancarlo, 1991] m [Galil and Giancarlo, 1992] 4 log m+2 n + (n m) m [Breslauer and Galil, 1993] (n m) 8 n + (n m) 3(m+1) [Cole et alii, 1995] Thierry Lecroq Text Searching 12/77

13 Known bounds on symbol comparisons (followed) Lower bounds Upper bounds search with a sliding window of size 1 2n 1 [Morris and Pratt, 1970] (2 1 )n (2 1 )n [Hancart, 1993] m m [Breslauer et alii, 1993] delay = maximum number of comp s on each text symbol m [Morris and Pratt, 1970] log Φ (m + 1) [Knuth, MP, 1977] min(log Φ (m + 1), c) [Simon, 1989] min(1 + log 2 m, c) min(1 + log 2 m, c) [Hancart, 1993] log min(1 + log 2 m, c) [Hancart, 1996] Thierry Lecroq Text Searching 13/77

14 Methods prefix of text in Σ pattern pattern sequential searches (window size = one symbol) adapted to telecommunications based on efficient implementations of automata [Knuth, Morris, Pratt, 1976], [Simon, 1989], [Hancart, 1993], [Breslauer, Colussi, Toniolo, 1993] Thierry Lecroq Text Searching 14/77

15 Methods (followed) time-space optimal searches mainly of theoretical interest based on combinatorial properties of strings [Galil, Seiferas, 1983], [Crochemore, Perrin, 1991], [Crochemore, 1992], [G asieniec, Plandowski, Rytter, 1995], [Crochemore, G asieniec, Rytter, 1999] practically-fast searches used in text editors, data retrieval software based on combinatorics + automata (+ heuristics) [Boyer, Moore, 1977], [Galil, 1979], [Apostolico, Giancarlo, 1986], [Crochemore et alii, 1994] Thierry Lecroq Text Searching 15/77

16 Plan 1 Introduction 2 Left-to-right scan shift 3 Right-to-left scan shift Thierry Lecroq Text Searching 16/77

17 Left-to-right scan shift text y u b pattern x u a period(u) border(u) Mismatch situation: a b period(u) = u border(u) Optimal shift length = period(ub) Valid if u = x Thierry Lecroq Text Searching 17/77

18 Periods and borders Non-empty string u, integer p, 0 < p u p is a period of u if any of these equivalent conditions is satisfied: u[i] = u[i + p], for 1 i u p u is a prefix of some y k, k > 0, y = p u = yw = wz, for some strings y, z, w with y = z = p String w is called a border of u The period of u, period(u), is its smallest period (can be u ) The border of u, border(u), is its longest border (can be empty) Thierry Lecroq Text Searching 18/77

19 Periods and borders (followed) Example Periods and borders of abacabacaba 4 abacaba 8 aba 10 a 11 empty string Thierry Lecroq Text Searching 19/77

20 Sequential search Simple online search Length of shift = period Memorization of borders Algorithm while window on text do u longest common prefix of window and pattern if u = pattern then report a match shift window period(u) places to right memorize border(u) Thierry Lecroq Text Searching 20/77

21 MP algorithm text y 0 pos u b j n 1 pattern x u a 0 i m 1 0 MPnext[i] m 1 Thierry Lecroq Text Searching 21/77

22 MP algorithm (followed) Algorithm MP(string x, y; integer m, n) i 0; j 0 while j < n do while (i = m) or (i 0 and x[i] y[j]) do i MPnext[i] i i + 1; j j + 1 if i = m then output( x occurs in y at position, j i) Thierry Lecroq Text Searching 22/77

23 Computing borders of prefixes A border of a border of u is a border of u A border of u is either border(u) or a border of it Algorithm Border[i] = border(x[0.. i 1]) j runs through decreasing lengths of borders ComputeBorders(string x; integer m) Border[0] 1 for i 0 to m 1 do j Border[i] while j 0 and x[i] x[j] do j Border[j] Border[i + 1] j + 1 return Border MPnext[i] = Border[i] for i = 0,..., m Thierry Lecroq Text Searching 23/77

24 Improvement x u σ x w τ p w Interrupted periods strict borders Changes only the preprocessing of MP algorithm Thierry Lecroq Text Searching 24/77

25 Improvement (followed) Algorithm while window on text do u longest common prefix of window and pattern if u = pattern then report a match shift window interruptpperiod(u) places to the right memorize strictborder(u) Thierry Lecroq Text Searching 25/77

26 Interrupted periods and strict borders Fixed string x, non-empty prefix u of x w is a strict border of u if both: w is a border of u wτ is a prefix of x, but uτ is not p is an interrupted period of u if p = u w for some strict border w of u Example Prefix abacabacaba of abacabacabacc Interrupted periods and strict borders of abacabacaba 10 a 11 empty string Thierry Lecroq Text Searching 26/77

27 KMP preprocessing k = MPnext[i] { k, if x[i] x[k] or if i = m, KMPnext[i] = KMPnext[k], if x[i] = x[k]. Algorithm ComputeKMPnext(string x; integer m); KMPnext[0] 1; j 0 for i 1 to m 1 do if x[i] = x[j] then KMPnext[i] KMPnext[j] else KMPnext[i] j do j KMPnext[j] while j 0 and x[i] x[j] j j + 1 KMPnext[m] j return KMPnext Thierry Lecroq Text Searching 27/77

28 Running times of MP and KMP Theorem On a text of length n, MP and KMP string-searching algorithm run in time O(n). They make less than 2n symbol comparisons. Proof. Positive comparisons increase the value of j Negative comparisons increase the value of j i (shift) Thierry Lecroq Text Searching 28/77

29 Running times of MP and KMP (followed) Delay = maximum number of comparisons on a text symbol Theorem Pattern of length m. The delay for MP algorithm is no more than m. The delay for KMP algorithm is no more than log Φ (m + 1), where Φ is the golden ratio, (1 + 5)/2. Proof. For KMP, use the periodicity theorem A worst-case pattern of length 19: abaababaabaababaaba Thierry Lecroq Text Searching 29/77

30 Periodicities Theorem (Fine & Wilf 1965) If p and q are periods of a word x and satisfy p + q GCD(p, q) x then GCD(p, q) is a period of x. Used in the analysis of KMP algorithm and in the analysis of many other pattern matching algorithms. Theorem (Weak version) If p and q are periods of a word x and satisfy p + q x then GCD(p, q) is a period of x. Proof. If p and q are periods of x, p > q, then p q is also a period of x. Thierry Lecroq Text Searching 30/77

31 Proof of the weak statement p and q periods of x with p + q x and p > q p q period of x because: p a b c p q q p c a b q p q rest of the proof like Euclid s induction Thierry Lecroq Text Searching 31/77

32 Searching with an automaton Uses the string-matching automaton SMA(x): smallest deterministic automaton accepting Σ x Example x = abaa Search for abaa in: b a b b a a b a a b a a b b a state Thierry Lecroq Text Searching 32/77

33 Searching algorithm Simple parsing of the text with the string-matching automaton SMA(x) Algorithm SEARCH(string x, y; integer m, n) (Q, Σ, initial, {terminal }, δ) is the automaton SMA(x) q initial state if q = terminal then report an occurrence of x in y while not end of y do σ next symbol of y q δ(q, σ) if q = terminal then report an occurrence of x in y Thierry Lecroq Text Searching 33/77

34 Construction of SMA(x) Unwinding arcs From SMA(abaa) to SMA(abaab) Thierry Lecroq Text Searching 34/77

35 Construction algorithm Algorithm automaton SMA(string x) let initial be a new state Q {initial } terminal initial for all σ in Σ do δ(initial, σ) initial while not end of x do τ next symbol of x r δ(terminal, τ) add new state s to Q δ(terminal, τ) s for all σ in Σ do δ(s, σ) δ(r, σ) terminal s return (Q, Σ, initial, {terminal }, δ) Thierry Lecroq Text Searching 35/77

36 Significant arcs Example Complete SMA(ananas) n, s a a a a 0 a 1 n 2 a 3 n 4 a 5 s 6 s n n, s s n, s n, s Thierry Lecroq Text Searching 36/77

37 Significant arcs (followed) Lemma Forward arcs: spell the pattern Backward arcs: arcs going backwards without reaching the initial state a a a a 0 a 1 n 2 a 3 n 4 a 5 s 6 n SMA(x) contains at most x backward arcs. Consequence The implementation of SMA(x) can be done in O( x ) time and space, independently of the alphabet size Thierry Lecroq Text Searching 37/77

38 Backward arcs in SMA States of SMA(x) identified with prefixes of x A backward arc is of the form (u, τ, vτ) (u, v prefixes of x, τ symbol) with vτ is the longest suffix of uτ that is a prefix of x, and u v Note: uτ is not a prefix of x Let p(u, τ) = u v (a period of u) Backward arcs to periods Two backward arcs (u, τ, vτ), (u, τ, v τ ) If p(u, τ) = p(u, τ ) = p, then vτ = v τ. Otherwise, if for instance vτ is a proper prefix of v τ, vτ occurs at position p like v does, uτ is a prefix of x, a contradiction Thus v = v, τ = τ, and then u = u Thierry Lecroq Text Searching 38/77

39 Backward arcs in SMA (followed) Each period p, 1 p x, corresponds to at most one backward arc, thus there are at most x such arcs A worst case SMA(ab m 1 ) has m backward arcs (a b) Thierry Lecroq Text Searching 39/77

40 Complexity of searching with SMA Pattern x of length m, text y of length n With complete SMA implemented by transition matrix Preprocessing on pattern x time O(m card Σ) space O(m card Σ) Search on text y time O(n) space O(m card Σ) Delay time constant With SMA implemented by lists of backward arcs Preprocessing on pattern x time O(m) space O(m) Search on text y time O(n) space O(m) Delay comparisons min(card Σ, log 2 m) Improves on KMP algorithm Thierry Lecroq Text Searching 40/77

41 Plan 1 Introduction 2 Left-to-right scan shift 3 Right-to-left scan shift Thierry Lecroq Text Searching 41/77

42 Right-to-left scan shift Example text c g c t c g c g c t a t c g pattern c g c t a g c c g c t a g c c g c t a g c Match shift: good-suffix rule (function d) Occurrence heuristics: bad-character rule Extra rules if memorization Thierry Lecroq Text Searching 42/77

43 Right-to-left scan shift (followed) Algorithm while window on text do u longest common suffix of window and pattern if u = pattern then report a match shift window d(u) places to the right Thierry Lecroq Text Searching 43/77

44 Match shift text y pos τ u pattern x text y pos σ i? τ u u u shift pattern x σ i u shift Precomputation of rightmost occurrences of u s: O(m) Second shift length = a period of x Table D implements the good-suffix rule: shift = d(u) = D[i] Thierry Lecroq Text Searching 44/77

45 BM algorithm text y τ u 0 pos j n 1 pattern x σ u 0 i m 1 No memorization of previous matches Table D implements the good-suffix rule Thierry Lecroq Text Searching 45/77

46 BM algorithm (followed) Algorithm BM(string x, y; integer m, n) pos 0 while pos n m do i m 1 while i 0 and x[i] = y[pos + i] do i i 1 if i = 1 then output( x occurs in y at position, pos) pos pos + D[i + 1] Thierry Lecroq Text Searching 46/77

47 Suffix displacement Displacement function d: d(u) = min{ z > 0 (x suffix of uz) or (τuz suffix of x and τu not suffix of x, for τ Σ)} Displacement table D: D[i] = d(x[i.. m 1]), for i = 0,.., m 1 D[m] = 1 τ u z x σ u u z Thierry Lecroq Text Searching 47/77

48 Suffix displacement (followed) Note 1: u is a (strict) border of uz Note 2: z is a period of uz (thus, z is a period of x) Lemma Table D can be computed in linear time. Thierry Lecroq Text Searching 48/77

49 Suffixes Elements for computing the table Suf i given: Suf [j] known for i < j m g = min{f Suf [f] i < f. m} pattern x τ u τ u σ u τ u σ u 0 g i f i + m f 1 m 1 If i Suf [i + m f 1] k: Suf [i] = min{suf [i + m f 1], i g} Thierry Lecroq Text Searching 49/77

50 Suffixes pattern x τ u τ u σ u τ u σ u 0 g i f i + m f 1 m 1 scan scan If i Suf [i + m f 1] = g: scan to find Suf [i] Yields new g and new f Thierry Lecroq Text Searching 50/77

51 Computing Suf Algorithm ComputeSUF(string x; integer m) Suf [m 1] m; g m 1 for i m 2 downto 0 do if i > g and Suf [i + m 1 f] i g then Suf [i] min{suf [i + m f 1], i g} else f i; g min(g, i) while g 0 and x[g] = x[g + m 1 f] do g g 1 Suf [i] f g return Suf Thierry Lecroq Text Searching 51/77

52 Computing Suf i x a z m j 1 x c z j Suf[j] D[i] = D[m 1 Suf [j]] = m j 1 Thierry Lecroq Text Searching 52/77

53 Computing Suf i x z m j 1 x j D[i] = m j 1 Thierry Lecroq Text Searching 53/77

54 BM preprocessing Algorithm ComputeD(string x; integer m; table Suf ) i 0 for j m 2 downto 1 do if j = 1 or Suf [j] = j + 1 then while i m 1 j do D[i] m 1 j i i + 1 for j 0 to m 2 do D[m 1 Suf [j]] m 1 j return D Thierry Lecroq Text Searching 54/77

55 Occurrence shift text y pos τ u pattern x σ τ u DA[τ] shift Table DA implements the bad-character rule: DA[σ] = min({ z > 0 σz suffix of x} { x }) shift = DA[τ] u = DA[τ] m + i Thierry Lecroq Text Searching 55/77

56 Occurrence shift (followed) Algorithm ComputeDA(string x; integer m) for all σ in Σ do DA[σ] = m for i 0 to m 2 do DA[x[i]] = m i 1 return DA Thierry Lecroq Text Searching 56/77

57 BM with occurrence shift text y τ u 0 pos j n 1 pattern x σ u 0 i m 1 Use of DA in addition to D Thierry Lecroq Text Searching 57/77

58 BM with occurrence shift (followed) Algorithm BM(string x, y; integer m, n); pos 0 while pos n m do i m 1 while i 0 and x[i] = y[pos + i] do i i 1 if i = 1 then output( x occurs in y at position, pos) pos pos + max(d[i + 1], DA[y[pos + i]] m + i + 1) Thierry Lecroq Text Searching 58/77

59 Complexity of BM Preprocessing phase match shift O(m) occurrence shift O(m + card Σ) Search phase (finding all occurrences) running time O(n m) minimum number of comparisons n/m maximum number of comparisons n m Extra space for shift functions O(m + card Σ) can be reduced to O(m) Thierry Lecroq Text Searching 59/77

60 Symbol comparisons of BM For finding the first occurrence [Knuth, Morris, Pratt, 1977] [Guibas, Odlysko, 1980] [Cole, 1994] 7 n 4 n 3 n Theorem If period(x) > m/2, BM searching algorithm performs at most 3n n/m symbol comparisons. The bound is tight. Proof. difficult Thierry Lecroq Text Searching 60/77

61 Symbol comparisons in variants of BM For finding all occurrences [Galil, 1979] prefix memorization [Apostolico, Giancarlo, 1986] all-suffix memorization [Crochemore et alii, 1994] last-suffix memorization Turbo-BM O(n) O(1) extra space 1.5 n O(m) extra space 2 n O(1) extra space Thierry Lecroq Text Searching 61/77

62 Turbo-BM method text y pattern x pos memory u u u match-shift z Features Stores the last match in case of match shift (memory) Jumps on memory Uses turbo-shifts Thierry Lecroq Text Searching 62/77

63 Turbo-BM method (followed) Preprocessing same as BM algorithm Search O(1) extra space to store memory: (length, right position) Note match-shift is a period of uz Thierry Lecroq Text Searching 63/77

64 Turbo-shift text y pos memory σ match τ pattern x σ turbo-shift σ turbo-shift = memory match Use in Turbo-BM: shift max(match-shift, occ-shift, turbo-shift); if (shift = match-shift) then set memory; else { shift max(shift, match+1); no memory; } Thierry Lecroq Text Searching 64/77

65 Tight number of comparisons for Turbo-BM Theorem The Turbo-BM searching algorithm runs in time O(n). It makes no more than 2n symbol comparisons. Proof. difficult Thierry Lecroq Text Searching 65/77

66 AG method text y previous matches pattern x current match Thierry Lecroq Text Searching 66/77

67 AG method Features Stores all previous matches (suffixes of pattern) Uses the table Suf Suf [i] = longest suffix of x ending at i in x pattern x Suf [i] i Search O(m) extra space to store the matches Thierry Lecroq Text Searching 67/77

68 Rules for shifts in AG match text y pattern x Suf [i] i current match mismatch text y pattern x Suf [i] i current match Thierry Lecroq Text Searching 68/77

69 Rules for shifts in AG mismatch text y pattern x Suf [i] i current match jump text y pattern x? Suf [i] i current match Thierry Lecroq Text Searching 69/77

70 Tight number of comparisons for AG Lemma The AG algorithm makes at most n 2 characters previously compared. comparisons on text Theorem The AG searching algorithm runs in time O(n). It makes no more than 1.5n symbol comparisons. Proof. rather difficult Thierry Lecroq Text Searching 70/77

71 Worst-case example Example a a b a a a b a a b a a a b a a b a a a b... a a b a a a b a a b a a a b a a b a a a b a a b a a a b a a b a a a b a a b a a a b a a b a a a b a a b a a a b a a b a a a b pattern = a m 1 ba m b, text = (a m 1 ba m b) e number of comparisons = 2m (3m + 1)e = 3m+1 2m+1 n m 1.5n Thierry Lecroq Text Searching 71/77

72 References A. Apostolico and M. Crochemore. Optimal Canonization of All Substrings of a String. Inf. Comput., 95(1):76-95, A. Apostolico et Z. Galil, editors. Pattern matching algorithms. Oxford University Press, A. Apostolico and R. Giancarlo. The Boyer-Moore-Galil string searching strategies revisited. SIAM J. Comput., 15(1):98 105, R. S. Boyer and J. S. Moore. A fast string searching algorithm. Commun. ACM, 20(10): , D. Breslauer, L. Colussi, and L. Toniolo. Tight comparison bounds for the string prefix-matching problem. Inf. Process. Lett., 47(1):51 57, Thierry Lecroq Text Searching 72/77

73 References D. Breslauer and Z. Galil. Efficient Comparison Based String Matching. J. Complexity, 9(3): , M. Crochemore, A. Czumaj, L. G asieniec, S. Jarominek, T. Lecroq, W. Plandowski, and W. Rytter. Speeding up two string matching algorithms. Algorithmica, 12(4/5): , M. Crochemore, L. G asieniec, and W. Rytter. Constant-space string-matching in sublinear average time. Theor. Comput. Sci., 218(1): , M. Crochemore, C. Hancart and T. Lecroq. Algorithms on Strings. Cambridge University Press, 20017, to appear. M. Crochemore and T. Lecroq. Tight bounds on the complexity of the Apostolico-Giancarlo algorithm. Inf. Process. Lett., 63(4): , Thierry Lecroq Text Searching 73/77

74 References M. Crochemore and T. Lecroq. Text Searching and Indexing. Recent Advances in Formal Languages and Applications, Series: Studies in Computational Intelligence, Volume 25, (2006), Z. Esik, C. Martin-Vide and V. Mitrana editors, pages 43-80, Springer Verlag. M. Crochemore and D. Perrin. Two-way string-matching. J. Assoc. Comput. Mach., 38(3): , M. Crochemore et W. Rytter. Jewels of Stringology. World Scientific, R. Cole. Tight bounds on the complexity of the Boyer-Moore string matching algorithm. SIAM J. Comput., 23(5): , R. Cole, R. Hariharan, M. Paterson, and U. Zwick. Tighter lower bounds on the exact complexity of string matching. SIAM J. Comput., 24(1):30 45, Thierry Lecroq Text Searching 74/77

75 References Z. Galil. On improving the worst case running time of the Boyer-Moore string searching algorithm. Commun. ACM, 22(9): , Z. Galil and R. Giancarlo. On the exact complexity of string matching: lower bounds. SIAM J. Comput., 20(6): , Z. Galil and R. Giancarlo. On the Exact Complexity of String Matching: Upper Bounds. SIAM J. Comput., 21(3): , Z. Galil and J. Seiferas. Time-space optimal string matching. J. Comput. Syst. Sci., 26(3): , L. G asieniec, W. Plandowski, and W. Rytter. Constant-space string matching with smaller number of comparisons: sequential sampling. In Z. Galil and E. Ukkonen, editors, Proceedings of the 6th Annual Thierry Lecroq Text Searching 75/77

76 References L. J. Guibas and A. M. Odlyzko. A new proof of the linearity of the Boyer-Moore string searching algorithm. SIAM J. Comput., 9(4): , D. Gusfield. Algorithms on strings, trees and sequences: computer science and computational biology. Cambridge University Press, C. Hancart. On Simon s string searching algorithm. Inf. Process. Lett., 47(2):95 99, C. Hancart, Personal communication. D. E. Knuth, J. H. Morris, Jr, and V. R. Pratt. Fast pattern matching in strings. SIAM J. Comput., 6(1): , Thierry Lecroq Text Searching 76/77

77 References J. H. Morris, Jr and V. R. Pratt. A linear pattern-matching algorithm. Report 40, University of California, Berkeley, I. Simon. Sequence comparison: some theory and some practice. In M. Gross and D. Perrin, editors, Electronic, Dictionaries and Automata in Computational Linguistics, number 377 in Lecture Notes in Computer Science, pages Springer-Verlag, Berlin, G. A. Stephen. String searching algorithms. World Scientific Press, A. C. Yao. The complexity of pattern matching for a random string. SIAM J. Comput., 8(3): , U. Zwick and M. S. Paterson. Lower bounds for string matching in the sequential comparison model. Manuscript, Thierry Lecroq Text Searching 77/77