M. Fischer and M. Paterson. String matching and other products. In Complexity of computation. SIAM-AMS proceedings, ed. R.M. Karp, pages 113--125, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....alternative is to simulate the automata in nondeterministic form, obtaining O(mn) search time [65] There are, however, particular cases where specialized solutions exist. On the theoretical side, there exist solutions to search for patterns with classes of characters and simple wild cards [24, 60, 1]. Although theoretically interesting, these are not relevant in practice. Bit parallelism, on the other hand, yields extremely simple and ecient solutions to this kind of problems. For example, classes of characters are easily handled in the Shift And and BNDM algorithms. We only need to set ....

M. Fischer and M. Paterson. String matching and other products. In Proc. 7th SIAM-AMS Complexity of Computation, pages 113-125. American Mathematical Society, 1974.

....of same length u w if and only if u[i] w[i] The problem of pattern matching with don t cares consists in nding all positions j such that P T [j : j m 1] Denote by IntMult(n) the time to multiply two n bit binary numbers. The following fact has been shown by Fischer and Paterson, see [8,5]. Lemma 1 The problem of pattern matching with don t cares for a pattern P and a text T of length n over an alphabet can be solved in time O(log j j IntMult(n) We show that for small alphabet matching is at least as dicult as matching with don t cares. Theorem 2 For binary alphabets fa; ....

M.J. Fischer, M.S. Paterson, String-matching and other products, in: R. Karp, ed., Complexity of Computation: Proceedings of a Symposium in Applied Mathematics of the AMS and the SIAM, (1974) 113-125. 11

....temporary location list T x = S x 0 =x : x 0 2M 0 occ x 0 for each distinct x 2 M thus obtained. Lemma 4. Each motif x 2 B satis es T x = L x . Step 3. Select M M, where M = fx 2 M : T x = L x g. In order to build M , we employ the Fischer Paterson algorithm based on convolution [5] for string matching with don t cares to compute the whole list of occurrences L x for each merge x 2 M. Its cost is O( jxj n) log n log j j) time for each merge x. Since jxj n and there are at most n 1 motifs x 2 M, we obtain O(n log n log j j) time to construct all lists L x . We can ....

M. Fischer and M. Paterson. String matching and other products. In R. Karp, editor, SIAM AMS Complexity of Computation, pages 113-125, 1974.

....0 occ x 0 for each distinct x 2 M . Unfortunately, it may be T x L x and so we are missing some occurrences of x. We therefore need to compute the whole list of occurrences L x for all merges x 2 M . Given one such merge x, we employ the algorithm by Fischer and Paterson based on convolution [4] for string matching with don t cares. Its cost is O( jxj n) log n log j j) time. Since jxj n and there are at most n such x s, we obtain a total of O(n log n log j j) time to construct all lists L x , where x 2 M . We then proceed by processing the lists L x to lter M , that is, we ....

....0 j k 1) Lemma 6 The positions of occurrences of a pattern x in a string of length n can be computed in time O(k n) Proof : This is a mere application of matching a pattern with don t cares inside a text without don t cares. Using for instance the Fischer and Paterson s algorithm [4] is not necessary. Instead, positions of u i s are computed by a multiple string matching algorithm, such as the Aho Corasick algorithm [1] For each position p, a counter associated with position p on s is incremented, where is the position of u i in x ( is the o set of u i in x) Counters ....

M. Fischer and M. Paterson. String matching and other products. In R. Karp, editor, SIAM AMS Complexity of Computation, pages 113-125, 1974.

....motifs in M B. For the previous example string FADABCXFADCYZEADCEADCFADC, one such motif is x = ADC with L x = f8; 14; 18; 22g while T x = f8; 18g. Step 3. Select M M, where M = fx 2 M : T x = L x g. In order to build M , we employ the Fischer Paterson algorithm based on convolution [7] for string matching with don t cares to compute the whole list of occurrences L x for each merge x 2 M. Its cost is O( jxj n) log n log j j) time for each merge x. Since jxj n and there are at most n 1 motifs x 2 M, we obtain log n log j j) time to construct all lists L x . We can compute ....

.... k 1) Proposition 3.11 The positions of the occurrences of a pattern x in a string of length n can be computed in time O(k n) Proof : This is a mere application of matching a pattern with don t cares inside a text without don t cares. Using for instance the Fischer and Paterson s algorithm [7] is not necessary. Instead, the positions of the subwords u i are computed by a multiple string matching algorithm, such as the Aho Corasick algorithm [1] For each position p, a counter associated with position p on s is incremented, where is the position of u i in x ( is the o set of u i in ....

M. Fischer and M. Paterson. String matching and other products. In R. Karp, editor, SIAM AMS Complexity of Computation, pages 113-125, 1974.

....of the pattern starting at an odd location l in the text, we determine dp Gamma oe(l) the dot product of p e and the overlapping portion of t o , and similarly, dp Gamma eo(l) the dot product of p o and the overlapping portion of t e . This can be done in O(m log m) time, using convolution [4]. We now have the following claim. Lemma 5 Consider a particular placement of the pattern starting at an odd location l in the text. If the pattern matches in this location, then the following hold. 1. dp Gamma eo(l) rand e (L ) rand o (R) right Gamma eo(l) 2. dp Gamma oe(l) rand ....

M. Fischer, M. Paterson. String Matching and Other Products, R.M. Karp (editor), SIAMAMS Proceedings, 7, pp. 113-125, 1974.

....of fields. Recently, various modifications of the exact string matching problem have been considered. Applications in computational biology and computer vision have motivated the study of approximate string matching [5, 6] where di#erent matching relations like swapped matching [2] don t cares [8] and overlap matching [3] have been proposed. There is another interesting variation of exact string matching, which we call context sensitive string matching. Here, the pattern is allowed to have variables and the goal is to map variables to strings, such that, when the variables in the pattern ....

M.J. Fischer and M.S. Paterson. String matching and other products. In Complexity of Computation, SIAM-AMS Proceedings, pages 7:113--125, 1974.

....for the string matching with don t cares problem. Based on the simple ngerprint method of Karp and Rabin for ordinary string matching [4] our algorithm runs in time O(n log m) for a text of length n and a pattern of length m and is simpler and slightly faster than the previous algorithms [3, 5, 1]. 1 Introduction. We extend the simple randomized ngerprinting algorithm of Karp and Rabin [4] to the problem of string matching with don t cares. Our algorithm uses a single, simple convolution. This is optimal in the sense that the string matching with don t cares problem is at least as hard ....

....sense that the string matching with don t cares problem is at least as hard as the boolean convolution problem [6] Thus, to improve our run time of O(n log m) on text of length n and pattern of length m, one would have to improve on the Fast Fourier Transform. Fischer and Paterson s algorithm [1] runs in time O(n log m log j j) 1 . Since their deterministic algorithm in 1974, the only improvements were by Muthukrishan and Palem [5] who reduced the constant factor, and Indyk [3] who gave a randomized algorithm that also involved convolutions, running in time O(n log n) In addition to ....

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M.J. Fischer and M.S. Paterson, String Matching and other Products, in Complexity of Computation, R.M. Karp (editor), SIAM AMS Proceedings, 7, pp. 113-125, 1974.

....sense that the text x is preprocessed to build the suffix array associated with it. This operation costs O(n log n) time in the worst case. Once this is done, the problem reduces to one of efficient implementation of 2 dimensional orthogonal range queries. A landmark paper by Fischer and Paterson [76] exposed the similarity of string searching to multiplication, thereby obtaining a number of interesting algorithms for exact string searching and some of its variants. It is not difficult to see that string matching problems can be rendered as special cases of a general linear product. Given two ....

M.J. Fischer and M.S. Paterson. String matching and other products. Complexity of Computation, R.M. Karp (editor), SIAMAMS Proceedings, 7:113--125, 1974.

....of this string corresponds to an initial palindrome of w. Two copies of the string w w R are aligned with each other shifted by some offset and the overlapping parts are identical if and only if the overlapping part is an initial palindrome of w. This reduction was used by Fischer and Paterson [5]. 2 3 Example: The string abaab has an initial palindrome aba. This initial palindrome corresponds to the period abaab ba of the string abaab baaba. Proof of Theorem 2.2: The algorithm will proceed in independent stages which are all computed simultaneously and described in the next section. ....

Fischer, M. J. and Paterson, M. S. (1974), String-Matching and other products, SIAM-AMS proceedings, Vol 7, 113-125.

....complexity of approximating polynomial ze ros. The latter bound can be slightly decreased [by the factor log(bn) if one performs the computations using the special techniques of binary segmentation, that is, splitting the 0 1 string that represents a binary value into several substrings (compare [FP], Sc82] P92a] and section 9 of chapter 3 of [BP94] on this subject) The arithmetic complexity bound O(n 2 log n log(bn) close to the bound of this paper) can be also obtained by making a hybrid of our modication of Weyl s geometric construc tion and the techniques of [Sc82] for splitting ....

M. J. Fischer, M. S. Paterson, String Matching and Other Products, SIAM-AMS Proc., 7(1974), pp. 113125.

....Classes of Characters As in the Shift Or algorithm, we allow that each position in the pattern matches not only a single character but an arbitrary set of characters. Some solutions for the case of don t care characters (i.e. pattern positions that match any character) have been presented in [14, 22, 1], but these have been shown to be only of theoretical interest in [3] Simple attempts to extend classical algorithms such as KMP or BM do not work well. To the best of our knowledge, the fastest algorithm for this problem is Shift Or. This type of patterns is called limited expressions in ....

M. J. Fischer and M. Paterson. String matching and other products. In R. M. Karp, editor, Proceedings SIAM-AMS Complexity of Computation, pages 113--125, Providence, RI, 1974.

....do not seem to be easily solvable in linear time. We show below how string pattern matching with don t care symbols can be reduced in linear time to ordered path inclusion problems. The time complexity of the best known algorithm for string matching with don t care symbols is O(polylog(m) n) FP74] and the existence of a faster algorithm has been an open problem for almost 20 years. The idea of the reduction was presented in [Kos89] and it was also noted in [Ver92] Let P be a pattern string p 1 ; pm and T a target string t 1 ; t n where m n, both over the alphabet ....

M. J. Fischer and M. S. Paterson. String-matching and other products. In Complexity of Computation, pages 113--125. SIAMAMS, 1974.

....Classical algorithms include Knuth Morris Pratt and Boyer Moore for exact matching and dynamic programming for the case with errors. We refer the reader to [GBY91, Chapter 7] for more details. For string matching problems, non comparison based algorithms include the use of matrix multiplication [FP74, Kar93], or bit wise techniques as in this paper [Abr87, Der95, Wri94, WMM95] The approach taken here tries to use the same technique for different problems. We use a RAM machine with word size w log 2 n, for any text size n. We use the uniform cost model for a restricted set of operations including ....

M. Fischer and M. Paterson. String matching and other products. In R. Karp, editor, Complexity of Computation (SIAM-AMS Proceedings 7), volume 7, pages 113--125. American Mathematical Society, Providence, RI, 1974.

....performed, is equal to the running time of the fastest sequential algorithm. Note that there exists a trivial constant time CRCW PRAM algorithm that finds all palindromes in a string using O(n 2 ) processors. However, the large number of processors leaves much to be desired. Fischer and Paterson [9] noticed that any string matching algorithm that finds all overhanging occurrences of a string in another can also find all initial palindromes. This observation has been used by Apostolico, Breslauer and Galil [1] to construct an optimal O(log log n) time parallel algorithm that finds all initial ....

M.J. Fischer and M.S. Paterson. String matching and other products. In R.M. Karp, editor, Complexity of Computation, pages 113--125. American Mathematical Society, Prividence, RI., 1974.

.... string matching [4, 18] Running times are O(kn) for Baeza Yates and Gonnet [4] and O(kn( 1 m Gammak k c ) for Tarhio and Ukkonen [18] The problem of approximate matching of a class of patterns was also studied [2, 1, 5] especially in the case of patterns with don t care symbols [10, 17, 16, 3, 8, 14]. Fisher et Paterson [10] developed an O(n log c log 2 m log log m) time algorithm based on the linear product. Abrahamson [1] extended this method for generalized string pattern. Pinter [17] has used the Aho and Corasick automaton [2] for searching a set of patterns. Other algorithms have ....

.... are O(kn) for Baeza Yates and Gonnet [4] and O(kn( 1 m Gammak k c ) for Tarhio and Ukkonen [18] The problem of approximate matching of a class of patterns was also studied [2, 1, 5] especially in the case of patterns with don t care symbols [10, 17, 16, 3, 8, 14] Fisher et Paterson [10] developed an O(n log c log 2 m log log m) time algorithm based on the linear product. Abrahamson [1] extended this method for generalized string pattern. Pinter [17] has used the Aho and Corasick automaton [2] for searching a set of patterns. Other algorithms have considered the problem of ....

M. Fischer and M. Paterson. String-matching and other products. In R. Karp, editor, Complexity of Computation (SIAM-AMS Proceedings 7), volume 7, pages 113--125. American Mathematical Society, Providence, R.I., 1974.

....of this string corresponds to an initial palindrome of w. Two copies of the string w w R are aligned with each other shifted by some offset and the overlapping parts are identical if and only if the overlapping part is an initial palindrome of w. This reduction was used by Fischer and Paterson [43]. See Figure 5.1 2 a b a a b b a a b a a b a a b b a a b a Figure 5.1: The string abaab has an initial palindrome aba . This initial palindrome corresponds to the period abaab ba of the string abaab baaba . 5.3 An Omega Gamma 15 log n) Lower Bound Given a string S[0: n] we say that it ....

M. J. Fischer and M. S. Paterson. String matching and other products. In R. M. Karp, editor, Complexity of Computation, pages 113--125. American Mathematical Society, Prividence, RI., 1974.

....Element (non)distinctness: L dup : f x 1 #x 2 # : #xm : 9i; j) i j x i = x j g: c) List intersection: L int : f x 1 #x 2 # : #xm , y 1 #y 2 # : #ym : 9i; j) x i = y j g: d) Triangle: L Delta : fG : G is an undirected graph which contains a triangleg. L pat belongs to DLIN [FP74] (cf. FMR72, GS83] and was recently shown not to be recognizable by a one way multihead DFA [JL93] L dup and L int can be solved in linear time by a RAM which treats list elements as cell addresses. L Delta is not believed recognizable in linear time on a RAM at all the best method known is ....

M. Fischer and M. Paterson. String matching and other products. In R. Karp, editor, Complexity of Computation, volume 7 of SIAM-AMS Proceedings, pages 113--125. Amer. Math. Soc., 1974.

....j=0 f(t i j )g(p j )ffi t i j ;p j ; where f and g are complex valued functions of the alphabet. The naive algorithm to compute the exact score vector has a time complexity of O( N Gamma M 1)M ) When the alphabet size is O(1) hence much smaller than M ) the algorithm of Fisher and Paterson [9] uses convolution to solve the problem in O(N log M) time. However, if the assumption of small alphabet size is dropped, then another approach is needed. This version of the problem (i.e. for possibly large alphabets) was posed by Apostolico and Galil in their book [2] where it is mentioned that ....

M.J. Fischer and M.S. Paterson, "String Matching and Other Products," Complexity of Computation, SIAM-AMS Proceedings, 7, 1974, pp. 113--125.

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M. Fischer and M. Paterson. String matching and other products. In Complexity of computation. SIAM-AMS proceedings, ed. R.M. Karp, pages 113--125, 1974.

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M. Fischer and M. Paterson. String matching and other products. In Complexity of computation. SIAM-AMS proceedings, ed. R.M. Karp, pages 113--125, 1974.

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M. Fischer and M. Paterson. String matching and other products. In Complexity of computation. SIAM-AMS proceedings, ed. R.M. Karp, pages 113--125, 1974. 17

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M. J. Fischer and M. S. Paterson, String Matching and Other Products, pages 113-125. In R. M. Karp, editor, Complexity of Computation. SIAM-AMS 1974.

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Michael J. Fischer, Michael S. Paterson, String-matching and other products, in [55] (1974), 113-125. MR 53 #4612.

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M. Fischer, M. Paterson. String Matching and Other Products, R.M. Karp (editor), SIAMAMS Proceedings, 7, pp. 113-125, 1974.

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