Ukkonen, E. (1985), `Algorithms for approximate string matching', Information and Control Vol 64.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....2 f Gamma1; 0; 1g Deltav i;j = M i;j Gamma M i Gamma1;j 2 f Gamma1; 0; 1g Deltad i;j = M i;j Gamma M i Gamma1;j Gamma1 2 f0; 1g the horizontal, vertical, and diagonal differences among consecutive cells. Their range of values come from the properties of the dynamic programming matrix [19]. We present a version [8] that differs slightly from that of [10] Although both perform the same number of operations per text character, the one we present is easier to understand and more convenient for our purposes. Let us introduce the following boolean variables. The first four refer to ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

....where jx i j, i = 1; 2, denotes the length of the string x i [32] The algorithm is a special instance of a single source shortest paths algorithm applied to a directed graph expanded dynamically. Ukkonen improved that algorithm by reducing the size of the directed graph that needs to be expanded [31]. The complexity of his algorithm is O(dmax fjx 1 j; jx 2 jg) where d is the edit distance of x 1 and x 2 . The algorithm is more ecient for strings such that the distance d is small with respect to jx 1 j and jx 2 j. We refer the reader to [6, 12] for general surveys of edit distance and ....

Esko Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100-118, 1985.

....index as an update. Consider the number of inverted index update oper ations (insert and delete postings) generated by our update SThe keyword with the smallest frequency. 3.2 The Complexity of the diff Operation The use of edit transcripts (dill output) is a key idea in our method. Ukkonen [20] has showed that an edit tran script of two documents with size rs and n words can be computed in O(D min n, rs ) time using O(D min n, space, where D is the minimum edit distance. The UNIX cliff program uses an output sensitive heuristic algorithm similar to that of [20] so that the running ....

....in our method. Ukkonen [20] has showed that an edit tran script of two documents with size rs and n words can be computed in O(D min n, rs ) time using O(D min n, space, where D is the minimum edit distance. The UNIX cliff program uses an output sensitive heuristic algorithm similar to that of [20], so that the running time is near linear when D is small. Since diff is near linear for small updates and the up dates are usual small between two consecutive samples [15] diff is not a bottleneck in the processing in most cases. If the diff operation did form a bottleneck, we could represent ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64, 100-118, 1985.

....is initialized at C i i and updated to C after reading text character T j using C i if (P i = T j ) then C i Gamma1 else 1 min(C i Gamma1 ; C i ; C i Gamma1 ) for all i 0, and hence we report every end position j where C i k. Several properties of the matrix M are discussed in [19]. The most important for us is that adjacent cells in M differ at most by 1, that is, both M i;j Gamma M i Sigma1;j and M i;j Gamma M i;j Sigma1 are in the range f Gamma1; 0; 1g. Also, M i 1;j 1 Gamma M i;j is in the range f0; 1g. Fig. 1 shows examples of edit distance computation and ....

....2 f Gamma1; 0; 1g Deltav i;j = M i;j Gamma M i Gamma1;j 2 f Gamma1; 0; 1g Deltad i;j = M i;j Gamma M i Gamma1;j Gamma1 2 f0; 1g the horizontal, vertical, and diagonal differences among consecutive cells. Their range of values come from the properties of the dynamic programming matrix [19]. We present a version [8] that differs slightly from that of [10] Although both perform the same number of operations per text character, the one we present is easier to understand and more convenient for our purposes. Let us introduce the following boolean variables. The first four refer to ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

....between the number of excluded suggestions and the time gained in the execution. Queries have been implemented using standard SQL code and Java Stored Procedures. In particular, the edit distance function has been implemented as the optimized diagonal transition algorithms described by Ukkonen [12]: the computation of only certain elements of the dynamic programming matrix allows the improvement of the worst case computational complexity which is O(d ) for checking whether the distance is d or not. The computation and storage of q grams of the translation memory sentences in the ....

E. Ukkonen. Algorithms for Approximate String Matching. In Information and Control, volume 64, pages 100--118, 1985.

....appended files x 1 which are concatenation of all source code files x 1 , x 2 , and y 1 , y 2 , for systems X and Y , respectively. Then we extract the longest common subsequence (LCS) between by some tool, say di# [13] which implements an LCS finding algorithm[14] 15] [16]. The extracted LCS is used as the correspondence. However, this method is fragile to the change of file concatenation order caused by renaming files and reorganizing file structures, since di# cannot follow line block movement to di#erent positions in the files. For example, for two files x 1 ....

E. Ukkonen, "Algorithms for approximate string matching," INFCTRL: Information and Computation (formerly Information and Control), vol. 64, pp. 100--118, 1985.

....j, for which exists such h 0 that ed(pat; text[j h: j] k. The oldest, but most exible in terms of permitting di erent edit operations and or edit operation costs, algorithms for computing edit distance (for example [WF74] are based on dynamic programming and run in time O(mn) Ukkonen [Ukk85a] has later proposed two O(dm) methods, and Myers [Mye86] an O(n d ) method. The latter is based on using a sux tree and is not viewed as being practical (e.g. Ste94] With fairly little modi cations these methods can also be used in computing the Damerau edit distance without a ecting the ....

....has presented an O(dm=wen) algorithm for approximate string matching under the Levenshtein edit distance. In [Hyy01] the algorithm was extended for computing the Damerau edit distance. In this paper we will present an initial study on combining one of the O(dm) edit distance algorithms of Ukkonen [Ukk85a] with the bit parallel algorithm of Myers [Mye99] to obtain a faster algorithm. We begin by reviewing these underlying algorithms in the next section. 2 Preliminaries In the following discussion let A be a string of length m and B a string of length n. We also use the notation A[u] to denote the ....

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E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100-118, 1985.

....C 0: m , which is initialized at C i i and updated to C after reading text character T j using C i if (P i = T j ) then C i 1 else 1 min(C i 1 ; C i ; C i 1 ) for all i 0, and hence we report every end position j where C i k. Several properties of the matrix M are discussed in [19]. The most important for us is that adjacent cells in M di er at most by 1, that is, both M i;j M i 1;j and M i;j M i;j 1 are in the range f1; 0; 1g. Also, M i 1;j 1 M i;j is in the range f0; 1g. Fig. 1 shows examples of edit distance computation and approximate string matching. 0 1 2 3 4 5 6 ....

....of the NFA states. Let us call h i;j = M i;j M i;j 1 2 f1; 0; 1g v i;j = M i;j M i 1;j 2 f1; 0; 1g d i;j = M i;j M i 1;j 1 2 f0; 1g the horizontal, vertical, and diagonal di erences among consecutive cells. Their range of values come from the properties of the dynamic programming matrix [19]. We present a version [8] that di ers slightly from that of [10] Although both perform the same number of operations per text character, the one we present is easier to understand and more convenient for our purposes. Let us introduce the following boolean variables. The rst four refer to ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100-118, 1985.

....a dictionary of words of equal length. With some additional care our method can be generalized to the cases in which different word lengths, and larger alphabets are allowed for the dictionary. Restricting the dynamic programming computation to a diagonal band of edit graph was used by Ukkonen [12]. We have essentially incorporated this idea in our method. For the purpose of developing new methods for d queries, the idea of using sets to keep track of partial results may also be used in conjunction with suitable edit distance algorithms such as by Myers [11] and Kim et. al [8] The ....

E. Ukkonen. Algorithms for Approximate String Matching. Information and Control, 64, 100-118, 1985.

....and Paterson [MP80] Their approach is based on the Four Russians algorithm for computing transitive closure. We can split matrix D into small submatrices of size Theta(log n) These submatrices can be precomputed and the relevant part of matrix D recovered in time O(n = log n) Ukkonen [Ukk85] formulates the problem of finding longest common sub 3 2 3 1 2 2 1 3 3 2 3 3 1 3 R R R R R R R R R R R q q q q q q q q q q q q q q Figure ....

....O( n R) log n) O(R n) Hirschberg [Hir77] O(Ln n log n) O(Ln) Hirschberg [Hir77] O(L(m Gamma L) log n) O( m Gamma L) n) Masek Paterson [MP80] O(nmaxf1;m= log ng) O(n = log n) Nakatsu at al. NKY82] O(n(m Gamma L) O(m Hsu Du [HD84, Apo87] O(Lm log(n=L) Lm) O(Lm) Ukkonen [Ukk85] O(Em) O(E minfm;Eg) Apostolico [Apo86] O(n m log n D log(mn=D) O(R n) Kumar Rangan [KR87] O(n(m Gamma L) O(n) Apostolico Guerra [AG87] O(Lm n) O(D n) Wu at al. WMMM90] O(n(m Gamma L) O(n) Chin Poon [CP90] O(n minfD; Lmg) O(D n) Apostolico at al. ABG92] O(n(m Gamma L) ....

Esko Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

....KST94] In this paper, we take the Levenshtein distance as a basis. The standard algorithm for computing the Levenshtein distance between two words by Wagner and Fisher [WF74] uses a dynamic programming scheme that leads to quadratic time complexity. Even with more sophisticated algorithms (cf. Ukk85] it is not realistic to compute the Levenshtein distance between the input word W and each of the words in the dictionary, already for dictionaries of a modest size. The problem becomes even more serious when using dictionaries for highly in ectional or agglutinating languages (e.g. Russian, ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100-118, 1985.

.... very elegant linear algorithms have been developed for the string matching problem [1] Automata are also widely used in fields such as metric lexical analysis [3] or computational biology, where approximate string matching is at the core of most algorithms that deal with genetic sequences [11], 4] In these fields, the huge amount of data to be processed sometimes billions of 1 characters calls for algorithms that are better than linear. Given a deterministic finite automaton, and an input sequence e 1 : e m , we are interested in the output sequence r 1 : r m of ....

E. Ukkonen, Algorithms for Approximate String Matching, Information and Control 64(1-3) (1985), 100-118. 13

....Therefore, when searching occurrences with k mismatches, only E1 and ID should be allowed to be used, while in the case of k di erences the use of the operations E2 and E3 should be permitted, as well. The occurrences of P within text S can be located by using dynamic programming technique [20]. Dynamic programming tabulates edit distances between all the possible pre xes of A and any factor of B: d ij = minfD(pre j (A) b r : b i ) j 1 r i 1g. Examples of such calculations can be found in Figs. 1 (p.13) and 2 (p.15) corresponding to the cases of k mismatches and ....

....only if A P;S j= k d , where k d is the sentence obtained from d by replacing the constant symbol for k by its de nition in FO. Here we shall prove that the kdf problem is expressible already in FO(DTC) In the proof we harness the fact that the accepting path must be in a diagonal band [20]. The width of such a band is restricted by k; the smaller the k, the narrower the band. The expressibility of this case in FO(DTC) is a consequence of the fact that the width of the band is known beforehand, and it suces to compute the values of d ij only relative to the diagonal band. More ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100-118, 1985.

....problems, respectively. Thus, when searching occurrences with k mismatches, only E1 and ID should be allowed, while in the case of k differences the operations E2 and E3 should be allowed, as well. The occurrences of P within text S can be located by using dynamic programming technique [15]. The dynamic programming tabulates the edit distances between all the possible prefixes of A and any factor of B: d ij = minfD(pre j (A) b r : b i ) j 1 r i 1g. Examples of such calculations can be found in Figs. 1 and 2 (corresponding to the cases of k mismatches and k differences, ....

....on the model we use; the model AP;S is otherwise identical to AP;S;k , but k is no more given as a constant. Note however, that since k is now a fixed integer, it can be defined by an FO formula. In the resulting formula we harness the fact that the accepting path must be in a diagonal band [15]. The width of such a band is restricted by k; the smaller k is, the narrower the band is. The expressibility of this case in FO(DTC) is a consequence of the fact, that the width of the band is known beforehand. 0 0 0 0 0 0 0 a 1 1 1 0 0 0 1 b 4 3 3 3 2 1 0 b 5 4 3 4 3 2 1 b 6 5 4 4 4 3 2 a ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

....transposition and di erent tempi. There is still much to do. For instance, consider the example given in Fig. 3 where the two excerpts are musically very similar. The traditional way to computationally measure the similarity between the two excerpts is to use the edit distance concept (see e.g. [13]) However, in this case at least 4 deletion (or insertion) operations would be required, and still the remaining RIS values would not be the same (unless they are calculated after the deletions, and the remaining durations are normalised) We propose two di erent heuristic to be considered. The ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100118, 1985.

....n, and an integer k, find all positions of B where A occurs with at most k differences. Many algorithms have been developed for the edit distance problem and the k differences problem [BN96, GG88] When the edit distance t between A and B is small, an O(t min(m, n) time algorithm due to Ukkonen [Uk85] is the best one for the edit distance problem. When the given difference k is small, O(kn) time algorithms due to Landau and Vishkin [LV89] Galil and Park [GP90] and Ukkonen and Wood [UW93] are best for the k differences problem. In this paper we consider an additional edit operation: iv) ....

....proposed an O(mn) time algorithm for the extended edit distance problem including the swap operation. For the k differences problem the swap operation has never been considered. The k differences problem including the swap operation will be called the extended k differences problem. Ukkonen [Uk85] considered transpositions in the edit distance problem. Galil and Park [GP90] also considered transpositions in the k differences problem. A transposition is a correction of a difference that two adjacent characters in A correspond to two adjacent characters in B. However, the swap operation is ....

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Ukkonen, E. (1985), Algorithms for approximate string matching, Inform. and Control 64, 100#118.

....Therefore, when searching occurrences with k mismatches, only E1 and ID should be allowed to use, while in the case of k differences the use of the operations E2 and E3 should be permitted, as well. The occurrences of P within text S can be located by using dynamic programming technique [20]. The dynamic programming tabulates the edit distances between all the possible prefixes of A and any factor of B: d ij = minfD(pre j (A) b r : b i ) j 1 r i 1g. Examples of such calculations can be found in Figs. 1 (p.13) and 2 (p.15) corresponding to the cases of k mismatches and k ....

.... Phi k d is the sentence obtained from Phi d by replacing the constant symbol for k by its definition in FO. Here we will prove that the k differences problem with fixed k is expressible already in FO(DTC) In the proof we harness the fact that the accepting path must be in a diagonal band [20]. The width of such a band is restricted by k; the smaller the k, the narrower the band. The expressibility of this case in FO(DTC) is a consequence of the fact, that the width of the band is known beforehand. 0 0 0 0 0 0 0 a 1 1 1 0 0 0 1 b 4 3 3 3 2 1 0 b 5 4 3 4 3 2 1 b 6 5 4 4 4 3 2 a 2 ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

....substitutions. Second, we analyze the case when the weights are rational, i.e. they are integral multiples of a fixed real number, but not necessarily uniform. These and more restricted cases of cost functions have been studied in the literature in the context of ordinary edit distances [12, 15, 10, 8]. The previously suggested algorithms for NED do not specialize to faster than O(mn 2 ) time algorithms for the cases we analyze in this paper. We propose provably Efficient Algorithms For Normalized Edit Distance 3 better algorithms for NED computation in these cases. Our algorithms are ....

....algorithm A in this case, the algorithm in [17] turns out to be the only feasible one. Next we discuss the reasons for this. Ordinary edit distance algorithms which assume a fixed cost function (e.g. fl u = 1; 1; 0; 1) and achieve fast running times by using this constant nature of the weights [15, 10] are evidently not suitable as algorithm A in UniformNED. This is because each execution of Step 6 of the algorithm uses a different cost function fl u 0 . Masek and Paterson gave a C(m;n) O(mn= log n) time algorithm [8] which can be used as algorithm A here, but it is no longer true that the ....

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E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

.... where the classical solution for a single pattern is based on dynamic programming and has a running time of O(mn) 26] In recent years several algorithms have improved the classical one [22] Some improve the worst or average case by using the properties of the dynamic programming matrix [30, 11, 16, 31, 9]. Others filter the text to quickly eliminate uninteresting parts [29, 28, 10, 14, 24] some of them being sublinear on average for moderate ff (i.e. they do not inspect all the text characters) Yet other approaches use bit parallelism [3] in a computer word of w bits to reduce the number of ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100-- 118, 1985.

....down considerably. We then tested the approximate string matching algorithm against two other algorithms, one by Ukkonen [Uk85a] which we implemented) and one by Galil and Park [GP90] labeled MN2; the program was provided to us by W. I. Chang) which is based on another technique by Ukkonen [Uk85b]. The last algorithm was found by Chang and Lawler [CL90] to be the fastest among the algorithms they tested. We used random text (of size 1,000,000) and pattern (of size 20) and two different alphabet sizes. In this case, since we use the idea of partitioning the pattern, the size of the ....

Ukkonen E., "Algorithms for approximate string matching," Information and Control, 64, (1985), pp. 100-118.

....in the case when the cost function is uniform; i.e. weight of an edit operation does not depend on the symbols involved in the operation, but only on its type. Uniform and even more restricted cases of cost functions have been studied in the literature in the context of ordinary edit distances [12, 15, 10]. We assume four types of operations: insertion, deletion, matching and non matching substitutions. The previously suggested algorithms for NED do not specialize to a faster than O(mn 2 ) time algorithm when the cost function is uniform. We propose an algorithm UniformNED which iteratively ....

....distance calculation may be a suitable candidate for A, if it is not general enough to meet the conditions in corollary 1. For example edit distance algorithms which assume a fixed cost function (e.g. fl = 1; 1; 0; 1) and achieve fast running times by using this constant nature of the weights [15, 10] are evidently not suitable for UniformNED. This is because each execution of Step 6 of the algorithm uses a different cost function fl 0 . Masek and Paterson [8] gave a C(m;n) O(mn= log n) time algorithm which can be used as algorithm A here, but it is no longer true that the running time of ....

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E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

....align conserved sequence regions, such as protein coding regions, or mitochondrial tRNA genes [58] However, aligning sequences where insertions and deletions are more frequent is often inevitable. Pairwise alignment problems have been thoroughly studied in the past few decades [25] 30] 60] [85], but multiple sequence alignment remains an open research area [32] 88] see [2] 8] 57] for surveys) In addition to the ambiguity inherent in selecting gap cost and alignment cost criteria, multiple sequence alignment has also been proven to be computationally intractable [41] 87] Step 3: ....

E. Ukkonen, Algorithms for Approximate String Matching, Information and Control, 64 (1985), pp. 100--118.

....Thus a sequence of CVs hypothesized is obtained for the given name utterance. This hypothesized sequence is matched with actual CV sequences of names (given in Table 6.1) to recognize the input name utterance. The longest common subsequence algorithm is used for matching the CV sequences [78]. The algorithm gives a similarity measure in terms of the number of matched CVs and the number of substitution errors, in presence of insertion and deletion errors in the hypothesized CV sequence. The name of the actual CV sequence that gives the highest similarity measure is assigned to the ....

....operations in the sequence is minimized. The minimum cost is denoted by D (A; B) and is called the edit distance from A to B. The problem of computing D (A; B) is also known as 117 string to string correction problem. The implementations details of the string matching algorithms are described in [78,79]. In our studies the UNIX di command is used to compute the edit distance between the hypothesized sequence of CVs and actual sequence of CVs in names. 6.5.2 Performance of the name recognition system The speech signal waveform of an utterance of name Visakhapattanam and derivative of the ....

E. Ukkonen, \Algorithms for approximate string matching," Information and control, vol. 64, pp. 100-118, 1985.

....In this paper, we take the Levenshtein distance as a basis. 1 The standard algorithm for computing the Levenshtein distance between two words by Wagner and Fisher [WF74] uses a dynamic programming scheme that leads to quadratic time complexity. Even with more sophisticated algorithms (cf. Ukk85, Mye99] it is not realistic to compute the Levenshtein distance between the input word W and each of the words in the dictionary, already for dictionaries of a modest size. Even if there are linear algorithms for computing n gram distances [Ukk92] the situation is basically the same for these ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100-118, 1985.

....University of Hong Kong Ming Li x University of Waterloo Abstract Multiple sequence alignment is a fundamental problem in computational biology. Because of its notorious difficulties, aligning sequences within a constant band is a popular practice in bioinformatics with good practical results [17, 13, 14, 15, 1, 3, 6, 20, 18]. However, the problem is still NP hard for multiple sequences. In this paper, we present a theoretical study of this problem. In particular, we present polynomial time approximation schemes (PTAS) for multiple sequence alignment within a constant band, under standard models of SP alignment and ....

.... in [17] Alignment within a band is used in the final stage of the FASTA program for rapid searching of protein and DNA sequence databases [13, 14] Pearson has shown that alignment within a band gives very good results for lots of protein superfamilies [15] Other references can be found in [1, 3, 6, 20]. Spouge gives a survey on this topic in [18] We first define our problem. 2 Definition 3 c Diagonal Alignment Let S = fs 1 ; s 2 ; s n g be a set of n sequences, each of length m, and M an alignment of the n sequences. The length of the alignment M be M . A c diagonal alignment M is ....

E. Ukkonen, Algorithms for approximate string matching, Inform. Control, 64, pp. 100-118, 1985.

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E. Ukkonen. Algorithms for approximate string matching. Information and Control 64(13):100118, 1985.

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E. Ukkonen. Algorithms for approximate string matching. Information and Control 64(13):100118, 1985.

....) time for searching a pattern of length m that is run length compressed to length m , in a run length compressed text of length n . We also study the LCS calculation. First, we give a greedy algorithm for the LCS that works in O(m ) time. Adapting the well known diagonal method [24], we are able to improve the greedy method to work in O(d ; m ) time, where d is the edit distance between the two strings (under insertions and deletions with the unit cost model) Then we present improvements for the greedy method for the LCS, which do not however a ect the worst case, ....

....time, and in equal letter boxes we can trace an optimal path to a corner in O(m ) time. Thus, we can calculate all the corner values in O(m ) time . It turns out that we can improve the greedy algorithm signi cantly by fairly simple means. We notice that the diagonal method of [24] can be applied, and yields an ) algorithm, where d = D ID (A; B) We also give other improvements Apostolico et al. 4] also gave a basic O(m ) algorithm for the LCS, which they then improved to O(m ) Their basic algorithm di ers from our greedy algorithm in that ....

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E. Ukkonen. Algorithms for approximate string matching. Information and Control 64(1-3):100-118, 1985.

....data generally does not exist in an ion linej form, but is always created from some richer source, such as a set of MIDI les. Furthermore, the representations in Chapters 7 and 8 can be said to resemble an index in some respects. 4. 2 A Generalization of the Problem A useful generalization [35] of the approximate string matching problem is that we have a distance function D = T; ffi) in a discrete alphabet Sigma. T ae Sigma Theta Sigma , where Sigma is the set of all strings over Sigma, is the set of allowed editing operations (x; y) usually written as x y. An element x y ....

E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100118, 1985. 80

....0 , in a run length compressed text of length n 0 . This algorithm works for both Levenshtein and LCS distance models. We also study the LCS calculation. First, we give a greedy algorithm for the LCS that works in O(m 0 n 0 (m 0 n 0 ) time. Adapting the well known diagonal method [17], we are able to improve the greedy method to work in O(d 2 min(n 0 ; m 0 ) time, where d is the edit distance between the two strings (under insertions and deletions with the unit cost model) Then we present improvements for the greedy method for the LCS, that do not however a ect the ....

....boxes we can trace an optimal path to a corner in O(m 0 n 0 ) time. Thus, we can calculate all the corner values in O(m 0 n 0 (m 0 n 0 ) time 1 . It turns out that we can improve the greedy algorithm signi cantly by fairly simple means. We notice that the diagonal method of [17] can be applied, and achieve an O(d 2 min(n 0 m 0 ) algorithm. We give also other improvements that do not a ect the worst case, but are signi cant in the average case and in practice. We end the section conjecturing that our improved algorithm runs in O(m 0 n 0 ) time in the average. ....

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E. Ukkonen. Algorithms for approximate string matching. Information and Control 64(13):100118, 1985.

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Ukkonen, E. (1985), `Algorithms for approximate string matching', Information and Control Vol 64.

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E. Ukkonen, Algorithms for approximate string matching., Information and Control, 64:100-118, 1985.

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Ukkonen, E.: Algorithms for approximate string matching. Inform. and Control 64, 1985, 100--118.

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Esko Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

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E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

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E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64(1--3):100--118, 1985.

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E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

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E. Ukkonen, Algorithms for approximate string matching, Information and Control, 64 (1985) 100-118

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E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64(1--3):100--118, 1985.

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E. Ukkonen. Algorithms for approximate string matching. Inf. Control, 64(1{ 3):100-118, 1985.

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Esko Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100-- 118, 1985.

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Esko Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

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E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100--118, 1985.

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E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100-- 118, 1985.

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E. Ukkonen. Algorithms for approximate string matching. Information and Control, 64(1--3):100--118, 1985.

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Esko Ukkonen. Algorithms for approximate string matching. Information and Control, 64:100-118, 1985.

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Ukkonen, E., Algorithms for approximate string matching. Information and Control 64 (1985) 100--118

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Esko Ukkonen, `Algorithms for approximate string matching', Information and Control, 64, 100-- 118 (1985).

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E. Ukkonen, Algorithms for approximate string matching, Inform. Control 64, 100 ---118 (1985).

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E. Ukkonen, Algorithms for approximate string matching, Inform. Control, 64, pp. 100-118, 1985.

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