Towards Approximate Matching in Compressed Strings: Local Subsequence Recognition

Tiskin, Alexander

doi:10.1007/978-3-642-20712-9_32

Cited by 13 publications

(8 citation statements)

References 19 publications

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“…There has also been a significant amount of recent research effort on developing string processing algorithms that operate directly on grammar-compressed data -i.e., without prior decompression. To date this work includes algorithms for computing subword complexity [3,19], online subsequence matching and approximate matching [4,47], faster edit distance computation [17,22], and computation of various kinds of biologically relevant repetitions [1,27,24,25]. We will often use a DAG (directed acyclic graph) representation of the grammar with one source (corresponding to the unique non-terminal that generates the whole string) and σ sinks (corresponding to the terminals).…”

Section: Introductionmentioning

confidence: 99%

Access, Rank, and Select in Grammar-compressed Strings

Belazzougui

Cording

Puglisi

et al. 2015

Algorithms - ESA 2015

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Given a string S of length N on a fixed alphabet of σ symbols, a grammar compressor produces a context-free grammar G of size n that generates S and only S. In this paper we describe data structures to support the following operations on a grammar-compressed string: rankc(S, i) (return the number of occurrences of symbol c before position i in S); selectc(S, i) (return the position of the ith occurrence of c in S); and access(S, i, j) (return substring S[i, j]). For rank and select we describe data structures of size O(nσ log N ) bits that support the two operations in O(log N ) time. We propose another structure that uses O(nσ log(N/n)(log N ) 1+ ) bits and that supports the two queries in O(log N/ log log N ), where > 0 is an arbitrary constant. To our knowledge, we are the first to study the asymptotic complexity of rank and select in the grammar-compressed setting, and we provide a hardness result showing that significantly improving the bounds we achieve would imply a major breakthrough on a hard graph-theoretical problem. Our main result for access is a method that requires O(n log N ) bits of space and O(log N + m/ log σ N ) time to extract m = j − i + 1 consecutive symbols from S. Alternatively, we can achieve O(log N/ log log N +m/ log σ N ) query time using O(n log(N/n)(log N ) 1+ ) bits of space. This matches a lower bound stated by Verbin and Yu for strings where N is polynomially related to n.

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Section: Introductionmentioning

confidence: 99%

Access, Rank, and Select in Grammar-compressed Strings

Belazzougui

Cording

Puglisi

et al. 2015

Algorithms - ESA 2015

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“…We presented a number of such algorithmic applications in [41,42,[61][62][63]65], and two biological applications in [7,52]. Our new distance multiplication algorithm implies immediate improvements in running time for a number of string comparison and graph algorithms: semi-local longest common subsequences between permutations; longest increasing subsequence in a cyclic permutation; maximum clique in a circle graph; longest common subsequence between a grammar-compressed string and an uncompressed string.…”

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confidence: 99%

Fast Distance Multiplication of Unit-Monge Matrices

Tiskin

2013

Algorithmica

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Monge matrices play a fundamental role in optimisation theory, graph and string algorithms. Distance multiplication of two Monge matrices of size n can be performed in time O(n 2 ). Motivated by applications to string algorithms, we introduced in previous works a subclass of Monge matrices, that we call simple unit-Monge matrices. We also gave a distance multiplication algorithm for such matrices, running in time O(n 1.5 ). Landau asked whether this problem can be solved in linear time. In the current work, we give an algorithm running in time O(n log n), thus approaching an answer to Landau's question within a logarithmic factor. The new algorithm implies immediate improvements in running time for a number of algorithms on strings and graphs. In particular, we obtain an algorithm for finding a maximum clique in a circle graph in time O(n log 2 n), and a surprisingly efficient algorithm for comparing compressed strings. We also point to potential applications in group theory, by making a connection between unit-Monge matrices and Coxeter monoids. We conclude that unit-Monge matrices are a fascinating object and a powerful tool, that deserve further study from both the mathematical and the algorithmic viewpoints.

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“…They presented O(nm 2 log m) time algorithms for solving the problems for an SLP of size n and subsequence pattern of length m. Later, an improved algorithm running in time O(nm 1.5 ) was presented by Tiskin [14]. Later, Tiskin improved the running time to O(nm log m) [15]. In this paper, we further reduce the time complexities to O(nm).…”

Section: Introductionmentioning

confidence: 94%

Faster Subsequence and Don’t-Care Pattern Matching on Compressed Texts

Yamamoto

Bannai

Inenaga

et al. 2011

Combinatorial Pattern Matching

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Subsequence pattern matching problems on compressed text were first considered by Cégielski et al. (Window Subsequence Problems for Compressed Texts, Proc. CSR 2006, LNCS 3967, pp. 127-136), where the principal problem is: given a string T represented as a straight line program (SLP) T of size n, a string P of size m, compute the number of minimal subsequence occurrences of P in T. We present an O(nm) time algorithm for solving all variations of the problem introduced by Cégielski et al.. This improves the previous best known algorithm of Tiskin (Towards approximate matching in compressed strings: Local subsequence recognition, Proc. CSR 2011), which runs in O(nm log m) time. We further show that our algorithms can be modified to solve a wider range of problems in the same O(nm) time complexity, and present the first matching algorithms for patterns containing VLDC (variable length don't care) symbols, as well as for patterns containing FLDC (fixed length don't care) symbols, on SLP compressed texts.

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Towards Approximate Matching in Compressed Strings: Local Subsequence Recognition

Cited by 13 publications

References 19 publications

Access, Rank, and Select in Grammar-compressed Strings

Access, Rank, and Select in Grammar-compressed Strings

Fast Distance Multiplication of Unit-Monge Matrices

Faster Subsequence and Don’t-Care Pattern Matching on Compressed Texts

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