An efficient algorithm to test square-freeness of strings compressed by straight-line programs

Bannai, Hideo; Gagie, Travis; Tomohiro, I; Inenaga, Shunsuke; Landau, Gad M.; Lewenstein, Moshe

doi:10.1016/j.ipl.2012.06.017

Cited by 6 publications

(5 citation statements)

References 12 publications

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“…As we have shown, proving this one way or the other would lead to progress on the path counting problem in DAGs, an old and interesting problem in graph theory. Relatedly, similar approaches may prove fruitful in judging the hardness of other problems on grammar-compressed strings, many solutions to which currently seem to be loose upperbounds [4,1,27,24,25].…”

Section: Discussionmentioning

confidence: 99%

“…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%

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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: Discussionmentioning

confidence: 99%

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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“…Our algorithm is based on the divide-and-conquer method used in [3] and also [9], which detects squares crossing the boundary of each variable X i . Roughly speaking, in order to detect such squares we take some substrings of val( X i ) as seeds each of which is in charge of distinct squares, and for each seed we detect squares by using LS and LCE a constant number of times.…”

Section: Finding Runsmentioning

confidence: 99%

“…Roughly speaking, in order to detect such squares we take some substrings of val( X i ) as seeds each of which is in charge of distinct squares, and for each seed we detect squares by using LS and LCE a constant number of times. There is a difference between [3] and [9] in how the seeds are taken, and our approach is similar to [3]. In the next subsection, we briefly describe our basic algorithm which runs in O (n 3 h log N) time.…”

Section: Finding Runsmentioning

confidence: 99%

“…Space complexities will be determined by the number of computer words (not bits). Given an SLP whose size is n and the height of its derivation tree is h, Bannai et al [3] showed how to test whether the string s is square-free or not, in O (n 3 h log N) time and O (n 2 ) space. Independently, Khvorost [9] presented an algorithm for computing a compact representation of all squares in s in O (n 3 h log 2 N) time and O (n 2 ) space.…”

Section: Introductionmentioning

confidence: 99%

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Detecting regularities on grammar-compressed strings

Tomohiro

Matsubara

Shimohira

et al. 2015

Information and Computation

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We address the problems of detecting and counting various forms of regularities in a string represented as a straight-line program (SLP) which is essentially a context free grammar in the Chomsky normal form. Given an SLP of size n that represents a string s of length N, our algorithm computes all runs and squares in s inh is the height of the derivation tree of the SLP. We also show an algorithm to compute all gapped-palindromes in O (n 3 h + gnh log N) time and O (n 2 ) space, where g is the length of the gap. As one of the main components of the above solution, we propose a new technique called approximate doubling which seems to be a useful tool for a wide range of algorithms on SLPs. Indeed, we show that the technique can be used to compute the periods and covers of the string in O (n 2 h) time and O (nh(n + log 2 N)) time, respectively.

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