Sublinear Space Algorithms for the Longest Common Substring Problem

Kociumaka, Tomasz; Starikovskaya, Tatiana; Vildhøj, Hjalte Wedel

doi:10.1007/978-3-662-44777-2_50

Cited by 22 publications

(18 citation statements)

References 20 publications

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“…Given two strings S, T ∈ Σ n of length n, the LONGEST COMMON SUBSTRING problem asks for the length of the longest string that appears in both S and T as a contiguous substring. The problem can be solved in optimal Θ(n) time using Suffix-Trees [Gus97] (See [KSV14] for a recent spaceefficient algorithm). We consider the variant in which the strings S and T are over (Σ ∪ { }) n , where 's correspond to "don't care" characters.…”

Section: Our Resultsmentioning

confidence: 99%

More Applications of the Polynomial Method to Algorithm Design

Abboud¹,

Williams²,

Yu³

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

232

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In low-depth circuit complexity, the polynomial method is a way to prove lower bounds by translating weak circuits into low-degree polynomials, then analyzing properties of these polynomials. Recently, this method found an application to algorithm design: Williams (STOC 2014) used it to compute all-pairs shortest paths in n 3 /2 Ω(√ log n) time on dense n-node graphs. In this paper, we extend this methodology to solve a number of problems in combinatorial pattern matching and Boolean algebra, considerably faster than previously known methods. First, we give an algorithm for BOOLEAN ORTHOGONAL DETECTION, which is to detect among two sets A, B ⊆ {0, 1} d of size n if there is an x ∈ A and y ∈ B such that x, y = 0. For vectors of dimension d = c(n) log n, we solve BOOLEAN ORTHOGONAL DETECTION in n 2−1/O(log c(n)) time by a Monte Carlo randomized algorithm. We apply this as a subroutine in several other new algorithms:

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Section: Our Resultsmentioning

confidence: 99%

More Applications of the Polynomial Method to Algorithm Design

Abboud¹,

Williams²,

Yu³

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

232

View full text Add to dashboard Cite

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“…Other results on the LCS problem include the linear-time computation of an LCS of several strings over an integer alphabet [46], trade-offs between the time and the working space for computing an LCS of two strings [13,53,60], and the dynamic maintenance of an LCS [2,3,27]. Very recently, a strongly sublinear-time quantum algorithm and a lower bound for the quantum setting were shown [41].…”

Section: Other Related Workmentioning

confidence: 99%

Faster Algorithms for Longest Common Substring

Charalampopoulos,

Kociumaka,

Pissis

et al. 2021

Preprint

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In the classic longest common substring (LCS) problem, we are given two strings S and T , each of length at most n, over an alphabet of size σ, and we are asked to find a longest string occurring as a fragment of both S and T . Weiner, in his seminal paper that introduced the suffix tree, presented an O(n log σ)-time algorithm for this problem [SWAT 1973]. For polynomially-bounded integer alphabets, the linear-time construction of suffix trees by Farach yielded an O(n)-time algorithm for the LCS problem [FOCS 1997]. However, for small alphabets, this is not necessarily optimal for the LCS problem in the word RAM model of computation, in which the strings can be stored in O(n log σ/ log n) space and read in O(n log σ/ log n) time. We show that, in this model, we canWe then lift our ideas to the problem of computing a k-mismatch LCS, which has received considerable attention in recent years. In this problem, the aim is to compute a longest substring of S that occurs in T with at most k mismatches. Flouri et al. showed how to compute a 1-mismatch LCS in O(n log n) time [IPL 2015]. Thankachan et al. extended this result to computing a k-mismatch LCS in O(n log k n) time for k = O(1) [J. Comput. Biol. 2016]. We show an O(n log k−1/2 n)-time algorithm, for any constant k > 0 and irrespective of the alphabet size, using O(n) space as the previous approaches. We thus notably break through the well-known n log k n barrier, which stems from a recursive heavy-path decomposition technique that was first introduced in the seminal paper of Cole et al. [STOC 2004] for string indexing with k errors.

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“…Recently, Kociumaka et al [7] showed that for any tradeoff parameter 1 ≤ τ ≤ n, the LCF problem can be solved in O(τ ) space and O(n 2 /τ ) time. Applying this to the LCAF problem, we obtain an algorithm using O(τ σn 2 ) time and O(σn/τ ) space, for any 1 ≤ τ ≤ σn.…”

Section: Reducing Space Usage In Alatabbi Et Al's Algorithmmentioning

confidence: 99%

Longest Common Abelian Factors and Large Alphabets

Badkobeh

Gagie

Grabowski

et al. 2016

String Processing and Information Retrieval

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Two strings X and Y are considered Abelian equal if the letters of X can be permuted to obtain Y (and vice versa). Recently, Alatabbi et al. (2015) considered the longest common Abelian factor problem in which we are asked to find the length of the longest Abelian-equal factor present in a given pair of strings. They provided an algorithm that uses O(σn 2) time and O(σn) space, where n is the length of the pair of strings and σ is the alphabet size. In this paper we describe an algorithm that uses O(n 2 log 2 n log * n) time and O(n log 2 n) space, significantly improving Alatabbi et al.'s result unless the alphabet is small. Our algorithm makes use of techniques for maintaining a dynamic set of strings under split, join, and equality testing (Melhorn et al., Algorithmica 17(2), 1997).

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Sublinear Space Algorithms for the Longest Common Substring Problem

Cited by 22 publications

References 20 publications

More Applications of the Polynomial Method to Algorithm Design

More Applications of the Polynomial Method to Algorithm Design

Faster Algorithms for Longest Common Substring

Longest Common Abelian Factors and Large Alphabets

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