Locally Maximal Common Factors as a Tool for Efficient Dynamic String Algorithms

Amir, Amihood; Boneh, Itai

doi:10.4230/lipics.cpm.2018.11

Cited by 4 publications

(4 citation statements)

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“…Recall that a τ -run is a run of length at least 3τ − 1 with period at most 1 3 τ . Suppose, towards a contradiction, that two distinct τ -runs overlap by more than 2 3 τ positions. By the fact that their periods are at most 1 3 τ and an application of the periodicity lemma (Lemma 5), we obtain that these two τ -runs cannot be distinct as they share the same shortest period; a contradiction.…”

Section: Preliminariesmentioning

confidence: 97%

“…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%

“…By the fact that their periods are at most 1 3 τ and an application of the periodicity lemma (Lemma 5), we obtain that these two τ -runs cannot be distinct as they share the same shortest period; a contradiction. This implies that two distinct τ -runs can overlap by no more than 2 3 τ positions. In turn, this implies that T contains O(n/τ ) runs.…”

Section: Preliminariesmentioning

confidence: 99%

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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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Section: Preliminariesmentioning

confidence: 97%

Section: Other Related Workmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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Faster Algorithms for Longest Common Substring

Charalampopoulos,

Kociumaka,

Pissis

et al. 2021

Preprint

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“…Abedin et al [1] improved the tradeoffs for this problem by designing a more efficient solution for the so-called heaviest induced ancestors problem. Amir and Boneh [3] investigated some special cases of the partially dynamic LCS problem (in which one of the strings is assumed to be static); namely, the case where the static string is periodic and the case where the substitutions in the dynamic string are substitutions with some letter # ∈ Σ. Finally, Amir et al [7] presented the first algorithm for the fully dynamic LCS problem (in which both strings are subject to updates) that needs only sublinear time per edit operation (insertion or deletion of a letter), namely Õ(n 2/3 ) time.…”

Section: Problem: Longest Common Substringmentioning

confidence: 99%

Dynamic Longest Common Substring in Polylogarithmic Time

Charalampopoulos,

Gawrychowski,

Pokorski

2020

Preprint

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The longest common substring problem consists in finding a longest string that appears as a (contiguous) substring of two input strings. We consider the dynamic variant of this problem, in which we are to maintain two dynamic strings S and T , each of length at most n, that undergo substitutions of letters, in order to be able to return a longest common substring after each substitution. Recently, Amir et al. [ESA 2019] presented a solution for this problem that needs only Õ(n 2/3 ) time per update. This brought the challenge of determining whether there exists a faster solution with polylogarithmic update time, or (as is the case for other dynamic problems), we should expect a polynomial (conditional) lower bound. We answer this question by designing a significantly faster algorithm that processes each substitution in amortized log O(1) n time with high probability. Our solution relies on exploiting the local consistency of the parsing of a collection of dynamic strings due to Gawrychowski et al. [SODA 2018], and on maintaining two dynamic trees with labeled bicolored leaves, so that after each update we can report a pair of nodes, one from each tree, of maximum combined weight, which have at least one common leaf-descendant of each color. We complement this with a lower bound of Ω(log n/ log log n) for the update time of any polynomial-size data structure that maintains the LCS of two dynamic strings, and the same lower bound for the update time of any data structure of size Õ(n) that maintains the LCS of a static and a dynamic string. Both lower bounds hold even allowing amortization and randomization.

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