Longest Common Factor After One Edit Operation

Amir, Amihood; Charalampopoulos, Panagiotis; Iliopoulos, Costas S.; Pissis, Solon P.; Radoszewski, Jakub

doi:10.1007/978-3-319-67428-5_2

Cited by 14 publications

(27 citation statements)

References 16 publications

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“…Amir et al [11] observed that the problem of computing an LCS after a single edit operation at position i can be decomposed into two queries out of which we choose the one with the maximal answer: an occurrence of an LCS either avoids i or it covers i. The former case can be precomputed.…”

Section: Lemma 4 ( [50]mentioning

confidence: 99%

“…A solution to a special case of three SubStringS LCS queries with W = T was already implicitly presented by Amir et al in [11]. It is based on the heaviest induced ancestors (HIA) problem on trees applied to the suffix tree of T. We generalize the HIA queries and use them to answer general three SubStringS LCS queries.…”

Section: Three Substrings Lcs Queriesmentioning

confidence: 99%

“…We use the following lemma that encapsulates one of the main techniques of [11]. It involves computing so-called ranges of substrings in the generalized suffix array of S and T and it relies on a result by Fischer et al [45].…”

Section: Lcs Contains a Changed Position In Exactly One Of The Stringsmentioning

confidence: 99%

“…Amir et al [11] introduced a restricted dynamic variant, where any single edit operation is allowed and is reverted afterwards. We call this problem LCS after One edit.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic and Internal Longest Common Substring

et al. 2020

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Given two strings S and T, each of length at most n, the longest common substring (LCS) problem is to find a longest substring common to S and T. This is a classical problem in computer science with an $$\mathcal {O}(n)$$ O ( n ) -time solution. In the fully dynamic setting, edit operations are allowed in either of the two strings, and the problem is to find an LCS after each edit. We present the first solution to the fully dynamic LCS problem requiring sublinear time in n per edit operation. In particular, we show how to find an LCS after each edit operation in $$\tilde{\mathcal {O}}(n^{2/3})$$ O ~ ( n 2 / 3 ) time, after $$\tilde{\mathcal {O}}(n)$$ O ~ ( n ) -time and space preprocessing. This line of research has been recently initiated in a somewhat restricted dynamic variant by Amir et al. [SPIRE 2017]. More specifically, the authors presented an $$\tilde{\mathcal {O}}(n)$$ O ~ ( n ) -sized data structure that returns an LCS of the two strings after a single edit operation (that is reverted afterwards) in $$\tilde{\mathcal {O}}(1)$$ O ~ ( 1 ) time. At CPM 2018, three papers (Abedin et al., Funakoshi et al., and Urabe et al.) studied analogously restricted dynamic variants of problems on strings; specifically, computing the longest palindrome and the Lyndon factorization of a string after a single edit operation. We develop dynamic sublinear-time algorithms for both of these problems as well. We also consider internal LCS queries, that is, queries in which we are to return an LCS of a pair of substrings of S and T. We show that answering such queries is hard in general and propose efficient data structures for several restricted cases.

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Section: Lemma 4 ( [50]mentioning

confidence: 99%

Section: Three Substrings Lcs Queriesmentioning

confidence: 99%

Section: Lcs Contains a Changed Position In Exactly One Of The Stringsmentioning

confidence: 99%

“…Amir et al [11] introduced a restricted dynamic variant, where any single edit operation is allowed and is reverted afterwards. We call this problem LCS after One edit.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic and Internal Longest Common Substring

et al. 2020

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“…As for the LCS problem itself, Amir et al [6] initiated the study of this question in the dynamic setting by considering the problem of constructing a data structure over two strings that returns the LCS after a single edit operation in one of the strings. However, in their solution, after each edit operation, the string is immediately reverted to its original version.…”

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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Searching for a Modified Pattern in a Changing Text

Amir

Kondratovsky

2018

String Processing and Information Retrieval

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Longest Common Factor After One Edit Operation

Cited by 14 publications

References 16 publications

Dynamic and Internal Longest Common Substring

Dynamic and Internal Longest Common Substring

Dynamic Longest Common Substring in Polylogarithmic Time

Searching for a Modified Pattern in a Changing Text

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