Circular pattern matching with k mismatches

Charalampopoulos, Panagiotis; Kociumaka, Tomasz; Pissis, Solon P.; Radoszewski, Jakub; Rytter, Wojciech; Straszyński, Juliusz; Waleń, Tomasz; Zuba, Wiktor

doi:10.1016/j.jcss.2020.07.003

Cited by 4 publications

(7 citation statements)

References 39 publications

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“…Thus, a relatively large body of work has been devoted to practically fast algorithms for approximate CPM; see [2,4,5,7,8,31,37,38] and references therein. All previous results for approximate CPM are average-case upper bounds or heuristics, except for the work of Charalampopoulos et al [17].…”

Section: Circular Pattern Matching (Cpm)mentioning

confidence: 99%

“…In this section, we give a solution to PeriodicSubstringMatch. Let us recall the notion of misperiods that was introduced in [17].…”

Section: The Reporting Version Of K-mismatch Cpmmentioning

confidence: 99%

“…The following problem is a simpler version of PeriodicSubstringMatch that was considered in [17]. (Actually, [17] considered a slightly more restricted problem which required that no two misperiods in U (p) and V (x) are aligned and computed a superset of its solution that corresponds exactly to the statement below.) PeriodicPeriodicMatch(U, V ) Input: Same as in PeriodicSubstringMatch.…”

Section: The Reporting Version Of K-mismatch Cpmmentioning

confidence: 99%

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Approximate Circular Pattern Matching

Charalampopoulos¹,

Kociumaka²,

Radoszewski³

et al. 2022

Preprint

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We consider approximate circular pattern matching (CPM, in short) under the Hamming and edit distance, in which we are given a length-n text T , a length-m pattern P , and a threshold k > 0, and we are to report all starting positions of fragments of T (called occurrences) that are at distance at most k from some cyclic rotation of P . In the decision version of the problem, we are to check if any such occurrence exists. All previous results for approximate CPM were either average-case upper bounds or heuristics, except for the work of Charalampopoulos et al. [CKP + , JCSS'21], who considered only the Hamming distance. For the reporting version of the approximate CPM problem, under the Hamming distance we improve upon the main algorithm of [CKP + , JCSS'21] from O(n + (n/m) • k 4 ) to O(n + (n/m) • k 3 log log k) time; for the edit distance, we give an O(nk 2 )-time algorithm. We also consider the decision version of the approximate CPM problem. Under the Hamming distance, we obtain an O(n + (n/m) • k 2 log k/ log log k)-time algorithm, which nearly matches the algorithm by Chan et al. [CGKKP, STOC'20] for the standard counterpart of the problem. Under the edit distance, the O(nk log 3 k) runtime of our algorithm nearly matches the O(nk) runtime of the Landau-Vishkin algorithm [LV, J. Algorithms'89]. As a stepping stone, we propose an O(nk log 3 k)-time algorithm for the Longest Prefix k -Approximate Match problem, proposed by Landau et al. [LMS, SICOMP'98], for all k ∈ {1, . . . , k}.We give a conditional lower bound that suggests a polynomial separation between approximate CPM under the Hamming distance over the binary alphabet and its non-circular counterpart. We also show that a strongly subquadratic-time algorithm for the decision version of approximate CPM under edit distance would refute the Strong ETH.

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Section: Circular Pattern Matching (Cpm)mentioning

confidence: 99%

“…In this section, we give a solution to PeriodicSubstringMatch. Let us recall the notion of misperiods that was introduced in [17].…”

Section: The Reporting Version Of K-mismatch Cpmmentioning

confidence: 99%

Section: The Reporting Version Of K-mismatch Cpmmentioning

confidence: 99%

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Approximate Circular Pattern Matching

Charalampopoulos¹,

Kociumaka²,

Radoszewski³

et al. 2022

Preprint

Self Cite

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“…In this case we also obtain a set of O(n/ℓ) anchors, where ℓ is the length of k-LCS. If the common substring is far from highly periodic, we use a synchronising set for τ = Θ(ℓ), and otherwise we generate anchors using a technique of misperiods that was initially introduced for k-mismatch pattern matching [19,29]. Now the families P, Q need to consist not simply of substrings of S and T , but rather of modified substrings generated by an approach that resembles k-errata trees [32].…”

Section: Solution To Lcsmentioning

confidence: 99%

“…A τsynchronising set can be computed in O(n) time by Theorem 7 and all the τ -runs, together with the position of the first occurrence of their Lyndon root, can be computed in O(n) time [8]. After an O(n)-time preprocessing, for every τ -run, we can compute the set of the k + 1 misperiods of its period to either side in O(1) time; see [29,Claim 18]. ◀…”

Section: Proof For Two Stringsmentioning

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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Circular Pattern Matching with k Mismatches

Charalampopoulos

Kociumaka

Pissis

et al. 2019

Fundamentals of Computation Theory

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The k-mismatch problem consists in computing the Hamming distance between a pattern P of length m and every length-m substring of a text T of length n, if this distance is no more than k. In many real-world applications, any cyclic shift of P is a relevant pattern, and thus one is interested in computing the minimal distance of every length-m substring of T and any cyclic shift of P . This is the circular pattern matching with k mismatches (k-CPM) problem. A multitude of papers have been devoted to solving this problem but, to the best of our knowledge, only average-case upper bounds are known. In this paper, we present the first non-trivial worst-case upper bounds for the k-CPM problem. Specifically, we show an O(nk)-time algorithm and an O(n + n m k 5 )-time algorithm. The latter algorithm applies in an extended way a technique that was very recently developed for the k-mismatch problem [Bringmann et al., SODA 2019].

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Circular pattern matching with k mismatches

Cited by 4 publications

References 39 publications

Approximate Circular Pattern Matching

Approximate Circular Pattern Matching

Faster Algorithms for Longest Common Substring

Circular Pattern Matching with k Mismatches

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