Polynomial-Time Approximation Algorithms for Weighted LCS Problem

Cygan, Marek; Kubica, Marcin; Radoszewski, Jakub; Rytter, Wojciech; Waleń, Tomasz

doi:10.1007/978-3-642-21458-5_38

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…than two strings (e.g., [1,25]), with variations such as string consensus (e.g., [11,12]) and more (e.g., [9,19,35,41,42,61]). Since natural language texts are well compressible, researchers also considered solving LCS directly on compressed strings, using either run-length encoding (e.g., [15,31,34,57]) or straight-line programs and other Lempel-Ziv-like compression schemes (e.g., [40,44,63,80]).…”

Section: Referencementioning

confidence: 99%

Multivariate Fine-Grained Complexity of Longest Common Subsequence

Bringmann¹,

Künnemann²

2018

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

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We revisit the classic combinatorial pattern matching problem of finding a longest common subsequence (LCS). For strings x and y of length n, a textbook algorithm solves LCS in time O(n 2 ), but although much effort has been spent, no O(n 2−ε )-time algorithm is known. Recent work indeed shows that such an algorithm would refute the Strong Exponential Time Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams FOCS'15; Bringmann, Künnemann FOCS'15].Despite the quadratic-time barrier, for over 40 years an enduring scientific interest continued to produce fast algorithms for LCS and its variations. Particular attention was put into identifying and exploiting input parameters that yield strongly subquadratic time algorithms for special cases of interest, e.g., differential file comparison. This line of research was successfully pursued until 1990, at which time significant improvements came to a halt. In this paper, using the lens of fine-grained complexity, our goal is to (1) justify the lack of further improvements and (2) determine whether some special cases of LCS admit faster algorithms than currently known.To this end, we provide a systematic study of the multivariate complexity of LCS, taking into account all parameters previously discussed in the literature: the input size n := max{|x|, |y|}, the length of the shorter string m := min{|x|, |y|}, the length L of an LCS of x and y, the numbers of deletions δ := m − L and ∆ := n − L, the alphabet size, as well as the numbers of matching pairs M and dominant pairs d. For any class of instances defined by fixing each parameter individually to a polynomial in terms of the input size, we prove a SETH-based lower bound matching one of three known algorithms (up to lower order factors of the form n o(1) ). Specifically, we determine the optimal running time for LCS under SETH as (n + min{d, δ∆, δm}) 1±o(1) . Polynomial improvements over this running time must necessarily refute SETH or exploit novel input parameters. We establish the same lower bound for any constant alphabet of size at least 3. For binary alphabet, we show a SETH-based lower bound of (n + min{d, δ∆, δM/n}) 1−o(1) and, motivated by difficulties to improve this lower bound, we design an O(n+δM/n)-time algorithm, yielding again a matching bound.We feel that our systematic approach yields a comprehensive perspective on the well-studied multivariate complexity of LCS, and we hope to inspire similar studies of multivariate complexity landscapes for further polynomialtime problems.

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Section: Referencementioning

confidence: 99%

Multivariate Fine-Grained Complexity of Longest Common Subsequence

Bringmann¹,

Künnemann²

2018

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

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“…This problem is a natural variant of LCS that, e.g., came up in a SETH-hardness proof of LCS [1]. It is not to be confused with other weighted variants of LCS that have been studied in the literature, such as a statistical distance measure where given the probability of every symbol's occurrence at every text location the task is to find a long and likely subsequence [6,18], a variant of LCS that favors consecutive matches [36], or edit distance with given operation costs [13].…”

Section: Wlcs: In Between Min-quadratic and Rectangular Timementioning

confidence: 99%

Sketching, Streaming, and Fine-Grained Complexity of (Weighted) LCS

Bringmann

Chaudhury

2018

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We study sketching and streaming algorithms for the Longest Common Subsequence problem (LCS) on strings of small alphabet size |Σ|. For the problem of deciding whether the LCS of strings x, y has length at least L, we obtain a sketch size and streaming space usage of O(L |Σ|−1 log L). We also prove matching unconditional lower bounds.As an application, we study a variant of LCS where each alphabet symbol is equipped with a weight that is given as input, and the task is to compute a common subsequence of maximum total weight. Using our sketching algorithm, we obtain an O(min{nm, n + m |Σ| })-time algorithm for this problem, on strings x, y of length n, m, with n ≥ m. We prove optimality of this running time up to lower order factors, assuming the Strong Exponential Time Hypothesis.

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“…A great deal of research has been conducted on weighted strings for pattern matching [3,4], for computing various types of regularities [5,6,7,8], for indexing [3,9], and for alignments [10,11]. The efficiency of most of the proposed algorithms relies on the assumption of a given constant cumulative weight threshold defining the minimal probability of occurrence of factors in the weighted string.…”

Section: Introductionmentioning

confidence: 99%

Linear-time computation of prefix table for weighted strings & applications

Barton

Liu

Pissis

2016

Theoretical Computer Science

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The prefix table of a string is one of the most fundamental data structures of algorithms on strings: it determines the longest factor at each position of the string that matches a prefix of the string. It can be computed in time linear with respect to the size of the string, and hence it can be used efficiently for locating patterns or for regularity searching in strings. A weighted string is a string in which a set of letters may occur at each position with respective occurrence probabilities. Weighted strings, also known as position weight matrices or uncertain strings, naturally arise in many biological contexts; for example, they provide a method to realise approximation among occurrences of the same DNA segment. In this article, given a weighted string x of length n and a constant cumulative weight threshold 1/z, defined as the minimal probability of occurrence of factors in x, we present an O(n)-time algorithm for computing the prefix table of x. Furthermore, we outline a number of applications of this result for solving various problems on non-standard strings, and present some preliminary experimental results.

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Polynomial-Time Approximation Algorithms for Weighted LCS Problem

Cited by 4 publications

References 15 publications

Multivariate Fine-Grained Complexity of Longest Common Subsequence

Multivariate Fine-Grained Complexity of Longest Common Subsequence

Sketching, Streaming, and Fine-Grained Complexity of (Weighted) LCS

Linear-time computation of prefix table for weighted strings & applications

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