Time-Space Trade-Offs for the Longest Common Substring Problem

Starikovskaya, Tatiana; Vildhøj, Hjalte Wedel

doi:10.1007/978-3-642-38905-4_22

Cited by 17 publications

(14 citation statements)

References 12 publications

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“…The suffix tree of T 1 and T 2 , a data structure containing all suffixes of T 1 and T 2 , allows to solve this problem in linear time and space [36,17,21], which is optimal as any algorithm needs Ω(n) time to read and Ω(n) space to store the strings. However, if we only account for "additional" space, the space the algorithm uses apart from the space required to store the input, then the suffix tree-based solution is not optimal and has been improved in a series of publications [5,26,32].…”

Section: Related Workmentioning

confidence: 99%

Longest Common Substring with Approximately k Mismatches

2019

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In the longest common substring problem, we are given two strings of length n and must find a substring of maximal length that occurs in both strings. It is well known that the problem can be solved in linear time, but the solution is not robust and can vary greatly when the input strings are changed even by one character. To circumvent this, Leimeister and Morgenstern introduced the problem of the longest common substring with k mismatches. Lately, this problem has received a lot of attention in the literature. In this paper, we first show a conditional lower bound based on the SETH hypothesis implying that there is little hope to improve existing solutions. We then introduce a new but closely related problem of the longest common substring with approximately k mismatches and use locality-sensitive hashing to show that it admits a solution with strongly subquadratic running time. We also apply these results to obtain a strongly subquadratic-time 2-approximation algorithm for the longest common substring with k mismatches problem and show conditional hardness of improving its approximation ratio. * This is a full and extended version of the conference paper [31].

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Section: Related Workmentioning

confidence: 99%

Longest Common Substring with Approximately k Mismatches

2019

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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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“…Accurate fragment reassembly requires not only a powerful shape description method but also an efficient shape matching method. The simplest matching method is to compare all the elements of two strings one by one and find the longest common substring (LCS) [17], but it is not so applicable when fragments have partial edge features missing. Another method is to look for the longest common subsequence (TLCS) [18][19].…”

Section: Curve Contour Matching Algorithm Based On Ddtwmentioning

confidence: 99%

Automatic Reassembly Method of 3D Thin-wall Fragments Based on Derivative Dynamic Time Warping

Qing

et al. 2020

IJEM

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In order to address the automatic virtual reassembling of 3D thin-wall fragments, this paper proposes a 3D fragment reassembly method based on derivative dynamic time warping. Firstly, a calculation method of discrete curvature and torsion is designed to solve the difficulty of calculating curvature and torsion of discrete data points and eliminate effectively the noise interferences in the calculation process. Then, it takes curvature and torsion as the feature descriptors of the curve, searches the candidate matching line segments by the derivative dynamic time warping (DDTW) method with the feature descriptors, and records the positions of the starting and ending points of each candidate matching segment. After that, it designs a voting mechanism with the geometric invariant as the constraint information to select further the optimal matching line segments. Finally, it adopts the least squares method to estimate the rotation and transformation matrices and uses the iterative closest point (ICP) method to complete the reassembly of fragments. The experimental results show that the reassembly error is less than 1mm and that the reassembly effect is good. The method can solve the 3D curve matching in case there are partial feature defects, and can achieve the virtual restoration of the broken thin-wall fragment model quickly and effectively.

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Time-Space Trade-Offs for the Longest Common Substring Problem

Cited by 17 publications

References 12 publications

Longest Common Substring with Approximately k Mismatches

Longest Common Substring with Approximately k Mismatches

Faster Algorithms for Longest Common Substring

Automatic Reassembly Method of 3D Thin-wall Fragments Based on Derivative Dynamic Time Warping

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