Multivariate Fine-Grained Complexity of Longest Common Subsequence

Bringmann, Karl; Künnemann, Marvin

doi:10.1137/1.9781611975031.79

Cited by 46 publications

(57 citation statements)

References 84 publications

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“…Notably, many conditional lower bounds in P target problems with natural DP algorithms that are proven to be near-optimal under some plausible assumption (see, e.g., [15,3,9,10,1,16,11,23,34] and [45] for an introduction to the field). Even if we restrict our attention to problems that find optimal sequence alignments under some restrictions, such as LCS, Edit Distance and LCIS, the currently known hardness proofs differ significantly, despite seemingly 1 We mention in passing that a systematic study of the complexity of LCS in terms of such input parameters has been performed recently in [17]. 2 We refer to [47] for a simple quadratic-time DP formulation for LCIS.…”

Section: Discussion Outline and Technical Contributionsmentioning

confidence: 99%

Tight Conditional Lower Bounds for Longest Common Increasing Subsequence

2018

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We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called k-LCIS: Given k integer sequences X1, . . . , X k of length at most n, the task is to determine the length of the longest common subsequence of X1, . . . , X k that is also strictly increasing. Especially for the case of k = 2 (called LCIS for short), several algorithms have been proposed that require quadratic time in the worst case.Assuming the Strong Exponential Time Hypothesis (SETH), we prove a tight lower bound, specifically, that no algorithm solves LCIS in (strongly) subquadratic time. Interestingly, the proof makes no use of normalization tricks common to hardness proofs for similar problems such as LCS. We further strengthen this lower bound (1) to rule out O (nL) 1−ε time algorithms for LCIS, where L denotes the solution size, (2) to rule out O n k−ε time algorithms for k-LCIS, and (3) to follow already from weaker variants of SETH. We obtain the same conditional lower bounds for the related Longest Common Weakly Increasing Subsequence problem.

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Section: Discussion Outline and Technical Contributionsmentioning

confidence: 99%

Tight Conditional Lower Bounds for Longest Common Increasing Subsequence

2018

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“…SETH is one of the most fruitful conjectures in the Fine-Grained Complexity. There are numerous conditional lower bounds based on it for problems in P among different areas, including: dynamic data structures [61,7,45,55,3,46,41], computational geometry [24,37,75,67], pattern matching [8,22,21,25,26], graph algorithms [66,40,9,56]. See [72] for a recent survey on SETH-based lower bounds (and more).…”

Section: Related Workmentioning

confidence: 99%

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Chen¹

2020

Theory of Comput.

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In this paper we study the (Bichromatic) Maximum Inner Product Problem (Max-IP), in which we are given sets A and B of vectors, and the goal is to find a ∈ A and b ∈ B maximizing inner product a • b. Max-IP is a basic question and serves as the base problem in the recent breakthrough of [Abboud et al., FOCS 2017] on hardness of approximation for polynomial-time problems. It is also used (implicitly) in the argument for hardness of exact 2-Furthest Pair (and other important problems in computational geometry) in poly-loglog dimensions in [Williams, SODA 2018]. We have three main results regarding this problem. * Supported by an Akamai Fellowship.

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“…Due to the technological advances, MDSs are generated in different application areas such as smart buildings, smart cities, wireless sensor networks, Internet of things (IoTs), scientific experiments, ECG signals and DNA analysis, stock markets, multimedia, and industrial domains etc., [14], [15]. A tightly coupled issue with these data sets is how to determine their similarity indexes with a minimum possible computational time of the resources [16]- [18]. In the literature, different methods were proposed to solve the longest common subsequence problem particularly for multivariate data sets [19].…”

Section: Literature Reviewmentioning

confidence: 99%

A Heuristic Approach for Finding Similarity Indexes of Multivariate Data Sets

et al. 2020

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Multivariate data sets (MDSs), with enormous size and certain ratio of noise/outliers, are generated routinely in various application domains. A major issue, tightly coupled with these MDSs, is how to compute their similarity indexes with available resources in presence of noise/outliers-which is addressed with the development of both classical and non-metric based approaches. However, classical techniques are sensitive to outliers and most of the non-classical approaches are either problem/application specific or overlay complex. Therefore, the development of an efficient and reliable algorithm for MDSs, with minimum time and space complexity, is highly encouraged by the research community. In this paper, a non-metric based similarity measure algorithm, for MDSs, is presented that solves the aforementioned issues, particularly, noise and computational time, successfully. This technique finds the similarity indexes of noisy MDSs, of both equal and variable sizes, through utilizing minimum possible resources i.e., space and time. Experiments were conducted with both benchmark and real time MDSs for evaluating the proposed algorithm's performance against its rival algorithms, which are traditional dynamic programming based and sequential similarity measure algorithms. Experimental results show that the proposed scheme performs exceptionally well, in terms of time and space, than its counterpart algorithms and effectively tolerates a considerable portion of noisy data. INDEX TERMS Similarity index, multivariate data set, outliers, the longest common subsequence. I. INTRODUCTION Recent technological advancements, particularly in sensors and actuators, lead to the generation of enormous multivariate data sets (MDSs) in different application areas i.e., wireless sensor networks, internet of things (IoT), scientific experiments, industrial control processes, educational purpose testbeds, web and databases [1]. An MDS is defined as a set of related numbers or values associated with a specific entity in an organization. In other words, a group of univariate data sets in columns form is known as MDS [2]. Mathematically, it is represented as a matrix X m , n , where m and n corresponds to the rows and columns respectively. These MDSs are thor-The associate editor coordinating the review of this manuscript and approving it for publication was Chongsheng Zhang. oughly examined, using various classical and non-classical approaches, to discover valuable information that is used to determine the correlating or distinguishing factor of entities. One of the major issue, closely linked with MDS, is to find their similarity indexes in the presence of noise/outliers that is not possible with existing techniques. Generally, two MDSs, X i , j and Y m , n , are believed similar if most of their elements are highly correlated [3]. MDSs similarity problem is an active research area, both in computer science and mathematics, that is due to its existence in different real world application environments i.e., DNA analysis, sensors-based real...

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Multivariate Fine-Grained Complexity of Longest Common Subsequence

Cited by 46 publications

References 84 publications

Tight Conditional Lower Bounds for Longest Common Increasing Subsequence

Tight Conditional Lower Bounds for Longest Common Increasing Subsequence

Untitled

A Heuristic Approach for Finding Similarity Indexes of Multivariate Data Sets

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