Tight Hardness Results for LCS and Other Sequence Similarity Measures

Abboud, Amir; Bačkurs, Artūrs; Williams, Virginia Vassilevska

doi:10.1109/focs.2015.14

Cited by 179 publications

(514 citation statements)

References 43 publications

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“…It is not difficult to see that checking whether a property expressible by a first-order formula with k + 1 quantifiers holds on a given structure with m records can be done in O(m k ) time, and if Strong Exponential Time Hypothesis (SETH) is true, there are such properties that require m k−o(1) time to decide. 1 For k = 1, this is linear time and so cannot be improved. For each such problem with k ≥ 2, we give a probabilistic algorithm that solves it in m k /2 Θ( √ log m) time.…”

Section: Introductionmentioning

confidence: 99%

“…Thus far, fine-grained complexity has remained focused on specific problems, rather than organizing problems into classes as in traditional complexity. As the field has grown, many fundamental relationships between problems have been discovered, making the graph of known results a somewhat tangled web of reductions ( [36,5,9,11,12,1,13,2,27]). …”

Section: Introductionmentioning

confidence: 99%

“…For many of the known SETH-hard problems of interest such as Edit Distance [9], Longest Common Subsequence [5,1,13], Dynamic Time Warping [1,13], Fréchet Distance [12], Succinct Stable Matching [27], etc. the reduction from SAT passes through the Orthogonal Vectors problem.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

Gao¹,

Impagliazzo²,

Kolokolova³

et al. 2017

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

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Properties definable in first-order logic are algorithmically interesting for both theoretical and pragmatic reasons. Many of the most studied algorithmic problems, such as Hitting Set and Orthogonal Vectors, are first-order, and the first-order properties naturally arise as relational database queries. A relatively straightforward algorithm for evaluating a property with k + 1 quantifiers takes time O(m k ) and, assuming the Strong Exponential Time Hypothesis (SETH), some such properties require O(m k− ) time for any > 0. (Here, m represents the size of the input structure, i.e. the number of tuples in all relations.)We give algorithms for every first-order property that improves this upper bound to m k /2 Θ( √ log n) , i.e., an improvement by a factor more than any poly-log, but less than the polynomial required to refute SETH. Moreover, we show that further improvement is equivalent to improving algorithms for sparse instances of the well-studied Orthogonal Vectors problem. Surprisingly, both results are obtained by showing completeness of the Sparse Orthogonal Vectors problem for the class of first-order properties under fine-grained reductions. To obtain improved algorithms, we apply the fast Orthogonal Vectors algorithm of [3,16].While fine-grained reductions (reductions that closely preserve the conjectured complexities of problems) have been used to relate the hardness of disparate specific problems both within P and beyond, this is the first such completeness result for a standard complexity class.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

Gao¹,

Impagliazzo²,

Kolokolova³

et al. 2017

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

View full text Add to dashboard Cite

show abstract

“…The longest possible well-formed gMSAs correspond to the concatenation of the N sequences. (1) and thus in particular ϒ p (0) = 0. We write π here as a function of the row index in the alignment; hence it can be viewed as a vector of length L. …”

Section: Notation and Basic Propertiesmentioning

confidence: 99%

Partially Local Multi-way Alignments

Retzlaff

Stadler

2018

Math.Comput.Sci.

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Multiple sequence alignments are an essential tool in bioinformatics and computational biology, where they are used to represent the mutual evolutionary relationships and similarities between a set of DNA, RNA, or protein sequences. More recently they have also received considerable interest in other application domains, in particular in comparative linguistics. Multiple sequence alignments can be seen as a generalization of the stringto-string edit problem to more than two strings. With the increase in the power of computational equipment, exact, dynamic programming solutions have become feasible in practice also for 3-and 4-way alignments. For the pairwise (2-way) case, there is a clear distinction between local and global alignments. As more sequences are considered, this distinction, which can in fact be made independently for both ends of each sequence, gives rise to a rich set of partially local alignment problems. So far these have remained largely unexplored. Here we introduce a general formal framework that gives raise to a classification of partially local alignment problems. This leads to a generic NR gratefully acknowledges the hospitality of the Santa Fe Institute, where much of this work was performed. scheme that guides the principled design of exact dynamic programming solutions for particular partially local alignment problems.

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“…Except for some polylogarithmic improvements, the best known algorithm is still the textbook quadratic time dynamic program. Recent results show that strongly subquadratic algorithms for LCS would violate the Strong Exponential Time Hypothesis [1,7,8].…”

Section: Introductionmentioning

confidence: 99%

Accurate and Nearly Optimal Sublinear Approximations to Ulam Distance

Naumovitz¹,

Saks²,

Seshadhri³

2017

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

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The Ulam distance between two permutations of length n is the minimum number of insertions and deletions needed to transform one sequence into the other. Equivalently, the Ulam distance d is n minus the length of the longest common subsequence (LCS) between the permutations. Our main result is an algorithm, that for any fixed ε > 0, provides a (1 + ε)-multiplicative approximation for d iñ Oε(n/d + √ n) time, which has been shown to be optimal up to polylogarithmic factors. This is the first sublinear time algorithm (provided that d = (log n)ω (1) ) that obtains arbitrarily good multiplicative approximations to the Ulam distance. The previous best bound is an O(1)-approximation (with a large constant) by Andoni and Nguyen (2010) with the same running time bound (ignoring polylogarithmic factors).The improvement in the approximation factor from O(1) to (1 + ε) allows for significantly more powerful sublinear algorithms. For example, for any fixed δ > 0, we can get additive δn approximations for the LCS between permutations inÕ δ ( √ n) time. Previous sublinear algorithms require δ to be at least 1−1/C, where C is the approximation factor, which is close to 1 when C is large.Our algorithm is obtained by abstracting the basic algorithmic framework of Andoni and Nguyen, and combining it with the sublinear approximations for the longest increasing subsequence by Saks and Seshadhri (2010).

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Tight Hardness Results for LCS and Other Sequence Similarity Measures

Cited by 179 publications

References 43 publications

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

Partially Local Multi-way Alignments

Accurate and Nearly Optimal Sublinear Approximations to Ulam Distance

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