Consequences of Faster Alignment of Sequences

Abboud, Amir; Williams, Virginia Vassilevska; Weimann, Oren

doi:10.1007/978-3-662-43948-7_4

Cited by 99 publications

(125 citation statements)

References 47 publications

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“…This motivates the following hypothesis used as the basis of hardness in prior work (see e.g. [BT16,AVW14]). …”

Section: Preliminariesmentioning

confidence: 89%

“…The Min Weight k-Clique Hypothesis has been considered for instance in [BT16] and [AVW14] to show hardness for improving upon the Viterbi algorithm, and for Local Sequence Alignment. The (unweighted) k-Clique problem is NP-Complete, but can be solved in O(n ωk/3 ) time when k is fixed [NP85] 3 where ω < 2.373 [Vas12,Gal14] is the matrix multiplication exponent.…”

Section: Hypothesis 11 (Min Weight K-clique)mentioning

confidence: 99%

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Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Lincoln

Williams²,

Williams³

2018

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

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Fine-grained reductions have established equivalences between many core problems withÕ(n 3 )-time algorithms on n-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also haveÕ(mn)-time algorithms on m-edge n-node weighted graphs, and such algorithms have wider applicability. Are these mn bounds optimal when m n 2 ?Starting from the hypothesis that the minimum weight (2 + 1)-Clique problem in edge weighted graphs requires n 2 +1−o(1) time, we prove that for all sparsities of the form m = Θ(n 1+1/ ), there is no O(n 2 + mn 1−ε ) time algorithm for ε > 0 for any of the below problems• Minimum Weight (2 + 1)-Cycle in a directed weighted graph, • Shortest Cycle in a directed weighted graph, • APSP in a directed or undirected weighted graph, • Radius (or Eccentricities) in a directed or undirected weighted graph, • Wiener index of a directed or undirected weighted graph, • Replacement Paths in a directed weighted graph, • Second Shortest Path in a directed weighted graph, • Betweenness Centrality of a given node in a directed weighted graph. That is, we prove hardness for a variety of sparse graph problems from the hardness of a dense graph problem. Our results also lead to new conditional lower bounds from several related hypothesis for unweighted sparse graph problems including k-cycle, shortest cycle, Radius, Wiener index and APSP. * andreali@mit.edu. Supported by the EECS Merrill Lynch Fellowship.†

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“…This motivates the following hypothesis used as the basis of hardness in prior work (see e.g. [BT16,AVW14]). …”

Section: Preliminariesmentioning

confidence: 89%

Section: Hypothesis 11 (Min Weight K-clique)mentioning

confidence: 99%

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Lincoln

Williams²,

Williams³

2018

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

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“…Other examples of this approach [1,2,44] include the famous results on 3-SUM hardness starting with the work of Gajentaan and Overmars [21].…”

Section: Introductionmentioning

confidence: 99%

Subcubic Equivalences Between Graph Centrality Problems, APSP and Diameter

Abboud¹,

Grandoni²,

Williams³

2014

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

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315

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Measuring the importance of a node in a network is a major goal in the analysis of social networks, biological systems, transportation networks etc. Different centrality measures have been proposed to capture the notion of node importance. For example, the center of a graph is a node that minimizes the maximum distance to any other node (the latter distance is the radius of the graph). The median of a graph is a node that minimizes the sum of the distances to all other nodes. Informally, the betweenness centrality of a node w measures the fraction of shortest paths that have w as an intermediate node.Finally, the reach centrality of a node w is the smallest distance r such that any s-t shortest path passing through w has either s or t in the ball of radius r around w.The fastest known algorithms to compute the center and the median of a graph, and to compute the betweenness or reach centrality even of a single node take roughly cubic time in the number n of nodes in the input graph. It is open whether these problems admit truly subcubic algorithms, i.e. algorithms with running timeÕ(n 3−δ ) for some constant δ > 0 1 . We relate the complexity of the mentioned centrality problems to two classical problems for which no truly subcubic algorithm is known, namely All Pairs Shortest Paths (APSP) and Diameter. We show that Radius, Median and Betweenness Centrality are equivalent under subcubic reductions to APSP, i.e. that a truly subcubic algorithm for any of these problems implies a truly subcubic algorithm for all of them. We then show that Reach Centrality is equivalent to Diameter under subcubic reductions. The same holds for the problem of approximating Betweenness Centrality within any constant factor. Thus the latter two centrality problems could potentially be solved in truly subcubic time, even if APSP requires essentially cubic time.

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“…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%

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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Consequences of Faster Alignment of Sequences

Cited by 99 publications

References 47 publications

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Subcubic Equivalences Between Graph Centrality Problems, APSP and Diameter

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

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