Extremal Distances in Directed Graphs: Tight Spanners and Near-Optimal Approximation Algorithms

Choudhary, Keerti; Gold, Omer

doi:10.1137/1.9781611975994.30

Cited by 4 publications

(2 citation statements)

References 31 publications

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“…Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Subsequent work. Our approximation algorithms for eccentricities in Theorems 3 and 4 have recently been improved to a true 2-approximation in \Õ (m) time by Choudhary and Gold [CG20]. From the lower bounds side, there have been several very recent improvements for diameter.…”

Section: Runtimementioning

confidence: 99%

Toward Tight Approximation Bounds for Graph Diameter and Eccentricities

Bačkurs¹,

Roditty²,

Segal³

et al. 2021

SIAM J. Comput.

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Among the most important graph parameters is the diameter, the largest distance between any two vertices. There are no known very efficient algorithms for computing the diameter exactly. Thus, much research has been devoted to how fast this parameter can be approximated. Chechik et al. [Proceedings of SODA 2014, Portland, OR, 2014, pp. 1041--1052 showed that the diameter can be approximated within a multiplicative factor of 3/2 in \Õ (m 3/2 ) time. Furthermore, Roditty and Vassilevska W. [Proceedings of STOC '13, New York, ACM, 2013, pp. 515--524] showed that unless the strong exponential time hypothesis (SETH) fails, no O(n 2 - \varepsi ) time algorithm can achieve an approximation factor better than 3/2 in sparse graphs. Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than 3/2. It was, however, completely plausible that a 3/2-approximation is possible in linear time. In this work we conditionally rule out such a possibility by showing that unless SETH fails no O(m 3/2 - \varepsi ) time algorithm can achieve an approximation factor better than 5/3. Another fundamental set of graph parameters is the eccentricities. The eccentricity of a vertex v is the distance between v and the farthest vertex from v. Chechik et al. [Proceedings of SODA 2014, Portland, OR, 2014, pp. 1041--1052 showed that the eccentricities of all vertices can be approximated within a factor of 5/3 in \Õ (m 3/2 ) time and Abboud, Vassilevska W., and Wang [Proceedings of SODA 2016, Arlington, VA, 2016 showed that no O(n 2 - \varepsi ) algorithm can achieve better than 5/3 approximation in sparse graphs. We show that the runtime of the 5/3 approximation algorithm is also optimal by proving that under SETH, there is no O(m 3/2 - \varepsi ) algorithm that achieves a better than 9/5 approximation. We also show that no near-linear time algorithm can achieve a better than 2 approximation for the eccentricities. This is the first lower bound in fine-grained complexity that addresses near-linear time computation. We show that our lower bound for near-linear time algorithms is essentially tight by giving an algorithm that approximates eccentricities within a 2 + \delta factor in \Õ (m/\delta ) time for any 0 < \delta < 1. This beats all eccentricity algorithms in Cairo, Grossi, and Rizzi [Proceedings of SODA 2016, Arlington, VA, 2016, pp. 363--376] and is the first constant factor approximation for eccentricities in directed graphs. To establish the above lower bounds we study the S-T diameter problem: Given a graph and two subsets S and T of vertices, output the largest distance between a vertex in S and a vertex in T . We give new algorithms and show tight lower bounds that serve as a starting point for all other hardness results. Our lower bounds apply only to sparse graphs. We show that for dense graphs, there are near-linear time algorithms for S-T diameter, diameter, and eccentricities, with almost the same approximation guarantees as their \Õ (m 3/2 ) counterparts, improving upon the best known al...

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Section: Runtimementioning

confidence: 99%

Toward Tight Approximation Bounds for Graph Diameter and Eccentricities

Bačkurs¹,

Roditty²,

Segal³

et al. 2021

SIAM J. Comput.

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“…Several lines of work have recently attacked the problem of computing the diameter in different settings. For example, Choudhary and Gold [16] constructed diameter spanners, which are subgraphs that approximately preserve the diameter of the original graph, Ancona et al [6] developed algorithms for computing the diameter in dynamic scenarios and proved matching conditional lower bounds, and Bonnet [10] proved that, for any constant ε > 0, computing a (7/4 − ε)-approximation of the diameter of a sparse graph n vertices and m = n 1+o (1) edges requires m 4/3−o (1) time, unless the Strong Exponential Time Hypothesis fails.…”

Section: Introductionmentioning

confidence: 99%

Space-Efficient Fault-Tolerant Diameter Oracles

Bilò¹,

Cohen²,

Friedrich³

et al. 2021

Preprint

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We design f -edge fault-tolerant diameter oracles (f -FDO, or simply FDO if f = 1). For a given directed or undirected and possibly edge-weighted graph G with n vertices and m edges and a positive integer f , we preprocess the graph and construct a data structure that, when queried with a set F of edges, whereFor the case of a single edge failure (f = 1) in an unweighted directed graph, there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch (1 + ε), constant query time, space O(m), and a combinatorial preprocessing time of O(mn + n 1.5 Dm/ε ), where D is the diameter.We present an FDO for directed graphs with the same stretch, query time, and space. It has a preprocessing time of O(mn + n 2 /ε), which is better for constant ε > 0. The preprocessing time nearly matches a conditional lower bound for combinatorial algorithms, also by Henzinger et al. With fast matrix multiplication, we achieve a preprocessing time of O(n 2.5794 + n 2 /ε). We further prove an information-theoretic lower bound showing that any FDO with stretch better than 3/2 requires Ω(m) bits of space. Thus, for constant 0 < ε < 3/2, our combinatorial (1 + ε)-approximate FDO is near-optimal in all parameters.In the case of multiple edge failures (f > 1) in undirected graphs with non-negative edge weights, we give an f -FDO with stretch (f + 2), query time O(f 2 log 2 n), O(f n) space, and preprocessing time O(f m). We complement this with a lower bound excluding any finite stretch in o(f n) space.Many real-world networks have polylogarithmic diameter. We show that for those graphs and up to f = o(log n/ log log n) failures one can swap approximation for query time and space. We present an exact combinatorial f -FDO with preprocessing time mn 1+o(1) , query time n o(1) , and space n 2+o(1) . When using fast matrix multiplication instead, the preprocessing time can be improved to n ω+o(1) , where ω < 2.373 is the matrix multiplication exponent. ACM Subject ClassificationTheory of computation → Shortest paths; Theory of computation → Data structures design and analysis; Theory of computation → Cell probe models and lower bounds; Theory of computation → Pseudorandomness and derandomization

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