Near-Optimal Decremental Approximate Multi-Source Shortest Paths

09/17/2020
by   Jakub Łącki, et al.
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We provide new algorithms for maintaining approximate distances in a weighted undirected graph G = (V, E) subject to edge deletions. Our first result is an algorithm that maintains (1+ϵ)-approximate distances from a set of s sources in Õ(sm) total update time, assuming that s= n^Ω(1), ϵ = Ω(1) and |E|= n^1+Ω(1). This matches the best known static algorithm, up to polylogarithmic factors for a wide range of settings. The currently best known algorithm for the problem is obtained by running the single-source algorithm of [Henzinger, Krinninger and Nanongkai, FOCS'14] independently from each source. Our result improves over the update time bound of this solution by removing a 2^Õ(log^3/4 n) factor. Additionally, we can maintain a (1+ϵ)-approximate single-source shortest paths with amortized update time of 2^Õ(√(log n)), when 0< ϵ<1 is a constant and |E|= n2^Ω̃(√(log n)). This improves over the best known update time of 2^Õ(log^3/4 n) by [Henzinger, Krinninger and Nanongkai, FOCS'14]. Furthermore, for any integer k ≥ 1 we give an algorithm for maintaining (2k-1)(1+ϵ)-approximate all-pairs-shortest-paths, in Õ(mn^1/k) total update time and O(k) query time[Throughout this paper we use the notation Õ(f(n)) to hide factors of O(polylog (f(n))).]. This improves over the result of [Chechik, FOCS'18] in a twofold way. Namely, we improve the total update time bound by removing an n^o(1) factor and reduce the query time from O(loglog (nW)) to O(k). Our results are based on a new decremental hopset construction that may be of independent interest.

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