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arXiv · 1308.0776

Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization

Monika Henzinger·Sebastian Krinninger·Danupon Nanongkai
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Abstract

We study dynamic $(1+ε)$-approximation algorithms for the all-pairs shortest paths problem in unweighted undirected $n$-node $m$-edge graphs under edge deletions. The fastest algorithm for this problem is a randomized algorithm with a total update time of $\tilde O(mn/ε)$ and constant query time by Roditty and Zwick [FOCS 2004]. The fastest deterministic algorithm is from a 1981 paper by Even and Shiloach [JACM 1981]; it has a total update time of $O(mn^2)$ and constant query time. We improve these results as follows: (1) We present an algorithm with a total update time of $\tilde O(n^{5/2}/ε)$ and constant query time that has an additive error of $2$ in addition to the $1+ε$ multiplicative error. This beats the previous $\tilde O(mn/ε)$ time when $m=Ω(n^{3/2})$. Note that the additive error is unavoidable since, even in the static case, an $O(n^{3-δ})$-time (a so-called truly subcubic) combinatorial algorithm with $1+ε$ multiplicative error cannot have an additive error less than $2-ε$, unless we make a major breakthrough for Boolean matrix multiplication [Dor et al. FOCS 1996] and many other long-standing problems [Vassilevska Williams and Williams FOCS 2010]. The algorithm can also be turned into a $(2+ε)$-approximation algorithm (without an additive error) with the same time guarantees, improving the recent $(3+ε)$-approximation algorithm with $\tilde O(n^{5/2+O(\sqrt{\log{(1/ε)}/\log n})})$ running time of Bernstein and Roditty [SODA 2011] in terms of both approximation and time guarantees. (2) We present a deterministic algorithm with a total update time of $\tilde O(mn/ε)$ and a query time of $O(\log\log n)$. The algorithm has a multiplicative error of $1+ε$ and gives the first improved deterministic algorithm since 1981. It also answers an open question raised by Bernstein [STOC 2013].

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Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai. 2015-06-16. Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization. https://doi.org/10.1137/140957299

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