Dynamic Approximate AllPairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization
Abstract
We study dynamic $(1+\epsilon)$approximation algorithms for the allpairs 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 $\tilde O(mn/\epsilon)$ 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 $\tilde O(n^{5/2}/\epsilon)$ and constant query time that has an additive error of $2$ in addition to the $1+\epsilon$ multiplicative error. This beats the previous $\tilde O(mn/\epsilon)$ time when $m=\Omega(n^{3/2})$. Note that the additive error is unavoidable since, even in the static case, an $O(n^{3\delta})$time (a socalled truly subcubic) combinatorial algorithm with $1+\epsilon$ multiplicative error cannot have an additive error less than $2\epsilon$, unless we make a major breakthrough for Boolean matrix multiplication [Dor et al. FOCS 1996] and many other longstanding problems [Vassilevska Williams and Williams FOCS 2010]. The algorithm can also be turned into a $(2+\epsilon)$approximation algorithm (without an additive error) with the same time guarantees, improving the recent $(3+\epsilon)$approximation algorithm with $\tilde O(n^{5/2+O(\sqrt{\log{(1/\epsilon)}/\log n})})$ 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 $\tilde O(mn/\epsilon)$ and a query time of $O(\log\log n)$. The algorithm has a multiplicative error of $1+\epsilon$ and gives the first improved deterministic algorithm since 1981. It also answers an open question raised by Bernstein [STOC 2013].
 Publication:

arXiv eprints
 Pub Date:
 August 2013
 arXiv:
 arXiv:1308.0776
 Bibcode:
 2013arXiv1308.0776H
 Keywords:

 Computer Science  Data Structures and Algorithms;
 F.2.0;
 G.2.2
 EPrint:
 A preliminary version was presented at the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS 2013)