Dynamic Approximate All-Pairs Shortest Paths: Breaking the $O(mn)$ Barrier and Derandomization
Monika Rauch Henzinger, Sebastian Forster, Danupon Nanongkai · SIAM Journal on Computing · 2016
We study dynamic $(1+\epsilon)$-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 $\tilde O(mn/\epsilon)$ and constant query time by Roditty and Zwick [SIAM J. Comput., 41 (2012), pp. 670--683]. The fastest deterministic algorithm is from a 1981 paper by Even and Shiloach [J. ACM, 28 (1981), pp. 1--4]; 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 so-called 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 [D. Dor, S. Halrepin, and U. Zwick, SIAM J. Comput., 29 (2000), pp. 1740--1759] and many other long-standing problems [V. Vassilevska Williams and R. Williams, Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, 2010, pp. 645--654]. 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 [Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011, pp. 1355--1365] 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 in [Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, 2013, pp. 725--734]. The deterministic algorithm can be turned into a deterministic fully dynamic $ (1+\epsilon) $-approximation with an amortized update time of $ \tilde O (m n / (\epsilon t))$ and a query time of $ \tilde O (t) $ for every $ t \leq \sqrt{n} $. In order to achieve our results, we introduce two new techniques: (i) A monotone Even--Shiloach tree algorithm which maintains a bounded-distance shortest-paths tree on a certain type of emulator called a locally persevering emulator. (ii) A derandomization technique based on moving Even--Shiloach trees as a way to derandomize the standard random set argument. These techniques might be of independent interest.