Dynamic approximate all-pairs shortest paths: Breaking the O(mn) barrier and derandomization

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 𝑂̃ (𝑚𝑛/𝜖) 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(𝑚𝑛2) and constant query time. We improve these results as follows: (1) We present an algorithm with a total update time of 𝑂̃ (𝑛5/2/𝜖) and constant query time that has an additive error of 2 in addition to the 1+𝜖 multiplicative error. This beats the previous 𝑂̃ (𝑚𝑛/𝜖) time when 𝑚=Ω(𝑛3/2). Note that the additive error is unavoidable since, even in the static case, an 𝑂(𝑛3−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 [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+𝜖)-approximation algorithm (without an additive error) with the same time guarantees, improving the recent (3+𝜖)-approximation algorithm with 𝑂̃ (𝑛5/2+𝑂(log(1/𝜖)/log𝑛√)) 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 𝑂̃ (𝑚𝑛/𝜖) and a query time of 𝑂(loglog𝑛). 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 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+𝜖)-approximation with an amortized update time of 𝑂̃ (𝑚𝑛/(𝜖𝑡)) and a query time of 𝑂̃ (𝑡) for every 𝑡≤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.