Deterministic Partially Dynamic Single Source Shortest Paths in Weighted Graphs.

In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest distances between s and all other nodes in G under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem in unweighted graphs with only O(mn) total update time over all edge deletions. Their classic was the state of the art for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed.The first improvement over the Even-Shiloach was given by Bernstein and Roditty [SODA 2011], who for the case of an unweighted and undirected graph presented a (1+epsilon)-approximate with constant query time and a total update time of O(n^{2+o(1)}). This work triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a (1+epsilon)-approximate for undirected weighted graphs whose total update time is near linear: O(m^{1+o(1)} log(W)), where W is the ratio of the heaviest to the lightest edge weight in the graph. In this paper they posed as a major open problem the question of derandomizing their result.Until very recently, all known improvements over the Even-Shiloach were randomized and required the assumption of a non-adaptive adversary. In STOC 2016, Bernstein and Chechik showed the first deterministic algorithm to go beyond O(mn) total update time: the is also (1+epsilon)-approximate, and has total update time tilde{O}(n^2). In SODA 2017, the same authors presented an with total update time tilde{O}(mn^{3/4}). However, both algorithms are restricted to undirected, unweighted graphs. We present the first deterministic for weighted undirected graphs to go beyond the O(mn) bound. The total update time is tilde{O}(n^2 log(W)).