Deterministic decremental single source shortest paths: Beyond the O(mn) bound

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 paths 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 with only O (mn) total update time over all edge deletions. Their classic algorithm was the best known result for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. The first improvement over the Even-Shiloach algorithm was given by Bernstein and Roditty [SODA 2011], who for the case of an unweighted and undirected graph presented an approximate (1 + ϵ) algorithm with constant query time and a total update time of O(n^{2+O(1/√log n)}). This work triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a (1 + ϵ)-approximate algorithm whose total update time is near linear O(m^{1+O(1/√log n)}). In this paper they posed as a major open problem the question of derandomizing their result. In fact, all known improvements over the Even-Shiloach algorithm are randomized. All these algorithms maintain some truncated shortest path trees from a small subset of nodes. While in the randomized setting it is possible to "hide" these nodes from the adversary, in the deterministic setting this is impossible: the adversary can delete all edges touching these nodes, thus forcing the algorithm to choose a new set of nodes and incur a new computation of shortest paths. In this paper we present the first deterministic decremental SSSP algorithm that breaks the Even-Shiloach bound of O(mn) total update time, for unweighted and undirected graphs. Our algorithm is (1 + ϵ) approximate and achieves a total update time of Õ(n²). Our algorithm can also achieve the same bounds in the incremental setting. It is worth mentioning that for dense instances where m = Ω(n^{2-1/√log(n)}), our algorithm is also faster than all existing randomized algorithms.