Single-source shortest paths in the CONGEST model with improved bounds

Computing shortest paths from a single source is one of the central problems studied in the CONGEST model of distributed computing. After many years in which no algorithmic progress was made, Elkin [STOC ‘17] provided the first improvement over the distributed Bellman-Ford algorithm. Since then, several improved algorithms have been published. The state-of-the-art algorithm for weighted directed graphs (with polynomially bounded non-negative integer weights) requires O~(min{nD1/2,nD1/4+n3/5+D}) rounds [Forster and Nanongkai, FOCS ‘18], which is still quite far from the known lower bound of Ω~(n+D) rounds [Elkin, STOC ‘04]; here D is the diameter of the underlying network and n is the number of vertices in it. For the (1 + o(1)) -approximate version of this problem and the same class of graphs, Forster and Nanongkai [FOCS ‘18] obtained a better upper bound of O~(nD1/4+D) rounds. In the same paper, they stated that achieving the same bound for the exact case remains a major open problem. In this paper we resolve the above mentioned problem by devising a new randomized algorithm for computing shortest paths from a single source in O~(nD1/4+D) rounds. Our algorithm is based on a novel weight-modifying technique that allows us to compute approximate distances that preserve a certain form of the triangle inequality for the edges in the graph.