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 \(\tilde{O}(\min \{\sqrt{n}D^{1/2} ,\sqrt{n}D^{1/4} + n^{3/5} + D\})\) rounds [Forster and Nanongkai, FOCS ‘18], which is still quite far from the known lower bound of \(\tilde{\Omega }(\sqrt{n} + D)\) 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 \(\tilde{O}(\sqrt{n}D^{1/4} + D)\) 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 \(\tilde{O}(\sqrt{n}D^{1/4} + D)\) 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.

从单一源计算最短路径是分布式计算的 CONGEST 模型中研究的核心问题之一。在多年没有取得算法进展之后，Elkin [ STOC '17 ] 提供了对分布式 Bellman-Ford 算法的第一个改进。从那时起，已经发布了几种改进的算法。加权有向图（具有多项式有界非负整数权重）的最新算法需要\(\tilde{O}(\min \{\sqrt{n}D^{1/2} ,\ sqrt{n}D^{1/4} + n^{3/5} + D\})\)轮 [ Forster and Nanongkai, FOCS '18 ]，这离已知的下界\( \tilde{\Omega }(\sqrt{n} + D)\)轮 [ Elkin, STOC '04 ]; 这里D是底层网络的直径，n是其中的顶点数。对于\((1+o(1))\) - 这个问题的近似版本和同类型的图，Forster 和 Nanongkai [ FOCS '18 ] 获得了更好的\(\tilde{O}(\ sqrt{n}D^{1/4} + D)\)轮。在同一篇论文中，他们表示，对确切案例实现相同的界限仍然是一个主要的开放问题。在本文中，我们通过设计一种新的随机算法来解决上述问题，用于计算\(\tilde{O}(\sqrt{n}D^{1/4} + D)\)回合。我们的算法基于一种新颖的权重修改技术，该技术允许我们计算近似距离，从而为图中的边保留某种形式的三角不等式。