Distributed Exact Shortest Paths in Sublinear Time

Abstract

The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it inO(n) time, wherenis the number of vertices in the input graphG. Peleg and Rubinovich [49] showed a lower bound of ˜Ω(D+ √n) for this problem, whereDis the hop-diameter ofG.Whether or not this problem can be solved inO(n) time whenDis relatively small is a major open question. Despite intensive research [10, 17, 33, 41, 45] that yielded near-optimal algorithms for theapproximatevariant of this problem, no progress was reported for the original problem.In this article, we answer this question in the affirmative. We devise an algorithm that requiresO((nlogn)5/6) time, forD=O(√nlogn), andO(D1/3⋅ (nlogn)2/3) time, for largerD. This running time is sublinear innin almost the entire range of parameters, specifically, forD=o(n/ log2n).We also generalize our result in two directions. One is when edges have bandwidthb≥ 1, and the other is thes-sources shortest paths problem. For both problems, our algorithm provides bounds that improve upon the previous state-of-the-art in almost the entire range of parameters. In particular, we provide an all-pairs shortest paths algorithm that requiresO(n5/3⋅ log2/3n) time, even forb= 1, for all values ofD.We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in themultipass semi-streamingmodel of computation.From the technical viewpoint, our distributed algorithm computes a hopsetG′′of a skeleton graphG′ofGwithout first computingG′itself. We then conduct a Bellman-Ford exploration inG′∪G′′, while computing the required edges ofG′on the fly. As a result, our distributed algorithm computesexactlythose edges ofG′that it really needs, rather than computing approximately the entireG′.