Efficient Construction of Directed Hopsets and Parallel Approximate Shortest Paths

• Published in: arXiv.org
• Main Authors: Cao, Nairen, Fineman, Jeremy T, Russell, Katina
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 11.12.2019
• Subjects: Algorithms; Graph theory; Graphs; Run time (computers); Shortest-path problems
• ISSN: 2331-8422

The approximate single-source shortest-path problem is as follows: given a graph with nonnegative edge weights and a designated source vertex \(s\), return estimates of the distances from~\(s\) to each other vertex such that the estimate falls between the true distance and \((1+\epsilon)\) times the distance. This paper provides the first nearly work-efficient parallel algorithm with sublinear span (also called depth) for the approximate shortest-path problem on \emph{directed} graphs. Specifically, for constant \(\epsilon\) and polynomially-bounded edge weights, our algorithm has work \(\tilde{O}(m)\) and span \(n^{1/2+o(1)}\). Several algorithms were previously known for the case of \emph{undirected} graphs, but none of the techniques seem to translate to the directed setting. The main technical contribution is the first nearly linear-work algorithm for constructing hopsets on directed graphs. A \((\beta,\epsilon)\)-hopset is a set of weighted edges (sometimes called shortcuts) which, when added to the graph, admit \(\beta\)-hop paths with weight no more than \((1+\epsilon)\) times the true shortest-path distances. There is a simple sequential algorithm that takes as input a directed graph and produces a linear-cardinality hopset with \(\beta=O(\sqrt{n})\), but its running time is quite high---specifically \(\tilde{O}(m\sqrt{n})\). Our algorithm is the first more efficient algorithm that produces a directed hopset with similar characteristics. Specifically, our sequential algorithm runs in \(\tilde{O}(m)\) time and constructs a hopset with \(\tilde{O}(n)\) edges and \(\beta = n^{1/2+o(1)}\). A parallel version of the algorithm has work \(\tilde{O}(m)\) and span \(n^{1/2+o(1)}\).