# Efficient Construction of Directed Hopsets and Parallel Approximate Shortest Paths

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+ϵ) 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 directed graphs. Specifically, for constant ϵ and polynomially-bounded edge weights, our algorithm has work Õ(m) and span n^1/2+o(1). Several algorithms were previously known for the case of 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 (β,ϵ)-hopset is a set of weighted edges (sometimes called shortcuts) which, when added to the graph, admit β-hop paths with weight no more than (1+ϵ) 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 β=O(√(n)), but its running time is quite high—specifically Õ(m√(n)). Our algorithm is the first more efficient algorithm that produces a directed hopset with similar characteristics. Specifically, our sequential algorithm runs in Õ(m) time and constructs a hopset with Õ(n) edges and β = n^1/2+o(1). A parallel version of the algorithm has work Õ(m) and span n^1/2+o(1).

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