Centralized and Parallel Multi-Source Shortest Paths via Hopsets and Fast Matrix Multiplication.

Consider an undirected weighted graph G=(V,E,w). We study the problem of computing (1+ϵ)-approximate shortest paths for S×V, for a subset S⊆V of |S|=nr sources, for some 0<r≤1. We devise a significantly improved algorithm for this problem in the entire range of parameter r, in both the classical centralized and the parallel (PRAM) models of computation, and in a wide range of r in the distributed (Congested Clique) model. Specifically, our centralized algorithm for this problem requires time O~(|E|⋅n^o(1)+n^ω(r)), where nω(r) is the time required to multiply an n^r×n matrix by an n×n one. Our PRAM algorithm has polylogarithmic time (logn)O(1/ρ), and its work complexity is O~(|E|⋅nρ+n^ω(r)), for any arbitrarily small constant ρ>0.In particular, for r≤0.313…, our centralized algorithm computes S×V (1+ϵ)-approximate shortest paths in n2+o(1) time. Our PRAM polylogarithmic-time algorithm has work complexity O(|E|⋅n^ρ+n^2+o(1)), for any arbitrarily small constant ρ>0. Previously existing solutions either require centralized time/parallel work of O(|E|⋅|S|) or provide much weaker approximation guarantees. In the Congested Clique model, our algorithm solves the problem in polylogarithmic time for |S|=n^r sources, for r≤0.655, while previous state-of-the-art algorithms did so only for r≤1/2. Moreover, it improves previous bounds for all r>1/2. For unweighted graphs, the running time is improved further to poly(loglogn).