Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication
Michael Elkin, Ofer Neiman · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2022
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| = n^r sources, for some 0 0. In particular, for r ≤ 0.313…, our centralized algorithm computes S × V (1+ε)-approximate shortest paths in n^{2 + 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(log log n) for r ≤ 0.655. Previously this running time was known for r ≤ 1/2.