Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication

TL;DR

This paper presents improved algorithms for approximate shortest paths in graphs, achieving faster centralized, parallel, and distributed solutions for multiple sources, with better approximation guarantees and efficiency across various models and parameters.

Contribution

The paper introduces novel algorithms that significantly improve the efficiency and approximation quality for multi-source shortest paths in centralized, parallel, and distributed models, especially for larger source sets.

Findings

Centralized algorithm runs in near-linear time for certain parameters.

PRAM algorithm achieves polylogarithmic time with low work complexity.

Distributed algorithm improves bounds for larger source sets, especially in the Congested Clique model.

Abstract

Consider an undirected weighted graph $G = (V,E,w)$. We study the problem of computing $(1+\epsilon)$-approximate shortest paths for $S \times V$, for a subset $S \subseteq V$ of $|S| = n^r$ sources, for some $0 < r \le 1$. 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 $\tilde{O}(|E| \cdot n^{o(1)} + n^{\omega(r)})$, where $n^{\omega(r)}$ is the time required to multiply an $n^r \times n$ matrix by an $n \times n$ one. Our PRAM algorithm has polylogarithmic time $(\log n)^{O(1/\rho)}$, and its work complexity is $\tilde{O}(|E| \cdot n^\rho + n^{\omega(r)})$, for any arbitrarily small constant $\rho >0$. In particular, for $r \le 0.313\ldots$, our centralized algorithm computes $S \times V$ $(1+\epsilon)$-approximate shortest paths in $n^{2 + o(1)}$ time. Our PRAM polylogarithmic-time algorithm has work complexity $O(|E| \cdot n^\rho + n^{2+o(1)})$, for any arbitrarily small constant $\rho >0$. Previously existing solutions either require centralized time/parallel work of $O(|E| \cdot |S|)$ 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 $r \le 0.655$, while previous state-of-the-art algorithms did so only for $r \le 1/2$. Moreover, it improves previous bounds for all $r > 1/2$. For unweighted graphs, the running time is improved further to $poly(\log\log n)$.