A forward-backward single-source shortest paths algorithm

TL;DR

This paper introduces a novel forward-backward SSSP algorithm that outperforms traditional forward-only methods on complete graphs with exponential edge weights, achieving optimal or near-optimal expected running times.

Contribution

The paper presents a new forward-backward SSSP algorithm that improves the worst-case and expected running times, demonstrating a separation from forward-only algorithms and providing a new all-pairs shortest paths method.

Findings

Achieves $O(n)$ time on complete graphs with high probability

Expected $O(n^2)$ running time matching recent algorithms

Probability of exceeding $O(n^2)$ time is exponentially small

Abstract

We describe a new forward-backward variant of Dijkstra's and Spira's Single-Source Shortest Paths (SSSP) algorithms. While essentially all SSSP algorithm only scan edges forward, the new algorithm scans some edges backward. The new algorithm assumes that edges in the outgoing and incoming adjacency lists of the vertices appear in non-decreasing order of weight. (Spira's algorithm makes the same assumption about the outgoing adjacency lists, but does not use incoming adjacency lists.) The running time of the algorithm on a complete directed graph on $n$ vertices with independent exponential edge weights is $O(n)$, with very high probability. This improves on the previously best result of $O(n\log n)$, which is best possible if only forward scans are allowed, exhibiting an interesting separation between forward-only and forward-backward SSSP algorithms. As a consequence, we also get a new all-pairs shortest paths algorithm. The expected running time of the algorithm on complete graphs with independent exponential edge weights is $O(n^2)$, matching a recent algorithm of Demetrescu and Italiano as analyzed by Peres et al. Furthermore, the probability that the new algorithm requires more than $O(n^2)$ time is exponentially small, improving on the $O(n^{-1/26})$ probability bound obtained by Peres et al.