A Faster Undirected Single-Source Shortest Path Algorithm

2026-09-14

AI Summary

本文针对加权无向图中的单源最短路径问题，提出了一种新的算法，将运行时间改进至O(m log^1/2 n log log^1/4 n log log log^1/4 n)，通过计算每个顶点到随机样本中最近顶点的距离来实现。

Abstract

The single-source shortest paths (SSSP) problem in graphs with non-negative edge weights is one of the most classic problems in algorithms. For decades, the best known running time in the comparison-addition model was the $O(m+n\log n)$ bound of Dijkstra's algorithm with Fibonacci heaps. Recently, Duan, Mao, Shu, and Yin (FOCS'23) gave a randomized $O(m\log^{1/2} n \log\log^{1/2} n)$-time algorithm for SSSP in weighted undirected graphs. For weighted directed graphs, Duan, Mao, Mao, Shu, and Yin (STOC'25) gave an $O(m\log^{2/3} n)$-time algorithm for SSSP. Very recently, Duan, Mao, Shu, and Yin (ICALP'26) obtained an algorithm for directed graphs whose running time matches the $O(m\log^{1/2} n \log\log^{1/2} n)$ time of the undirected case. In this paper, we present a faster algorithm for SSSP in weighted undirected graphs, giving the first improvement in running time since the FOCS'23 breakthrough of Duan, Mao, Shu, and Yin. Our algorithm runs in $O(m\log^{1/2} n \log\log^{1/4} n \log\log\log^{1/4} n)$ time, improving the previous running time by a factor of $(\frac{\log\log n}{\log\log\log n})^{1/4}$. Our main contribution is a simple and efficient tool that computes, for every vertex, its distance to the nearest vertex in a random sample; this tool may be of independent interest.

Problem

• SSSP
• weighted undirected graphs
• running time

Innovation

• SSSP
• weighted undirected graphs
• random sample