一种更快的无向单源最短路径算法

中文摘要 AI 辅助

边权非负的图中的单源最短路径（SSSP）问题是算法领域最经典的问题之一。数十年来，比较-加法模型下已知的最优运行时间是采用斐波那契堆的Dijkstra算法的$O(m+n\log n)$界。\n 近期，Duan、Mao、Shu和Yin（FOCS'23）提出了一种针对加权无向图SSSP的随机算法，运行时间为$O(m\log^{1/2} n \log\log^{1/2} n)$。对于加权有向图，Duan、Mao、Mao、Shu和Yin（STOC'25）提出了运行时间为$O(m\log^{2/3} n)$的SSSP算法。就在最近，Duan、Mao、Shu和Yin（ICALP'26）得到了一种有向图算法，其运行时间与无向图情形的$O(m\log^{1/2} n \log\log^{1/2} n)$持平。\n 本文提出了一种更快的加权无向图SSSP算法，实现了自Duan、Mao、Shu和Yin的FOCS'23突破性成果以来的首次运行时间提升。我们的算法运行时间为$O(m\log^{1/2} n \log\log^{1/4} n \log\log\log^{1/4} n)$，将此前的运行时间提升了$(\frac{\log\log n}{\log\log\log n})^{1/4}$倍。\n 我们的核心贡献是一个简单高效的工具，它能计算每个顶点到随机样本中最近顶点的距离；该工具本身也具备独立研究价值。

英文摘要

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.

发表机构

• Bar Ilan University（巴伊兰大学）

机构由 AI 辅助整理，请以论文原文为准。