All-Pairs Shortest Paths with Few Weights per Node
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3717823.3718240.pdf
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Author(s)
Abboud, Amir
Fischer, Nick
Jin, Ce
Williams, Virginia Vassilevska
Xi, Zoe
Date Issued
June 15, 2025
Publisher
ACM|Proceedings of the 57th Annual ACM Symposium on Theory of Computing
Citation
Amir Abboud, Nick Fischer, Ce Jin, Virginia Vassilevska Williams, and Zoe Xi. 2025. All-Pairs Shortest Paths with Few Weights per Node. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC '25). Association for Computing Machinery, New York, NY, USA, 1956–1964.
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Final published version
Abstract
We study the central All-Pairs Shortest Paths (APSP) problem under the restriction that there are at most d distinct weights on the outgoing edges from every node.
For d=n this is the classical (unrestricted) APSP problem that is hypothesized to require cubic time n3−o(1), and at the other extreme, for d=1, it is equivalent to the Node-Weighted APSP problem.
We present new algorithms that achieve the following results:
* Node-Weighted APSP can be solved in time Õ(n(3+ω)/2) = Õ(n2.686), improving on the 15-year-old subcubic bounds Õ(n(9+ω)/4) = Õ(n2.843) [Chan; STOC ’07] and Õ(n2.830) [Yuster; SODA ’09]. This positively resolves the question of whether Node-Weighted APSP is an ”intermediate” problem in the sense of having complexity n2.5+o(1) if ω=2, in which case it also matches an n2.5−o(1) conditional lower bound.
* For up to d ≤ n3−ω−є distinct weights per node (where є > 0), the problem can be solved in subcubic time O(n3−f(є)) (where f(є) > 0). In particular, assuming that ω = 2, we can tolerate any sublinear number of distinct weights per node d ≤ n1−є, whereas previous work [Yuster; SODA ’09] could only handle d ≤ n1/2−є in subcubic time. This promotes our understanding of the APSP hypothesis showing that the hardest instances must exhaust a linear number of weights per node. With the current bounds on ω, we achieve a subcubic algorithm for d ≤ n0.628 whereas previously a subcubic running time could only be achieved for d ≤ n0.384. Our result also applies to the All-Pairs Exact Triangle problem, thus generalizing a result of Chan and Lewenstein on “Clustered 3SUM” from arrays to matrices. Notably, our technique constitutes a rare application of additive combinatorics in graph algorithms.
We complement our algorithmic results with simple hardness reductions extending the n2.5−o(1) conditional lower bound for Node-Weighted APSP to undirected graphs. Interestingly, under fine-grained assumptions, the complexity in the undirected case jumps from O(nω) for d=1 to n2.5−o(1) for d ≥ 2.
Description
STOC ’25, Prague, Czechia
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1145/3717823.3718240
