From Hop Reduction to Sparsification for Negative Length Shortest Paths

TL;DR

This paper advances algorithms for negative-length shortest paths by introducing sparsification and recursive techniques, achieving faster randomized running times than previous methods, with practical improvements in graph processing efficiency.

Contribution

It develops new sparsification and recursive algorithms that significantly improve the running time for negative-length shortest path computations.

Findings

Achieved randomized time complexity of O(mn^{0.7193}) for certain graph densities.

Improved the construction of sparse shortcut graphs for faster algorithms.

Enhanced recursive sparsification methods to reduce running times below previous bounds.

Abstract

The textbook algorithm for real-weighted single-source shortest paths takes $O(mn)$ time on a graph with $m$ edges and $n$ vertices. A recent breakthrough algorithm by [Fin24] takes $\tilde{O}(mn^{8/9})$ randomized time. The running time was subsequently improved to $\tilde{O}(mn^{4/5})$ [HJQ25] and then $\tilde{O}(mn^{3/4}+m^{4/5}n)$ [HJQ26]. We build on the algorithms of [Fin24; HJQ25; HJQ26] to obtain faster strongly-polynomial randomized-time algorithms for negative-length shortest paths. An important new technique in this algorithm repurposes previous "hop-reducers" from [Fin24; HJQ26] into "negative edge sparsifiers", reducing the number of negative edges by essentially the same factor by which the "hops" were previously reduced. A simple recursive algorithm based on sparsifying the layered hop reducers of [Fin24] already gives an $\tilde{O}(mn^{\sqrt{3}-1})<O(mn^{.7321})$ randomized running time, improving [HJQ26] uniformly. We also improve the construction of the bootstrapped hop reducers in [HJQ26] by proposing new sparse shortcut graphs replacing the dense shortcut graphs in [HJQ26]. Integrating all three of layered sparsification, recursion, and sparse bootstrapping into the algorithm of [HJQ26] gives new upper bounds of $O(mn^{.7193})$ randomized time for $m\geq n^{1.03456}$ and $O((mn)^{.8620})$ randomized time for $m<n^{1.03456}$. Lastly, concurrent work by [LLRZ25] obtained an $\tilde{O}(n^{2.5})$ randomized time algorithm for the same problem, and along the way improved the running time of the "betweenness reduction" step in Fineman's framework. Dropping in this subroutine as a black box improves the running time of the simple recursive sparsification algorithm to $\tilde{O}(mn^{1/\sqrt{2}})<O(mn^{.70711})$, and a slightly modified recursive sparsification algorithm runs in $O(mn^{.69562})$ randomized time for $m\geq n^{1.0274}$ and $O((mn)^{.85})$ for $m<n^{1.0274}$.