Fully-Dynamic All-Pairs Shortest Paths: Likely Optimal Worst-Case Update Time
Xiao Mao
TL;DR
This work studies the fully dynamic APSP problem under vertex insertions/deletions on an $n$-vertex graph with no negative cycles. It proposes a Monte Carlo data structure achieving randomized worst-case update time $ ilde O(n^{2.5})$ and space $ ilde O(n^{2})$, against adaptive adversaries. The key innovation is the hop-dominant path framework, which constrains paths by hop counts and enables robust concatenations through centers, augmented by randomization and a multi-layer Gutenberg–Wulff-Nilsen framework. A core challenge—replacing hop-restricted paths with hop-dominant paths—is addressed via a two-step approach: (i) approximate congested vertex sets with hop-restricted paths using an oracle, and (ii) remove the oracle by efficient concatenation in a final data structure with Type I/II candidates. The result tightens the known barrier around $ ilde O(n^{2.5})$ for worst-case update time and maintains $ ilde O(n^{2})$ space, advancing both theoretical understanding and practical prospects for dynamic APSP without negative cycles.
Abstract
The All-Pairs Shortest Paths (APSP) problem is one of the fundamental problems in theoretical computer science. It asks to compute the distance matrix of a given $n$-vertex graph. We revisit the classical problem of maintaining the distance matrix under a fully dynamic setting undergoing vertex insertions and deletions with a fast worst-case running time and efficient space usage. Although an algorithm with amortized update-time $\tilde O(n ^ 2)$ has been known for nearly two decades [Demetrescu and Italiano, STOC 2003], the current best algorithm for worst-case running time with efficient space usage runs is due to [Gutenberg and Wulff-Nilsen, SODA 2020], which improves the space usage of the previous algorithm due to [Abraham, Chechik, and Krinninger, SODA 2017] to $\tilde O(n ^ 2)$ but fails to improve their running time of $\tilde O(n ^ {2 + 2 / 3})$. It has been conjectured that no algorithm in $O(n ^ {2.5 - ε})$ worst-case update time exists. For graphs without negative cycles, we meet this conjectured lower bound by introducing a Monte Carlo algorithm running in randomized $\tilde O(n ^ {2.5})$ time while keeping the $\tilde O(n ^ 2)$ space bound from the previous algorithm. Our breakthrough is made possible by the idea of ``hop-dominant shortest paths,'' which are shortest paths with a constraint on hops (number of vertices) that remain shortest after we relax the constraint by a constant factor.