Fully dynamic all-pairs shortest paths with worst-case update-time revisited

TL;DR

This paper presents a new randomized algorithm for fully dynamic all-pairs shortest paths with improved worst-case update time, providing probabilistic correctness and surpassing previous deterministic bounds after over a decade.

Contribution

The paper introduces a simple randomized algorithm achieving better worst-case update times for dynamic shortest paths, improving upon the deterministic algorithms for the first time in over ten years.

Findings

Achieves worst-case update time of O(cn^{2+2/3} log^{4/3} n) with high probability

Provides probabilistic correctness against adaptive adversaries in graphs without negative cycles

First significant improvement in over a decade for this problem

Abstract

We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates). Our main result is a simple randomized algorithm that for any parameter $c>1$ has a worst-case update time of $O(cn^{2+2/3} \log^{4/3}{n})$ and answers distance queries correctly with probability $1-1/n^c$, against an adaptive online adversary if the graph contains no negative cycle. The best deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time of $\tilde O(n^{2+3/4})$ and assumes non-negative weights. This is the first improvement for this problem for more than a decade. Conceptually, our algorithm shows that randomization along with a more direct approach can provide better bounds.