All-Pairs Shortest Paths in $O(n^2)$ time with high probability

We present an all-pairs shortest path algorithm whose running time on a complete directed graph on $n$ vertices whose edge weights are chosen independently and uniformly at random from $[0,1]$ is $O(n^2)$, in expectation and with high probability. This resolves a long standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano. The analysis relies on a proof that the number of \emph{locally shortest paths} in such randomly weighted graphs is $O(n^2)$, in expectation and with high probability. We also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in $O(\log^{2}n)$ expected time.

💡 Research Summary

The paper tackles the classic all‑pairs shortest‑paths (APSP) problem under a probabilistic model: a complete directed graph on n vertices where each edge weight is drawn independently and uniformly from the interval