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.
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