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

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 $O(c{n}^{2+2\mathrm{/}3}{\log }^{4\mathrm{/}3}n)$ and answers distance queries correctly with probability $1-1\mathrm{/}{n}^c$, 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 $\tilde{O}(n^{2+3\mathrm{/}4})$ 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.