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(cn2+2/3 log4/3 n) and answers distance queries correctly with probability 1 − 1/nc, 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 O(n2+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.

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