For any fixed 1 > [epsilon] > 0 we present a fully dynamic algorithm for maintaining (2 + [epsilon])-approximate all-pairs shortest paths in undirected graphs with positive edge weights. We use a randomized (Las Vegas) update algorithm (but a deterministic query procedure), so the time given is the expected amortized update time. Our query time O(log log log n). The update time is O[over ~](mnO(1/[sqrt](log n)) log (nR)), where R is the ratio between the heaviest and the lightest edge weight in the graph (so R = 1 in unweighted graphs). Unfortunately, the update time does have the drawback of a super-polynomial dependence on e. it grows as (3/[epsilon])[sqrt]log n/log(3/[epsilon]) = n [sqrt]log (3/[epsilon])/log n. Our algorithm has a significantly faster update time than any other algorithm with sub-polynomial query time. For exact distances, the state of the art algorithm has an update time of O[over ~](n[superscript 2]). For approximate distances, the best previous algorithm has a O(kmn[superscript 1/k]) update time and returns (2 k - 1) stretch paths. Thus, it needs an update time of O(m[sqrt](n)) to get close to our approximation, and it has to return O([sqrt](log n)) approximate distances to match our update time.
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