Fully Dynamic (2 + epsilon) Approximate All-Pairs Shortest Paths with Fast Query and Close to Linear Update Time

For any fixed 1 > ¿ > 0 we present a fully dynamic algorithm for maintaining (2 + ¿)-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¿(mn^{O(1/¿(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/¿)^{(¿(log n/log(3/¿))}) = n^{(¿(log(3/¿)/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¿(n²). For approximate distances, the best previous algorithm has a O(kmn^1/k) update time and returns (2 k - 1) stretch paths. Thus, it needs an update time of O(m¿(n)) to get close to our approximation, and it has to return O(¿(log n)) approximate distances to match our update time.