Fully dynamic (2 + ∈) approximate all-pairs shortest paths with fast query and close to linear update time

For any fixed 1 > e > 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 Õ(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 ∈: 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 Õ(n²). For approximate distances, the best previous algorithm has a O(kmn¹/^k) update time and returns (2k-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.