Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels

This paper considers fully dynamic (1+ε) distance oracles and (1+ε) forbidden-set labeling schemes for planar graphs. For a given n-vertex planar graph G with edge weights drawn from [1,M] and parameter ε>0, our forbidden-set labeling scheme uses labels of length λ = O(ε ^-1 log ²n log(nM) · maxlogn). Given the labels of two vertices s and t and of a set F of faulty vertices/edges, our scheme approximates the distance between s and t in G \ F with stretch (1+ε), in O(|F| ² λ) time. We then present a general method to transform (1+ε) forbidden-set labeling schemas into a fully dynamic (1+ε) distance oracle. Our fully dynamic (1+ε) distance oracle is of size O(n log{n} · maxlogn) and has Õ(n ^1/2) query and update time, both the query and the update time are worst case. This improves on the best previously known (1+ε) dynamic distance oracle for planar graphs, which has worst case query time Õ(n ^2/3) and amortized update time of Õ(n ^2/3). Our (1+ε) forbidden-set labeling scheme can also be extended into a forbidden-set labeled routing scheme with stretch (1+ε).