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

Abstract

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This paper considers fully dynamic (1+e) distance oracles and (1+e) forbidden-set labeling schemes for planar graphs. For a given n-vertex planar graph G with edge weights drawn from [1,M] and parameter e>0, our forbidden-set labeling scheme uses labels of length λ = O(e-1 log2n 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+e), in O(|F|2 λ) time. We then present a general method to transform (1+e) forbidden-set labeling schemas into a fully dynamic (1+e) distance oracle. Our fully dynamic (1+e) distance oracle is of size O(n log{n} • maxlogn) and has ~O(n1/2) query and update time, both the query and the update time are worst case. This improves on the best previously known (1+e) dynamic distance oracle for planar graphs, which has worst case query time ~O(n2/3) and amortized update time of ~O(n2/3). Our (1+e) forbidden-set labeling scheme can also be extended into a forbidden-set labeled routing scheme with stretch (1+e).