Many Distances in Planar Graphs

We show how to compute in O(n4/3log 1/3n+n2/3k2/3log n) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/log n)5/6≤k≤n2/log 6n. As an application, we speed up previous algorithms for computing the dilation of geometric planar graphs.

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