Exact Distance Oracles for Planar Graphs with Failing Vertices
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We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex u, a target vertex v and a set X of k failed vertices, such an oracle returns the length of a shortest u-to-v path that avoids all vertices in X. We propose oracles that can handle any number k of failures. More specifically, for a directed weighted planar graph with n vertices, any constant k, and for any q ∈ [1,√(n)], we propose an oracle of size Õ(n^k+3/2/q^2k+1) that answers queries in Õ(q) time. In particular, we show an Õ(n)-size, Õ(√(n))-query-time oracle for any constant k. This matches, up to polylogarithmic factors, the fastest failure-free distance oracles with nearly linear space. For single vertex failures (k=1), our Õ(n^5/2/q^3)-size, Õ(q)-query-time oracle improves over the previously best known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for q = Ω(n^t), t ∈ (1/4,1/2]. For multiple failures, no planarity exploiting results were previously known.
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