Exact Distance Oracles for Planar Graphs with Failing Vertices

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. We show several tradeoffs between space, query time, and preprocessing time. In particular, for a directed weighted planar graph with n vertices and any constant k, we show an Õ(n)-size, Õ(√ n)-query-time oracle.1 We then present a space vs. query time tradeoff: for any q ε [ 1,√ n ], we propose an oracle of size nk+1+o(1)/q2k that answers queries in Õ(q) time. For single vertex failures (k = 1), our n2+o(1)/q2-size, Õ(q)-query-time oracle improves over the previously best known tradeoff of Baswana et al. SODA 2012 by polynomial factors for q ≥ nt, for any t ∈ (0,1/2]. For multiple failures, no planarity exploiting results were previously known.