Exact Distance Oracles for Planar Graphs with Failing Vertices

Abstract

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 n k+1+o(1) / q 2k that answers queries in Õ( q ) time. For single vertex failures ( k = 1), our n 2+o(1) / q 2 -size, Õ( q )-query-time oracle improves over the previously best known tradeoff of Baswana et al. SODA 2012 by polynomial factors for q ≥ n t , for any t ∈ (0,1/2]. For multiple failures, no planarity exploiting results were previously known.