Almost Optimal Exact Distance Oracles for Planar Graphs

We consider the problem of preprocessing a weighted directed planar graph in order to quickly answer exact distance queries. The main tension in this problem is between space S and query time Q, and since the mid-1990s all results had polynomial time-space tradeoffs, e.g., Q = (Theta) over tilde (n/root S) or Q = (Theta) over tilde (n(5/2)/S-3/2). In this article we show that there is no polynomial tradeoff between time and space and that it is possible to simultaneously achieve almost optimal space n(1+o(1)) and almost optimal query time n(o(1)). More precisely, we achieve the following space-time tradeoffs: n(1+o(1)) space and log(2+o(1)) n query time, n log(2+o(1)) n space and n(o(1)) query time, n(4/3+o(1)) space and log(1+o(1)) n query time. We reduce a distance query to a variety of point location problems in additively weighted Voronoi diagrams and develop new algorithms for the point location problem itself using several partially persistent dynamic tree data structures.