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 = ~ Θ(n/√ S) or Q = ~Θ(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.