Fast and Compact Exact Distance Oracle for Planar Graphs

Abstract

For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5/3</sup> )-space distance oracle which answers exact distance queries in O(log n) time for n-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space (i.e., space O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2-ϵ)</sup> for some constant ϵ > 0) either required query time polynomial in n or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any S ≥ n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sup> , we show how to obtain an S-space distance oracle that answers queries in time O( n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5/2/S3/2</sup> logn). This is a polynomial improvement over the previous planar distance oracles with o(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/4</sup> ) query time.