Planar distance oracles with better time-space tradeoffs

Abstract

In a recent breakthrough, Charalampopoulos, Gawrychowski, Mozes, and Weimann [9] showed that exact distance queries on planar graphs could be answered in no(1) time by a data structure occupying n1+o(1) space, i.e., up to o(1) terms, optimal exponents in time (0) and space (1) can be achieved simultaneously. Their distance query algorithm is recursive: it makes successive calls to a point-location algorithm for planar Voronoi diagrams, which involves many recursive distance queries. The depth of this recursion is non-constant and the branching factor logarithmic, leading to (log n)ω(1) = no(1) query times. In this paper we present a new way to do point-location in planar Voronoi diagrams, which leads to a new exact distance oracle. At the two extremes of our space-time tradeoff curve we can achieve either n1+o(1) space and log2+o(1) n query time, or n log2+o(1) n space and no(1) query time. All previous oracles with Õ(1) query time occupy space n1+Ω(1), and all previous oracles with space Õ(n) answer queries in nΩ(1) time.