In a recent breakthrough, Charalampopoulos, Gawrychowski, Mozes, and Weimann (STOC 2019) showed that exact distance queries on planar graphs could be answered in n^o(1) time by a data structure occupying n^1+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.

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... e depth of this recursion is non-constant and the branching factor logarithmic, leading to (log n)^ω(1) = n^o(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 n^1+o(1) space and log^2+o(1)n query time, or nlog^2+o(1)n space and n^o(1) query time. All previous oracles with Õ(1) query time occupy space n^1+Ω(1), and all previous oracles with space Õ(n) answer queries in n^Ω(1) time.