Optimal Approximate Distance Oracle for Planar Graphs

A (
$1+\epsilon$
) -approximate distance oracle of an edge-weighted graph is a data structure that returns an approximate shortest path distance between any two query vertices up to a (
$1+\epsilon$
) factor. Thorup (FOCS 2001, JACM 2004) and Klein (SODA 2002) independently constructed a (
$1+\epsilon$
) -approximate distance oracle with
$O(n\log n)$
space, measured in number of words, and
$O(1)$
query time when
$G$
is an undirected planar graph with
$n$
vertices and
$\epsilon$
is a fixed constant. Many follow-up works gave (
$1+\epsilon$
) -approximate distance oracles with various trade-offs between space and query time. However, improving
$O(n\log n)$
space bound without sacrificing query time remains an open problem for almost two decades. In this work, we resolve this problem affirmatively by constructing a (
$1+\epsilon$
) approximate distance oracle with optimal
$O(n)$
space and
$O(1)$
query time for undirected planar graphs and fixed
$\epsilon$
. We also make substantial progress for planar digraphs with non-negative edge weights. For fixed
$\epsilon > 0$
, we give a (
$1+\epsilon$
) -approximate distance oracle with space
$o(n\log(Nn))$
and
$O(\log\log(Nn)$
query time; here
$N$
is the ratio between the largest and smallest positive edge weight. This improves Thorup's (FOCS 2001, JACM 2004)
$O(n\log(Nn)\log n)$
space bound by more than a logarithmic factor while matching the query time of his structure. This is the first improvement for planar digraphs in two decades, both in the weighted and unweighted setting.