Approximate Distance Oracles for Planar Graphs with Subpolynomial Error Dependency

Abstract

Thorup [FOCS'01, JACM'04] and Klein [SODA'01] independently showed that there exists a $(1+\epsilon )$-approximate distance oracle for planar graphs with $O(n(\log n){\epsilon }^{-1})$ space and $O({\epsilon }^{-1})$ query time. While the dependency on $n$ is nearly linear, the space-query product of their oracles depend quadratically on $1/\epsilon$. Many follow-up results either improved the space \emph{or} the query time of the oracles while having the same, sometimes worst, dependency on $1/\epsilon$. Kawarabayashi, Sommer, and Thorup [SODA'13] were the first to improve the dependency on $1/\epsilon$ from quadratic to nearly linear (at the cost of ${\log }^{\ast }(n)$ factors). It is plausible to conjecture that the linear dependency on $1/\epsilon$ is optimal: for many known distance-related problems in planar graphs, it was proved that the dependency on $1/\epsilon$ is at least linear. In this work, we disprove this conjecture by reducing the dependency of the space-query product on $1/\epsilon$ from linear all the way down to \emph{subpolynomial} $(1/\epsilon {)}^{o(1)}$. More precisely, we construct an oracle with $O(n\log (n)({\epsilon }^{-o(1)}+{\log }^{\ast }n))$ space and ${\log }^{2+o(1)}(1/\epsilon )$ query time. Our construction is the culmination of several different ideas developed over the past two decades.

34 pages, 10 figures