# Constant Query Time $(1 + \epsilon)$-Approximate Distance Oracle for Planar Graphs

## TL;DR

This paper introduces a novel $(1+psilon)$-approximate distance oracle for planar graphs that achieves constant query time, with size and preprocessing nearly linear in the number of vertices, improving upon previous methods.

## Contribution

The paper presents the first $(1+psilon)$-approximate distance oracle for planar graphs with $O(1)$ query time independent of psilon, and nearly linear size and preprocessing time.

## Key findings

- Achieves $O(1)$ query time for approximate distances in planar graphs.
- Size and preprocessing time are nearly linear in the number of vertices.
- Improves upon previous $(1+psilon)$-approximate distance oracles with higher query times.

## Abstract

We give a $(1+\epsilon)$-approximate distance oracle with $O(1)$ query time for an undirected planar graph $G$ with $n$ vertices and non-negative edge lengths. For $\epsilon>0$ and any two vertices $u$ and $v$ in $G$, our oracle gives a distance $\tilde{d}(u,v)$ with stretch $(1+\epsilon)$ in $O(1)$ time. The oracle has size $O(n\log n ((\log n)/\epsilon+f(\epsilon)))$ and pre-processing time $O(n\log n((\log^3 n)/\epsilon^2+f(\epsilon)))$, where $f(\epsilon)=2^{O(1/\epsilon)}$. This is the first $(1+\epsilon)$-approximate distance oracle with $O(1)$ query time independent of $\epsilon$ and the size and pre-processing time nearly linear in $n$, and improves the query time $O(1/\epsilon)$ of previous $(1+\epsilon)$-approximate distance oracle with size nearly linear in $n$.