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

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 $n$.