Spanners for Geodesic Graphs and Visibility Graphs

Abstract

Let $\mathcal{P}$ be a set of n points inside a polygonal domain $\mathcal{D}$. A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set $\mathcal{P}$ with respect to the geodesic distance function $\pi$ where for any two points p and q, $\pi (p,q)$ is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., $h=0$), we construct a ($\sqrt{10}+\epsilon$)-spanner that has $O(n \log ^2 n)$ edges. For a case where there are h holes, our construction gives a ($5+\epsilon$)-spanner with the size of $O(n\sqrt{h}\log ^2 n)$. Moreover, we study t-spanners for the visibility graph of $\mathcal{P}$ ($VG(\mathcal{P})$, for short) with respect to a hole-free polygonal domain $\mathcal{D}$. The graph $VG(\mathcal{P})$ is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a ($3+\epsilon$)-spanner of size $O(n^{4/3+\delta })$ for some $\delta >0$. In addition, we show that there is a set $\mathcal{P}$ of n points such that any $(3-\epsilon )$-spanner of $VG(\mathcal{P})$ must contain $\varOmega (n^2)$ edges.