Geometric spanners of bounded tree-width

Abstract

Given a point set $P$ in the Euclidean space, a geometric $t$-spanner $G$ is a graph on $P$ such that for every pair of points, the shortest path in $G$ between those points is at most a factor $t$ longer than the Euclidean distance between those points. The value $t\geq 1$ is called the dilation of $G$. Commonly, the aim is to construct a $t$-spanner with additional desirable properties. In graph theory, a powerful tool to admit efficient algorithms is bounded tree-width. We therefore investigate the problem of computing geometric spanners with bounded tree-width and small dilation $t$. Let $d$ be a fixed integer and $P \subset \mathbb{R}^d$ be a point set with $n$ points. We give a first algorithm to compute an $\mathcal{O}(n/k^{d/(d-1)})$-spanner on $P$ with tree-width at most $k$. The dilation obtained by the algorithm is asymptotically worst-case optimal for graphs with tree-width $k$: We show that there is a set of $n$ points such that every spanner of tree-width $k$ has dilation $\mathcal{O}(n/k^{d/(d-1)})$. We further prove a tight dependency between tree-width and the number of edges in sparse connected planar graphs, which admits, for point sets in $\mathbb{R}^2$, a plane spanner with tree-width at most $k$ and small maximum vertex degree. Finally, we show an almost tight bound on the minimum dilation of a spanning tree of $n$ equally spaced points on a circle, answering an open question asked in previous work.