Degree four plane spanners: Simpler and better

Abstract

Let $\mathcal{P}$ be a set of $n$ points embedded in the plane, and let $\mathcal{C}$ be the complete Euclidean graph whose point-set is $\mathcal{P}$. Each edge in $\mathcal{C}$ between two points $p$, $q$ is realized as the line segment $[pq]$ and is assigned a weight equal to the Euclidean distance $|pq|$. In this paper, we show how to construct in $O(n \lg n)$ time a plane spanner of $\mathcal{C}$ of maximum degree at most $4$ and of stretch factor at most $20$. This improves a long sequence of results on the construction of bounded degree plane spanners of $\mathcal{C}$. Our result matches the smallest known upper bound of $4$ by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from $156.82$ to $20$. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner and reveals useful structural properties of Delaunay triangulations defined with respect to the equilateral-triangle distance.

The structure of the constructed spanner implies that when $\mathcal{P}$ is in convex position, the maximum degree of the spanner is at most $3$. Combining the above degree upper bound with the fact that $3$ is a lower bound on the maximum degree of any plane spanner of $\mathcal{C}$ when the point-set $\mathcal{P}$ is in convex position, the results in this paper give a tight bound of $3$ on the maximum degree of plane spanners of $\mathcal{C}$ for point-sets in convex position.