Degree Four Plane Spanners: Simpler and Better

TL;DR

This paper presents a simple, efficient method to construct plane spanners with maximum degree 4 and stretch factor at most 20, improving previous bounds and revealing structural properties of Delaunay triangulations based on equilateral-triangle distance.

Contribution

It introduces a new approach using equilateral-triangle distance Delaunay triangulations to build plane spanners with optimal degree and stretch factor bounds, simplifying previous methods.

Findings

Constructs a plane spanner with degree at most 4 and stretch factor at most 20 in O(n log n) time.

Matches the smallest known maximum degree bound of 4, improving the stretch factor from 156.82 to 20.

When points are in convex position, the maximum degree is at most 3, which is tight.

Abstract

Let ${\cal P}$ be a set of $n$ points embedded in the plane, and let ${\cal C}$ be the complete Euclidean graph whose point-set is ${\cal P}$. Each edge in ${\cal C}$ 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 ${\cal C}$ of maximum degree at most 4 and stretch factor at most 20. This improves a long sequence of results on the construction of plane spanners of ${\cal C}$. Our result matches the smallest known upper bound of 4 by Bonichon et al. on the maximum degree of plane spanners of ${\cal C}$, 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 the Delaunay triangulations defined with respect to the equilateral-triangle distance. The structure of the constructed spanner implies that when ${\cal P}$ is in convex position, the maximum degree of this 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 ${\cal C}$ when the point-set ${\cal P}$ is in convex position, the results in this paper give a tight bound of 3 on the maximum degree of plane spanners of ${\cal C}$ for point-sets in convex position.