Lower bounds on the dilation of plane spanners

TL;DR

This paper establishes new lower bounds on the dilation of plane spanners, including specific point set constructions with high stretch factors and improved bounds for worst-case dilation.

Contribution

It provides the first explicit point set with dilation at least 1.4308 and demonstrates degree-specific dilation bounds for plane spanners, improving previous lower bounds.

Findings

A set of 23 points with dilation ≥ 1.4308.

Existence of n-point sets with degree 3 dilation = 2.7321.

Existence of n-point sets with degree 4 dilation = 2.1755.

Abstract

(I) We exhibit a set of 23 points in the plane that has dilation at least $1.4308$, improving the previously best lower bound of $1.4161$ for the worst-case dilation of plane spanners. (II) For every integer $n\geq13$, there exists an $n$-element point set $S$ such that the degree 3 dilation of $S$ denoted by $\delta_0(S,3) \text{ equals } 1+\sqrt{3}=2.7321\ldots$ in the domain of plane geometric spanners. In the same domain, we show that for every integer $n\geq6$, there exists a an $n$-element point set $S$ such that the degree 4 dilation of $S$ denoted by $\delta_0(S,4) \text{ equals } 1 + \sqrt{(5-\sqrt{5})/2}=2.1755\ldots$ The previous best lower bound of $1.4161$ holds for any degree. (III) For every integer $n\geq6 $, there exists an $n$-element point set $S$ such that the stretch factor of the greedy triangulation of $S$ is at least $2.0268$.