The Stretch Factor of $L_1$- and $L_\infty$-Delaunay Triangulations

TL;DR

This paper precisely determines the stretch factor of $L_1$- and $L_$-Delaunay triangulations as approximately 2.61, improving previous bounds and being the first exact calculation for any $L_p$-Delaunay triangulation.

Contribution

It provides the first exact calculation of the stretch factor for $L_p$-Delaunay triangulations for any real $p  1$, improving longstanding bounds.

Findings

Stretch factor is approximately 2.61 for $L_1$- and $L_$-Delaunay triangulations.

This is the first exact determination of the stretch factor for any $L_p$-Delaunay triangulation.

The bound improves the previous 25-year-old bound of .  by Chew.

Abstract

In this paper we determine the stretch factor of the $L_1$-Delaunay and $L_\infty$-Delaunay triangulations, and we show that this stretch is $\sqrt{4+2\sqrt{2}} \approx 2.61$. Between any two points $x,y$ of such triangulations, we construct a path whose length is no more than $\sqrt{4+2\sqrt{2}}$ times the Euclidean distance between $x$ and $y$, and this bound is best possible. This definitively improves the 25-year old bound of $\sqrt{10}$ by Chew (SoCG '86). To the best of our knowledge, this is the first time the stretch factor of the well-studied $L_p$-Delaunay triangulations, for any real $p\ge 1$, is determined exactly.