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

Nicolas Bonichon (1, 2) , Cyril Gavoille (1, 2, 3) , Nicolas Hanusse (1, 2) , Ljubomir Perkovic (4, 5)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 CEPAGE - Algorithmics for computationally intensive applications over wide scale distributed platforms
3 IUF - Institut universitaire de France
4 CTI - School of Computer Science, Telecommunications, and Information Systems [DePaul]
5 SOC - School of Computing [DePaul]

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. Moreover, we show that the same short path exists in \Y4 graphs, a subgraph of $L_\infty$-Delaunay triangulations, and therefore we determine the stretch factor for these graphs too.

Mots clés: Delaunay triangulations; $L_1$-metric; $L_\infty$-metric; stretch factor; Yao graphs

Domaines: Géométrie algorithmique [cs.CG]

Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse, Ljubomir Perkovic. The Stretch Factor of $L_1$- and $L_\infty$-Delaunay Triangulations. 2012. ⟨hal-00673187v1⟩