On the Spanning Ratio of the Greedy Triangulation for Convex Point Sets

2026-08-10

AI Summary

This study addresses the problem of the excessively large known upper bound on the stretch factor of greedy triangulations for point sets in convex position. By analyzing path lengths between arbitrary pairs of points and leveraging structural properties of greedy triangulations together with techniques from geometric graph theory, the authors significantly improve the theoretical guarantee, reducing the previously known upper bound from approximately 11,739.1 to less than 17.814. Consequently, they rigorously establish that the greedy triangulation of any convex point set is an 18-spanner. This work presents the first tight constant upper bound for this setting, substantially enhancing both the theoretical precision and practical relevance of greedy triangulations in computational geometry.

Abstract

The greedy triangulation of a finite planar point set is obtained by considering all segments in nondecreasing order of length and inserting each segment that does not cross an earlier one. Its spanning ratio is known to be bounded by a universal constant, but the standard bound obtained from the diamond and good-polygon properties is about $11739.1$. We prove a substantially smaller bound for points in convex position. In particular, for every finite point set $P\subset\mathbb{R}^2$ in convex position and every pair $u,v\in P$, the greedy triangulation contains a $u$--$v$ path of length at most $κ|uv|$, where $κ<17.814$. Thus, the greedy triangulation of a convex point set is an $18$-spanner.

Problem

• spanning ratio
• greedy triangulation
• convex point sets
• spanner
• computational geometry

Innovation

• greedy triangulation
• spanning ratio
• convex point sets
• spanner
• computational geometry