2023/12/21 by Prosenjit Bose, Bose, Prosenjit, Jean-Lou De Carufel +3 · 1 citation
Computer Science · Environmental Science · #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #Data Management and Algorithms #FOS: Computer and information sciences #Remote Sensing and LiDAR Applications
paper · pdf · doi:10.48550/arxiv.2312.14305
openalex publication_date 2023/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Finding the exact spanning ratio of a Delaunay graph has been one of the longstanding open problems in Computational Geometry. Currently there are only four convex shapes for which the exact spanning ratio of their Delaunay graph is known: the equilateral triangle, the square, the regular hexagon and the rectangle. In this paper, we show the exact spanning ratio of the parallelogram Delaunay graph, making the parallelogram the fifth convex shape for which an exact bound is known. The worst-case spanning ratio is exactly (√(2)√(1+A2+2Acos(θ0)+(A+cos(θ0))√(1+A2+2Acos(θ0))))/(sin(θ0)) . where A is the aspect ratio and θ0 is the non-obtuse angle of the parallelogram. Moreover, we show how to construct a parallelogram Delaunay graph whose spanning ratio matches the above mentioned spanning ratio.