Improved stretch factor of Delaunay triangulations of points in convex position

Xuehou Tan (), Charatsanyakul Sakthip, Bo Jiang () and Shimao Liu
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Xuehou Tan: Tokai University
Charatsanyakul Sakthip: Tokai University
Bo Jiang: Dalian Maritime University
Shimao Liu: Tokai University

Journal of Combinatorial Optimization, 2023, vol. 45, issue 1, No 14, 12 pages

Abstract: Abstract Let S be a set of n points in the plane, and let DT(S) be the planar graph of the Delaunay triangulation of S. For a pair of points $$a, b \in S$$ a , b ∈ S , denote by |ab| the Euclidean distance between a and b. Denote by DT(a, b) the shortest path in DT(S) between a and b, and let |DT(a, b)| be the total length of DT(a, b). Dobkin et al. were the first to show that DT(S) can be used to approximate the complete graph of S in the sense that the stretch factor $$\frac{|DT(a, b)|}{|a b|}$$ | D T ( a , b ) | | a b | is upper bounded by $$((1 + \sqrt{5})/2) \pi \approx 5.08$$ ( ( 1 + 5 ) / 2 ) π ≈ 5.08 . Recently, Xia improved this factor to 1.998. Amani et al. have also shown that if the points of S are in convex position (i.e., they form the vertices of a convex polygon), then a planar graph with these vertices can be constructed such that its stretch factor is 1.88. In this paper, we prove that if the points of S are in convex position, then the stretch factor of DT(S) is less than 1.84, improving upon the previously known factors of Delaunay triangulations or planar graphs in the convex case.

Keywords: Computational geometry; Delaunay triangulations; Stretch factor; Convex polygons (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10878-022-00940-4

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