Algorithms and Complexity

Volume 6078 of the series Lecture Notes in Computer Science pp 217-228

On the Number of Higher Order Delaunay Triangulations

  • Dieter MitscheAffiliated withCentre de Recerca Matemàtica, Universitat Autònoma de Barcelona
  • , Maria SaumellAffiliated withDept. Matemàtica Aplicada II, Universitat Politècnica de Catalunya
  • , Rodrigo I. SilveiraAffiliated withDept. Matemàtica Aplicada II, Universitat Politècnica de Catalunya

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Higher order Delaunay triangulations are a generalization of the Delaunay triangulation which provides a class of well-shaped triangulations, over which extra criteria can be optimized. A triangulation is order-k Delaunay if the circumcircle of each triangle of the triangulation contains at most k points. In this paper we study lower and upper bounds on the number of higher order Delaunay triangulations, as well as their expected number for randomly distributed points. We show that arbitrarily large point sets can have a single higher order Delaunay triangulation, even for large orders, whereas for first order Delaunay triangulations, the maximum number is 2 n − 3. Next we show that uniformly distributed points have an expected number of at least \(2^{\rho_1 n(1+o(1))}\) first order Delaunay triangulations, where ρ 1 is an analytically defined constant (ρ 1 ≈ 0.525785), and for k > 1, the expected number of order-k Delaunay triangulations (which are not order-i for any i < k) is at least \(2^{\rho_k n(1+o(1))}\), where ρ k can be calculated numerically.