Optimal higher order Delaunay triangulations of polygons

Abstract

a r t i c l e i n f o a b s t r a c t This paper presents an algorithm to triangulate polygons optimally using order-k Delaunay triangulations, for a number of quality measures. The algorithm uses properties of higher order Delaunay triangulations to improve the O (n 3 ) running time required for normal triangulations to O (k 2 n log k + kn log n) expected time, where n is the number of vertices of the polygon. An extension to polygons with points inside is also presented, allowing to compute an optimal triangulation of a polygon with h 1 components inside in O (kn log n) + O (k) h+2 n expected time. Furthermore, through experimental results we show that, in practice, it can be used to triangulate point sets optimally for small values of k. This represents the first practical result on optimization of higher order Delaunay triangulations for k > 1.

Towards a definition of higher order constrained Delaunay triangulations

“…Another useful operation related to higher order Delaunay triangulations is computing all the order-k triangles. For example, this is a fundamental step when triangulating polygons optimally for order-k Delaunay triangulations [21]. Table 1 summarizes our results for the constrained order definitions.…”

Generalized Delaunay Triangulations:Graph-Theoretic Properties and Algorithms

“…The lemma follows since we can store all the order-k possible triangles in order-k triangulations given by Silveira and van Kreveld [97] in a hash table. Then, for each order-k triangle we subdivide the problem into three subsets of S and count all possible triangulations of each subset without repeating triangles already visited.…”

“…Note that an order-0 triangulation is a standard Delaunay triangulation. Order-k triangulations have been used for terrain modeling, minimum interference networks and triangulation of polygons [17,97,91]. Note that for k ≥ 1, there might be more than one order-k triangulation.…”

“…In this chapter we study the flip graph of higher order Delaunay triangulations. Most previous work on such triangulations focused on algorithmic questions related to finding order-k triangulations that are optimal with respect to extra criteria [104,97,98], or on evaluating their effectiveness in practical settings [20,45]. One of the few theoretical aspects studied is the asymptotic number of order-k triangulations [82].…”

“…Theorem 2.2.3 also implies an enumeration algorithm for all order-k triangulations. However, using the preprocessing method given by Silveira and van Kreveld [97] where all order-k triangles and order-k edges in all order-k triangulation of a point set in general position can be computed in O(k n log k + kn log n) expected time, implies a faster algorithm. Lemma 2.2.5.…”