Abstract: Let P be a simple polygon with n vertices. The dual graphT of a triangulation T of P is the graph whose vertices correspond to the bounded faces of T and whose edges connect those faces of T that share an edge. We consider triangulations of P that minimize or maximize the diameter of their dual graph. We show that both triangulations can be constructed in O(n3logn) time using dynamic programming. If P is convex, we show that any minimizing triangulation has dual diameter exactly 2log2(n/3) or 2log2(n/3)1, depending on n. Trivially, in this case any maximizing triangulation has dual diameter n2. Furthermore, we investigate the relationship between the dual diameter and the number of ears (triangles with exactly two edges incident to the boundary of P) in a triangulation. For convex P, we show that there is always a triangulation that simultaneously minimizes the dual diameter and maximizes the number of ears. In contrast, we give examples of general simple polygons where every triangulation that maximizes the number of ears has dual diameter that is quadratic in the minimum possible value. We also consider the case of point sets in general position in the plane. We show that for any such set of n points there are triangulations with dual diameter in O(logn) and in (n).

Abstract: We present a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time O(2^O(d)(n log n + m)), where n is the input size, m is the output point set size, and d is the ambient dimension. The constants only depend on the desired element quality bounds. To gain this new efficiency, the algorithm approximately maintains the Voronoi diagram of the current set of points by storing a superset of the Delaunay neighbors of each point. By retaining quality of the Voronoi diagram and avoiding the storage of the full Voronoi diagram, a simple exponential dependence on d is obtained in the running time. Thus, if one only wants the approximate neighbors structure of a refined Delaunay mesh conforming to a set of input points, the algorithm will return a size 2^O(d)m graph in 2^O(d)(n log n + m) expected time. If m is superlinear in n, then we can produce a hierarchically well-spaced superset of size 2^O(d)n in 2^O(d)n log n expected time.