A comparison of sequential Delaunay triangulation algorithms

Abstract

This paper presents an experimental comparison of a number of different algorithms for computing the Deluanay triangulation. The algorithms examined are: Dwyer's divide and conquer algorithm, Fortune's sweepline algorithm, several versions of the incremental algorithm (including one by Ohya, Iri, and Murota, a new bucketing-based algorithm described in this paper, and Devillers's version of a Delaunay-tree based algorithm that appears in LEDA), an algorithm that incrementally adds a correct Delaunay triangle adjacent to a current triangle in a manner similar to gift wrapping algorithms for convex hulls, and Barber's convex hull based algorithm.Most of the algorithms examined are designed for good performance on uniformly distributed sites. However, we also test implementations of these algorithms on a number of non-uniform distibutions. The experiments go beyond measuring total running time, which tends to be machine-dependent. We also analyze the major high-level primitives that algorithms use and do an experimental analysis of how often implementations of these algorithms perform each operation.

Design and Implementation of a Practical Parallel Delaunay Algorithm

“…We therefore considered distributions motivated by different domains, such as the distribution of stars in a flat galaxy (Kuzmin) and point sets originating from mesh generation problems. The density of our distributions may vary greatly across the domain, but we observed that the resulting triangulations tend to contain relatively few bad aspect-ratio triangles-especially when contrasted with some artificial distributions, such as points along the diagonals of a square [1]. Indeed, for the case of uniform distribution, Bern et al [31] provided bounds on the expected worst aspect ratio.…”

“…The data files are available upon request if a different definition of work is of interest. Although floating-point operations certainly do not account for all costs in an algorithm they have the important advantage of being machine independent (at least for machines that implement the standard IEEE floating-point instructions) and seem to have a strong correlation to running time [1], at least for algorithms with similar structure.…”

“…Dwyer's algorithm is a variation of Guibas and Stolfi's divide-and-conquer algorithm, which is careful about the cuts in the divide-and-conquer phase so that for quasi-uniform distributions the expected run time is O(n log log n), and on other distributions is at least as efficient as the original algorithm. In a recent paper, Su and Drysdale [1] experimented with a variety of sequential Delaunay algorithms, and Dwyer's algorithm performed as well or better than all others across a variety of distributions. It is therefore a good target to compare with.…”

“…As a consequence, there are many sequential algorithms available for Delaunay triangulation, along with efficient implementations. Su and Drysdale [1] present an excellent experimental comparison of several such algorithms. Since these algorithms are time and memory intensive, parallel implementations are important both for improved performance and to allow the solution of problems that are too large for sequential machines.…”

“…This is relative to Dwyer's algorithm, which is the best of the sequential Delaunay algorithms studied by Su and Drysdale [1]. Figure 1 shows a comparison of floating-point operations performed by our algorithm and Dwyer's algorithm for the four distributions (see Section 3.2 for full results).…”