A divide-and-conquer Delaunay triangulation algorithm with a vertex array and flip operations in two-dimensional space

Abstract

Divide-and-conquer algorithms provide computational efficiency for constructing Delaunay triangulations; however, their implementation is complicated. Most divide-and-conquer algorithms for Delaunay triangulation utilize edge-based structures, because triangles are frequently deleted and created during the merge process. However, as our proposed divide-and-conquer algorithm does not require existing edges to be deleted in the merge process, a simple array-based data structure can be used for the representation of the triangulation topology. Rather than deleting the edges of the conflicting triangles, which was used in previous methods, the Delaunay property is also preserved with a new flip propagation method in the merge phase. This array-based data structure is much simpler than the commonly used edge-based data structures and requires less memory allocation. The proposed algorithm arranges sites into a permutation vector that represents a kdtree with an array; thus, the space partitioning information is internally represented in the array without any additional data. Since the proposed divide-and-conquer algorithm is compact, the implementation complexity of conventional divideand-conquer triangulation algorithms can be avoided. Despite of the simplicity of this new algorithm, the experimental results indicate that the computational efficiency is comparable to the previous divide-and-conquer algorithms.