An incremental algorithm for simultaneous construction of 2D Voronoi diagram and Delaunay triangulation based on a face-based data structure

Abstract

In this paper, a new incremental algorithm with an overall complexity of O(n logn) for constructing both the 2D Voronoi diagram (VD) and Delaunay triangulation (DT) is proposed. Moreover, a new face-based data structure is proposed which is more cost-effective in terms of memory than the previous edge-based data structures. The proposed data structure maintains the VD and DT in separate matrices, while establishing a connection between these two, allowing extracting neighborhoods and geometric information without the need for further processing in many applications. Moreover, the proposed data structure can be easily converted to the other edge-based data structures such as the Doubly Connected Edge List. The proposed algorithm for finding the nearest neighbor to the added point, in this incremental algorithm, has O(logn) complexity. Evaluations performed by points with different numbers and distributions confirm the high robustness of the proposed algorithm against different point distributions and its O(n logn) overall complexity.