Fast segment insertion and incremental construction of constrained delaunay triangulations

Abstract

The most commonly implemented method of constructing a constrained Delaunay triangulation (CDT) in the plane is to first construct a Delaunay triangulation, then incrementally insert the input segments one by one. For typical implementations of segment insertion, this method has a Θ(kn2) worst-case running time, where n is the number of input vertices and k is the number of input segments. We give a randomized algorithm for inserting a segment into a CDT in expected time linear in the number of edges the segment crosses, and demonstrate with a performance comparison that it is faster than gift-wrapping for segments that cross many edges. A result of Agarwal, Arge, and Yi implies that randomized incremental construction of CDTs by our segment insertion algorithm takes expected O(n log n + n log2 k) time. We show that this bound is tight by deriving a matching lower bound. Although there are CDT construction algorithms guaranteed to run in O(n log n) time, incremental CDT construction is easier to program and competitive in practice. Moreover, the ability to incrementally update a CDT by inserting a segment is useful in itself.