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Shewchuk, Jonathan Richard; Brown, Brielin C.
Computational geometry : theory and applications, 09/2015, Volume: 48, Issue: 8Journal Article
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. We demonstrate with a performance comparison that for segments that cross many edges, our algorithm is faster than gift-wrapping. We also show that a simple algorithm for segment location, which precedes segment insertion, is fast enough never to be a bottleneck in CDT construction. A result of Agarwal, Arge, and Yi implies that randomized incremental construction of CDTs by our segment insertion algorithm takes expected O(nlogn+nlog2k) 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(nlogn) time, incremental CDT construction is easier to program and competitive in practice. Lastly, we partly extend the analysis (albeit not the linear-time insertion algorithm) to randomized incremental CDT construction in three dimensions.
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