We present results of an empirical investigation into the performance of two O(nlogn) worst-case optimal Delaunay triangulation algorithms: a divide-andconquer algorithm and a plane-sweep algorithm. We present improvements which give a factor of 4-5 speedup to the divide-and-conquer algorithm and a factor of 13-16 speed-up to the plane-sweep algorithm. Experiments using our improved implementations of both algorithms show the plane-sweep algorithm to be slightly faster (about 20%) than the divide-andconquer algorithm across a range of distributions. Using our fastest implementation of the plane-sweep algorithm a set of points can be triangulated in 7-8 times the time it takes to (merge) sort them.

We show that the wavefront approach to Voronoi diagrams (a deterministic line sweep algorithm that does not use geometric transform) can be generalized to distance measures more general than the Euclidean metric. In fact, we provide the first worstcase optimal (O(n log time, O(n) space) algorithm that is valid for the full class of what has been called nice metrics in the plane. This also solves the previously open problem of providing an time plane-sweep algorithm for arbitrary L k -metrics.