Progress in Hierarchical Clustering & Minimum Weight Triangulation

Abstract: In this thesis we study efficient computational methods for geometrical problems of practical importance and theoretical interest. The problems that we consider are primarily complete linkage clustering, minimum spanning trees, and approximating minimum weight triangulation. Below is a list of the main results proved in the thesis. The complete linkage clustering of n points in the plane can be computed in O(n log2 n) time and linear space. If the points lie in Rd, the complete linkage clustering can be computed in optimal O(n log n) time, under the L1 and Loo-metrics. We also design efficient algorithms for approximating the complete linkage clustering. A minimum spanning tree of n points in Rd can be obtained in optimal O(Td(n,n)) time, where Td(n,m) denotes the time to find a closest bichromatic pair between n red points and m blue points. The greedy triangulation of n points in the plane has length O( sqrt(n)) times that of a minimum weight triangulation, and can be computed in linear time, given the Delaunay triangulation. A triangulation of length at most a constant times that of a minimum weight triangulation can be obtained in polynomial time (in fact, O(n log n) time suffices). If the points are corners of their convex hull, we show that linear time suffices to find a triangulation of length at most 1+e times that of a minimum weight triangulation, where e is an arbitrarily small positive constant.