Computational Geometry II D. T. Lee Department of Electrical and Computer Engineering Northwestern University Evanston, IL 60208 e-mail:
[email protected]
1 Introduction This is a follow up on the previous Chapter dealing with geometric problems and their ecient solutions. The classes of problems that we address in this Chapter include proximity, optimization, intersection, searching, point location, and some discussions of geometric software that is under development.
2 Proximity Geometric problems pertaining to the questions of how close two geometric entities are among a collection of objects or how similar two geometric patterns match each other abound. For example, in pattern classi cation and clustering, features that are similar according to some metric, are to be clustered in a group. The two aircrafts that are closest at any time instant in the air space will have the largest likelihood of collision with each other. In some cases one may be interested in how far apart or how dissimilar the objects are. Some of these proximity related problems will be addressed in this section.
2.1 Closest Pair
Consider a set S of n points in 0, nd an annulus containing S whose width is upper bounded by . They presented an O(n log n) algorithm for 2-dimensions and a near quadratic-time algorithm for 3-dimensions.
4 Geometric Matching Matching in general graphs is one of the classical subjects in combinatorial optimization and has applications in operations research, pattern recognition and VLSI design. Only geometric versions of the matching problem are discussed here. For graph-theoretic matching problems see [82]. Given a weighted undirected complete graph on a set of 2n vertices, a complete matching is a set of n edges such that each vertex has exactly one edge incident on it. The weight of a matching is the sum of the weights of the edges in the matching. In a metric space, the vertices are points in the plane and the weight of an edge between two points is the distance between them. The Euclidean minimum weight P matching problem is that given 2n points, nd n matching pairs of points (pi ; qi ) such that d(pi ; qi ) is minimized. It was not known if geometric properties can be exploited to obtain an algorithm that is faster than the (n3) algorithm for general graphs (see [82]). Vaidya[96] settled this question in the armative. His algorithm is based on a well-studied primal-dual algorithm for weighted matching. Making use of additive weighted Voronoi diagram discussed in Section 2.2.4 and the range search tree structure (see Section 7.1), Vaidya solved the problem in O(n2:5 log4 n) time. This algorithm also generalizes to
