Chapter 20 - Closest-Point Problems in Computational Geometry

Abstract

This chapter discusses the closest pair problem, the exact and approximate post-office problem, and the problem of constructing spanners. The closest pair problem and its generalizations arise in areas such as statistics, pattern recognition, and molecular biology. The complexity of the closest pair problem heavily depends on the machine model. The closest pair problem has an Ω(n log n) lower bound in the algebraic computation tree model. An optimal data structure can be used for maintaining the closest pair under insertions. To estimate the running time, a constant number of point location queries and at most one split operation can be performed. Because the tree T may have a linear height, the worst-case running time of the insertion algorithm is O(n). There are two possibilities to improve the running time. One of them is to use a “centroid decomposition” to represent the tree T as a balanced tree.