Approximating extent measures of points

Abstract

We present a general technique for approximating various descriptors of the extent of a set P of n points in R d when the dimension d is an arbitrary fixed constant. For a given extent measure μ and a parameter ε > 0, it computes in time O ( n + 1/ε O (1) ) a subset Q ⊆ P of size 1/ε O (1) , with the property that (1 − ε)μ( P ) ≤ μ( Q ) ≤ μ( P ). The specific applications of our technique include ε-approximation algorithms for (i) computing diameter, width, and smallest bounding box, ball, and cylinder of P , (ii) maintaining all the previous measures for a set of moving points, and (iii) fitting spheres and cylinders through a point set P . Our algorithms are considerably simpler, and faster in many cases, than previously known algorithms.