Approximating extent measures of points

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

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