Practical Methods for Shape Fitting and Kinetic Data Structures using Coresets

Written with Pankaj Agarwal, Raghunath Poreddy, and Kasturi Varadarajan.

Algorithmica, to appear.

Also in Proc. 20th Annual ACM Symposium on Computational Geometry, pages 263-272, 2004.

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Abstract: The notion of ε-kernel was introduced by Agarwal, Har-Peled and Varadarajan to set up a unified framework for computing various extent measures of a point set P approximately. Roughly speaking, a subset Q⊆P is an ε-kernel of P if for every slab W containing Q, the expanded slab (1+ε)W contains P. They illustrated the significance of an ε-kernel by showing that it yields approximation algorithms for a wide range of problems. We present a simpler and more practical algorithm for computing the ε-kernel of a set P of points in R^d. We demonstrate the practicality of our algorithm by showing its empirical performance on various inputs. We then describe an incremental algorithm for fitting various shapes and use the ideas of our algorithm for computing ε-kernel to analyze the performance of this algorithm. We illustrate the versatility and practicality of this technique by implementing approximation algorithms for minimum enclosing cylinder, minimum-volume bounding box, and minimum-width annulus. Finally, we show that ε-kernels can be effectively used to expedite the algorithms for maintaining extents of moving points.