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Practical Methods for Shape Fitting and Kinetic Data Structures using Coresets

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Abstract

The notion of ε-kernel was introduced by Agarwal et al. (J. ACM 51:606–635, 2004) 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 ε-kernel by showing that it yields approximation algorithms for a wide range of geometric optimization problems.

We present a simpler and more practical algorithm for computing the ε-kernel of a set P of points in ℝd. 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-kernels 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.

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Authors and Affiliations

  1. Department of Computer Science, Duke University, Durham, NC, 27708-0129, USA

    Hai Yu & Pankaj K. Agarwal

  2. Department of Computer Science, University of Iowa, Iowa City, IA, 52242-1419, USA

    Raghunath Poreddy & Kasturi R. Varadarajan

Authors
  1. Hai Yu
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  2. Pankaj K. Agarwal
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  3. Raghunath Poreddy
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  4. Kasturi R. Varadarajan
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Corresponding author

Correspondence to Hai Yu.

Additional information

A preliminary version of the paper appeared in Proceedings of the 20th Annual ACM Symposium on Computational Geometry, 2004, pp. 263–272. Research by the first two authors is supported by NSF under grants CCR-00-86013, EIA-98-70724, EIA-01-31905, and CCR-02-04118, and by a grant from the US–Israel Binational Science Foundation. Research by the fourth author is supported by NSF CAREER award CCR-0237431.

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Yu, H., Agarwal, P.K., Poreddy, R. et al. Practical Methods for Shape Fitting and Kinetic Data Structures using Coresets. Algorithmica 52, 378–402 (2008). https://doi.org/10.1007/s00453-007-9067-9

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  • DOI: https://doi.org/10.1007/s00453-007-9067-9

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