## Practical Methods for Shape Fitting and Kinetic Data Structures using Core Sets (2004)

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Venue: | In Proc. 20th Annu. ACM Sympos. Comput. Geom |

Citations: | 27 - 8 self |

### BibTeX

@INPROCEEDINGS{Yu04practicalmethods,

author = {Hai Yu and Pankaj K. Agarwal and Raghunath Poreddy and Kasturi R. Varadarajan},

title = {Practical Methods for Shape Fitting and Kinetic Data Structures using Core Sets},

booktitle = {In Proc. 20th Annu. ACM Sympos. Comput. Geom},

year = {2004},

pages = {263--272}

}

### Years of Citing Articles

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### Abstract

The notion of ε-kernel was introduced by Agarwal et al. [5] 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 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-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. 1

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Citation Context ...raditionally P is assumed to be stationary (insertion/deletion of points at discrete times have been considered), recently there has been work on maintaining extent measures of a set of moving points =-=[4]-=-. In the latter, the goal is to maintain the extent measure as the points move, e.g., using the kinetic data structure framework [12]. ∗ A preliminary version of the paper appeared in Proceedings of t... |

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Citation Context ...ng time were previously known for these problems. Many subsequent papers have used a similar approach for other geometric optimization problems, including clustering and other extent-measure problems =-=[6, 8, 13, 14, 22, 23, 24, 25, 26]-=-. These approaches compute a subset Q ⊆ P of small size and solve the underlying optimization problem for Q. The term coreset is now commonly used to refer to such a subset. Although the algorithm by ... |

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Citation Context ...lgorithm extends readily to the case where Aopt is replaced by an approximation algorithm.) This algorithm requires a procedure for computing smallest enclosing disks, and we have used Gärtner’s code =-=[20]-=- for this purpose. We chose the initial subset R as follows: let p1 be an arbitrary point from the input, p2 be the point farthest from p1, and p3 be the point farthest from the line p1p2; we set R = ... |

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Citation Context ...ng time were previously known for these problems. Many subsequent papers have used a similar approach for other geometric optimization problems, including clustering and other extent-measure problems =-=[6, 8, 13, 14, 24, 25, 26, 27, 28]-=-. These approaches compute a subset Q ` P of small size and solve the underlying optimization problem for Q. The term coreset is now commonly used to refer to such a subset. Although the algorithm by ... |