A space-optimal data-stream algorithm for coresets in the plane
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
No dimension independent core-sets for containment under homothetics
Proceedings of the twenty-seventh annual symposium on Computational geometry
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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 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.