Abstract
Let P be a set of n points in ℝd. A subset \(\mathcal {S}\) of P is called a (k,ε)-kernel if for every direction, the directional width of \(\mathcal {S}\) ε-approximates that of P, when k “outliers” can be ignored in that direction. We show that a (k,ε)-kernel of P of size O(k/ε (d−1)/2) can be computed in time O(n+k 2/ε d−1). The new algorithm works by repeatedly “peeling” away (0,ε)-kernels from the point set.
We also present a simple ε-approximation algorithm for fitting various shapes through a set of points with at most k outliers. The algorithm is incremental and works by repeatedly “grating” critical points into a working set, till the working set provides the required approximation. We prove that the size of the working set is independent of n, and thus results in a simple and practical, near-linear ε-approximation algorithm for shape fitting with outliers in low dimensions.
We demonstrate the practicality of our algorithms by showing their empirical performance on various inputs and problems.
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A preliminary version of this paper appeared in Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 182–191. P.A. and H.Y. are supported by NSF under grants CCR-00-86013, EIA-01-31905, CCR-02-04118, and DEB-04-25465, by ARO grants W911NF-04-1-0278 and DAAD19-03-1-0352, and by a grant from the U.S.–Israel Binational Science Foundation. S.H.-P. is supported by a NSF CAREER award CCR-0132901.
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Agarwal, P.K., Har-Peled, S. & Yu, H. Robust Shape Fitting via Peeling and Grating Coresets. Discrete Comput Geom 39, 38–58 (2008). https://doi.org/10.1007/s00454-007-9013-2
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DOI: https://doi.org/10.1007/s00454-007-9013-2