Abstract
Given a set P of n points in ℝd, an ε-kernel K ⊆ P approximates the directional width of P in every direction within a relative (1 − ε) factor. In this paper we study the stability of ε-kernels under dynamic insertion and deletion of points to P and by changing the approximation factor ε. In the first case, we say an algorithm for dynamically maintaining a ε-kernel is stable if at most O(1) points change in K as one point is inserted or deleted from P. We describe an algorithm to maintain an ε-kernel of size O(1/ε (d − 1)/2) in O(1/ε (d − 1)/2 + logn) time per update. Not only does our algorithm maintain a stable ε-kernel, its update time is faster than any known algorithm that maintains an ε-kernel of size O(1/ε (d − 1)/2). Next, we show that if there is an ε-kernel of P of size κ, which may be dramatically less than O(1/ε (d − 1)/2), then there is an (ε/2)-kernel of P of size \(O(\min\{ 1/\varepsilon^{(d-1)/2}, \kappa^{\lfloor d/2 \rfloor} \log^{d-2} (1/\varepsilon)\})\). Moreover, there exists a point set P in ℝd and a parameter ε> 0 such that if every ε-kernel of P has size at least κ, then any (ε/2)-kernel of P has size \(\Omega(\kappa^{\lfloor d/2 \rfloor})\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Agarwal, P.K., Har-Peled, S., Varadarajan, K.: Approximating extent measure of points. Journal of ACM 51(4), 606–635 (2004)
Agarwal, P.K., Har-Peled, S., Varadarajan, K.: Geometric approximations via coresets. In: Combinatorial and Computational Geometry, pp. 1–31 (2005)
Agarwal, P.K., Phillips, J.M., Yu, H.: Stability of ε-kernels. arXiv:1003.5874
Agarwal, P.K., Yu, H.: A space-optimal data-stream algorithm for coresets in the plane. In: SoCG, pp. 1–10 (2007)
Barequet, G., Har-Peled, S.: Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. Journ. of Algs. 38, 91–109 (2001)
Chan, T.: Faster core-set constructions and data-stream algorithms in fixed dimensions. Computational Geometry: Theory and Applications 35, 20–35 (2006)
Chan, T.: Dynamic coresets. In: SoCG, pp. 1–9 (2008)
Har-Peled, S.: Approximation Algorithm in Geometry, ch. 22 (2010), http://valis.cs.uiuc.edu/~sariel/teach/notes/aprx/
Hershberger, J., Suri, S.: Adaptive sampling for geometric problems over data streams. Computational Geometry: Theory and Applications 39, 191–208 (2008)
Yu, H., Agarwal, P.K., Poreddy, R., Varadarajan, K.: Practical methods for shape fitting and kinetic data structures using coresets. Algorithmica 52, 378–402 (2008)
Zarrabi-Zadeh, H.: An almost space-optimal streaming algorithm for coresets in fixed dimensions. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 817–829. Springer, Heidelberg (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Agarwal, P.K., Phillips, J.M., Yu, H. (2010). Stability of ε-Kernels. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15775-2_42
Download citation
DOI: https://doi.org/10.1007/978-3-642-15775-2_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15774-5
Online ISBN: 978-3-642-15775-2
eBook Packages: Computer ScienceComputer Science (R0)