Abstract
Given a set P of n points on the real line and a (potentially infinite) family of functions, we investigate the problem of finding a small (weighted) subset \({\mathcal{S}} \subseteq P\), such that for any \(f \in {\mathcal{F}}\), we have that f(P) is a (1±ε)-approximation to \(f({\mathcal{S}})\). Here, f(Q) = ∑ q ∈ Q w(q) f(q) denotes the weighted discrete integral of f over the point set Q, where w(q) is the weight assigned to the point q.
We study this problem, and provide tight bounds on the size \({\mathcal{S}}\) for several families of functions. As an application, we present some coreset constructions for clustering.
The latest version of this paper is avaiable online [1].
This is a preview of subscription content, log in via an institution to check access.
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
Har-Peled, S.: Coresets for discrete integration and clustering (2006), Available from: http://www.uiuc.edu/~sariel/papers/06/integrate
Matoušek, J.: Geometric Discrepancy. Springer, Heidelberg (1999)
Har-Peled, S., Kushal, A.: Smaller coresets for k-median and k-means clustering. In: Proc. 21st Annu. ACM Sympos. Comput. Geom., pp. 126–134 (2005)
Har-Peled, S., Mazumdar, S.: Coresets for k-means and k-median clustering and their applications. In: Proc. 36th Annu. ACM Sympos. Theory Comput., pp. 291–300 (2004)
Chen, K.: On k-median clustering in high dimensions. In: Proc. 17th ACM-SIAM Sympos. Discrete Algorithms, pp. 1177–1185 (2006)
Agarwal, P.K., Har-Peled, S., Varadarajan, K.: Geometric approximation via coresets. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry. Math. Sci. Research Inst. Pub., Cambridge (2005)
Feldman, D., Fiat, A., Sharir, M.: Coresets for weighted facilities and their applications (manuscript 2006)
Alon, N., Dar, S., Parnas, M., Ron, D.: Testing of clustering. In: Proc. 41st Annu. IEEE Sympos. Found. Comput. Sci., pp. 240–250 (2000)
Agarwal, P.K., Procopiuc, C.M.: Approximation algorithms for projective clustering. In: Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, pp. 538–547 (2000)
Bădoiu, M., Clarkson, K.: Smaller coresets for balls. In: Proc. 14th ACM-SIAM Sympos. Discrete Algorithms, pp. 801–802 (2003)
Eppstein, D.: Fast hierarchical clustering and other applications of dynamic closest pairs. In: Proc. 9th ACM-SIAM Sympos. Discrete Algorithms, pp. 619–628 (1998)
Feder, T., Greene, D.H.: Optimal algorithms for approximate clustering. In: Proc. 20th Annu. ACM Sympos. Theory Comput., pp. 434–444 (1988)
Gonzalez, T.: Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci. 38, 293–306 (1985)
Indyk, P.: A sublinear time approximation scheme for clustering in metric spaces. In: Proc. 40th Annu. IEEE Sympos. Found. Comput. Sci., pp. 100–110 (1999)
Mishra, N., Oblinger, D., Pitt, L.: Sublinear time approximate clustering. In: Proc. 12th ACM-SIAM Sympos. Discrete Algorithms, pp. 439–447 (2001)
Ostrovsky, R., Rabani, Y.: Polynomial time approximation schemes for geometric k-clustering. In: Proc. 41st Symp. Foundations of Computer Science, pp. 349–358. IEEE, Los Alamitos (2000)
Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k-means clustering. Comput. Geom. Theory Appl. 28, 89–112 (2004)
Inaba, M., Katoh, N., Imai, H.: Applications of weighted voronoi diagrams and randomization to variance-based k-clustering. In: Proc. 10th Annu. ACM Sympos. Comput. Geom., pp. 332–339 (1994)
Har-Peled, S.: How to get close to the median shape. In: Proc. 22nd Annu. ACM Sympos. Comput. Geom. (to appear, 2006), Available from: http://www.uiuc.edu/~sariel/papers/05/l1_fitting/
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Har-Peled, S. (2006). Coresets for Discrete Integration and Clustering. In: Arun-Kumar, S., Garg, N. (eds) FSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2006. Lecture Notes in Computer Science, vol 4337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11944836_6
Download citation
DOI: https://doi.org/10.1007/11944836_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-49994-7
Online ISBN: 978-3-540-49995-4
eBook Packages: Computer ScienceComputer Science (R0)