ABSTRACT
(MATH) Let P be a set of n points in $\Red, and for any integer 0 ≤ k ≤ d--1, let $\RDk(P) denote the minimum over all k-flats $\FLAT$ of maxpεP Dist(p,\FLAT). We present an algorithm that computes, for any 0 < ε < 1, a k-flat that is within a distance of (1 + $egr;) \RDk(P) from each point of P. The running time of the algorithm is dnO(k/ε5log(1/ε)). The crucial step in obtaining this algorithm is a structural result that says that there is a near-optimal flat that lies in an affine subspace spanned by a small subset of points in P. The size of this "core-set" depends on k and ε but is independent of the dimension.This approach also extends to the case where we want to find a k-flat that is close to a prescribed fraction of the entire point set, and to the case where we want to find j flats, each of dimension k, that are close to the point set. No efficient approximation schemes were known for these problems in high-dimensions, when k>1 or j>1.
- D. Achlioptas. Database-friendly random projections. In Proc. 20th ACM Sympos. Principles Database Syst., pages 274--281, 2001. Google ScholarDigital Library
- N. Alon, S. Dar, M. Parnas, and D. Ron. Testing of clustering. In Proc. 41th Annu. IEEE Sympos. Found. Comput. Sci., 2000. Google ScholarDigital Library
- P.K. Agarwal and C.M. Procopiuc. Approximation algorithms for projective clustering. In Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, pages 538--547, 2000. Google ScholarDigital Library
- I. Barany and Z. Furedi Computing the volume is difficult. In Discrete Comput. Geom. 2, 319--326, 1987.Google ScholarDigital Library
- M. Bern and D. Eppstein. Approximation algorithms for geometric problems. In D.S. Hochbaum, editor, Approximationg algorithms for NP-Hard problems. PWS Publishing Company, 1997. Google ScholarDigital Library
- G. Barequet and S. Har-Peled. Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. J. Algorithms, 38:91--109, 2001. Google ScholarDigital Library
- M. Badoiu, S. Har-Peled, and P. Indyk. Approximate clustering via core-sets. To appear in STOC 2002. Google ScholarDigital Library
- A. Brieden and P. Gritzmann and V. Klee. Inapproximability of some geometric and quadratic optimization problems. In P. M. Pardalos, editor, Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems, pages 96--115, Kluwer, 2000.Google ScholarCross Ref
- R.O. Duda, P.E. Hart, and D.G. Stork. Pattern Classification. Wiley-Interscience, New York, 2nd edition, 2001. Google ScholarDigital Library
- P. Gritzmann and V. Klee. Inner and outer j-radii of convex bodies in finite-dimensional normed spaces. Discrete Comput. Geom., 7:255--280, 1992.Google ScholarDigital Library
- P. Gritzmann and V. Klee. Computational complexity of inner and outer $j$-radii of polytopes in finite-dimensional normed spaces. Math. Program., 59:163--213, 1993. Google ScholarDigital Library
- P. Gritzmann and V. Klee. On the complexity of some basic problems in computational convexity: I. containment problems. Discrete Math., 136:129--174, 1994. Google ScholarDigital Library
- M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer-Verlag, Berlin Heidelberg, 2nd edition, 1988. 2nd edition 1994.Google Scholar
- S. Har-Peled. Clustering motion. In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci., pages 84--93, 2001. Google ScholarDigital Library
- S. Har-Peled and K.R. Varadarajan. Approximate shape fitting via linearization. In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci., pages 66--73, 2001. Google ScholarDigital Library
- P. Indyk. Algorithmic applications of low-distortion geometric embeddings. In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci., pages 10--31, 2001. Google ScholarDigital Library
- P. Indyk and M. Thorup. Approximate 1-median. manuscript, 2000.Google Scholar
- W.B. Johnson and J. Lindenstrauss. Extensions of lipshitz mapping into hilbert space. Contemporary Mathematics, 26:189--206, 1984.Google ScholarCross Ref
- A. Magen. Dimensionality reductions that preserve volumes and distance to affine spaces, and its algorithmic applications. Manuscript, 2001.Google Scholar
- N. Mishra, D. Oblinger, and L. Pitt. Sublinear time approximate clustering. In SODA 12, 2001. Google ScholarDigital Library
- N. Megiddo and A. Tamir. On the complexity of locating linear facilities in the plane. Oper. Res. Lett., 1:194--197, 1982.Google ScholarDigital Library
Index Terms
- Projective clustering in high dimensions using core-sets
Recommendations
Approximate clustering via core-sets
STOC '02: Proceedings of the thiry-fourth annual ACM symposium on Theory of computingIn this paper, we show that for several clustering problems one can extract a small set of points, so that using those core-sets enable us to perform approximate clustering efficiently. The surprising property of those core-sets is that their size is ...
Approximate minimum enclosing balls in high dimensions using core-sets
We study the minimum enclosing ball (MEB) problem for sets of points or balls in high dimensions. Using techniques of second-order cone programming and "core-sets", we have developed (1 + ε)-approximation algorithms that perform well in practice, ...
High-dimensional shape fitting in linear time
SCG '03: Proceedings of the nineteenth annual symposium on Computational geometryLet P be a set of n points in Rd. The radius of a k-dimensional flat F with respect to P, denoted by RD(F,P), is defined to be maxp ? P dist(F,p), where dist(F,p) denotes the Euclidean distance between p and its projection onto F. The k-flat radius of P,...
Comments