Affiliation:
1. Stony Brook University, NY
2. The Open University of Israel, Israel
3. California Institute of Technology, CA
Abstract
A set of
k
balls
B
1
, …,
B
k
in a Euclidean space is said to cover a collection of lines if every line intersects some ball. We consider the
k
-
center problem for lines
in high-dimensional space: Given a set of
n
lines
l
= {
l
1
,…,
l
n
in R
d
, find
k
balls of minimum radius which cover
l
. We present a 2-approximation algorithm for the cases
k
= 2, 3 of this problem, having running time quasi-linear in the number of lines and the dimension of the ambient space.
Our result for 3-clustering is strongly based on a new result in discrete geometry that may be of independent interest: a Helly-type theorem for collections of axis-parallel “crosses” in the plane. The family of crosses does not have finite Helly number in the usual sense. Our Helly theorem is of a new type: it depends on ε-contracting the sets.
In statistical practice, data is often incompletely specified; we consider lines as the most elementary case of incompletely specified data points. Clustering of data is a key primitive in nonparametric statistics. Our results provide a way of performing this primitive on incomplete data, as well as imputing the missing values.
Funder
National Science Foundation
NSA
Division of Computing and Communication Foundations
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Reference25 articles.
1. Agarwal P. Har-Peled S. and Varadarajan K. R. 2005a. Geometric approximation via coresets. In Current Trends in Combinatorial and Computational Geometry. Cambridge University Press. Agarwal P. Har-Peled S. and Varadarajan K. R. 2005a. Geometric approximation via coresets. In Current Trends in Combinatorial and Computational Geometry. Cambridge University Press.
2. Agarwal P. K. Arge L. and Yi K. 2005b. An optimal dynamic interval stabbing-max data structure? In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'05). SIAM 803--812. Agarwal P. K. Arge L. and Yi K. 2005b. An optimal dynamic interval stabbing-max data structure? In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'05). SIAM 803--812.
3. A (1+ε)-approximation algorithm for 2-line-center
4. Allison P. D. 2002. Missing Data. Sage Publications. Allison P. D. 2002. Missing Data. Sage Publications.
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