Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
On the complexity of some geometric problems in unbounded dimension
Journal of Symbolic Computation
Computational complexity of inner and outer j-radii of polytopes in finite-dimensional normed spaces
Mathematical Programming: Series A and B
On the complexity of some basic problems in computational convexity: I.: containment problems
Discrete Mathematics - Special issue: trends in discrete mathematics
Note on the computational complexity of j-radii of polytopes in Rn
Mathematical Programming: Series A and B
Approximate clustering via core-sets
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Projective clustering in high dimensions using core-sets
Proceedings of the eighteenth annual symposium on Computational geometry
Lectures on Discrete Geometry
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
On Approximating the Radii of Point Sets in High Dimensions
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Approximate Shape Fitting via Linearization
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Faster core-set constructions and data stream algorithms in fixed dimensions
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Faster core-set constructions and data-stream algorithms in fixed dimensions
Computational Geometry: Theory and Applications
Faster core-set constructions and data-stream algorithms in fixed dimensions
Computational Geometry: Theory and Applications
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
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Let 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, which we denote by Rkopt(P), is the minimum, over all k-dimensional flats F, of RD(F,P). We consider the problem of computing Rkopt(P) for a given set of points P. We are interested in the high-dimensional case where d is a part of the input and not a constant. This problem is NP-hard even for k = 1. We present an algorithm that, given P and a parameter 0 , returns a k-flat F such that RD(F,P) = (1 + e) Rkopt(P). The algorithm runs in O(nd Ce,k) time, where Ce,k is a constant that depends only on e and k. Thus the algorithm runs in time linear in the size of the point set and is a substantial improvement over previous known algorithms, whose running time is of the order of d nO(k/ec), where c is an appropriate constant.