## Abstract

Let P be a set of n points in ℝ^{d}. The radius of a k-dimensional flat F with respect to P, which we denote by RD(F, P), is defined to be max_{pεP} 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 R_{k}^{opt}(P), is the minimum, over all k-dimensional flats F, of RD(F, P). We consider the problem of computing R _{k}^{opt}(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 < ε ≤ 1, returns a k-flat F such that RD(F, P) ≤ (1 + ε)R_{k}^{opt}(P). The algorithm runs in O (ndC_{ε,k}) time, where C_{ε,k} is a constant that depends only on ε 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 dn^{O(k/εc)} where c is an appropriate constant.

Original language | English (US) |
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Pages (from-to) | 269-288 |

Number of pages | 20 |

Journal | Discrete and Computational Geometry |

Volume | 32 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2004 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics