High-dimensional shape fitting in linear time

Sariel Har-Peled, Kasturi R. Varadarajan

Research output: Contribution to journalArticlepeer-review


Let P be a set of n points in ℝd. The radius of a k-dimensional flat F with respect to P, which we denote by RD(F, P), is defined to be maxpεP 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 R kopt(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)Rkopt(P). The algorithm runs in O (ndCε,k) time, where Cε,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 dnO(k/εc) where c is an appropriate constant.

Original languageEnglish (US)
Pages (from-to)269-288
Number of pages20
JournalDiscrete and Computational Geometry
Issue number2
StatePublished - Sep 2004

ASJC Scopus subject areas

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


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