### Abstract

In this paper, we study the problem of ^{L1}-fitting a shape to a set of n points in R ^{d} (where d is a fixed constant), where the target is to minimize the sum of distances of the points to the shape, or alternatively the sum of squared distances. We present a general technique for computing a (1+)-approximation for such a problem, with running time O(n+poly(logn,1/)), where poly(logn,1/) is a polynomial of constant degree of logn and 1/ (the power of the polynomial is a function of d). This is a linear time algorithm for a fixed >0, and is the first subquadratic algorithm for this problem. Applications of the algorithm include best fitting either a circle, a sphere or a cylinder to a set of points when minimizing the sum of distances (or squared distances) to the respective shape.

Original language | English (US) |
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Pages (from-to) | 39-51 |

Number of pages | 13 |

Journal | Computational Geometry: Theory and Applications |

Volume | 36 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2007 |

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### Keywords

- Approximation algorithms
- Shape fitting

### ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics