### Abstract

Let S be a set of m convex polygons in the plane with a total number of n vertices. Each polygon has a positive weight. This paper presents algorithms to solve the weighted minmax approximation and the weighted minsum approximation problems. For the first problem, a line minimizing the maximum weighted distance to the polygons can be found in O(n^{2} log n) time and O(n^{2}) space. The time and space complexities can be reduced to O(n log n) and O(n), respectively, when the weights are equal. For the second problem, a line minimizing the sum of the weighted distances to the polygons can be found in O(n^{2} log n) time and O(n) space. For both problems, we also obtain similar results for sets of n circles or line segments.

Original language | English (US) |
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Title of host publication | STACS 1992 - 9th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings |

Publisher | Springer-Verlag |

Pages | 233-244 |

Number of pages | 12 |

ISBN (Print) | 9783540552109 |

State | Published - Jan 1 1992 |

Event | 9th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1992 - Cachan, France Duration: Feb 13 1992 → Feb 15 1992 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 577 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 9th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1992 |
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Country | France |

City | Cachan |

Period | 2/13/92 → 2/15/92 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*STACS 1992 - 9th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings*(pp. 233-244). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 577 LNCS). Springer-Verlag.

**Linear approximation of simple objects.** / Robert, Jean Marc; Toussaint, Godfried.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STACS 1992 - 9th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 577 LNCS, Springer-Verlag, pp. 233-244, 9th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1992, Cachan, France, 2/13/92.

}

TY - GEN

T1 - Linear approximation of simple objects

AU - Robert, Jean Marc

AU - Toussaint, Godfried

PY - 1992/1/1

Y1 - 1992/1/1

N2 - Let S be a set of m convex polygons in the plane with a total number of n vertices. Each polygon has a positive weight. This paper presents algorithms to solve the weighted minmax approximation and the weighted minsum approximation problems. For the first problem, a line minimizing the maximum weighted distance to the polygons can be found in O(n2 log n) time and O(n2) space. The time and space complexities can be reduced to O(n log n) and O(n), respectively, when the weights are equal. For the second problem, a line minimizing the sum of the weighted distances to the polygons can be found in O(n2 log n) time and O(n) space. For both problems, we also obtain similar results for sets of n circles or line segments.

AB - Let S be a set of m convex polygons in the plane with a total number of n vertices. Each polygon has a positive weight. This paper presents algorithms to solve the weighted minmax approximation and the weighted minsum approximation problems. For the first problem, a line minimizing the maximum weighted distance to the polygons can be found in O(n2 log n) time and O(n2) space. The time and space complexities can be reduced to O(n log n) and O(n), respectively, when the weights are equal. For the second problem, a line minimizing the sum of the weighted distances to the polygons can be found in O(n2 log n) time and O(n) space. For both problems, we also obtain similar results for sets of n circles or line segments.

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UR - http://www.scopus.com/inward/citedby.url?scp=84936647933&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84936647933

SN - 9783540552109

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 233

EP - 244

BT - STACS 1992 - 9th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings

PB - Springer-Verlag

ER -