### Abstract

Efficient online algorithms are presented for maintaining the (almost-exact) width and diameter of a dynamic planar point-set, 5. Let n be the number of points currently in S, let W and D denote the width and diameter of S, respectively, and let α and β be positive, integer-valued parameters. The algorithm for the width problem uses O(αn) space, supports updates in O(αlog^{2} n) time, and reports in O(αlog^{2} n) time an approximation, Ŵ, to the width such that Ŵ/W ≤ √l + tan^{2} π/4α. The algorithm for the diameter problem uses O(βn) space, supports updates in O(βlogn) time, and reports in O(βn) time an approximation, D, to the diameter such that (formula presented). Thus, for instance, even for a as small as 5, Ŵ/W ≤ 1.01, and for β as small as 11, D/D ≥ .99. All bounds stated are worst-case. Both algorithms, but especially the one for the diameter problem, use well-understood data structures and should be simple to implement. The diameter result yields a fast implementation of the greedy heuristic for maximum-weight Euclidean matching and an efficient online algorithm to maintain approximate convex hulls in the plane.

Original language | English (US) |
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Title of host publication | ISA 1991 Algorithms - 2nd International Symposium on Algorithms, Proceedings |

Editors | R.C.T. Lee, Wen-Lian Hsu |

Publisher | Springer- Verlag |

Pages | 137-149 |

Number of pages | 13 |

ISBN (Print) | 9783540549451 |

DOIs | |

State | Published - Jan 1 1991 |

Event | 2nd Annual International Symposium on Algorithms, ISA 1991 - Taipei, China Duration: Dec 16 1991 → Dec 18 1991 |

### 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 | 557 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 2nd Annual International Symposium on Algorithms, ISA 1991 |
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Country | China |

City | Taipei |

Period | 12/16/91 → 12/18/91 |

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## Cite this

*ISA 1991 Algorithms - 2nd International Symposium on Algorithms, Proceedings*(pp. 137-149). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 557 LNCS). Springer- Verlag. https://doi.org/10.1007/3-540-54945-5_57