### Abstract

This paper considers the problem of granting a dynamic data structure the capability of remembering the situation it held at previous times. We present a new scheme for recording a history of h updates over an ordered set S of n objects, which allows fast neighbor computation at any time in the history. This scheme requires O(n + h) space and O(log n log h) query response-time, which saves a factor of log n space over previous structures. Aside from its improved performance, the novelty of our method is to allow the set S to be only partially ordered with respect to queries and the time-measure to be multi-dimensional. The generality of our method makes it useful to a number of problems in three-dimensional geometry. For example, we are able to give fast algorithms for locating a point in a 3d-complex, using linear space, or for finding which of n given points is closest to a query plane. Using a simpler, yet conceptually similar technique, we show that with only O(n^{2}) preprocessing, we can determine in O(log^{2} n) time which of n given points in E^{3} is closest to an arbitrary query point.

Original language | English (US) |
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Title of host publication | Foundations of Computation Theory - Proceedings of the 1983 International FCT-Conference |

Editors | Marek Karpinski |

Publisher | Springer Verlag |

Pages | 52-63 |

Number of pages | 12 |

ISBN (Print) | 9783540126898 |

DOIs | |

State | Published - Jan 1 1983 |

Externally published | Yes |

Event | International Symposium on Fundamentals of Computation Theory, FCT 1983 - Borgholm, Sweden Duration: Aug 21 1983 → Aug 27 1983 |

### 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 | 158 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | International Symposium on Fundamentals of Computation Theory, FCT 1983 |
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Country | Sweden |

City | Borgholm |

Period | 8/21/83 → 8/27/83 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Foundations of Computation Theory - Proceedings of the 1983 International FCT-Conference*(pp. 52-63). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 158 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-12689-9_93