### Abstract

We present efficient algorithms for approximately answering distance queries in disk graphs. Let G be a disk graph with n vertices and m edges. For any fixed ε > 0, we show that G can be preprocessed in O(m√nε^{-1} + mε^{-2} log S) time, constructing a data structure of size O(n^{3/2}ε^{-1} + nε^{-2} log S), such that any subsequent distance query can be answered approximately in O(√nε^{-1}+ε^{-2}log S) time. Here S is the ratio between the largest and smallest radius. The estimate produced is within an additive error which is only e times the longest edge on some shortest path. The algorithm uses an efficient subdivision of the plane to construct a sparse graph having many of the same distance properties as the input disk graph. Additionally, the sparse graph has a small separator decomposition, which is then used to answer distance queries. The algorithm extends naturally to the higher dimensional ball graphs.

Original language | English (US) |
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Title of host publication | Approximation and Online Algorithms - 4th International Workshop, WAOA 2006, Revised Papers |

Pages | 174-187 |

Number of pages | 14 |

Publication status | Published - Dec 1 2007 |

Event | 4th Workshop on Approximation and Online Algorithms, WAOA 2006 - Zurich, Switzerland Duration: Sep 14 2006 → Sep 15 2006 |

### 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 | 4368 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 4th Workshop on Approximation and Online Algorithms, WAOA 2006 |
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Country | Switzerland |

City | Zurich |

Period | 9/14/06 → 9/15/06 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Approximation and Online Algorithms - 4th International Workshop, WAOA 2006, Revised Papers*(pp. 174-187). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4368 LNCS).