### 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 |

State | 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).

}

*Approximation and Online Algorithms - 4th International Workshop, WAOA 2006, Revised Papers.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4368 LNCS, pp. 174-187, 4th Workshop on Approximation and Online Algorithms, WAOA 2006, Zurich, Switzerland, 9/14/06.

**Approximate distance queries in disk graphs.** / Furer, Martin; Kasiviswanathan, Shiva Prasad.

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

TY - GEN

T1 - Approximate distance queries in disk graphs

AU - Furer, Martin

AU - Kasiviswanathan, Shiva Prasad

PY - 2007/12/1

Y1 - 2007/12/1

N2 - 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(n3/2ε-1 + nε-2 log S), such that any subsequent distance query can be answered approximately in O(√nε-1+ε-2log 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.

AB - 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(n3/2ε-1 + nε-2 log S), such that any subsequent distance query can be answered approximately in O(√nε-1+ε-2log 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.

UR - http://www.scopus.com/inward/record.url?scp=38149038705&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38149038705&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:38149038705

SN - 9783540695134

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

SP - 174

EP - 187

BT - Approximation and Online Algorithms - 4th International Workshop, WAOA 2006, Revised Papers

ER -