### Abstract

A disk graph is an intersection graph of a set of disks with arbitrary radii in the plane. In this paper, we consider the problem of efficient construction of sparse spanners of disk (ball) graphs with support for fast distance queries. These problems are motivated by issues arising from topology control and routing in wireless networks. We present the first algorithm for constructing spanners of ball graphs. For a ball graph in ℝ^{k}, we construct a (1 + ε)-spanner with O(nε^{-k+1}) edges in O(n^{2ℓ+ε}ε^{-k} log^{ℓ} S) expected time, using an efficient partitioning of the space into hypercubes and solving intersection problems. Here ℓ = 1-1/(⌊k/2⌋+2), ε is any positive constant, and S is the ratio between the largest and smallest radius. For the special case where all the balls have the same radius, we show that the spanner construction has complexity almost equivalent to the construction of a Euclidean minimum spanning tree. Previously known constructions of spanners of unit ball graphs have time complexity much closer to n^{2}. Additionally, these spanners have a small vertex separator (hereditary), which is then exploited for fast answering of distance queries. The results on geometric graph separators might be of independent interest.

Original language | English (US) |
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Title of host publication | Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings |

Pages | 312-324 |

Number of pages | 13 |

State | Published - Dec 1 2007 |

Event | 10th International Workshop on Algorithms and Data Structures, WADS 2007 - Halifax, Canada Duration: Aug 15 2007 → Aug 17 2007 |

### 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 | 4619 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 10th International Workshop on Algorithms and Data Structures, WADS 2007 |
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Country | Canada |

City | Halifax |

Period | 8/15/07 → 8/17/07 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings*(pp. 312-324). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4619 LNCS).

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*Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4619 LNCS, pp. 312-324, 10th International Workshop on Algorithms and Data Structures, WADS 2007, Halifax, Canada, 8/15/07.

**Spanners for geometric intersection graphs.** / Furer, Martin; Kasiviswanathan, Shiva Prasad.

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

TY - GEN

T1 - Spanners for geometric intersection graphs

AU - Furer, Martin

AU - Kasiviswanathan, Shiva Prasad

PY - 2007/12/1

Y1 - 2007/12/1

N2 - A disk graph is an intersection graph of a set of disks with arbitrary radii in the plane. In this paper, we consider the problem of efficient construction of sparse spanners of disk (ball) graphs with support for fast distance queries. These problems are motivated by issues arising from topology control and routing in wireless networks. We present the first algorithm for constructing spanners of ball graphs. For a ball graph in ℝk, we construct a (1 + ε)-spanner with O(nε-k+1) edges in O(n2ℓ+εε-k logℓ S) expected time, using an efficient partitioning of the space into hypercubes and solving intersection problems. Here ℓ = 1-1/(⌊k/2⌋+2), ε is any positive constant, and S is the ratio between the largest and smallest radius. For the special case where all the balls have the same radius, we show that the spanner construction has complexity almost equivalent to the construction of a Euclidean minimum spanning tree. Previously known constructions of spanners of unit ball graphs have time complexity much closer to n2. Additionally, these spanners have a small vertex separator (hereditary), which is then exploited for fast answering of distance queries. The results on geometric graph separators might be of independent interest.

AB - A disk graph is an intersection graph of a set of disks with arbitrary radii in the plane. In this paper, we consider the problem of efficient construction of sparse spanners of disk (ball) graphs with support for fast distance queries. These problems are motivated by issues arising from topology control and routing in wireless networks. We present the first algorithm for constructing spanners of ball graphs. For a ball graph in ℝk, we construct a (1 + ε)-spanner with O(nε-k+1) edges in O(n2ℓ+εε-k logℓ S) expected time, using an efficient partitioning of the space into hypercubes and solving intersection problems. Here ℓ = 1-1/(⌊k/2⌋+2), ε is any positive constant, and S is the ratio between the largest and smallest radius. For the special case where all the balls have the same radius, we show that the spanner construction has complexity almost equivalent to the construction of a Euclidean minimum spanning tree. Previously known constructions of spanners of unit ball graphs have time complexity much closer to n2. Additionally, these spanners have a small vertex separator (hereditary), which is then exploited for fast answering of distance queries. The results on geometric graph separators might be of independent interest.

UR - http://www.scopus.com/inward/record.url?scp=38149119050&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38149119050&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:38149119050

SN - 3540739483

SN - 9783540739487

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

SP - 312

EP - 324

BT - Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings

ER -