## Abstract

We address the following problem: Given a complete k-partite geometric graph K whose vertex set is a set of n points in Rdbl; ^{d}, compute a spanner of K that has a "small" stretch factor and "few" edges. We present two algorithms for this problem. The first algorithm computes a (5+ε)- spanner of K with O(n) edges in O(n log n) time. The second algorithm computes a (3 + ε)-spanner of K with O(n log n) edges in O(n log n) time. The latter result is optimal: We show that for any 2 ≤ k ≤ n - Θ( √ n log n), spanners with O(n log n) edges and stretch factor less than 3 do not exist for all complete k-partite geometric graphs.

Original language | English |
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Pages (from-to) | 1803-1820 |

Number of pages | 18 |

Journal | SIAM Journal on Computing |

Volume | 38 |

Issue number | 5 |

DOIs | |

State | Published - 1 Dec 2008 |

Externally published | Yes |

## Keywords

- Computational geometry
- K-partite geometric graphs
- Spanners