## Abstract

We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs (p,r) where p is a point in the plane and r is a real number. The distance between two points (^{pi},^{ri}) and (^{pj},^{rj}) is defined as |^{pipj}|-^{ri}-^{rj}. We show that in the case where all ^{ri} are positive numbers and |^{pi} ^{pj}|≥^{ri}+^{rj} for all i, j (in which case the points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a (1+ε -spanner that has a linear number of edges. We also show that the Additively Weighted Delaunay graph (the face-dual of the Additively Weighted Voronoi diagram) has a spanning ratio bounded by a constant. The straight-line embedding of the Additively Weighted Delaunay graph may not be a plane graph. Given the Additively Weighted Delaunay graph, we show how to compute a plane straight-line embedding that also has a spanning ratio bounded by a constant in O(nlogn) time.

Original language | English |
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Pages (from-to) | 287-298 |

Number of pages | 12 |

Journal | Journal of Discrete Algorithms |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - 1 Sep 2011 |

Externally published | Yes |

## Keywords

- Delaunay triangulation
- Geometric spanners
- Yao-graph

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics