## Abstract

We consider a geometric optimization problem that arises in network design. Given a set P of n points in the plane, source and destination points s, t ∈ P, and an integer k > 0, one has to locate k Steiner points, such that the length of the longest edge of a bottleneck path between s and t is minimized. In this paper, we present an O(n log^{2} n)-time algorithm that computes an optimal solution, for any constant k. This problem was previously studied by Hou et al. [11], who gave an O(n^{2} log n)-time algorithm. We also study the dual version of the problem, where a value λ > 0 is given (instead of k), and the goal is to locate as few Steiner points as possible, so that the length of the longest edge of a bottleneck path between s and t is at most λ. Our algorithms are based on two new geometric structures that we develop - an (α, β)-pair decomposition of P and a floor (1 + ∈)-spanner of P. For real numbers β > α > 0, an (α, β)-pair decomposition of P is a collection W =-(A_{1},B_{1}),.. .(A_{m},B_{m})} of pairs of subsets of P, satisfying: (i) For each pair (A_{i},B_{i}) ∈ W, the radius of the minimum enclosing circle of A_{i} (resp. B _{i}) is at most α, and (ii) For any p, q ∈ P, such that |pq| ≤ β, there exists a single pair (A_{i},B_{i}) ∈ W, such that p ∈ A_{i} and q ∈ B_{i}, or vice versa. We construct (a compact representation of) an (α, β)-pair decomposition of P in time O((n/∈)^{3}n log n). Finally, for the complete graph with vertex set P and weight function w(p, q) = b|pq|c, we construct a (1 + ")-spanner of size O(n/∈^{4}) in time O((1/∈^{4}) n log^{2} n), even though w is not a metric.

Original language | English |
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Title of host publication | Proceedings of the 27th Annual Symposium on Computational Geometry, SCG'11 |

Pages | 440-447 |

Number of pages | 8 |

DOIs | |

State | Published - 15 Jul 2011 |

Event | 27th Annual ACM Symposium on Computational Geometry, SCG'11 - Paris, France Duration: 13 Jun 2011 → 15 Jun 2011 |

### Conference

Conference | 27th Annual ACM Symposium on Computational Geometry, SCG'11 |
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Country/Territory | France |

City | Paris |

Period | 13/06/11 → 15/06/11 |

## Keywords

- Bottleneck path
- Geometric networks
- Geometric optimization
- Geometric spanners
- Pair decomposition
- Steiner points

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics