## Abstract

We introduce a new structure for a set of points in the plane and an angle α, which is similar in flavor to a bounded-degree MST. We name this structure α-MST. Let P be a set of points in the plane and let 0 < α ≤ 2π be an angle. An α-ST of P is a spanning tree of the complete Euclidean graph induced by P, with the additional property that for each point p ∈ P, the smallest angle around p containing all the edges adjacent to p is at most α. An α-MST of P is then an α-ST of P of minimum weight. For α < π/3, an α-ST does not always exist, and, for α ≥ π/3, it always exists [1,2,9]. In this paper, we study the problem of computing an α-MST for several common values of α. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point p ∈ P, we associate a wedge W _{p} of angle α and apex p. The goal is to assign an orientation and a radius r_{p} to each wedge W_{p}, such that the resulting graph is connected and its MST is an α-MST. (We draw an edge between p and q if p ∈ W_{q}, q ∈ W_{p}, and |pq| ≤ r _{p}, r_{q}.) Unsurprisingly, the problem of computing an α-MST is NP-hard, at least for α = π and α = 2π/3. We present constant-factor approximation algorithms for α = π/2, 2π/3, π. One of our major results is a surprising theorem for α = 2π/3, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set P of 3n points in the plane and any partitioning of the points into n triplets, one can orient the wedges of each triplet independently, such that the graph induced by P is connected. We apply the theorem to the antenna conversion problem.

Original language | English |
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Title of host publication | Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings |

Publisher | Springer Verlag |

Pages | 387-398 |

Number of pages | 12 |

Edition | PART 2 |

ISBN (Print) | 9783662439500 |

DOIs | |

State | Published - 1 Jan 2014 |

Event | 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 - Copenhagen, Denmark Duration: 8 Jul 2014 → 11 Jul 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 2 |

Volume | 8573 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 |
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Country/Territory | Denmark |

City | Copenhagen |

Period | 8/07/14 → 11/07/14 |