## Abstract

We study the problem of assigning transmission ranges to radio stations in the plane such that any pair of stations can communicate within a bounded number of hops h and the cost of the network is minimized. We consider two settings of the problem: collinear station locations and arbitrary locations. For the case of collinear stations, we introduce the pioneer polynomial-time exact algorithm for any α≥ 1 and constant h, and thus conclude that the 1D version of the problem, where h is a constant, is in P. Furthermore, we provide a 3 / 2-approximation algorithm for the case where h is an arbitrary number and α= 1 , improving the previously best known approximation ratio of 2. For the case of stations placed arbitrarily in the plane and α= 1 , we first present a (1.5 + ε) -approximation algorithm for a case where a deviation of one hop (h+ 1 hops in total) is acceptable. Then, we show a (3 + ε) -approximation algorithm that complies with the exact hop bound. To achieve that, we introduce the following two auxiliary problems, which are of independent interest. The first is the bounded-hop multi-sink range problem, for which we present a PTAS which can be applied to compute a (1 + ε) -approximation for the bounded diameter minimum spanning tree, for any ε> 0. The second auxiliary problem is the bounded-hop dual-sink pruning problem, for which we show a polynomial-time algorithm. To conclude, we consider the initial bounded-hop all-to-all range assignment problem and show that a combined application of the aforementioned problems yields the (3 + ε) -approximation ratio for this problem, which improves the previously best known approximation ratio of 4(9h-2)/(2h-1).

Original language | English |
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Pages (from-to) | 3050-3077 |

Number of pages | 28 |

Journal | Algorithmica |

Volume | 80 |

Issue number | 11 |

DOIs | |

State | Published - 1 Nov 2018 |

## Keywords

- Approximation algorithms
- Computational geometry
- Wireless networks

## ASJC Scopus subject areas

- General Computer Science
- Computer Science Applications
- Applied Mathematics