## On the complexity of computing minimum energy consumption broadcast subgraphs (2001)

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Venue: | in Symposium on Theoretical Aspects of Computer Science |

Citations: | 96 - 11 self |

### BibTeX

@INPROCEEDINGS{Clementi01onthe,

author = {Andrea E. F. Clementi and Pilu Crescenzi and Paolo Penna and Gianluca Rossi and Paola Vocca},

title = {On the complexity of computing minimum energy consumption broadcast subgraphs},

booktitle = {in Symposium on Theoretical Aspects of Computer Science},

year = {2001},

pages = {121--131},

publisher = {Springer-Verlag}

}

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### Abstract

Abstract. We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, called Minimum Energy Consumption Broadcast Subgraph (in short, MECBS): Given a weighted directed graph and a specified source node, find a minimum cost range assignment to the nodes, whose corresponding transmission graph contains a spanning tree rooted at the source node. We first prove that MECBS is not approximable within a constant factor (unless P=NP). We then consider the restriction of MECBS to wireless networks and we prove several positive and negative results, depending on the geometric space dimension and on the distance-power gradient. The main result is a polynomial-time approximation algorithm for the NP-hard case in which both the dimension and the gradient are equal to 2: This algorithm can be generalized to the case in which the gradient is greater than or equal to the dimension. 1

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Citation Context ...α d ] with α<dand we summarize some questions left open by this paper. Prerequisites. We assume the reader to be familiar with the basic concepts of computational complexity theory (see, for example, =-=[4,20]-=-) and with the basic concepts of the theory of approximation algorithms (see, for example, [1]). 2 The Complexity of MECBS In this section, we prove that the MINIMUM ENERGY CONSUMPTION BROADCAST SUBGR... |

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Citation Context ...he reader to be familiar with the basic concepts of computational complexity theory (see, for example, [4,20]) and with the basic concepts of the theory of approximation algorithms (see, for example, =-=[1]-=-). 2 The Complexity of MECBS In this section, we prove that the MINIMUM ENERGY CONSUMPTION BROADCAST SUBGRAPH problem is not approximable within a sub-logarithmic factor (unless P= NP). To this aim, w... |

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Citation Context ...n of a station depends, in turn, on the energy power supplied to the station: In particular, the powerÈ×required by a station×to correctly transmit data to another stationØmust satisfy the inequality =-=[PL95]-=- (1) where�×�Øis the Euclidean distance between×andØ,«�is the distancepower gradient, and�is the transmission-quality parameter. In an ideal environment it holds that«�but it may vary from 1 to more ... |

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Citation Context ... (unless P=NP) [CPS99,KKKP00], (b) it is possible to compute a range assignment which is at most twice the optimal one (that is, the problem is 2-approximable), for multidimensional wireless networks =-=[KKKP00]-=-, (c) there exists a constantÖ� such that the problem is notÖ-approximable (unless P=NP), for�-dimensional networks with��[CPS99], and (d) the problem can be solved in polynomial time for one-dimensio... |

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Citation Context ...eration. Broadcast is a task initiated by the source station which transmits a message to all stations in the wireless network: This task constitutes a major part of real life multi-hop radio network =-=[2,3]-=-. The Optimization Problem. The broadcast range assignment problem described above is a special case of the following combinatorial optimization problem, called MINIMUM ENERGY CONSUMPTION BROADCAST SU... |

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Citation Context ...nge assigned to the stations can be modified at any time: Our algorithm can thus take advantage of all known techniques to dynamically maintain MSTs (see, for example,s124 Andrea E.F. Clementi et al. =-=[9,11,18]-=-). MSTs have already been used in order to develop approximation algorithms for range assignment problems in wireless networks: However, we believe that the analysis of the performance of our algorith... |

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Citation Context ...he problem is 2-approximable), for multidimensional wireless networks [KKKP00], (c) there exists a constantÖ� such that the problem is notÖ-approximable (unless P=NP), for�-dimensional networks with��=-=[CPS99]-=-, and (d) the problem can be solved in polynomial time for one-dimensional networks [KKKP00]. Another analyzed case consists in choosing�as follows: The diameter of the transmission graph has to be at... |

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Citation Context ... transmission graph has to be at most a fixed value�. In this case, while non-trivial negative results are not known, some tight bounds (depending on�) on the minimum energy power have been proved in =-=[CPS00]-=-, and an approximation algorithm for the onedimensional case has been given in [CFP 00]. Other trade-offs between connectivity and energy consumption have been obtained in [MM96,Pir91,UY96]. In this p... |

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Citation Context ...onnects all base stations by high-speed wired links. On the other hand, the multi-hop model [Lau95] requires neither fixed, wired infrastructure nor predetermined interconnectivity. Ad hoc networking =-=[HT98]-=- is the most popular type of multi-hop wireless networks because of its simplicity: Indeed, an ad hoc wireless network is constituted by a homogeneous system of mobile stations connected by wireless l... |

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Citation Context ... of the above theorem is again based on the computation of a MST of the input graph: Indeed, the algorithm is exactly the same. Unfortunately, the following result (whose proof is based on results in =-=[8,13,25]-=- and will be given in the full version of the paper) shows that the function f in the statement of the theorem grows exponentially with respect to d. Theorem 7. There exists a positive constant γ such... |

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Citation Context ...l negative results are not known, some tight bounds (depending on h) on the minimum energy power have been proved in [7], and an approximation algorithm for the one-dimensional case has been given in =-=[5]-=-. Other trade-offs between connectivity and energy consumption have been obtained in [16,21,24]. B D CsOn the Complexity of Computing Minimum Energy Consumption 123 In this paper we address the case i... |

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Citation Context ...nge assigned to the stations can be modified at any time: Our algorithm can thus take advantage of all known techniques to dynamically maintain MSTs (see, for example,s124 Andrea E.F. Clementi et al. =-=[9,11,18]-=-). MSTs have already been used in order to develop approximation algorithms for range assignment problems in wireless networks: However, we believe that the analysis of the performance of our algorith... |

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Citation Context ...gy power have been proved in [7], and an approximation algorithm for the one-dimensional case has been given in [5]. Other trade-offs between connectivity and energy consumption have been obtained in =-=[16,21,24]-=-. B D CsOn the Complexity of Computing Minimum Energy Consumption 123 In this paper we address the case in which π is defined as follows: Given a source station s, the transmission graph has to contai... |

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Citation Context ...eration. Broadcast is a task initiated by the source station which transmits a message to all stations in the wireless network: This task constitutes a major part of real life multi-hop radio network =-=[2,3]-=-. The Optimization Problem. The broadcast range assignment problem described above is a special case of the following combinatorial optimization problem, called MINIMUM ENERGY CONSUMPTION BROADCAST SU... |

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Citation Context ...ic nodes known as base stations. This type of networks relies on a fixed backbone infrastructure that interconnects all base stations by high-speed wired links. On the other hand, the multi-hop model =-=[Lau95]-=- requires neither fixed, wired infrastructure nor predetermined interconnectivity. Ad hoc networking [HT98] is the most popular type of multi-hop wireless networks because of its simplicity: Indeed, a... |

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Citation Context ...nge assigned to the stations can be modified at any time: Our algorithm can thus take advantage of all known techniques to dynamically maintain MSTs (see, for example,s124 Andrea E.F. Clementi et al. =-=[9,11,18]-=-). MSTs have already been used in order to develop approximation algorithms for range assignment problems in wireless networks: However, we believe that the analysis of the performance of our algorith... |

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Citation Context ... case in which�is defined as follows: Given a source station×, the transmission graph has to contain a directed spanning tree rooted at×. This case has been posed as an open question by Ephremides in =-=[Eph00]-=-: Its relevance is due to the fact that any transmission graph satisfying the above property allows the source station to perform a broadcast operation. Broadcast is a task initiated by the source sta... |

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Citation Context ... of the above theorem is again based on the computation of a MST of the input graph: Indeed, the algorithm is exactly the same. Unfortunately, the following result (whose proof is based on results in =-=[8,13,25]-=- and will be given in the full version of the paper) shows that the function f in the statement of the theorem grows exponentially with respect to d. Theorem 7. There exists a positive constant γ such... |