We study the power assignment problem in radio networks, where each radio station can transmit in one of two possible power levels, corresponding to two ranges - short and long. We show that this problem is NP-hard, and present an O(n^2)-time assignment algorithm, such that the number of transmitters that are assigned long range by the algorithm is at most (11/6) times the number of transmitters that are assigned long range by an optimal algorithm. We also present an (9/5)-approximation algorithm for this problem whose running time is considerably higher. Next, we formulate and study the Minimum-Area Spanning Tree (MAST)problem: Given a set P of n points in the plane, find a spanning tree of P of minimum "area," where the area of a spanning tree T is the area of the union of the n-1 disks whose diameters are the edges in T. We prove that the Euclidean minimum spanning tree of P is a constant-factor approximation for MAST. We then apply this result to obtain constant-factor approximations for the Minimum-Area Range Assignment (MARA) problem, for the Minimum-Area Connected Disk Graph (MACDG) problem, and for the Minimum-Area Tour (MAT) problem. The first problem is a variant of the power assignment problem in radio networks, the second problem is a related natural problem, and the third problem is a variant of the traveling salesman problem.
@InProceedings{carmi_et_al:DagSemProc.06481.3, author = {Carmi, Paz and Katz, Matthew}, title = {{Power Assignment in Radio Networks with Two Power Levels}}, booktitle = {Geometric Networks and Metric Space Embeddings}, pages = {1--18}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2007}, volume = {6481}, editor = {Joachim Gudmundsson and Rolf Klein and Giri Narasimhan and Michiel Smid and Alexander Wolff}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06481.3}, URN = {urn:nbn:de:0030-drops-10277}, doi = {10.4230/DagSemProc.06481.3}, annote = {Keywords: Radio networks, power assignment, approximation algorithms, minimum spanning tree, disk graph} }
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