Given a set N of radio stations located on an Euclidean space, a source station s and an integer h (1 less than or equal to h less than or equal to N - 1), the minimum bounded-hop broadcast range assignment problem consists in finding a range assignment for N of minimum total power consumption that allows broadcast operations from s to every station in N in at most h hops. The problem is known to be NP-hard on d-dimensional spaces for any d greater than or equal to 2 (18th Annual Symp. on Theoretical Aspects of Computer Science (STACS'01), Lecture Notes in Computer Science, Vol. 1770, 2000, pp. 651-660.) and some efficient approximation algorithms have been given in Clementi et al. and Wann et al. (18th Annual Symp. on Theoretical Aspects of Computer Science (STACS'01), Lecture Notes in Computer Science, Vol. 1770, 2000, pp. 651-660, IEEE INFOCOM'01, 2001). In this paper, we address the case in which the stations are arbitrarily located along a line (i.e., the linear case). We provide the first polynomial-time algorithm that returns an optimal solution for any instance of the linear case. The algorithm works in O(hN(2)) time. (C) 2002 Published by Elsevier Science B.V

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The paper studies the problem of computing a minimal energy cost range assignment in an ad-hoc wireless network which allows a station s to perform a broadcast operation in at most h hops. The general version of the problem (i.e. when transmission costs are arbitrary) is known to be log-APX hard even for h=2. The current paper considers the well-studied real case in which n stations are located in the plane and the cost to transmit from station i to station j is proporttional to the a-th power of the distance between i and j, where a is any positive constant. A polynomial-time algorithm is presented for finding an optimal range assignment to perform a 2-hop broadcast from a given source station. The algorithm relies on dynamic programming and operates in (worst case) O(n^7) time. Then, a polynomial-time approximation scheme (PTAS) is provided for the above problem for any fixed h>=1

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Range assignment problems in Ad-Hoc wireless networks have been the subject of several recent studies. All these studies deal with the homogeneous case, i.e., all stations share the same energy cost function. However, this assumption does not well model realistic scenarios in which the energy cost of a station varies dramatically depending on the particular enviroment conditions of its location. We introduce the weighted version of the range assignment problem in which the cost a station × pays to transmit to another station depends on the distance between the stations and on the energy cost of station ×. Most of the algorithm results for the unweighted range assignment problem can not be applied to the weighted version. We thus provide a set of algorithmic results for this version and discuss some interesting related open questions.