Efficient algorithms for low-energy bounded-hop broadcast in Ad-Hoc wireless networks

The paper studies the problem of computing a minimal energy cost range assignment in a 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 whicli n stations are located on the plane and the cost to transmit from station i to station j is proportional to the α-th power of the distance between station 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) time O(n⁷). Then, a polynomial-time approximation scheme (PTAS) is provided for the above problem for any fixed h ≥ 1. For fixed h ≥ 1 and ∈ > 0, the PTAS has time complexity O(n^μ) where μ = O((α2^αh^α/∈) ^αh).