We study the problem of assigning transmission ranges to radio stations placed in a d-dimensional (d-D) Euclidean space in order to achieve a strongly connected communication network with minimum total cost, where the cost of transmitting in range r is proportional to rα. While this problem can be solved optimally in 1D, in higher dimensions it is known to be NP-hard for any α ≥ 1. For the 1D version of the problem and α ≥ 1, we propose a new approach that achieves an exact O(n2)-time algorithm. This improves the running time of the best known algorithm by a factor of n. Moreover, we show that this new technique can be utilized for achieving a polynomialtime algorithm for finding the minimum cost range assignment in 1D whose induced communication graph is a t-spanner, for any t ≥ 1. In higher dimensions, finding the optimal range assignment is NPhard; however, it can be approximated within a constant factor. The best known approximation ratio is for the case α = 1, where the approximation ratio is 1.5. We show a new approximation algorithm that breaks the 1.5 ratio.