The Power Range Assignment Problem in Radio Networks on the Plane

Given a finite set S of points (i.e. the stations of a radio network) on the plane and a positive integer 1 ≤ h ≤ |S| − 1, the 2d Min h R. Assign. problem consists of assigning transmission ranges to the stations so as to minimize the total power consumption provided that the transmission ranges of the stations ensure the communication between any pair of stations in at most h hops. We provide a lower bound on the total power consumption opth(S) yielded by an optimal range assignment for any instance (S,h) of 2d Min h R. Assign., for any positive constant h > 0. The lower bound is a function of |S|, h and the minimum distance over all the pairs of stations in S. Then, we derive a constructive upper bound for the same problem as a function of |S|, h and the maximum distance over all the pairs of stations in S (i.e. the diameter of S). Finally, by combining the above bounds, we obtain a polynomial-time approximation algorithm for 2d Min h R. Assign. restricted to well-spread instances, for any positive constant h. Previous results for this problem were known only in special 1-dimensional configurations (i.e. when points are arranged on a line).