Minimum-energy broadcast and disk cover in grid wireless networks

The Minimum-Energy Broadcast problem is to assign a transmission range to every station of an ad hoc wireless networks so that (i) a given source station is allowed to perform broadcast operations and (ii) the overall energy consumption of the range assignment is minimized. We prove a nearly tight asymptotical bound on the optimal cost for the Minimum-Energy Broadcast problem on square grids. We also derive near-tight bounds for the Bounded-Hop version of this problem. Our results imply that the best-known heuristic, the MST-based one, for the Minimum-Energy Broadcast problem is far to achieve optimal solutions (even) on very regular, well-spread instances: its worst-case approximation ratio is about π and it yields View the MathML source hops, where n is the number of stations. As a by product, we get nearly tight bounds for the Minimum-Disk Cover problem and for its restriction in which the allowed disks must have non-constant radius. Finally, we emphasize that our upper bounds are obtained via polynomial time constructions.