Minimum energy broadcast on rectangular grid wireless networks

The minimum energy broadcast problem is to assign a transmission range to each node in an ad hoc wireless network to construct a spanning tree rooted at a given source node such that any non-root node resides within the transmission range of its parent. The objective is to minimize the total energy consumption, i.e., the sum of the δth powers of a transmission range (δ≥1). In this paper, we consider the case that δ=2, and that nodes are located on a 2-dimensional rectangular grid. We prove that the minimum energy consumption for an n-node k×l-grid with n=kl and k≤l is at most and at least . Our bounds close the previously known gap of upper and lower bounds for square grids. Moreover, our lower bound is for 3≤k≤18, which matches a naive upper bound within a constant term for .