Minimum Energy Broadcast on Rectangular Grid Wireless Networks

Abstract

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 $\frac{n}{\pi}+O(\frac{n}{k^{0.68}})$ and at least $\frac{n}{\pi}+\Omega(\frac{n}{k})-O(k)$.Our bounds close the previously known gap of upper and lower bounds for square grids. Moreover, our lower bound is $\frac{n}{3}-O(1)$ for 3 ≤ k ≤ 18, which matches a naive upper bound within a constant term for $k\equiv 0\pmod 3$.