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 δ th 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 = k l and k ≤ l is at most n π + O ( n k 0.68 ) and at least n π + Ω ( n k ) − O ( k ) . Our bounds close the previously known gap of upper and lower bounds for square grids. Moreover, our lower bound is n 3 − O ( 1 ) for 3 ≤ k ≤ 18 , which matches a naive upper bound within a constant term for k ≡ 0 ( mod 3 ) .