General lower and best case upper bounds on energy optimal multicasting in wireless ad hoc and sensor networks

Given a source node and a set of k destination nodes, what is the minimum total energy required for sending a message from source to destinations when multicasting along optimal placed relaying nodes is applied? In this work I will answer this question under the assumption that non-cooperative relaying is applied and that communication over a distance d requires energy proportional to d^α + β for some a > 1 and β > 0. I derive lower bound expressions for a MAC layer model which does not exploit the broadcast property of wireless communication and for a MAC layer model which exploits it. I show further, in case of optimal relay positions, how multicasts can be constructed whose energy consumption always stays below the derived lower bound plus an additive constant expression depending on α, β, k. The usefulness of the bound derived in this work is exemplified with three different example applications.