Lower and upper bounds for multicasting under distance dependent forwarding cost functions

Assume a forwarding cost function which depends on the sender receiver separation, and assume further that non cooperative relaying is applied. What is the minimum total forwarding cost required for sending a message from source to destinations when multicasting along optimal placed relaying nodes is applied? In my last year´s WoWMoM publication, this question was already answered for a specific energy cost function. In this work I generalize my previous findings for generalized classes of distance dependent forwarding cost functions. I define cost function properties from which I derive generalized lower bounds on multicasting costs. I consider again, a MAC layer model which does not exploit the broadcast property of wireless communication and a MAC layer model which exploits it. This work also generalizes the upper bound result from. For specific cost functions, I show that in case of optimal relay positions, multicasts can be constructed whose cost always stays below of one of the derived lower bound expressions plus an additive constant depending on the number of destinations. For both, lower and upper bounds, I define a general procedure to check if and if yes how my findings can be used to derive the specific lower and upper bound expressions for a given cost function. I explain the procedure with two cost function examples, the Euclidean distance and the energy cost function used in. For the latter, the bounds derived in follow immediately as corollaries in this work.