Min-cost multicast networks in Euclidean space

Space information flow is a new field of research recently proposed by Li and Wu [1], [2]. It studies the transmission of information in a geometric space, where information flows can be routed along any trajectories, and can be encoded wherever they meet. The goal is to satisfy given end-to-end unicast/multicast throughput demands, while minimizing a natural bandwidth-distance sum-product (network volume). Space information flow models the design of a blueprint for a minimum-cost network. We study the multicast version of the space information flow problem, in Euclidean spaces. We present a simple example that demonstrates the design of an information network is indeed different from that of a transportation network. We discuss properties of optimal multicast network embedding, prove that network coding does not make a difference in the basic case of 1-to-2 multicast, and prove upper-bounds on the number of relay nodes required in an optimal acyclic multicast network.