Minimal Range Assignments for Broadcasts

Abstract

Providing people with up-to-date information has become more and more crucial over the past centuries. Right now, this is mostly done via newspapers or television. While this is no big issue in densely populated areas like cities, where a reliable infrastructure exists, it is rather challenging in other regions like deserts or jungles. Distributing daily newspapers could involve enormous cost. In order to solve this problem for a region without a highly connected infrastructure, one could imagine a digital newspaper broadcasted via wireless links between houses—in the following called stations. This process starts at one station s acting as a server. To keep operation expenses as low as possible multi-hop strategies are used. In this chapter we consider a single-hop strategy to be in particular a multi-hop strategy; in that way this solution is implicitly regarded as well. The resulting question is then how to choose the transmission range of each station. To measure the quality of such a range assignment we use the sum of the energy needed to provide it: based upon experiences and theoretical research from communications engineering (see Section 2.3), the power P _u required by a station u to correctly transmit data to another station v must satisfy

$$\frac{P_u}{d(u,v)^\alpha} > \gamma$$

where d(u,v) is the Euclidean distance between u and v, α ≥ 1 is the path-loss exponent and γ ≥ 1 describes properties of the used technology (including antenna characteristics, transmission quality, etc.). Fixing the range assignment induces the transmission graph: edge (u,v) represents the ability of u to communicate with v. Note that this graph is directed since the transmission ranges of u and v may differ. The task is to find a range assignment that keeps the sum of the energy used by all stations as low as possible while guaranteeing s the ability to send messages to all other stations. We assume all stations to be embedded in the plane, i.e., the problem is 2-dimensional, and refer to [77] for a generalization of this Range Assignment Problem to d-dimensional space.