Abstract
The aim of the paper is to investigate the average case behavior of certain algorithms that are designed for connecting mobile agents in the two- or three-dimensional space. The general model is the following: let $X$ be a set of points in the $d$-dimensional Euclidean space $E_d$, $d\ge 2$, $r$ be a function that associates each element of $x\in X$ with a positive real number $r(x)$.
A graph $G(X,r)$ is an oriented graph with the vertex set $X$, in which $(x,y)$ is an edge if and only if $\rho(x,y)\le r(x)$, where $\rho(x,y)$ denotes the Euclidean distance in the space $E_d$. Given a set $X$, the goal is to find a function $r$ so that the graph $G(X,r)$ is strongly connected (note that the graph $G(X,r)$ need not be symmetric).
The function $r$ computed by the algorithm of the present paper is such that, given a random set $X$ of points, the average value of $r(x)^d$ (related to the average transmitter power) is almost surely constant.