Random Geometric Graphs and the Initialization Problem for Wireless Networks
Résumé
The initialization problem, also known as naming, assigns one unique identifier (ranging from $1$ to $n$) to a set of $n$ indistinguishable nodes (stations or processors) in a given wireless network $\mathcal{N}$. $\mathcal{N}$ is composed of $n$ nodes randomly deployed within a square (resp. a cube) $X$. We assume the time to be slotted and $\mathcal{N}$ to be synchronous; two nodes are able to communicate if they are within a distance at most $r$ of each other ($r$ is the transmitting/receiving range). Moreover, if two or more neighbors of a processor $u$ transmit concurrently at the same round, $u$ does not receive either messages. After the analysis of various critical transmitting/sensing ranges for connectivity and coverage of randomly deployed sensor networks, we design sub-linear randomized initialization and gossiping algorithms with running time $\BO\l(n^{1/2} \log{(n)}^{1/2}\r)$ and $\BO\l(n^{1/3} \log{(n)}^{2/3}\r)$) in the two-dimensional and the three-dimensional cases, respectively. Next, we propose energy-efficient initialization and gossiping algorithms running in time $\BO\l(n^{3/4} \log{(n)}^{1/4}\r)$, with no station being awake for more than $\BO\l(n^{1/4}\log{(n)}^{3/4}\r)$ rounds.
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