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 $N$. $N$ is composed of $n$ nodes randomly deployed within a square (or a cube) $X$. We assume the time to be slotted and $N$ 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 gossip algorithms achieving $O(n^1/2 \log(n)^1/2)$ and $O(n^1/3 \log(n)^2/3) rounds in the two-dimensional and the three-dimensional cases, respectively. Next, we propose energy-efficient initialization and gossip algorithms running in $O(n^3/4 \log (n)^1/4)$ rounds, with no station being awake for more than O(n^1/4 \log (n)^3/4)$ rounds.
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