## A note on uniform power connectivity in the SINR model

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Avin, Chen

Lotker, Zvi

Pasquale, Francesco

Pignolet, Yvonne-Anne

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In this paper we study the connectivity problem for wireless networks under
the Signal to Interference plus Noise Ratio (SINR) model. Given a set of radio
transmitters distributed in some area, we seek to build a directed strongly
connected communication graph, and compute an edge coloring of this graph such
that the transmitter-receiver pairs in each color class can communicate
simultaneously. Depending on the interference model, more or less colors,
corresponding to the number of frequencies or time slots, are necessary. We
consider the SINR model that compares the received power of a signal at a
receiver to the sum of the strength of other signals plus ambient noise . The
strength of a signal is assumed to fade polynomially with the distance from the
sender, depending on the so-called path-loss exponent $\alpha$.
We show that, when all transmitters use the same power, the number of colors
needed is constant in one-dimensional grids if $\alpha>1$ as well as in
two-dimensional grids if $\alpha>2$. For smaller path-loss exponents and
two-dimensional grids we prove upper and lower bounds in the order of
$\mathcal{O}(\log n)$ and $\Omega(\log n/\log\log n)$ for $\alpha=2$ and
$\Theta(n^{2/\alpha-1})$ for $\alpha<2$ respectively. If nodes are distributed
uniformly at random on the interval $[0,1]$, a \emph{regular} coloring of
$\mathcal{O}(\log n)$ colors guarantees connectivity, while $\Omega(\log \log
n)$ colors are required for any coloring.

Comment: 13 pages

Comment: 13 pages

##### Keywords

Computer Science - Discrete Mathematics, Computer Science - Performance