A note on uniform power connectivity in the SINR model

TL;DR

This paper investigates the minimum number of frequency or time slot colors needed to ensure strong connectivity in wireless networks under the SINR model, revealing bounds based on network dimension and path-loss exponent.

Contribution

It provides new bounds on the number of colors needed for uniform power connectivity in SINR models across different dimensions and path-loss conditions.

Findings

Constant number of colors suffices in 1D for α>1 and 2D for α>2.

Upper and lower bounds of O(log n) and Ω(log n / log log n) for α=2.

Regular coloring with O(log n) colors guarantees connectivity in random distributions.

Abstract

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.