Nearly Optimal Local Broadcasting in the SINR Model with Feedback

Abstract

We consider the SINR wireless model with uniform power. In this model the success of a transmission is determined by the ratio between the strength of the transmission signal and the noise produced by other transmitting processors plus ambient noise. The local broadcasting problem is a fundamental problem in this setting. Its goal is producing a schedule in which each processor successfully transmits a message to all its neighbors. This problem has been studied in various variants of the setting, where the best currently-known algorithm has running time $O(\bar{\Delta} + \log^2 n)$ in n-node networks with feedback, where $\bar{\Delta}$ is the maximum neighborhood size [9]. In the latter setting processors receive free feedback on a successful transmission. We improve this result by devising a local broadcasting algorithm with time $O(\bar{\Delta} + \log n \log \log n)$ in networks with feedback. Our result is nearly tight in view of the lower bounds $\Omega(\bar{\Delta})$ and Ω(logn) [13]. Our results also show that the conjecture that $\Omega(\bar{\Delta} + \log^2 n)$ time is required for local broadcasting [9] is not true in some settings.

We also consider a closely related problem of distant-k coloring. This problem requires each pair of vertices at geometrical distance of at most k transmission ranges to obtain distinct colors. Although this problem cannot be always solved in the SINR setting, we are able to compute a solution using an optimal number of Steiner points (up to constant factors). We employ this result to devise a local broadcasting algorithm that after a preprocessing stage of $O(\log^* n \cdot (\bar{\Delta} + \log n \log \log n))$ time obtains a local-broadcasting schedule of an optimal (up to constant factors) length $O(\bar{\Delta})$. This improves upon previous local-broadcasting algorithms in various settings whose preprocessing time was at least $O(\bar{\Delta} \log n$) [3,10,5,]. Finally, we prove a surprising phenomenon regarding the influence of the path-loss exponent α on performance of algorithms. Specifically, we show that in vacuum (α = 2) any local broadcasting algorithm requires $\Omega(\bar{\Delta} \log n)$ time, while on earth (α > 2) better results are possible as illustrated by our $O(\bar{\Delta} + \log n \log \log n)$-time algorithm.