Deterministic Digital Clustering of Wireless Ad Hoc Networks

We consider deterministic distributed communication in wireless ad hoc networks of identical devices in the SINR model without predefined infrastructure. Most algorithmic results in this model rely on additional features or capabilities, e.g., randomization, access to geographic coordinates, power control, carrier sensing with various precision of measurements, and interference cancellation. We study a pure scenario, when no such features are available. Several distributed algorithms have been presented in recent years for the local and global broadcast problems considered in this paper. However, all these solutions were either randomized or relied on the assumption that nodes of a network know their own coordinates in a given metric space or devices has signal sensitivity limited to some area (typically smaller than transmission range). In contrast, the aim of this paper is to check how efficient could be solutions without randomization, availability of locations, limited sensitivity, power control, carrier sensing, interference cancellation or other features. Our research objective is motivated twofolds. Firstly, examination of necessity and impacf of randomization is a natural research topic in algorithm design. Moreover, as wireless ad hoc networks are usually built from computationally limited devices run on batteries, it is desirable to use simple and energy efficient algorithms which do not need access to several sensing capabilities or true randomness. As a general tool, we develop a deterministic distributed clustering algorithm, which splits nodes of a multi-hop ad hoc network into clusters of constant diameter. Our solution relies on a new type of combinatorial structures (called witnessed strong selectors), which might be of independent interest. Using the clustering, we develop a deterministic distributed local broadcast algorithm accomplishing this task in O(Δ log N log N) rounds, where Δ is the density of a network. This is the first solution in pure scenario which is only polylog(n) away from the universal lower bound Ω(Δ), valid also for scenarios with randomization and other features. Therefore, we conclude that none of these features substantially helps for the local broadcast task. Using clustering, we also build a deterministic global broadcast algorithm that terminates within O(D(Δ + log? N) log N) rounds, where D is the diameter of the network. This result is complemented by a lower bound Ω(DΔ1-1/∝ ), where ∝ > 2 is the path-loss parameter of the environment. This lower bound, in view of previous work, shows that randomization or knowledge of own location help substantially (by a factor polynomial in Δ) in the global broadcast. Similar clusterbased techniques can be used to build efficient (comparing to the lower bound) solutions to the wake-up problem and the global leader election problem. Summarizing, our results prove that additional model/environment features may help substantially in design of time-efficient solutions for global communication problems, but not much in case of local problems.