text: Radio networks are a long-studied model for distributed system of devices which communicate wirelessly. When these devices are mobile or have limited capabilities, the system is often best modeled by the ad-hoc variant, in which the devices do not know the structure of the network. A large body of work has been devoted to designing algorithms for the ad-hoc model, particularly for fundamental communications tasks such as broadcasting. Most of these algorithms, however, assume that devices have some network knowledge (usually bounds on the number of nodes in the network @math , and the diameter @math ), which may not always be realistic in systems with weak devices or gradual deployment. Very little is known about what can be done when this information is not available. This is the issue we address in this work, by presenting the first broadcasting algorithms for networks in which nodes have no prior knowledge whatsoever. We demonstrate that lack of parameter knowledge can be overcome at only a small increase in running time. Specifically, we show that in networks without collision detection, broadcast can be achieved in @math time, almost reaching the @math lower bound. We also give an algorithm for directed networks with collision detection, which requires only @math time.
  @cite_9: We study two fundamental communication primitives: broadcasting and leader election in the classical model of multi-hop radio networks with unknown topology and without collision detection mechanisms. It has been known for almost 20 years that in undirected networks with n nodes and diameter D, randomized broadcasting requires Omega(D log n D + log^2 n) rounds in expectation, assuming that uninformed nodes are not allowed to communicate (until they are informed). Only very recently, Haeupler and Wajc (PODC'2016) showed that this bound can be slightly improved for the model with spontaneous transmissions, providing an O(D log n loglog n log D + log^O(1) n)-time broadcasting algorithm. In this paper, we give a new and faster algorithm that completes broadcasting in O(D log n log D + log^O(1) n) time, with high probability. This yields the first optimal O(D)-time broadcasting algorithm whenever D is polynomial in n. Furthermore, our approach can be applied to design a new leader election algorithm that matches the performance of our broadcasting algorithm. Previously, all fast randomized leader election algorithms have been using broadcasting as their subroutine and their complexity have been asymptotically strictly bigger than the complexity of broadcasting. In particular, the fastest previously known randomized leader election algorithm of Ghaffari and Haeupler (SODA'2013) requires O(D log n D min loglog n, log n D + log^O(1) n)-time with high probability. Our new algorithm requires O(D log n log D + log^O(1) n) time with high probability, and it achieves the optimal O(D) time whenever D is polynomial in n.
  @cite_14: 
  @cite_2: We present a randomized distributed algorithm that in radio networks with collision detection broadcasts a single message in O(D + log6 n) rounds, with high probability. This time complexity is most interesting because of its optimal additive dependence on the network diameter D. It improves over the currently best known O(Dlogn D + log2 n) algorithms, due to Czumaj and Rytter [FOCS 2003], and Kowalski and Pelc [PODC 2003]. These algorithms where designed for the model without collision detection and are optimal in that model. However, as explicitly stated by Peleg in his 2007 survey on broadcast in radio networks, it had remained an open question whether the bound can be improved with collision detection. We also study distributed algorithms for broadcasting k messages from a single source to all nodes. This problem is a natural and important generalization of the single-message broadcast problem, but is in fact considerably more challenging and less understood. We show the following results: If the network topology is known to all nodes, then a k-message broadcast can be performed in O(D + klog n + log2 n) rounds, with high probability. If the topology is not known, but collision detection is available, then a k-message broadcast can be performed in O(D + klog n + log6 n) rounds, with high probability. The first bound is optimal and the second is optimal modulo the additive O(log6 n) term.
summary: More recently, Ghaffari, Haupler and Khabbazian @cite_2 showed that collision detection can be used to surpass this lower bound, attaining an @math time algorithm. Work by Haeupler and Wajc @cite_14 demonstrated that even without collision detection, the lower bound could be beaten assuming were permitted; that is, nodes have access to a global clock and are allowed to transmit before receiving the source message. Czumaj and Davies @cite_9 extended this approach and obtained a running time of @math for the setting with spontaneous transmissions. However, these algorithms only work in undirected networks.