A communication network is called a radio network if its nodes exchange messages in the following restricted way. First, a send operation performed by a node delivers copies of the same message to all directly reachable nodes. Secondly, a node can successfully receive an incoming message only if exactly one of its neighbors sent a message in that step. It is this semantics of how ports at nodes send and receive messages that defines the networks rather than the fact that only radio waves are used as a medium of communication; but if that is the case then just a single frequency is used. We discuss algorithmic aspects of exchanging information in such networks, concentrating on distributed randomized protocols. Specific problems and solutions depend a lot on the topology of the underlying reachability graph and how much the nodes know about it. In single-hop networks each pair of nodes can communicate directly. This kind of networks is also known as the multiple access channel. Popular broadcasting protocols used on such channels are Aloha and the exponential backoff. Multi-hop networks may have arbitrary topology and packets need to be routed hopping through a sequence of adjacent nodes. Distributed protocols run by such networks are usually robust enough not to expect the nodes to know their neighbors. These ad-hoc networks and protocols model the situation when nodes are mobile and do not rely on a fixed infrastructure.
Diks, Kranakis, Krizanc and Pelc @cite_113 developed deterministic algorithms and lower bounds for broadcasting in the case of nodes located on a line, the nodes know their coordinates. Ravishankar and Singh @cite_110 @cite_44 studied broadcasting and gossiping when stations are randomly placed on a line. Kranakis, Krizanc and Pelc @cite_84 studied deterministic fault-tolerant broadcasting in radio networks with the stations located either on a line or on a two-dimensional grid, some of them faulty.
@cite_110 @cite_44 @cite_84 @cite_113
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We consider the problem of distributed deterministic broadcasting in radio networks whose nodes are located on a line. Nodes send messages in synchronous time-slots. Each node υ has a given transmission range. All nodes located within this range can receive messages from υ. However, a node situated in the range of two or more nodes that send messages simultaneously, cannot receive these messages and hears only noise. Each node knows only its own position and range, as well as the maximum of all ranges. Broadcasting is adaptive: nodes can decide on the action to take on the basis of previously received messages, silence or noise. We prove lower bounds on broadcasting time in this model and construct broadcasting protocols whose performance nearly matches these bounds for the simplest case when nodes are situated on a line. We also show that if nodes do not even know their own range, every broadcasting protocol must be hopelessly slow. While distributed randomized broadcasting algorithms, and, on the other hand, deterministic off-line broadcasting algorithms assuming full knowledge of the radio network, have been extensively studied in the literature, ours are the first results concerning broadcasting algorithms that are distributed and deterministic at the same time. We show that in this case, information available to nodes influences the efficiency of broadcasting in a significant way. We consider broadcasting in radio networks, modeled as undirected graphs, whose nodes know only their own label and labels of their neighbors. In every step every node acts either as a transmitter or as a receiver. A node acting as a transmitter sends a message which can potentially reach all of its neighbors. A node acting as a receiver in a given step gets a message if and only if exactly one of its neighbors transmits in this step. Bar-Yehuda, Goldreich, and Itai [J. Comput. System Sci., 45 (1992), pp. 104--126] considered broadcasting in this model. They claimed a linear lower bound on the time of deterministic broadcasting in such radio networks of diameter 3. This claim turns out to be incorrect in this model (although it is valid in a more pessimistic model [R. Bar-Yehuda, O. Goldreich, and A. Itai, Errata Regarding "On the time complexity of broadcast in radio networks: An exponential gap between determinism and randomization," http: www.wisdom.weizmann.ac.il mathusers oded p .html, 2002]). We construct an algorithm that broadcasts in logarithmic time on all graphs from the Bar-Yehuda, Goldreich, and Itai paper (BGI). Moreover, we show how to broadcast in sublinear time on all n-node graphs of diameter @math . On the other hand, we construct a class of graphs of diameter 4, such that every broadcasting algorithm requires time @math on these graphs. In view of the randomized algorithm from BGI, running in expected time @math on all @math -node graphs of diameter D (cf. also a recent @math -time algorithm from [D. Kowalski and A. Pelc, Proceedings of the 22nd Annual ACM Symposium on Principles of Distributed Computing, Boston, 2003, pp. 73--82; A. Czumaj and W. Rytter, Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, Cambridge, MA, 2003, pp. 492--501]), our lower bound gives the first correct proof of an exponential gap between determinism and randomization in the time of radio broadcasting, under the considered model of radio communication. In a seminal paper, Bar- (1992) considered broadcasting in radio networks whose nodes know only their own label and labels of their neighbors. They claimed a linear lower bound on the time of deterministic broadcasting in such radio networks, by constructing a class of graphs of diameter 3, with the property that every broadcasting algorithm requires linear time on one of these graphs. Due to a subtle error in the argument, this result is incorrect. We construct an algorithm that broadcasts in logarithmic time on all graphs from the work of Bar- Moreover, we show how to broadcast in sublinear time on all n-node graphs of diameter o(log log n). On the other hand, we construct a class of graphs of diameter 4, such that every broadcasting algorithm requires time spl Omega (4 spl radic n) on one of these graphs. In view of the randomized algorithm, running in expected time O(D log n + log sup 2 n) on all n-node graphs of diameter D, our lower bound gives the first correct proof of an exponential gap between determinism and randomization in the time of radio broadcasting. We consider deterministic broadcasting in geometric radio networks (GRN) whose nodes know only a limited part of the network. Nodes of a GRN are situated in the plane and each of them is equipped with a transmitter of some range r. A signal from this node can reach all nodes at distance at most r from it but if a node u is situated within the range of two nodes transmitting simultaneously, then a collision occurs at u and u cannot get any message. Each node knows the part of the network within knowledge radius s from it, i.e., it knows the positions, labels and ranges of all nodes at distance at most s. The aim of this paper is to study the impact of knowledge radius s on the time of deterministic broadcasting in a GRN with n nodes and eccentricity D of the source. Our results show sharp contrasts between the efficiency of broadcasting in geometric radio networks as compared to broadcasting in arbitrary graphs. They also show quantitatively the impact of various types of knowledge available to nodes on broadcasting time in GRN. Efficiency of broadcasting is influenced by knowledge radius, knowledge of individual positions when knowledge radius is zero, and awareness of collisions.