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.
The following are some results on centralized off-line algorithms computing broadcast schedules, the networks are known then, and are inputs to such sequential algorithms. Chlamtac and Weinstein @cite_33 developed a deterministic algorithm which finds in polynomial time a schedule giving broadcasting time @math . Gaber and Mansour @cite_82 showed that for any packet radio network there is a schedule of broadcasting giving time @math , and that it can be found by a deterministic polynomial algorithm. Chlamtac and Kutten @cite_26 and Sen and Hun @cite_10 showed the NP-completeness of finding optimal broadcasting schedules in packet radio networks, in particular it was shown in @cite_10 that the problem remains NP-complete even if the nodes are points in the plane and the reachability is determined by distances. Kirousis, Kranakis, Krizanc and Pelc @cite_102 studied the problem of assigning transmission ranges to nodes located on a line so as to minimize the total power consumption and preserve strong connectedness. This was later extended by Clementi, Ferreira, Penna, Perennes and Silvestri @cite_27 . Chlamtac and Kutten @cite_76 showed the NP-completeness of minimizing the average time per node during broadcasting when nodes are blocked from hearing messages other than those resulting from a broadcast algorithm.
@cite_26 @cite_33 @cite_102 @cite_27 @cite_76 @cite_10 @cite_82
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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. This paper concerns the communication primitives of broadcasting (one-to-all communication) and gossiping (all-to-all communication) in radio networks with known topology, i.e., where for each primitive the schedule of transmissions is precomputed based on full knowledge about the size and the topology of the network.The first part of the paper examines the two communication primitives in general graphs. In particular, it proposes a new (efficiently computable) deterministic schedule that uses O(D+Δ log n) time units to complete the gossiping task in any radio network with size n, diameter D and max-degree Δ. Our new schedule improves and simplifies the currently best known gossiping schedule, requiring time O(D+√[i+2]DΔ logi+1 n), for any network with the diameter D=Ω(logi+4n), where i is an arbitrary integer constant i ≥ 0, see [17]. For the broadcast task we deliver two new results: a deterministic efficient algorithm for computing a radio schedule of length D+O(log3 n), and a randomized algorithm for computing a radio schedule of length D+O(log2 n). These results improve on the best currently known D+O(log4 n) time schedule due to Elkin and Kortsarz [12].The second part of the paper focuses on radio communication in planar graphs, devising a new broadcasting schedule using fewer than 3D time slots. This result improves, for small values of D, on currently best known D+O(log3n) time schedule proposed by Elkin and Kortsarz in [12]. Our new algorithm should be also seen as the separation result between the planar and the general graphs with a small diameter due to the polylogarithmic inapproximability result in general graphs due to Elkin and Kortsarz, see [11]. 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. A comprehensive study of the problem of scheduling broadcast transmissions in a multihop, mobile packet radio network is provided that is based on throughput optimization subject to freedom from interference. It is shown that the problem is NP complete. A centralized algorithm that runs in polynomial time and results in efficient (maximal) schedules is proposed. A distributed algorithm that achieves the same schedules is then proposed. The algorithm results in a maximal broadcasting zone in every slot. >