Efficient communication in wireless networks is typically challenged by the possibility of interference among several transmitting nodes. Much important research has been invested in decreasing the number of collisions in order to obtain faster algorithms for communication in such networks. This paper proposes a novel approach for wireless communication, which embraces collisions rather than avoiding them, over an additive channel. It introduces a coding technique called Bounded-Contention Coding (BCC) that allows collisions to be successfully decoded by the receiving nodes into the original transmissions and whose complexity depends on a bound on the contention among the transmitters. BCC enables deterministic local broadcast in a network with n nodes and at most a transmitters with information of l bits each within O(a log n + al) bits of communication with full-duplex radios, and O((a log n + al)(log n)) bits, with high probability, with half-duplex radios. When combined with random linear network coding, BCC gives global broadcast within O((D + a + log n)(a log n + l)) bits, with high probability. This also holds in dynamic networks that can change arbitrarily over time by a worst-case adversary. When no bound on the contention is given, it is shown how to probabilistically estimate it and obtain global broadcast that is adaptive to the true contention in the network.
In the setting of a wireless network, deterministic global broadcast of a single message was studied in @cite_24 @cite_10 @cite_25 , the best results given being @math and @math , where @math is the diameter of the network. Bar- @cite_15 were the first to study randomized global broadcast algorithms. Kowalski and Pelc @cite_24 and Czumaj and Rytter @cite_25 presented randomized solutions based on selecting sequences, with complexities of @math . These algorithms match lower bounds of @cite_3 @cite_36 but in a model that is weaker than the one addressed in this paper. The algorithms mentioned above are all for global broadcast of one message from a known source. For multiple messages, a deterministic algorithm for @math messages with complexity @math appears in @cite_22 , while randomized global broadcast of multiple messages was studied in @cite_37 @cite_31 @cite_1 . We refer the reader to an excellent survey on broadcasting in radio networks in @cite_12 .
@cite_37 @cite_31 @cite_22 @cite_36 @cite_1 @cite_3 @cite_24 @cite_15 @cite_10 @cite_25 @cite_12
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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. In this paper we present new randomized and deterministic algorithms for the classical problem of broadcasting in radio networks with unknown topology. We consider directed n-node radio networks with specified eccentricity D (maximum distance from the source node to any other node). Bar- presented an algorithm that for any n-node radio network with eccentricity D completes the broadcasting in O(D log n + log2 n) time, with high probability. This result is almost optimal, since as it has been shown by Kushilevitz and Mansour and , every randomized algorithm requires Ω (D log(n D) + log2 n) expected time to complete broadcasting.Our first main result closes the gap between the lower and upper bound: we describe an optimal randomized broadcasting algorithm whose running time complexity is O(D log (n D + log2 n), with high probability. In particular, we obtain a randomized algorithm that completes broadcasting in any n-node radio network in time O(n), with high probability.The main source of our improvement is a better "selecting sequence" used by the algorithm that brings some stronger property and improves the broadcasting time. Two types of "selecting sequences" are considered: randomized and deterministic ones. The algorithm with a randomized sequence is easier (more intuitive) to analyze but both randomized and deterministic sequences give algorithms of the same asymptotic complexity. 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. In much of the theoretical literature on wireless algorithms, issues of message dissemination are considered together with issues of contention management. This combination leads to complicated algorithms and analysis, and makes it difficult to extend the work to harder communication problems. In this paper, we present results of a current project aimed at simplifying such algorithms and analysis by decomposing the treatment into two levels, using abstract "MAC layer" specifications to encapsulate the contention management. We use two different abstract MAC layers: the basic one of [14, 15] and a new probabilistic layer. We first present a typical randomized contention-manageent algorithm for a standard graph-based radio network model We show that it implements both abstract MAC layers. We combine this algorithm with greedy algorithms for single-message and multi-message global broadcast and analyze the combination, using both abstract MAC layers as intermediate layers. Using the basic MAC layer, we prove a bound of O(D log(n ∈) log Δ) for the time to deliver a single message everywhere with probability 1 -- ∈, where D is the network diameter, n is the number of nodes, and Δ is the maximum node degree. Using the probabilistic layer, we prove a bound of O((D + log(n ∈)) log Δ), which matches the best previously-known bound for single-message broadcast over the physical network model. For multi-message broadcast, we obtain bounds of O((D + kΔ) log(n ∈) log Δ) using the basic layer and O((D + kΔ log(n ∈)) log Δ) using the probabilistic layer, for the time to deliver a message everywhere in the presence of at most k concurrent messages.