The number of nodes of a network, called its size, is one of the most important network parameters. A radio network is a collection of stations, called nodes, with wireless transmission and receiving capabilities. It is modeled as a simple connected undirected graph whose nodes communicate in synchronous rounds. In each round, a node can either transmit a message to all its neighbors, or stay silent and listen. At the receiving end, a node @math hears a message from a neighbor @math in a given round, if @math listens in this round, and if @math is its only neighbor that transmits in this round. If @math listens in a round, and two or more neighbors of @math transmit in this round, a collision occurs at @math . Two scenarios are considered in the literature: if nodes can distinguish collision from silence (the latter occurs when no neighbor transmits), we say that the network has the collision detection capability, otherwise there is no collision detection. We consider the task of size discovery: finding the size of an unknown radio network with collision detection. All nodes have to output the size of the network, using a deterministic algorithm. Nodes have labels which are (not necessarily distinct) binary strings. The length of a labeling scheme is the largest length of a label. Our main result states that the minimum length of a labeling scheme permitting size discovery in the class of networks of maximum degree Delta is Theta( Delta).
There are two papers studying the size of advice in the context of radio networks. In @cite_21 , the authors studied radio networks for which it is possible to perform centralized broadcasting in constant time. They proved that @math bits of advice allow to obtain constant time in such networks, while @math bits are not enough. In @cite_12 , the authors considered the problem of topology recognition in wireless trees without collision detection. Similarly to the present paper, they investigated short labeling schemes permitting to accomplish this task. It should be noted that the results in @cite_12 and in the present paper are not comparable: @cite_12 studies a harder task (topology recognition) in a weaker model (no collision detection), but restricts attention only to trees, while the present paper studies an easier task (size discovery) in a stronger model (with collision detection) but our results hold for arbitrary networks.
@cite_21 @cite_12
2951371116 2181598850 2251394987 2048113794
We consider the problem of topology recognition in wireless (radio) networks modeled as undirected graphs. Topology recognition is a fundamental task in which every node of the network has to output a map of the underlying graph i.e., an isomorphic copy of it, and situate itself in this map. In wireless networks, nodes communicate in synchronous rounds. In each round a node can either transmit a message to all its neighbors, or stay silent and listen. At the receiving end, a node @math hears a message from a neighbor @math in a given round, if @math listens in this round, and if @math is its only neighbor that transmits in this round. Nodes have labels which are (not necessarily different) binary strings. The length of a labeling scheme is the largest length of a label. We concentrate on wireless networks modeled by trees, and we investigate two problems. What is the shortest labeling scheme that permits topology recognition in all wireless tree networks of diameter @math and maximum degree @math ? What is the fastest topology recognition algorithm working for all wireless tree networks of diameter @math and maximum degree @math , using such a short labeling scheme? We are interested in deterministic topology recognition algorithms. For the first problem, we show that the minimum length of a labeling scheme allowing topology recognition in all trees of maximum degree @math is @math . For such short schemes, used by an algorithm working for the class of trees of diameter @math and maximum degree @math , we show almost matching bounds on the time of topology recognition: an upper bound @math , and a lower bound @math , for any constant @math . Topology recognition is one of the fundamental distributed tasks in networks. Each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph, with all ports correctly marked. This task is usually unfeasible without any a priori information. Such information can be provided to nodes as advice. An oracle knowing the network can give a possibly different string of bits to each node, and all nodes must reconstruct the network using this advice, after a given number of rounds of communication. During each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations. The time of completing topology recognition is the number of rounds it takes, and the size of advice is the maximum length of a string given to nodes. We investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes. We provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time, in the class of all graphs of size n and diameter D≤αn, for any constant α<1. In most cases, our bounds are asymptotically tight. More precisely, if the allotted time is D-k, where 0<k≤D, then the optimal size of advice is i¾?n 2 logn D-k+1. If the allotted time is D, then this optimal size is i¾?n logn. If the allotted time is D+k, where 0<k≤D 2, then the optimal size of advice is i¾?1+logn k. The only remaining gap between our bounds is for time D+k, where D 2<k≤D. In this time interval our upper bound remains O1+logn k, while the lower bound that holds for any time is 1. This leaves a gap if D∈ologn. Finally, we show that for time 2D+1, one bit of advice is both necessary and sufficient. Our results show how sensitive is the minimum size of advice to the time allowed for topology recognition: allowing just one round more, from D to D+1, decreases exponentially the advice needed to accomplish this task. In topology recognition, each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph, with all ports correctly marked. This task is usually unfeasible without any a priori information. Such information can be provided to nodes as advice. An oracle knowing the network can give a (possibly different) string of bits to each node, and all nodes must reconstruct the network using this advice, after a given number of rounds of communication. During each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations. The time of completing topology recognition is the number of rounds it takes, and the size of advice is the maximum length of a string given to nodes.We investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes. We provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time, in the class of all graphs of size n and diameter D ? α n , for any constant α < 1 . In most cases, our bounds are asymptotically tight. 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.