In multi-hop wireless networks, flow demands mean that some nodes have routing demands of transmitting their data to other nodes with a certain level of transmission rate. When a set of nodes have been deployed with flow demands, it is worth to know how to construct paths to satisfy these flow demands with nodes placed as few as possible. In this paper, we study this flow demands oriented node placement problem that has not been addressed before. In particular, we divide and conquer the problem by three steps: calculating the maximal flow for single routing demand, calculating the maximal flow for multiple routing demands, and finding the minimal number of nodes for multiple routing demands with flow requirement. During the above solving procedure, we prove that the second and third step are NP-hard and propose two algorithms that have polynomial-time complexity. The proposed algorithms are evaluated under practical scenarios. The experiments show that the proposed algorithms can achieve satisfactory results on both flow demands and total number of wireless nodes.
Suppose that a set of nodes have been deployed in a plane. Some of these nodes may have routing demands of transmitting their data to other nodes with flow requirements that should be achieved. Then, relay nodes are needed to be placed to route and satisfy these flow demands. We assume that each node is equipped with a radio, and the radio has the its maximal transmission range @math . If two nodes, @math and @math , that are located with their distance @math can interfere the transmission between each other. With respect to a successful transmission of two nodes, we consider the . A transmission from node @math to @math is successful if and only if 1) @math ; 2) there does not exist a transmission node @math such that @math . This interference model is widely used in references like @cite_12 @cite_11 . Besides, we use a time slotted system. In the time slotted system, the time is divided into equal length slots, and the transmission between nodes are synchronized. We define @math as the maximal flow (or maximal transmission rate) that can be transmitted in a single time slot.
@cite_12 @cite_11
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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. 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. Wireless systems comprised of rechargeable nodes have a significantly prolonged lifetime and are sustainable. A distinct characteristic of these systems is the fact that the nodes can harvest energy throughout the duration in which communication takes place. As such, transmission policies of the nodes need to adapt to these harvested energy arrivals. In this paper, we consider optimization of point-to-point data transmission with an energy harvesting transmitter which has a limited battery capacity, communicating in a wireless fading channel. We consider two objectives: maximizing the throughput by a deadline, and minimizing the transmission completion time of the communication session. We optimize these objectives by controlling the time sequence of transmit powers subject to energy storage capacity and causality constraints. We, first, study optimal offline policies. We introduce a directional water-filling algorithm which provides a simple and concise interpretation of the necessary optimality conditions. We show the optimality of an adaptive directional water-filling algorithm for the throughput maximization problem. We solve the transmission completion time minimization problem by utilizing its equivalence to its throughput maximization counterpart. Next, we consider online policies. We use stochastic dynamic programming to solve for the optimal online policy that maximizes the average number of bits delivered by a deadline under stochastic fading and energy arrival processes with causal channel state feedback. We also propose near-optimal policies with reduced complexity, and numerically study their performances along with the performances of the offline and online optimal policies under various different configurations. We consider the problem of constructing a communication infrastructure from scratch, for a collection of identical wireless nodes. Combinatorially, this means a) finding a set of links that form a strongly connected spanning graph on a set of @math points in the plane, and b) scheduling it efficiently in the SINR model of interference. The nodes must converge on a solution in a distributed manner, having no means of communication beyond the sole wireless channel. We give distributed connectivity algorithms that run in time @math , where @math is the ratio between the longest and shortest distances among nodes. Given that algorithm without prior knowledge of the instance are essentially limited to using uniform power, this is close to best possible. Our primary aim, however, is to find efficient structures, measured in the number of slots used in the final schedule of the links. Our main result is algorithms that match the efficiency of centralized solutions. Specifically, the networks can be scheduled in @math slots using (arbitrary) power control, and in @math slots using a simple oblivious power scheme. Additionally, the networks have the desirable properties that the latency of a converge-cast and of any node-to-node communication is optimal @math time.