A decentralized algorithm to combine topology control with network coding

Abstract

Network coding and topology control techniques have been widely used to increase throughput and improve the lifetime of Wireless Sensor Networks (WSNs). This paper considers the simultaneous utilization of these techniques in a WSN and proposes convex non-linear programming. Since solving the problem for a large-scale and dynamic WSN is impractical and almost impossible, Lagrangian, sub-gradient and the decomposition methods are employed to provide a decentralized algorithm. In the proposed algorithm, a node makes the computations by acquiring local knowledge and information from its neighbors. This paper provides a mathematical language to build an analytic foundation for the design of a modularized and decentralized algorithm that provides transmission ranges and routes for a WSN. The simulation results show that increasing the number of sources, sinks, sensors, and traffic load leads to improving the lifetime which is acquired by the proposed algorithm.

Introduction

Network coding comprises two distinct sub-problems: coding and sub-graph selection. The coding sub-problem determines each intermediate node should do which operation. The sub-graph selection argues on the routes between the sources and destination nodes, which should be optimal in cost. Since network coding, specifically sub-graph selection, needs to know about the neighbors (this information is used to make the routes), topology control is in conflict with the network coding opportunity [20], [25]. In fact, topology control uses some relay nodes to assist transmitting the packets while in network coding the relay nodes may mix the packets with an operator and then send the coded packets [1], [34].

In the following, we illustrate an example of exchanging data in a WSN where topology control and network coding are used, simultaneously. See Fig. 1, where the nodes $v_1$ and $v_4$ want to exchange their own packets, $P_1$ and $P_4$, respectively. The first choice is direct transmission where the nodes $v_1$ and $v_4$ exchange their packets directly with no relay node (see Fig. 1-a). However, the nodes $v_1$ and $v_4$ can send their packets to the relay node $v_2$ that mixes the packets and then sends. The nodes $v_1$ and $v_4$ can retrieve each others’ packets by XOR-ing $P_1\oplus P_4$ with their own packets (see Fig. 1-b). Note that without network coding, $4$ transmissions are required because the node $v_2$ should send $P_1$ and $P_4$ independently. This transmission can be accomplished by the relay node $v_3$, too. Here, the node $v_1$ exhausts more energy to communicate with the farther node $v_3$, while the node $v_4$ uses less transmission range. Additionally, both nodes $v_2$ and $v_3$ may be employed as relays in which either $v_2$ or $v_3$ mix the packets. The best solution for this problem is affected by the various factors, such as generated traffic of nodes, the distance between them, and their residual energy.

Recently, some studies, see e.g. [18], [22], [23], have been introduced which combined network coding with topology control. These studies provided some mathematical theories and showed some advantages, such as increasing the lifetime, and decreasing energy consumption. To the best knowledge of authors, available approaches use either a heuristics method or a centralized solver to solve the problem. However, the heuristics methods obtain a sub-optimal solution (generally is not close to the optimal solution) and the centralized approaches are subject to some well-known disadvantages. By considering the advantages of combining network coding with topology control, a decentralized approach is required in which the nodes obtain the network topology and network-coding-based routes, independently.

In this paper, a convex optimization problem is proposed that considers multiple sources and combines network coding with topology control. Then, Lagrange dual, sub-gradient, and the decomposition methods are utilized to present a decentralized algorithm. The proposed algorithm repeatedly solves the simpler sub-problems until it achieves the global object that is the optimal lifetime. We assess the performance of the proposed approach through simulations and compare the obtained results with the available optimal solutions. The simulation outcomes show that considering multiple sources leads to more lifetime compared to considering the sources, separately. We also study the effect of network parameters on the convergence rate of the proposed algorithm. The main contributions of this paper are summarized as follows:

• It considers multiple sources and provides a convex optimization problem for optimal topology control in network-coding-based-multicast WSNs. The mathematical formulation can help us to assess the essence of this problem.

• For the first time, a decentralized and optimal algorithm is presented that combines network coding and topology control. The proposed algorithm creates an automated and dynamic structure for WSNs, which continuously changes the routes and transmission range of the nodes.

• We prove the efficiency of the proposed algorithm by comparing it with the optimal solution and state-of-art approaches.

The rest of this paper is organized as follows: Section 2 presents an overview of related literature. System model and problem definition are described in Section 3. Section 4 describes the proposed decentralized algorithm. The assessment of approaches is done through simulations and the results are reported in Section 5. Finally, conclusions and directions for future research are presented in Section 6.