Joint Coding/Routing Optimization for Correlated Sources in Wireless Visual Sensor Networks

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1 Joint Coding/Routing Optimization for Correlated Sources in Wireless Visual Sensor Networks Chenglin Li 1, Junni Zou 2, Hongkai Xiong 1, Yongsheng Zhang 1 1 Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai , P.R. China 2 Department of Communication Engineering, Shanghai University, Shanghai, , P.R. China Abstract This paper studies a joint coding/routing optimization between network lifetime and rate-distortion, by applying information theory to wireless visual sensor networks for correlated sources. Arbitrary coding (distributed source coding and network coding) from both combinatorial optimization and information theory could make significant progress towards the performance limit of information networks and tractable. Also, multipath routing can spread energy utilization across nodes within the entire network to keep a potentially longer lifetime, and solve the wireless contention issues by the splitting traffic. The objective function not only keeps a total energy consumption of encoding power, transmission power, and reception power minimized, but ensures the information received by sink nodes to approximately reconstruct the visual field. Based on the localized Slepian-Wolf coding and network coding-based multipath routing, the balance problem between distortion (capacity) and lifetime (costs) is modeled as an optimization formulation with a distributed solution. Through a primal decomposition, a two-level optimization is relaxed with Lagrangian dualization and solved with the gradient algorithm. The low-level optimization problem is decomposed into a secondary master dual problem (encoding, energy, and congestion prices update) with four cross-layer subproblems: a rate control problem, a channel contention problem, a distortion control problem, and an energy conservation problem. Numerical results validate the convergence and performance of the proposed algorithm. Keywords Wireless visual sensor network, distributed source coding, network coding, network lifetime, rate-distortion I. INTRODUCTION In wireless visual sensor network (WVSN), each sensor equipped with video capture and processing functionalities, is tasked to capture digital visual information about target events, and deliver the video data to a control unit for further data analysis and decision making [1]. As illustrated in Fig. 1, we observe that videos captured from WVSN are uniquely correlated under the multi-view geometry. The major concern is to extend the lifetime of WVSN, and at the same time, to optimize the network performance for correlated source under The work has been partially supported by the NSFC grants No , No , No , No and the National High Technology Research and Development Program of China (863 Program) (No. 2006AA01Z322). Fig.1. A wireless visual sensor network under the multi-view geometry energy consumption, rate-distortion, and bandwidth constraints. There has been considerable interest in applying information theory to data networks recently, and the traditional routing problems become joint coding/routing optimization problems. It was proved that either the minimum communication cost or energy consumption for correlated sensed data can be achieved using Distributed Source Coding (DSC) and network coding when the link communication cost is a convex function of link data rate [2]. DSC allows a sensor node to encode its data to the joint rate vector without explicit communication, and exploits the source statistics at decoder. The Slepian-Wolf and Wyner-Ziv theorems for lossless and lossy distributed source coding state that separate encoding of correlated sources can achieve similar information-theoretic bounds as joint encoding [3]. It signifies that, the encoding power consumption at each sensor node can be reduced by moving the computation burden from the encoder side to the decoder side; the transmission power consumption can still decrease as the rate required to be transmitted at sensor nodes decreases by utilizing the data correlation among spatially adjacent video sensor nodes [4]. Network coding (NC) as a generalization of routing, offers a best achievable rate region for multicasting in a communication network [5]. In practice, random linear network coding as an efficient distributed strategy, achieves this capacity with high probability [6]. It is shown that random linear network coding suffices for the network coding of correlated sources [7]. [8] provides a practical low complexity scheme of joint DSC and NC. Most of the works on joint DSC and NC coding all focus on the capacity aspect and ignore costs.

2 Network lifetime maximization for wireless sensor networks has been extensively studied [9, 10]. However, those omit the processing power consumption at the visual sensor nodes. Recently, He et al [11] reported the lifetime maximization problem with a unicast routing in WVSN based on an analytic power-rate-distortion (P-R-D) model [1]. It does not consider both a multicast capacity and a source correlation model over a generalized network model. To the best of our knowledge, joint coding/routing optimization of WVSN for correlated sources has not been investigated in the literature. The objective of this work is to make a joint coding/routing optimization of network lifetime and rate-distortion over WVSN for correlated sources. It is involved with network combinatorics and information theory over an arbitrary graph structure, aiming to investigate the performance limit of a generalized WVSN model. Also, multipath routing can spread energy utilization across nodes within the entire network to keep a potentially longer lifetime, and solve the wireless contention issues by the splitting traffic. In this context, the objective function not only keeps a total energy consumption of encoding power, transmission power, and reception power minimized, but ensures the information received by sink nodes to approximately reconstruct the visual field. Based on a localized Slepian-Wolf coding and network coding-based multipath routing, the balance problem between distortion (capacity) and lifetime (costs) is modeled as an optimization formulation with a distributed solution. Through a primal decomposition, a two-level optimization is relaxed with Lagrangian dualization and solved with the gradient algorithm. The low-level optimization problem is decomposed into a secondary master dual problem (encoding, energy, and congestion prices update) with four cross-layer subproblems: a rate control problem, a channel contention problem, a distortion control problem, and an energy conservation problem. The rest of the paper is organized as follows. Sec. II presents a generalized system model. Sec. III formulates the joint coding/routing problem of lifetime and rate-distortion optimization. A fully decentralized algorithm over lossy WVSNs is proposed in Sec. IV. Numerical results are presented in Sec. V. II. SYSTEM MODEL A. Network Model Supposing a wireless visual sensor network modeled as a directed graph. In the network model, is the set of directed wireless links, is the set of nodes, where and denotes the set of wireless video sensor nodes and sink nodes, respectively. Assuming that all sensor nodes have a fixed transmission range. Let denote the distance between node and node through link, then a directed wireless link exists if. B. Distributed Source Coding Model 1) Localized Slepian-Wolf coding Here we employ Slepian-Wolf coding, a fundamental research study in distributed source coding. The Slepian-Wolf region specifies the minimum encoding rates that the sensor nodes must meet in order to transmit all independent data from sensor nodes to the sink nodes. It is satisfied when any subset of sensor nodes encode their collected video data at a total rate exceeding their joint entropy. In [4], localized Slepian-Wolf coding is proposed to relax the Slepian-Wolf coding constraint, such that only local correlation information is required at each sensor node. When a sensor node is determining its data transmission rate, it only considers its data correlation with nearby sensor nodes, instead of the entire network. Generally, we can suppose each sensor node has complete information (distances between nodes and total weights to the sink nodes) only about local vicinity formed by its -hop neighbours on the network. The localized Slepian-Wolf coding scheme specifies that each sensor node should encode its data at a rate equal to the conditioned entropy where the conditioning is performed only on, i.e. the rate allocation for each sensor node is. 2) Rate Estimation Model For the estimation of the encoding rate at each video sensor nodes, we use rate-distortion theory [16] to analyze the problem. For correlated sensor nodes, suppose they are dependent Gaussian sources with the correlation matrix and, where is a correlation parameter that represents the amount of spatial correlation between two sensor nodes and, and should be less than one to make sure that is a semi-positive definite matrix. Let be a vector of samples measured by these sensor nodes, and let be a representation of, then the distortion can be written as with the Mean Square Error (MSE) as the distortion measure. Since is a spatially correlated Gaussian vector, the rate distortion function of is where are the ordered eigenvalues of the correlation matrix,, and is chosen to meet the condition that. Given any subset of nodes and the distortion per node, we can find its correlation matrix and therefore approximate its joint entropy by its rate distortion function:. In the context, we assume the set of four direct neighbour nodes which are closer to sensor node have the data correlation with the central node. Once the data correlation (1) (2)

3 model of either two nodes with known node or three nodes with known nodes and, we should allocate for node the encoding rate to sink node, respectively: C. Wireless Channel Contention Model (3) (4) Contention-based MAC protocols [12] are universally used as medium access control protocol in wireless sensor networks. Here, a p-persistent contention based protocol at the MAC layer is employed, i.e. each sensor node will contend for channel access with a certain persistence probability. Furthermore, assume that time is divided into slots and each sensor node can only start transmission at the beginning of each time slot. When node decides to transmit, it first chooses one link out of the set of its outgoing links with probability, then contend for channel access with persistence probability. Therefore, the transmission attempt probability of the link is given by, where.thus, the persistence probability is given by (5) where,. Consider the saturated wireless sensor network scenario where each link always has data to transmit, and assume the wireless channel at link is with the packet loss probability. Under such conditions, the probability for a packet transmission to be successful is (6) where is defined as the cluster of nodes whose transmissions will interfere with the transmission at link, assuming that any link originating from node will interfere with link if. Then the average throughput of link is (7) where is the instantaneous transmission rate at link. And the wireless channel contention constraint requiring the aggregate flow rate should not exceed the link capacity is (8) D. Network Coding Model With network coding, flows from the same video sensor node to different sink nodes are allowed to share network capacity by being coded together. Suppose a sensor node with transmission rate to sink node, information flow must flow at rate to sink nodes, while by network coding the actual physical flow on each link needs only to be the maximum of the individual sink node s information flow. For each link, let denote the information flow for sink node from sensor node, and denote the physical flow from sensor node, then these constraints can be expressed as 1) (9) 2) (10) Constraints 1) reflect the information flow balance equation at each node. Constraints 2) specify the network coding condition relating physical rate and information rate: 2-1) (11) The above constraints are obtained from ideal conditions that there is no channel contention in wireless links. When considering the wireless contention, we assume that there is no retransmission limit of MAC protocol, i.e. the MAC protocol will not stop transmitting a packet until this packet is successfully delivered. In such case, the average number of transmissions attempted by a link to successfully transmit a packet is. Therefore, given a physical flow rate on the link, the actual flow rate to make sure successful transmission through the transmitter to the receiver can achieve the rate of. Then the network coding constraints 2) can also be rewritten as 2-2) (12) E. Multipath Routing Model The main advantage of multi-path routing is for loading balancing, by means of splitting traffic between a source-destination pair across multiple partially or completely disjoint paths. In our multipath routing model, we choose the braided multi-path routing mechanism, since this kind of routing scheme is proved more energy efficient than disjoint multi-path routing. Furthermore, two major criterions are considered when choosing the multiple paths. First, the shortest distance path based on link weights is proved to be the best way to efficiently transmit data between each source-destination pair [4]. Applying this idea to our problem, we should find for each sensor node multiple shortest distance path to the set of sink nodes. Second, in order to achieve the bound of the multicast capacity from network coding, the multiple paths in a one sensor node to multiple sink nodes multicast session should be chosen such that the probability of path overlapping is high. Next, we take into account the notation to describe multi-path routing transmission structure. For each node, we use a matrix to reflect the relationship between its path and corresponding edges. If each wireless sensor node has alternative paths to the sink node, let if the path of node to the sink node uses link, and otherwise. In the network coding based routing, let denotes the information flow rate of wireless sensor node s th path to the sink node, represents the physical flow rate of wireless sensor node at link. F. Power Consumption Model According to the Power-Rate-Distortion (PRD) analytical model [1], the encoding power consumption of each sensor

4 node can be computed by: (13) where is the average input variance, is the encoding efficiency coefficient. The transmission power consumption [15] at link can be formulated as: and (14) where is the actual aggregate rate transmitted through link, is the transmission power consumption cost of link, is the energy cost of the transmit electronics, is the coefficient term corresponding to the energy cost of transmit amplifier, is the distance between the sender node and receiver node along link, and is the path-loss factor taking values between 2 and 4. The reception power consumption at node can be formulated as: (15) at each sensor node to be as much as possible, increasing the transmission load and thus transmission energy consumption at each wireless link accordingly. The corresponding objective can be expressed as:, where is the distortion based on the encoded video rate of each sensor node. Analyzing the two problems jointly, to achieve the maximal performance of correlated sources reconstruction in WVSN, there exist two important but conflicting objectives, i.e. maximize the lifetime of network and minimize the total video distortion at all the sink nodes. The tradeoff between these two constrained problems can be formulated as a multi-objective optimization problem. When introducing a new weighted system parameter, two objectives are combined together into a single objective function. With the constraints introduced above, the balance problem can be formulated as a constrained optimization problem P1: P1: (17) where and is the energy consumption cost of the radio receiver, is the actual aggregate rate received at node. Assuming that network coding at sensor nodes only employ the binary coding approach proposed in [13], which means only bit-wise addition operations with much lower complexity than other operations (e.g. multiplication operations on a large alphabet size rather than ) are needed when implementing network coding. Due to the simplicity and low complexity of network coding operation, the corresponding power consumption is very little in contrast with the main video encoding and transmission power. The total power dissipation at node can be expressed as: Assuming that each sensor node has initial energy the lifetime of sensor node can be stated as:. (16), then III. OPTIMIZATION PROBLEM In the proposed problem, one of our major concerns is to extend the lifetime of the entire video data gathering and transmission network, which means we should minimize the cost function in respect to the allocated rate for each sensor node and the transmission structure. Furthermore, all video sensor nodes are supposed to be of equal importance. After defining as the lifetime of sensor node, the lifetime of the visual sensor network can be defined as the time until the first sensor exhausts its battery energy. Thus, the objective is:. Another major concern is to improve the decoded video quality of each sink node, which prefers the encoded video rate However, since equality constraint 4) and 7) are not linear, the formulated optimization problem P1 is not a convex optimization problem, which is usually more difficult to solve. For the sake of simplicity, we will first take some efforts to transform them into linear equality. For constraint 4), it is reformulated by applying logarithmic transformation at both sides. As to constraint 7), we introduce a new variable, which can be interpreted as node s normalized power dissipation with respect to its initial energy, then. Since P1 contains the maximum function which is not differentiable and need the knowledge of global information of all sensor nodes, it is difficult to solve the problem in a fully distributed manner. Using the similar analysis in [14], we have (18) It can be verified that when is a sufficiently large integer, might approximate to. Slightly rewriting the objective function to, we can reformulated P1 as:

5 P2: (19) objective value of P2-1 for a given. The objective value of the low-level optimization is locally optimal, which approximates to the global optimality using the result of the high-level optimization. Furthermore, after the second level dual decomposition, the low-level optimization problem P2-1 can be decomposed into several subproblems the solution of which only need the local information instead of global information of the entire network. It can be observed that variable vectors, and are dummy variables as they can be expressed by functions of other variables, thus the optimization variable vectors in problem P2-1 are,, and. By associating coupling constraints 1), 2) and 5) with Lagrange multipliers, and respectively, the Lagrangian of problem P2-1 can be expressed as: IV. DISTRIBUTED ALGORITHM A. Two Level Decomposition Considering problem P2, one way to decouple this problem is by first taking a primal decomposition with respect to the coupling variable, and then a dual decomposition with respect to the coupling constraints 1), 2) and 6). Then a two-level optimization decomposition procedure is proposed: a master primal problem, a secondary master dual problem with several subproblems. In mathematical term, after the first level (high level) primal decomposition by fixing he coupling variable, the original large optimization problem P3 can be decoupled into two hierarchical problems shown as follows: P2-1: (20) and the corresponding dual function is (22) Then the Lagrange dual problem of P2-1 can be formulated as. And the low-level optimization problem P2-1 can be further decomposed to a secondary master dual problem P2-1a with four subproblems that can be solved in a distributed way: P2-1a: (23) P2-1b: P2-1c: P2-2: (21) where P2-1 performs a low-level optimization when the coupling variable vector is fixed, P2-2 performs a high-level optimization in charge of updating, and is the optimal

6 P2-1d: P2-1e: The relationship among the two-level decomposition is represented in Fig.2. In the high level, master primal problem P2-2 is a link capacity allocation problem at the transport layer. In the low level, subproblem P2-1b is a rate control problem at the transport layer, subproblem P2-1c is a contention resolution problem at the MAC layer, subproblem P2-1d and P2-1e are about distortion control and energy conservation taking into account impacts both from transport layer and MAC layer in wireless sensor networks, respectively. B. Low Level Optimization We propose the following primal-dual algorithm that updates the primal and dual variables simultaneously and moves together towards the optimal points asymptotically to solve the low-level optimization problem. 1) Rate Control Problem at Transport Layer In problem P2-1b, denote the objective function by (24) Since the objective function is differentiable with respect to variable, the rate control optimization problem P2-1b can be solved by the gradient algorithm as follows (25) where denotes the low-level iteration index, is corresponding positive step size, and denotes the projection onto the set of non-negative real numbers. Through mathematically deduction, the derivative of is: (26) 2) Contention Resolution Problem at the MAC Layer In problem P2-1c, define, then the objective function can be denoted by (27) Since the objective function is differentiable with respect to variable, the contention resolution optimization problem P2-1c can be solved by the gradient algorithm as follows (28) Fig.2. A hierarchical decomposition with two levels. onto the range. And the derivative of is: (29) where we define as the set of links whose transmissions can be corrupted by transmissions from node 3) Distortion Control Problem In problem P2-1d, denote the objective function by (30) Since the objective function is differentiable with respect to variable, the distortion control optimization problem P2-1d can be solved by the gradient algorithm as follows (31) where denotes the low-level iteration index, is corresponding positive step size, and the derivative of is: 4) Energy Conservation Problem In problem P2-1e, denote the objective function by (32) (33) Since the objective function is differentiable with respect to variable, the energy conservation optimization problem P2-1e can be solved by the gradient algorithm as follows (34) where denotes the low-level iteration index, is corresponding positive step size, and the derivative of is: where we have the below equality: (35) (36) where denotes the low-level iteration index, is corresponding positive step size, and denotes the projection 5) Secondary Master Dual Problem In problem P2-1a, since the objective function is

7 differentiable with respect to variable, the problem P2-1d can also be solved by the gradient algorithm. Under physical context, can be mapped to the encoding prices at sensor nodes. If the encoding rate demand exceeds the supply, then price will rise, which in problem P2-1b will lead the encoding rate supply to increase to meet the encoding rate demand, and vice versa. Similarly, and can be interpreted as congestion prices at wireless links and energy prices at sensor nodes, respectively. Furthermore, all updating steps are distributed and can be implemented at individual links and nodes using only local information. C. High Level Optimization Next, we discuss how to adjust in order to solve the high-level optimization problem P2-2. Suppose and are the optimal Lagrange price and optimal variable corresponding to the constraint in problem P2-1. Similarly as the solution of P2-1, first we define the Lagrangian of problem P2-2 as: (37) where is the Lagrange price. Then a primal-dual algorithm similar to P2-1 is applied to solve it. could be regarded as the aggregate congestion prices at wireless links. The update of can be performed individually by each link, only with knowledge of the congestion price ; while the update of simply uses the local information of each link. V. EXPERIMENTAL RESULTS We conduct numerical experiments over a wireless visual network in Fig.3 (a), where each node of the video sensor node array wants to transmit video sequence with encoding rate to both 2 sink nodes. The configuration of model parameters are set as follows: assume the transmission range and interference range of each node are equal, and set to. For each link, we set the distance to, the packet loss probability to and the transmission rate to. For power consumption model [1] [15], we set,,,. The initial energy of each sensor node is set to. For each sensor node-sink node transmission pair, we find two most suitable paths according to the multi-path routing model. For example, Fig.3(b) shows the multi-path routing from sensor node to both sink nodes, which are,, and. Red links belong to the route from to, green links belong to the route from to, and black links are shared by both two routes, while the thickness of each link illustrate the number of paths it belongs to. It can be seen that since s neighbouring nodes and are on the routes to and, the actual (a) (b) Fig.3. Wireless visual sensor network (a) topology with all nodes and links, (b) a multi-path routing example encoding rate requirement can be reduced and less than when using distributed source coding. Furthermore, network coding can be implemented on the overlapped (black) links to improve the multicast throughput. Fig.4 shows the convergence behaviour of the four optimization variables,, and during the low-level optimization with a fixed stepsize. Here we set in optimization objective and. It can be seen that all variables approach the optimal value after 100 iterations, which means the algorithm converges with a relatively fast speed. For example, variable and in Fig.4 (a) and (b) with a faster speed converge to the optimal value after 50 iterations, while in Fig.4 (c) and (d), the other two varibles and converge to the optimal value after 100 iterations. The convergence behavior of the high-level optimization is shown in Fig.5, where we present variables and for illustration. It can be seen that all the variables converge after 25 iterations. The impact of weighted system parameter on the tradeoff between network lifetime maximization and total video distortion minimization is illustrated in Fig.6, where ranges from to. It can be seen that as the weighted parameter decreases form to, the network lifetime maximization problem in the optimization problem is taken into more consideration than the total distortion minimization problem, which means the corresponding optimal network lifetime will increases with the tradeoff of the increment of the video distortion. Furthermore, we can observe from Fig.6 that when, network lifetime achieves its maximal value, since at this time the network lifetime maximization problem is the dominant problem that can approximate the original optimization problem. For the same reason, when the original optimization problem transforms to the total distortion minimization problem. Fig.7 illustrates the optimal value of each sensor node s lifetime and video distortion solved by the algorithm with independent source coding (i.e. without DSC), with single shortest path routing (i.e. without multi-path routing), and the proposed algorithm. It can be seen that the proposed algorithm can achieve both greater network lifetime and less video distortion for each sensor node, since the other two algorithms that require more encoding rate will cause the increment of encoding, transmission and reception power consumption. VI. CONCLUSION This paper investigates a joint coding/routing optimization of network costs (lifetime) and capacity (distortion) in WVSNs for

8 (a) (b) Fig.6. Impact of on the network lifetime/distortion tradeoff. (c) (d) Fig.4. The convergence behavior of low-level optimization (a) (b) Fig.5. The convergence behavior of high-level optimization correlated sources. Through distributed source coding and network coding, the network employs network combinatorics and information theory over an arbitrary graph structure to investigate its performance limit. The link rate allocation as well as network coding-based multipath routing jointly optimize a balance of the network lifetime and information distortion. The objective function not only keeps a total energy consumption of encoding power, transmission power, and reception power minimized, but ensures the information received by sink nodes to approximately reconstruct the visual field. A two-level optimization, using a fully decentralized algorithm, is decomposed as: a master primal problem, a secondary master dual problem with four cross-layer subproblems: a rate control problem, a channel contention problem, a distortion control problem, and an energy conservation problem. Numerical results validate the proposed algorithm from the convergence and performance optimization. For the future work, we will investigate the condition of dynamic network when sleep schedule or node mobility exists. REFERENCE [1] Z. He and D. Wu, Resource allocation and performance analysis of wireless video sensors, IEEE Transactions on Circuits and Systems for Video Technology, vol. 16, no. 5, pp , May [2] J. Liu, M. Adler, D. Towsley, C. Zhang, On Optimal Communication Cost for Gathering Correlated Data through Wireless Sensor Networks, in Proc. of ACM MobiCom, Sept [3] R. Puri and K. Ramchandran, PRISM: A new robust video coding architecture based on distributed compression principles, in Proc. Allerton Conference on Communication, Control, and Computing, Allerton, IL, Oct Fig.7. Comparison of optimal value with regard to lifetime and video distortion [4] R Cristescu, B Beferull-Lozano, M Vetterli, Networked Slepian-Wolf: theory, algorithms, and scaling laws, IEEE Transactions on Information Theory, Volume 51, Issue 12, Dec. 2005, pp [5] R. Ahlswede, N. Cai, S.-Y. Li, and R. W. Yeung, Network information flow, IEEE Trans. Inf. Theory, vol. 46, no. 4, pp , Jul [6] T. Ho, M. Médard, J. Shi, M. Effros, and D. R. Karger, On randomized network coding, in Proc. 41st Allerton Conf. Communication, Control and Computing, Monticello, IL, Sep [7] T. Ho, M. Médard, M. Effros, and R. Koetter. Network coding for correlated sources, In CISS, [8] Y. Wu, V. Stankovic, Z. Xiong, and S. Kung, On practical design for joint distributed source and network coding, In NETCOD, [9] R. Madan, S. Lall, Distributed algorithms for maximum lifetime routing in wireless sensor networks, IEEE Transactions on Wireless Communications, vol. 5, no. 8, pp , Aug [10] R. Madan, S. Cui, S. Lall, and A. Goldsmith, Cross-layer design for lifetime maximization in interference-limited wireless sensor networks sensor networks, in Proc. of IEEE INFOCOM, pp , [11] Y. He, I. Lee and L. Guan, Distributed Algorithms for Network Lifetime Maximization in Wireless Visual Sensor Networks, submitted to IEEE Trans. Circuits and Systems for Video Technology, [12] Junhua Zhu, Shan Chen, Bensaou, B., Ka-Lok Hung, Tradeoff Between Lifetime and Rate Allocation in Wireless Sensor Networks: A Cross Layer Approach, in Proc. of IEEE INFOCOM, pp , May [13] M. Xiao, M. Medard, T. Aulin, A Binary Coding Approach for Combination Networks and General Erasure Networks, IEEE International Symposium on Information Theory, pp , [14] R. Madan, Z. Q. Luo, S. Lall, A distributed algorithm with linear convergence for maximum lifetime routing in wireless sensor networks, Proceedings of the Allerton Conference on Communication, Control and Computing, [15] W. Heinzelman, A. Chandrakasan, and H. Balakrishnan, An application-specific protocol architecture for wireless microsensor networks, IEEE Transactions on Wireless Communications, vol. 1, no. 4, pp , [16] T. M. Cover and J. A. Thomas, Elements of Information Theory, New York: Wiley, 1991.