Energy-Efficient Node Deployment in Static and Mobile Heterogeneous Multi-Hop Wireless Sensor Networks
aa r X i v : . [ c s . I T ] J a n Energy-Efficient Node Deployment in Staticand Mobile Heterogeneous Multi-Hop WirelessSensor Networks
Saeed Karimi-Bidhendi, Jun Guo, and Hamid Jafarkhani
Abstract
We study a heterogeneous wireless sensor network (WSN) where N heterogeneous access points(APs) gather data from densely deployed sensors and transmit their sensed information to M heteroge-neous fusion centers (FCs) via multi-hop wireless communication. This heterogeneous node deploymentproblem is modeled as an optimization problem with total wireless communication power consumptionof the network as its objective function. We consider both static WSNs, where nodes retain theirdeployed position, and mobile WSNs where nodes can move from their initial deployment to theiroptimal locations. Based on the derived necessary conditions for the optimal node deployment in staticWSNs, we propose an iterative algorithm to deploy nodes. In addition, we study the necessary conditionsof the optimal movement-efficient node deployment in mobile WSNs with constrained movement energy,and present iterative algorithms to find such deployments, accordingly. Simulation results show that ourproposed node deployment algorithms outperform the existing methods in the literature, and achieves alower total wireless communication power in both static and mobile WSNs, on average. Index Terms
Node deployment, heterogeneous multi-hop wireless sensor networks, power optimization, networklifetime.
I. I
NTRODUCTION
Wireless sensor networks (WSNs) consist of small and low-cost sensor devices used to monitorthe environment and transfer the sensed information through wireless channels to dedicated fusion
Authors are with the Center for Pervasive Communications & Computing, University of California, Irvine, Irvine CA, 92697USA (e-mail: { skarimib, guoj4, hamidj } @uci.edu). This work was presented in part in 2020 IEEE International Conference onCommunications [1]. This work was supported in part by the NSF Award CCF-1815339. centers. WSNs can be classified into either homogeneous WSNs [1]–[5], in which sensors sharethe same characteristics such as storage, antennas, sensitivity etc., or heterogeneous WSNs wheresensor nodes have different characteristics [6]–[11]. Based on the network architecture, WSNscan be divided into either hierarchical WSNs, where sensors are often grouped into clusters withsome of them chosen to be cluster heads, or non-hierarchical WSNs where sensors have identicalfunctionality and multi-hop wireless communications is used to maintain the connectivity of thenetwork. Sensor nodes can also be classified as either static [8], [11]–[13], in which each noderemains at its deployed position, or mobile where nodes can move to their optimal locations toimprove the energy efficiency and sensing quality of the WSNs [14]–[19].Energy efficiency is a key determinant in longevity of the WSNs since sensors have limitedenergy resources and it is difficult or infeasible to recharge the batteries of densely deployedsensors. In general, many factors contribute to the energy consumption of the WSNs, e.g. commu-nication energy, movement energy, sensing energy and computation energy [20], [21]. Empiricalmeasurements in many applications have shown that the data processing and computation energyis negligible compared to communication energy [22], [23]. Moreover, the sensing energy forpassive sensors, such as light or thermal sensors, is considerably small. Therefore, wirelesscommunication dominates the energy consumption in static sensors in practice while movementenergy dominates the energy consumption in mobile wireless sensor networks [8], [24], [25].According to the study in [26], for the optimal angular velocity and acceleration, the movementenergy consumption is approximately linear to the distance that the sensor has to travel, and thislinear model is a widely adopted assumption in the literature [27]–[30].Several methods have been proposed in the literature to reduce the energy consumption ofwireless communication in WSNs. Topology control has been adopted in [31], [32] to circumventexcess energy consumption by appropriately switching sensors between awake and asleep states.Energy efficient routing protocols have been established in [5], [33] to find optimal paths to trans-fer data from sensors to fusion centers. Power control protocols reduce the energy consumptionof WSNs by calibrating the transmission power of sensors while a reliable communication ismaintained [34], [35]. Clustering methods [34], [36] iterate among cluster heads to balance theenergy consumption among the sensor nodes. The common assumption of these approachesis that the node deployment is assumed to be known and fixed while a proper deployment cansignificantly affect the energy consumption of the WSNs. Furthermore, the above MAC protocolsrequire a large number of message exchanges since the geometry and energy of the network is needed for the operation [36], [37]. Inspired by dynamics of swarm behavior, a populationbased iterative algorithm called particle swarm optimization (PSO) is proposed in [38] to findoptimal node locations. An iterative algorithm is proposed in [39] to determine the position ofnodes such that the average sensors’ distance to the nearest fusion center is minimized. Givena maximum length on the cluster diameter, a cluster formation algorithm is proposed in [40]to enhance the network lifetime and reduce the average number of hops for data packets toreach fusion centers. An efficient routing scheme is proposed in [41] to minimize the maximumenergy consumed by each fusion center. The optimal node deployment in two-tier WSNs hasbeen studied for heterogeneous networks in [8]; however, the WSN is restricted to a two-tierarchitecture while a multi-hop model can provide more degrees of freedom to optimize the datarouting. The optimal deployment and trajectory of UAVs with a fixed altitude is discussed in[42] to maintain a reliable communication with ground terminal stations.Many methods have been developed for mobile WSNs, where movement energy dominatesthe energy consumption of nodes, to find the optimal deployment given a constraint on availablemovement energy. The Lloyd − α and DEED algorithms proposed in [15] use a movement-dependent penalty term to implement centroidal Voronoi tessellation for sensor deployment. InLloyd − α algorithm, each movement iteration is scaled by a factor of α ∈ [0 , to compensate forlimited movement energy resources. In DEED, the gradient and Hessian matrix of the objectivefunction are used to optimize the sensor movements. Several virtual force based algorithms areproposed in [16]–[18] to determine virtual motion trajectories and the rate of sensor movementsusing a combination of attractive and repulsive forces. A minimum cost maximum weightedflow based algorithm is developed to determine the optimal movement plan of sensors andencourage a minimum number of sensor nodes in each region of a square field [19]; however, theproposed algorithm regards sensor movements as hops between neighboring grid points and lacksa rigorous formulation of movement distance. Similarly, other methods also lack a theoreticalframework for a movement efficient sensor deployment that prolongs the network lifetime, e.g.,the scaling parameter α ∈ [0 , in the Lloyd − α algorithm has to be specified empirically tomeet the movement energy constraints given an initial node deployment. Furthermore, many ofthe existing work explore the one-tier network architecture while a two-tier or multi-hop protocolprovides more flexibility on how sensory data from the physical environment is transferred tothe virtual information world.In this paper, we study the node deployment in heterogeneous multi-hop WSNs consisting of homogeneous densely deployed sensors, heterogeneous APs and heterogeneous FCs, to mini-mize the total wireless communication power consumption with and without movement energyconstraints. The energy efficient node deployment is studied in [8] for heterogeneous WSNs;however, the network is restricted to a two-tiered architecture. In [1], we studied the necessaryconditions for an optimal node deployment in homogeneous multi-hop WSNs; however, thehomogeneous setting in [1] does not address many challenges that is inherent in heterogeneousWSNs, e.g., non-convexity or discontinuity of cells in the optimal partitioning of the sensingenvironment. To the best of our knowledge, the energy efficient node deployment in hetero-geneous multi-hop WSNs is still an open problem. By deriving the necessary conditions ofthe optimal deployments that minimizes the total wireless communication power consumptionof such heterogeneous multi-hop WSNs, we design iterative algorithms to deploy nodes. Inaddition, we study the optimal node deployment in such networks with limited movement energyfor mobile nodes.The rest of this paper is organized as follows: In Section II, we provide the system modeland problem formulation. In Section III, we study the optimal node deployment in static het-erogeneous multi-hop WSNs, and propose an iterative algorithm based on the derived necessaryconditions. The analysis of optimal node deployment with network’s total movement energyconstraint is provided in Section IV. In Section V, we study an energy efficient node deploymentthat guarantees a given network’s lifetime in mobile WSNs. Experimental results are providedin Section VI and Section VII concludes the paper.II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
In this section, we study the system model of heterogeneous multi-hop WSNs, as shownin Fig. 1, consisting of three types of nodes: homogeneous sensors, heterogeneous APs andheterogeneous FCs. Given the target region Ω ∈ R which is a convex polygon including itsinterior, N APs and M FCs are deployed to collect information from densely deployed sensors.Let I A = { , · · · , N } and I F = { N + 1 , · · · , N + M } denote the set of node indices forAPs and FCs, respectively. If n ∈ I A , Node n refers to AP n ; however, when n ∈ I F , Node n refers to FC ( n − N ) . The location of Node n is denoted by p n ⊂ Ω and collectively thenode deployment is denoted by P = ( p , · · · , p N , p N +1 , · · · , p N + M ) . Throughout this paper, weassume that each sensor only sends data to one AP; therefore, for each n ∈ I A , AP n gathers datafrom sensors within the region W n ⊆ Ω , and W = ( W , · · · , W N ) provides a set partitioning of the target region. The density of sensors is denoted via a continuous and differentiable function f : Ω −→ R + . The total amount of data collected from sensors within the region W n in one timeunit is R b R W n f ( ω ) dω , where the bit-rate R b is a constant due to the homogeneity of sensors [2].For each n ∈ I A , the volume and centroid of the region W n is defined as v ( W n ) , R W n f ( ω ) dω and c ( W n ) , R Wn ωf ( ω ) dω R Wn f ( ω ) dω , respectively. The data gathered from each sensor is forwarded to othernodes in the network until it eventually reaches to one or more FCs.Fig. 1: System model.As shown in Fig. 1, the network can be regarded as a directed acyclic graph G ( I A S I F , E ) where APs and FCs are source and sink nodes, respectively, and E is the set of directed edges ( i, j ) such that i ∈ I A and j ∈ I A S I F [43]. Note that any cycle in the network’s graph can beremoved by reducing the flow of data along the cycle without changing the in-flow and out-flowlinks to that cycle. Let F = [ F i,j ] N × ( N + M ) be the flow matrix, where F i,j is the amount of datatransmitted through the link ( i, j ) in one time unit. Since the in-flow to each AP, say i , shouldbe equal to the out-flow, we have P Nj =1 F j,i + R b R W i f ( ω ) dω = P N + Mj =1 F i,j . For i ∈ I A , wedefine F i , P N + Mj =1 F i,j to be the total flow originated from AP i . Let S = [ s i,j ] N × ( N + M ) bethe normalized flow matrix, where s i,j , F i,j P N + Mj =1 F i,j is the ratio of the in-flow data to AP i thatis transmitted to node j . The normalized flow matrix S satisfies the following properties: (a) s i,j ∈ [0 , ; (b) P N + Mj =1 s i,j = 1 , ∀ i ∈ { , · · · , N } ; (c) No cycle: if there exists a path in thenetwork’s graph such as l → l → · · · → l K , i.e., Q Kk =1 s l k − ,l k > , then we have s l K ,l = 0 . For time-invariant routing algorithms, such as Bellman-Ford Algorithm [44], [45], the flows construct a tree-structured graphin which each node has only one successor. Under such circumstances, the normalized flow from Node i to Node j is either or , i.e., s i,j ∈ { , } . However, the time-variant routing algorithms, such as Flow Augmentation Algorithm [43], generatedifferent flows during different time periods. As a result, the overall normalized flow from Node i to Node j can be a realnumber between and , i.e., s i,j ∈ [0 , . In particular, we have s i.i = 0 , ∀ i ∈ { , · · · , N } . Since the flow matrix F can be uniquelydetermined by the set partitioning W and the normalized flow matrix S , in the remaining ofthis paper, we use the notation F ( W , S ) instead of F . The following example describes how tocalculate F ( W , S ) in terms of W and S . Example 1.
We consider a heterogeneous multi-hop WSN with three APs and one FC, i.e. N = 3 and M = 1 , and the bit-rate R b = 20 . For a cell partitioning W with cell volumes v ( W ) = v ( W ) = 0 . , v ( W ) = 0 . , and the normalized flow matrix S = [ s i,j ] N × ( N + M ) withnon-zero entries s , = 0 . , s , = 0 . , s , = 0 . , s , = 0 . and s , = 1 , the correspondingflow network is illustrated in Fig. 1. The amount of data generated from sensors within each cellcan be calculated as: Γ( W ) = R b v ( W ) = 6 , Γ( W ) = R b v ( W ) = 6 , and Γ( W ) = R b v ( W ) = 8 .AP does not receive data from any other AP, and only transmits its collected sensed data; thus, F ( W , S ) = Γ( W ) = 6 . The flows from AP are then F , ( W , S ) = s , × F ( W , S ) = 2 . and F , ( W , S ) = s , × F ( W , S ) = 3 . , respectively. AP ’s flows come from F , ( W , S ) and thedata gathered from the region W . Hence, F ( W , S ) = Γ( W )+ F , ( W , S ) = 8 . . Therefore, theflows from AP are F , ( W , S ) = s , × F ( W , S ) = 2 . and F , ( W , S ) = s , × F ( W , S ) = 6 . .Similarly, for AP , we have F ( W , S ) = Γ( W ) + F , ( W , S ) + F , ( W , S ) = 13 . ; hence, theunique flow from AP is F , ( W , S ) = s , × F ( W , S ) = 13 . .In what follows, we formulate the wireless communication power consumption of the network.Also, we focus on the power consumption of sensors and APs, since FCs are usually suppliedwith reliable energy sources and their power consumption is not the main concern. First, wefocus on the sensor’s power consumption. According to [2], due to the path-loss, the instanttransmission power is equal to the square of the distance between the two nodes multiplied bya constant that depends on the characteristics of both nodes, i.e., η × k p n − ω k for a sensorpositioned at ω that transmits its data to AP n , n ∈ I A . As shown in [21], the parameter η is given by η = P th (4 π ) R b G t G r λ c , where P th is the minimum receiver power threshold for successfulreception, R b is the bit-rate, G t and G r are the antenna gains of the transmitter and receiver,respectively, and λ c is the carrier signal wavelength. In the homogeneous setting, all nodes havethe same characteristics; thus, the parameter η is the same and will not affect the optimization.However, in a heterogeneous multi-hop WSN, AP nodes can have different antenna gains andSNR thresholds; hence, the parameter η will be a function of the node index. Therefore, the sensors’ transmission power consumption can be written as P T S ( P , W ) = N X n =1 Z W n η n k p n − ω k R b f ( ω ) dω. (1)Similarly, the instant transmission power from Node i to Node j can be written as β × k p i − p j k where the parameter β depends on the antenna gain and SNR threshold of Node j and the antennagain of Node i [21]. Therefore, it is the same for the homogeneous setting and will not affectthe optimization. However, in a heterogeneous multi-hop WSN, the heterogeneity of the nodescauses the parameter β to be a function of the node indices. Hence, the average transmissionpower through link ( i, j ) is equal to β i,j k p i − p j k F i,j ( W , S ) , and the APs’ total transmissionpower consumption can be written as P T A ( P , W , S ) = N X i =1 N + M X j =1 β i,j k p i − p j k F i,j ( W , S ) . (2)According to [7], power at the receiver of AP n can be modeled as P Ni =1 ρ n F i,n ( W , S ) + ρ n R b R W n f ( ω ) dω , where ρ n is the power consumption coefficient for receiving data at AP n ,and depends on digital coding, modulation and filtering of the signal before transmission [21].Therefore, the APs’ total receiver power consumption can be written as: P R A ( W , S ) = N X n =1 ρ n " N X i =1 F i,n ( W , S ) + R b Z W n f ( ω ) dω . (3)Thus, the total communication power consumption of the multi-hop WSN can be written as: D ( P , W , S ) = P T S ( P , W ) + λ h P T A ( P , W , S ) + P R A ( W , S ) i , (4)where the Lagrangian multiplier λ ≥ provides a trade-off between the sensor and AP powerconsumption. Our main objective in this paper is to minimize the multi-hop power consumptiondefined in (4) over the node deployment P , cell partitioning W , and the normalized flow matrix S in both static and mobile WSNs with constrained movement energy.III. O PTIMAL N ODE D EPLOYMENT IN S TATIC H ETEROGENEOUS M ULTI -H OP WSN S As shown in (4), the total power consumption depends on three variables P , W and S .Thus, our goal is to find the optimal AP and FC deployments, cell partitioning and normalizedflow matrix, denoted by P ∗ = (cid:0) p ∗ , · · · , p ∗ N , p ∗ N +1 , · · · , p ∗ N + M (cid:1) , W ∗ = ( W ∗ , · · · , W ∗ N ) and S ∗ = (cid:2) s ∗ i,j (cid:3) N × ( N + M ) , respectively, that minimizes the multi-hop power consumption. Note that not onlythe variables P , W and S are interdependent, i.e., the optimal value for each of them dependson the value of the other two variables, but also this optimization problem is NP-hard. Our aimis to design an iterative algorithm that optimizes the value of one variable while the other twovariables are held fixed. For this purpose, first we introduce a few concepts, and then we derivethe necessary conditions for optimal deployment at each step.Without loss of generality, we assume that AP n ’s gathered data goes through K n paths inthe network’s graph before it reaches to one or more fusion centers. We denote these paths by n L ( n ) k ( S ) o k ∈{ , ··· ,K n } , where L ( n ) k ( S ) = l ( n ) k, → l ( n ) k, → · · · → l ( n ) k,J ( n ) k , l ( n ) k, = n , l ( n ) k,i ∈ I A for i ∈ { , · · · , J ( n ) k − } , l ( n ) k,J ( n ) k ∈ I F and J ( n ) k is the number of nodes on the k -th path excludingNode n . The portion of the total flow originated from AP n that goes through the k -th path canthen be calculated as µ ( n ) k ( W , S ) = F n ( W , S ) J ( n ) k Y i =1 s l ( n ) k,i − ,l ( n ) k,i . (5)In particular, we have P K n k =1 µ ( n ) k ( W , S ) = F n ( W , S ) that indicates the data from AP n eventually reaches to one or more FCs. Next, for each link ( i, j ) in the network’s graph, wedefine the energy cost (Watt/bit) to be: e i,j ( P ) , β i,j k p i − p j k + ρ j , if j ∈ I A β i,j k p i − p j k , if j ∈ I F . (6)Hence, we define the path cost corresponding to the k -th path from AP n to FCs as: e ( n ) k ( P , S ) = J ( n ) k X i =1 e l ( n ) k,i − ,l ( n ) k,i ( P ) . (7)Now, AP n ’s power coefficient, denoted by g n ( P , S ) is defined to be the power consumption(Watt/bit) for transmitting bit data from AP n to the FCs, i.e., we have: g n ( P , S ) = P K n k =1 µ ( n ) k ( W , S ) e ( n ) k ( P , S ) F n ( W , S ) (8) = K n X k =1 J ( n ) k Y i =1 s l ( n ) k,i − ,l ( n ) k,i J ( n ) k X j =1 β l ( n ) k,j − ,l ( n ) k,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p l ( n ) k,j − − p l ( n ) k,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + J ( n ) k − X j =1 ρ l ( n ) k,j . (9)Note that the term F n ( W , S ) is canceled in (8), implying that power coefficient g n ( P , S ) is independent of W . Below we provide an example to clarify how to calculate the AP powercoefficients. Example 2.
Consider the WSN described in Example 1, and let P = ((0 , , (0 , , (1 , , (1 , , β i,j = 1 and ρ i = 1 for all i ∈ I A and j ∈ I A S I F . We aim to find AP ’s power coefficient g ( P , S ) . The link energy costs for this network can be calculated as e , ( P ) = e , ( P ) = 2 , e , ( P ) = 3 , and e , ( P ) = e , ( P ) = 1 . Note that AP ’s data goes through the following paths: L (1)1 ( S ) = 1 → → , L (1)2 ( S ) = 1 → → , and L (1)3 ( S ) = 1 → → → . The data rate through theabove paths are, respectively, µ (1)1 ( W , S ) = F ( W , S ) × s , × s , = 0 . F ( W , S ) , µ (1)2 ( W , S ) = F ( W , S ) × s , × s , = 0 . F ( W , S ) , and µ (1)3 ( W , S ) = F ( W , S ) × s , × s , × s , = 0 . F ( W , S ) .Moreover, we can calculate the path costs using (7) as follows: e (1)1 ( P ) = e , ( P ) + e , ( P ) = 3 , e (1)2 ( P ) = e , ( P ) + e , ( P ) = 3 , and e (1)3 ( P ) = e , ( P ) + e , ( P ) + e , ( P ) = 6 . Then, AP ’spower coefficient is g ( P , S ) = 0 . × . × . × . .To derive the necessary condition for an optimal cell partitioning, first, we need to rewrite theobjective function in (4). Lemma 1:
For the AP power coefficient defined in (8), we have: N X n =1 g n ( P , S ) R b Z W n f ( ω ) dω = N X i =1 " N + M X j =1 β i,j k p i − p j k F i,j ( W , S ) + N X j =1 ρ j F i,j ( W , S ) . (10)The proof is provided in Appendix A. Using Lemma 1, the objective function is: D ( P , W , S ) = N X n =1 Z W n (cid:0) η n k p n − ω k R b + λg n ( P , S ) R b + λρ n R b (cid:1) f ( ω ) dω. (11)Now, we study the properties of the optimal cell partitioning. For each n ∈ I A , the Voronoi cell V n for a node deployment P and normalized flow matrix S is defined to be: V n ( P , S ) , (cid:8) ω : η n k p n − ω k + λg n ( P , S )+ λρ n ≤ η k k p k − ω k + λg k ( P , S )+ λρ k , ∀ k = n (cid:9) . (12)Ties are broken in the favor of the smaller index to ensure that each Voronoi cell V n is a Borelset. For brevity, we write V n instead of V n ( P , S ) when it is clear from the context. The collection V ( P , S ) = ( V , V , · · · , V N ) (13)is referred to as the generalized Voronoi diagram [8]. Note that in contrast to the regularVoronoi diagrams, the Voronoi cells defined in (12) can be non-convex, not star-shaped and even disconnected. The following proposition indicates that given a node deployment P andnormalized flow matrix S , the generalized Voronoi diagram provides the optimal cell partitioning. Proposition 1:
For any node deployment P , cell partitioning W and normalized flow matrix S , we have: D ( P , W , S ) ≥ D ( P , V ( P , S ) , S ) . (14)The proof is provided in Appendix B. Now, given the link costs { e i,j ( P ) } s and generated sensingdata rate from each cell partition, the total multi-hop power consumption can be minimized byBellman-Ford Algorithm [44], [45]. For convenience, we show the functionality of Bellman-Ford Algorithm by R ( P , W ) , where P and W are inputs and S is the output, i.e., R ( P , W ) =arg min S h P T A ( P , W , S ) + P R A ( W , S ) i . Since the sensors’ power consumption is independentof S , we have: R ( P , W ) = arg min S P T S ( P , W ) + λ h P T A ( P , W , S ) + P R A ( W , S ) i = arg min S D ( P , W , S ) . (15)Hence, the optimal flow matrix for a given P and W is F ( W , R ( P , W )) . For notational brevity,we define the point z i ( P , W , S ) , or z i for short, to be: z i = η i R b v i c i + λ (cid:16)P N + Mj =1 β i,j F i,j p j + P Nj =1 β j,i F j,i p j (cid:17) η i R b v i + λ (cid:16)P N + Mj =1 β i,j F i,j + P Nj =1 β j,i F j,i (cid:17) , ∀ i ∈ I A (16) z i = P Nj =1 β j,i F j,i p j P Nj =1 β j,i F j,i . ∀ i ∈ I F (17)The following theorem provides the necessary conditions for the optimal deployment. Proposition 2:
The necessary conditions for the optimal deployments in heterogeneous multi-hop WSNs with communication power consumption defined in (4) are p ∗ i = z ∗ i , ∀ i ∈ I A [ I F (18) W ∗ = V ( P ∗ , S ∗ ) , (19) S ∗ = R ( P ∗ , W ∗ ) , (20)where z ∗ i = z i ( P ∗ , W ∗ , S ∗ ) is given by Eqs. (16) and (17).The proof of Proposition 2 is provided in Appendix C.Note that depending on the cell partitioning and normalized flow matrix, there may not be any flow through some links in the network’s graph. Let N Pi ( S ) , { j | F j,i ( W , S ) > } be the setof Node i ’s predecessors, and N Si ( S ) , { j | F i,j ( W , S ) > } be the set of Node i ’s successors.We can then simplify Eq. (18) as: p ∗ i = η i R b v ∗ i c ∗ i + λ P j ∈N Si ( S ∗ ) β i,j F ∗ i,j p ∗ j + P j ∈N Pi ( S ∗ ) β j,i F ∗ j,i p ∗ j ! η i R b v ∗ i + λ P j ∈N Si ( S ∗ ) β i,j F ∗ i,j + P j ∈N Pi ( S ∗ ) β j,i F ∗ j,i ! , ∀ i ∈ I A (21) p ∗ i = P j ∈N Pi ( S ∗ ) β j,i F ∗ j,i p ∗ j P j ∈N Pi ( S ∗ ) β j,i F ∗ j,i , ∀ i ∈ I F . (22)In other words, AP i ’s optimal location is a linear combination of its geometric centroid, prede-cessors, and successors while FC i ’s optimal location is a linear combination of its predecessors.In what follows, first, we quickly review the conventional Lloyd Algorithm [46], then wepropose an algorithm to optimize the communication power consumption defined in Eq. (4)for heterogeneous multi-hop WSNs. Lloyd Algorithm iterates between two steps: (i) Voronoipartitioning and (ii) Moving each node to the geometric centroid of its corresponding Voronoiregion. Although the conventional Lloyd Algorithm can be used for one-tier quantizers or one-tiernode deployment tasks [11], it cannot be applied to WSNs with multi-hop wireless communi-cations. Based on the properties explored in this section, we design a Routing-aware Lloyd(RL) Algorithm, as outlined in Algorithm 1, to optimize the node deployment in heterogeneousmulti-hop WSNs and minimize the objective function in (4). Proposition 3:
RL Algorithm is an iterative improvement algorithm, i.e., the objective functionis non-increasing and the algorithm converges.The proof of Proposition 3 is provided in Appendix D.IV. T HE N ODE D EPLOYMENT WITH A T OTAL E NERGY C ONSTRAINT IN M OBILE
WSN S A. Problem formulation
In Section III, we studied the scenario where nodes are directly placed at the optimal locationscalculated via RL Algorithm. However, here we study mobile heterogeneous multi-hop WSNsin which each node moves from its initial position to its optimal location that minimizes thecommunication power consumption in (4) while the total movement energy consumption of the Algorithm 1:
Routing-aware Lloyd Algorithm
Result:
Optimal node deployment P , cell partitioning W and normalized flow matrix S .Input: Convergence error threshold ǫ ∈ R + ; do – Calculate the objective function D old = D ( P , W , S ) ;1. Update the cell partitioning W according to the Eq. (19);2. Update the normalized flow matrix S using to the Bellman-Ford algorithm;3. Update the node deployment P as follows: p n = η n R b v n c n + λ (cid:16)P N + Mj =1 β n,j F n,j p j + P Nj =1 β j,n F j,n p j (cid:17) η n R b v n + λ (cid:16)P N + Mj =1 β n,j F n,j + P Nj =1 β j,n F j,n (cid:17) , ∀ n ∈ I A p n = P Nj =1 β j,n F j,n p j P Nj =1 β j,n F j,n , ∀ n ∈ I F – Calculate the objective function D new = D ( P , W , S ) ; while D old −D new D old ≥ ǫ ;network is constrained. More precisely, given the linear model for movement energy consumptionin [26], for each n ∈ I A S I F , Node n ’s movement energy can be modeled as: E n ( P ) = ζ n k p n − ˜ p n k , (23)where the moving cost parameter ζ n depends on Node n ’s energy efficiency, p n and ˜ p n are itsdestination and initial locations, respectively. Therefore, the total movement energy consumptionof the network is E ( P ) = N + M X n =1 E n ( P ) = N + M X n =1 ζ n k p n − ˜ p n k . (24)Our main objective in this section is to minimize the multi-hop communication power consump-tion in Eq. (4) while the total movement energy is limited, i.e., the constrained optimizationproblem is defined as minimize P , W , S D ( P , W , S ) , (25)s.t. E ( P ) ≤ γ (26)where γ ≥ is the maximum movement energy consumption of the network. B. The Optimal Node Deployment
Here, we aim to find the optimal node deployment P ∗ , cell partitioning W ∗ and normalizedflow matrix S ∗ that minimizes the total multi-hop communication power consumption whilethe movement energy consumption is constrained. Note that the movement energy in (26)is independent of the cell partitioning and normalized flow matrix; therefore, the generalizedVoronoi diagram and Bellman-Ford Algorithm, represented in Eqs. (13) and (15), respectively,still provide the optimal cell partitioning and normalized flow matrix. Now, we discuss theoptimal node deployment for the constrained optimization problem in Eqs. (25) and (26). Lemma 2:
Let P ∗ , W ∗ and S ∗ be the optimal node deployment, cell partitioning and normal-ized flow matrix for the constrained optimization problem in Eqs. (25) and (26). We have: p ∗ i = δ i ˜ p i + (1 − δ i ) × z ∗ i , ∀ i ∈ I A [ I F (27)where δ i ∈ [0 , and ˜ p i is the initial location of Node i .The proof is provided in Appendix E.Lemma 2 states that the optimal location for Node i is on the line connecting its initial positionto the point z ∗ i = z i ( P ∗ , W ∗ , S ∗ ) . Note that this is in contrast to the optimal node deploymentwithout movement energy constraint in Section III, i.e., p ∗ i = z ∗ i , as shown in Proposition 2. Thedifference is because of the constraint in Eq. (26). Intuitively, for γ = 0 we have δ i = 1 for all i ∈ I A S I F , i.e., each node will remain at its initial position since there is zero total availablemovement energy. However, for sufficiently large enough γ , we have δ i = 0 , i.e., p ∗ i = z ∗ i for all i ∈ I A S I F . In general, nodes can be classified into two groups based on whether they havepositive moving distance or they stand still. Let I d = { n | k p n − ˜ p n k > , ∀ n ∈ I A S I F } and I s = { n | k p n − ˜ p n k = 0 , ∀ n ∈ I A S I F } be the set of dynamic and static nodes, respectively.The following theorem provides the necessary condition for the optimal node deployment inmulti-hop WSNs with total movement energy constraint: Proposition 4:
Let P ∗ , W ∗ and S ∗ be the optimal node deployment, cell partitioning andnormalized flow matrix for the constrained optimization problem in Eqs. (25) and (26). Then: χ ∗ n = χ ∗ m ≥ χ ∗ k , ∀ n, m ∈ I d , k ∈ I s (28) p ∗ n = ˜ p n + Γ ∗ n × − max (cid:0) , P i ∈I d ζ i k Γ ∗ i k − γ (cid:1) k Γ ∗ n k × ψ ∗ n ζ n × P i ∈I d ζ i ψ ∗ i , ∀ n ∈ I d (29) where Γ ∗ n = z ∗ n − ˜ p n and ψ ∗ n is defined to be ψ ∗ n , η n R b v ∗ n + λ hP N + Mk =1 β n,k F ∗ n,k + P Nk =1 β k,n F ∗ k,n i , if n ∈ I A λ P Nk =1 β k,n F ∗ k,n , if n ∈ I F (30)and the moving efficiency χ ∗ n is defined as χ ∗ n = ψ ∗ n k p ∗ n − z ∗ n k ζ n k p ∗ n − z ∗ n k = ψ ∗ n ζ n k p ∗ n − z ∗ n k , ∀ n ∈ I A [ I F (31)to reflect Node n ’s ability to reduce the communication power consumption by movement.The proof is provided in Appendix F. Proposition 4 captures the intuition in Lemma 2 that in anoptimal deployment, Node n is located on the line connecting its initial position ˜ p n to the point z ∗ n , for all n ∈ I A S I F . Furthermore, for a sufficiently large enough available movement energy γ , say γ ≥ P i ∈I d ζ i k Γ ∗ i k , we have p ∗ n = z ∗ n for all n ∈ I d . Based on the necessary conditionsin Proposition 4, we propose a Movement-Efficient Routing-aware Lloyd (MERL) Algorithm,as outlined in Algorithm 2, to optimize the node deployment in heterogeneous multi-hop WSNswith constrained movement energy, and minimize the objective function in Eqs. (25) and (26). Algorithm 2:
Movement-Efficient Routing-aware Lloyd Algorithm
Result:
Optimal node deployment P , cell partitioning W and normalized flow matrix S .Input: Initial node deployment ˜ P , convergence error threshold ǫ ∈ R + ; do – Calculate the objective function D old = D ( P , W , S ) ;1. Update the cell partitioning W according to the Eq. (19);2. Update the normalized flow matrix S using to the Bellman-Ford algorithm;3. Set I d = { , · · · , N + M } and calculate r n , " − max ( , P i ∈I d ζ i k Γ i k− γ ) k Γ n k× ψnζn × P i ∈I d ζ iψi , ∀ n ∈ I d ;4. while ∃ n ∈ I d such that r n ≤ do I d = I d − S r n ≤ n ;4.2. Update { r n } n ∈I d ; end p n = ˜ p n + Γ n × " − max ( , P i ∈I d ζ i k Γ i k− γ ) k Γ n k× ψnζn × P i ∈I d ζ iψi × I d ( n ) , ∀ n ∈ I A S I F ;– Calculate the objective function D new = D ( P , W , S ) ; while D old −D new D old ≥ ǫ ; Proposition 5:
MERL Algorithm is an iterative improvement algorithm, i.e., the objectivefunction is non-increasing and the algorithm converges. The proof of Proposition 5 is provided in Appendix G.V. T HE N ODE D EPLOYMENT WITH A N ETWORK L IFETIME C ONSTRAINT IN M OBILE
WSN S A. Problem formulation
In Section IV, we studied the node deployment with a total movement energy constraint,which can be seen as a resource allocation problem. This is because we can calculate how muchmovement energy each node requires once an optimal deployment is obtained. In this section, wefocus on minimizing the communication power consumption given a constraint on the networklifetime. Let ν n be the residual movement energy on Node n , and α n be the power consumptionfor Node n after relocation. To ensure a network lifetime of T , the following condition ν n − E n ( P ) ≥ α n T, ∀ n ∈ I A [ I F (32)has to be satisfied. Hence, the network lifetime of T can be achieved by setting a maximumindividual movement energy consumption for each node. Here, our main objective is to find theoptimal node deployment for the following constrained optimization problem: minimize P , W , S D ( P , W , S ) (33)s.t. E n ( P ) ≤ γ n , ∀ n ∈ I A [ I F (34)where γ n = ν n − α n T is the maximum individual movement energy consumption of Node n . B. The Optimal Node Deployment
Here, our goal is to find the optimal node deployment P ∗ , cell partitioning W ∗ and normalizedflow matrix S ∗ that minimizes the multi-hop communication power consumption while eachindividual movement energy consumption is constrained. The following theorem provides thenecessary condition for optimal node deployment in the constrained optimization problem inEqs. (33) and (34). Proposition 6:
Let P ∗ , W ∗ and S ∗ be the optimal node deployment, cell partitioning andnormalized flow matrix for the constrained optimization problem in Eqs. (33) and (34). Then, p ∗ n = ˜ p n + Γ ∗ n × min (cid:18) , γ n ζ n k Γ ∗ n k (cid:19) , ∀ n ∈ I A [ I F (35)where Γ ∗ n = z ∗ n − ˜ p n . The proof of Proposition 6 is provided in Appendix H. Based on the optimal condition inProposition 6, we design the Lifetime-Optimized Routing-aware Lloyd (LORL) Algorithm, asoutlined in Algorithm 3, to optimize the node deployment in heterogeneous multi-hop WSNswith network lifetime constraint, and minimize the objective function in Eqs. (33) and (34).
Algorithm 3:
Lifetime-Optimized Routing-aware Lloyd Algorithm
Result:
Optimal node deployment P , cell partitioning W and normalized flow matrix S .Input: Initial node deployment ˜ P , convergence error threshold ǫ ∈ R + ; do – Calculate the objective function D old = D ( P , W , S ) ;1. Update the cell partitioning W according to the Eq. (19);2. Update the normalized flow matrix S using to the Bellman-Ford algorithm;3. p n = ˜ p n + Γ n × min (cid:16) , γ n ζ n k Γ n k (cid:17) , ∀ n ∈ I A S I F ;– Calculate the objective function D new = D ( P , W , S ) ; while D old −D new D old ≥ ǫ ; Proposition 7:
LORL Algorithm is an iterative improvement algorithm, i.e., the objectivefunction is non-increasing and the algorithm converges.The proof of Proposition 7 is provided in Appendix I.VI. E
XPERIMENTS
Simulations are carried out for a heterogeneous wireless sensor network consisting of APsand FCs. We consider a square field of size km × km, i.e., Ω = [0 , . Simulationsare performed for two different sensor density functions, i.e., a uniform distribution f ( ω ) = R Ω dω = 10 − , and a mixture of Gaussian where sensors are distributed according to: f ( ω ) = 12 × N e +33 e +3 , . e +6 00 1 . e +6 + 14 × N e +37 e +3 , e +6 00 2 e +6 + 14 × N . e +32 . e +3 , e +6 00 1 e +6 . (36)All homogeneous densely deployed sensors share the transmitter antenna gain of G t sensor = 1 . Weconsider a radio bit-rate of R b = 1 Mbps, and assume that the wavelength of the carrier signal is λ c = 0 . m. In order for APs and FCs to receive the signal without error, the received power ateach node n ∈ I A S I F should be greater than some threshold P th n . Moreover, the transceiver electronics in each AP n consumes ρ n Joules per bit for digital coding, modulation and filteringbefore signal transmission. Table I summarizes the values of P th n and ρ n for all nodes [21].TABLE I: Simulation parameters minimum received power (nW) electronics energy dissipation (nJ/bit) P th : P th : P th P th : ρ : ρ : ρ :
10 6 6 10 40 50 60
For each AP n , we denote its transmitter antenna gain via G t n . In addition, for each node n ∈ I A S I F , let G r n be its receiver antenna gain. Table II summarizes the values of thetransmitter and receiver antenna gains for all nodes [21].TABLE II: Transmitter and receiver antenna gains transmitter antenna gain receiver antenna gain G t : , : G t : , : G r : , : , : , : , : G r : , : , : , : , Note that parameters η i and β i,j , for all i ∈ I A and j ∈ I A S I F , can be calculated usingthe explained experimental setup. For example, we have η = P th × (4 π ) R b G t sensor G r λ c = − × (4 π ) × × × (0 . =8 . pJ/bit/m and β , = P th × (4 π ) R b G t G r λ c = × − × (4 π ) × × × (0 . = 2 . pJ/bit/m . For performanceevaluation, initial AP and FC deployments are generated randomly on Ω , i.e., the locationof each node is generated according to a uniform distribution on Ω . The maximum number ofiterations for all algorithms is set to and the Lagrangian multiplier is set to λ = 0 . . A. Static Heterogeneous Multi-Hop WSNs
We compare the total weighted communication power consumption of our proposed RLAlgorithm with Cluster-Formation Algorithm [40], Global Algorithm [39], HTTL Algorithm [8],PSO Algorithm [38], and SHMS Algorithm [41]. To reduce the number of hops that data packetshave to travel to reach the fusion centers, the Cluster-Formation algorithm employs a graphtheoretic approach to optimize both the number of clusters and their corresponding diameters.To reduce the communication distance between the nodes, the Global algorithm deploys nodessuch that the average Euclidean distance between access points and their corresponding fusioncenters is minimized. For a two-tier hierarchy of APs and FCs, the HTTL algorithm iterativelyupdates the node deployment, cell partitioning, and connections between APs and FCs while the flow of data from each sensor to its corresponding FC is mediated by exactly one accesspoint. PSO is a population based iterative algorithm for finding the optimal node deployment andminimizing the non-linear objective function. For a given node deployment, the SHMS algorithmdetermines the connections between APs and FCs such that the maximum energy consumed byeach node is minimized.The weighted power consumption of Cluster-Formation, Global, HTTL, PSO, SHMS andRL algorithms for the uniform sensor density function are summarized in Table III. The RLalgorithm outperforms other algorithms, and achieves a lower weighted communication powerconsumption. Note that although the HTTL algorithm proposed in [8] deploys nodes based on thenecessary conditions of optimality, the network architecture is restricted to a two-tier hierarchywhile the RL algorithm simultaneously optimizes over the node deployment and data routing.As a result, the node deployment based on the RL algorithm results in a WSN that saves about of the energy consumed by the node deployment based on HTTL Algorithm.TABLE III: Weighted power comparison for the uniform sensor density function Cluster-Formation Global HTTL PSO SHMS RL .
49 14 .
98 12 .
80 19 .
98 22 . . Table IV summarizes the weighted communication power consumption of Cluster-Formation,Global, HTTL, PSO, SHMS and RL algorithms for the mixture of Gaussian sensor densityfunction in Eq. (36). The RL algorithm results in a power consumption of . Watts, andoutperforms other methods. Furthermore, the RL algorithm leads to a network architecture thatexhaust its available communication energy in a time period that is longer by about ofthat of HTTL Algorithm. Figure 2 shows the optimal node deployment for different algorithmswhere APs and FCs are denoted by red squares and black circles, respectively.TABLE IV: Weighted power comparison for the mixture of Gaussian sensor density function
Cluster-Formation Global HTTL PSO SHMS RL .
07 6 .
81 6 .
23 9 .
97 16 . . B. Mobile Heterogeneous Multi-Hop WSNs with a Total Movement Energy Constraint
The underlying assumption in all deployment strategies studied in Section VI-A is that theoptimal node locations are calculated offline, then each node is placed at its corresponding (a) (b) (c)(d) (e) (f) Fig. 2:
Node deployment for different algorithms and the mixture of Gaussian sensor density function. (a) Cluster-Formation (b) Global (c) HTTL (d) PSO (e) SHMS (f) RL. position. However, in many applications, e.g. when the target region is a hostile environment,static deployment is not feasible. Instead, nodes are initially deployed in the target region, e.g.by airdropping them using small drones or manual placement in an accessible sub-region of thefield, then each node moves to its optimal location based on the initial deployment and availablemovement energy. When the total available movement energy is constrained, the optimizationproblem is translated into a resource allocation problem where the optimal energy supply for eachnode is determined such that the resulting total communication power consumption after optimaldeployment is minimized. The performance evaluation under this scenario is the focus of thissection. In Section VI-C, we study the performance evaluation when the available movementenergy for each node is predetermined, and the optimization problem is translated to that ofenhancing the network lifetime.The same experimental setup described at the beginning of Section VI and in Tables I and IIis used for the simulations. Furthermore, Table V provides the moving cost parameters ζ n for each node n ∈ I A S I F [47], [48]. We consider a total available movement energy of γ = 40000 Joules for the constrained objective function in Eqs. (25) and (26).TABLE V: Moving cost parameters (J/m) ζ : ζ : ζ : ζ ζ ζ We compare the total weighted communication power consumption of our proposed MERLAlgorithm with Lloyd- α Algorithm [15], OMF Algorithm [19], and VFA Algorithm [16]. TheLloyd- α algorithm applies a penalty term to the Lloyd algorithm to reduce the movement stepsand save traveling energy while guaranteeing the convergence property. The OMF algorithmoptimizes the movement plan for nodes such that each region in the network has a minimumnumber of nodes to relay the data to fusion centers while the sum of nodes’ traveling distancesis minimized. The VFA algorithm uses attractive and repulsive virtual forces on nodes such thatnot only every two nodes in the final deployment maintain a minimum distance from each other,but also the communication distances are minimized by avoiding nodes to be located very farfrom each other. For a fair comparison, the same initial deployment is used for all algorithms.The weighted communication power consumption of Lloyd- α , OMF, VFA, and MERL al-gorithms for the uniform sensor density function are summarized in Table VI. All algorithmsexhausted the available movement energy γ to move the AP and FC nodes from their initialdeployment to their designated optimal locations. The MERL algorithm leads to a deploymentthat consumes communication energy in a rate that is almost half of other algorithms. Thesuperior performance of the MERL algorithm is due to the optimal energy allocation amongnodes, as it is implicit in Eq. (29). Note that if the total movement energy γ is large enough, e.g. γ ≥ P N + Mi =1 ζ i k ˜ p i − z ∗ i k , then the performance of the MERL algorithm will converge to that ofthe RL algorithm. However, since the value of γ in our experiments is not large enough, nodeswill run out of their allocated movement energy, and MERL algorithm leads to a communicationpower consumption that is larger than that of the RL algorithm in Section VI-A.TABLE VI: Weighted power comparison for the uniform sensor density function Lloyd- α OMF VFA MERL .
12 27 .
35 27 . . Table VII also summarizes the weighted communication power consumption of Lloyd- α , OMF, VFA, and MERL algorithms for the mixture of Gaussian sensor density function in Eq. (36). TheMERL algorithm significantly outperforms other methods and leads to a communication powerconsumption that is less than half of what other algorithms achieve. This is because the MERLalgorithm can optimally adapt to any underlying sensor density function f ( ω ) and deploy nodesaccordingly, as we studied in Section IV.TABLE VII: Weighted power comparison for the mixture of Gaussian sensor density function Lloyd- α OMF VFA MERL .
38 17 .
29 18 . . Figure 3 shows the final deployment for different algorithms where APs and FCs are denotedby red squares and black circles, respectively. (a) (b) (c) (d)
Fig. 3:
Node deployment for different algorithms and the mixture of Gaussian sensor density function. (a) Lloyd- α (b) OMF (c) VFA (d) MERL. C. Mobile Heterogeneous Multi-Hop WSNs with a Network Lifetime Constraint
While in Section VI-B we studied the performance evaluation of mobile WSNs under a totalmovement energy constraint, here we focus on enhancing the network lifetime, which necessitatesnodes to have individual movement energy constraints, as formulated in Eqs. (33) and (34).We use the same experimental setup and node characterization as described at the beginning ofSection VI and in Tables I, II and V for performance evaluation. In addition, Table VIII providesthe maximum individual movement energy consumption γ n for all nodes n ∈ I A S I F .TABLE VIII: Movement energy constraints (J) γ : γ : γ : γ γ γ
800 1100 1400 2000 2400 2600 We compare the weighted communication power consumption of our proposed LORL Al-gorithm with those of Lloyd- α Algorithm, OMF Algorithm, and VFA Algorithm described inSection VI-B. For a fair comparison, the same initial deployment as in Section VI-B is used forall algorithms.The weighted communication power consumption of Lloyd- α , OMF, VFA, and LORL algo-rithms for the uniform sensor density function are provided in Table IX. The LORL algorithmoutperforms other methods, and achieves a significantly lower power consumption. For instance,the LORL algorithm leads to a node deployment in which the network consumes its residualenergy with a rate that is less than of that of the VFA algorithm. This in turn prolongs thenetwork lifetime, which is a prominent factor in wireless sensor networks.TABLE IX: Weighted power comparison for the uniform sensor density function Lloyd- α OMF VFA LORL .
64 30 .
12 25 . . Table X also summarizes the weighted power consumption of different algorithms for themixture of Gaussian sensor density function given in Eq. (36). The LORL algorithm achievesa power consumption of . Watts and outperforms other methods. Figure 4 shows the finalnode deployment for different algorithms.TABLE X: Weighted power comparison for the mixture of Gaussian sensor density function
Lloyd- α OMF VFA LORL .
24 20 .
12 14 . . The sum of individual movement energies in Table VIII, i.e. P N + Mi =1 γ i , is equal to the value of γ in Section VI-B. In other words, Table VIII represents one exemplary distribution of the totalmovement energy γ among the AP and FC nodes; however, it is different from the optimal energyallocation provided by the MERL algorithm in Section VI-B. The results in Tables VI, VII, IXand X verify that the MERL algorithm achieves a lower total power consumption compared tothe LORL algorithm although it does not guarantee any individual power constraint.VII. C ONCLUSION
In this work, a heterogeneous multi-hop wireless sensor network is discussed where data iscollected from densely deployed sensors and transferred to heterogeneous fusion centers using (a) (b) (c) (d) Fig. 4:
Node deployment for different algorithms and the mixture of Gaussian sensor density function. (a) Lloyd- α (b) OMF (c) VFA (d) LORL. heterogeneous access points as relay nodes. We modeled the minimum communication powerconsumption of such networks as an optimization problem, and studied the necessary conditionsof optimal deployment under both static and mobile network settings. A novel generalizedVoronoi diagram is proposed to provide the best cell partition for the heterogeneous multi-hopnetwork. When manual deployment is feasible, the necessary conditions of optimal deploymentare explored under the static network setup, and accordingly a Routing-aware Lloyd algorithm isproposed to deploy nodes. However, when static placement is not doable, the necessary conditionsof the optimal deployment are studied under a mobile network setting where nodes move fromtheir initial locations to their optimal positions. We consider both total and individual movementenergy constraints and formulate them as resource allocation and lifetime optimizations, re-spectively. Based on the derived necessary conditions, we propose Movement-Efficient Routing-aware Lloyd and Lifetime-Optimized Routing-aware Lloyd algorithms to deploy nodes undertotal and individual energy constraints, respectively. Simulation results show that our proposedRL, MERL, and LORL algorithms significantly save communication power in such networksand provide superior results compared to other methods in the literature.A PPENDIX
AProof of Lemma 1: The AP power coefficient g n ( P , S ) defined in Eq. (8) is the powerconsumption for transmitting bit data from AP n to the FCs. This includes both the transmissionpower at each node, including AP n , on the paths connecting AP n to the FCs, and thereceiver power at each node, excluding AP n , on the paths connecting AP n to the FCs. Since R b R W n f ( ω ) dω is the total amount of data collected by AP n from sensors within the region W n in a unit time, the term g n ( P , S ) R b R W n f ( ω ) dω is the required communication power for transmitting the sensory data collected within the region W n from AP n to the FCs. Hence,the left-hand-side of Eq. (10) is the required communication power for transmitting the sensorydata collected within the target region from APs to FCs. This can be decomposed into the APs’total transmission power in addition to the required receiver power for the data to reach FCsfrom AP nodes. This proves Eq. (10) since the right-hand-side of Eq. (10) can be rewritten as P T A + P Ni =1 P Nj =1 ρ j F i,j ( W , S ) , i.e. the sum of APs’ total transmission power and the receiverpower for all links ( i, j ) connecting AP i and AP j . (cid:4) A PPENDIX
BProof of Proposition 1: Using Eq. (11), we have: D ( P , W , S ) = N X n =1 Z W n (cid:0) η n k p n − ω k R b + λg n ( P , S ) R b + λρ n R b (cid:1) f ( ω ) dω ≥ N X n =1 Z W n min j (cid:0) η j k p j − ω k R b + λg j ( P , S ) R b + λρ j R b (cid:1) f ( ω ) dω = Z Ω min j (cid:0) η j k p j − ω k R b + λg j ( P , S ) R b + λρ j R b (cid:1) f ( ω ) dω = N X n =1 Z V n min j (cid:0) η j k p j − ω k R b + λg j ( P , S ) R b + λρ j R b (cid:1) f ( ω ) dω = N X n =1 Z V n (cid:0) η n k p n − ω k R b + λg n ( P , S ) R b + λρ n R b (cid:1) f ( ω ) dω = D ( P , V ( P , S ) , S ) . (37)Hence, the generalized Voronoi diagram provides the optimal cell partitioning for any given nodedeployment P and normalized flow matrix S . (cid:4) A PPENDIX
CProof of Proposition 2: Eq. (19) is a direct implication of Proposition 1. Eq. (20) is directlyfollowed from Eq. (15). Here, we prove Eq. (18) for the optimal locations of APs and FCs. First,we study the shape of the Voronoi regions in (12). Let B ( c, r ) = { ω |k ω − c k ≤ r } be a diskcentered at c with radius r in two-dimensional space. In particular, B ( c, r ) = ∅ when r < . Let HS ( a, b ) = { ω | a · ω + b ≤ } be a half space, where a ∈ R is a vector and b ∈ R is aconstant. For i, j ∈ I A , we define V ij ( P , S ) , { ω | η i k p i − ω k + λg i ( P , S ) + λρ i ≤ η j k p j − ω k + λg j ( P , S ) + λρ j } (38)to be the pairwise Voronoi region of AP i where only APs i and j are considered. Then, AP i ’sVoronoi region can be represented as V i ( P , S ) = hT j = i V ij ( P , S ) i T Ω . By expanding (38) andstraightforward algebraic calculations, the pairwise Voronoi region V ij is derived as: V ij = Ω ∩ HS ( a ij , b ij ) , η i = η j B ( c ij , r ij ) , η i > η j , L ij ≥ ∅ , η i > η j , L ij < B c ( c ij , r ij ) , η i < η j , L ij ≥ R , η i < η j , L ij < , (39)where a ij = η j p j − η i p i , b ij = ( η i k p i k − η j k p j k + λg i ( P , S )+ λρ i − λg j ( P , S ) − λρ j ) , c ij = η i p i − η j p j η i − η j , L ij = η i η j k p i − p j k ( η i − η j ) − λ × g i ( P , S )+ ρ i − g j ( P , S ) − ρ j ( η i − η j ) , r ij = p max ( L ij , , and B c ( c ij , r ij ) is the complementaryof B ( c ij , r ij ) . Note that for two distinct indices such as i, j ∈ I A , if η i > η j and L ij < , thentwo regions Ω ∩ B ( c ij , r ij ) and ∅ differ only in the point c ij . Similarly, for η i < η j and L ij < ,two regions Ω ∩ B c ( c ij , r ij ) and Ω differ only in the point c ij . If we define: V k = " \ i : η k >η i B ( c ki , r ki ) i : η k = η i HS ( a ki , b ki ) i : η k <η i B c ( c ki , r ki ) Ω , (40)then two regions V k and V k differ only in finite number of points. As a result, integrals over both V k and V k have the same value since the density function f is continuous and differentiable, andremoving finite number of points from the integral region does not change the integral value.Note that if V k is empty, the Proposition 1 in [11] holds since the integral over an empty regionis zero. If V k is not empty, the same arguments as in Appendix A of [11] can be replicated since V k in (40) is similar to (31) in [11].Using parallel axis theorem [49], the heterogeneous multi-hop communication power con- sumption can be written as: D ( P , W , S ) = N X n =1 Z W n η n k c n − ω k R b f ( ω ) dω + N X n =1 η n k p n − c n k R b v n + λ N X i =1 N + M X j =1 β i,j k p i − p j k F i,j ( W , S ) + λ N X n =1 ρ n " N X i =1 F i,n ( W , S ) + R b Z W n f ( ω ) dω , (41)where v n = v ( W n ) and c n are the volume and centroid of the region W n , respectively. UsingProposition 1 in [11], since the optimal deployment P ∗ should have a zero gradient, we take thepartial derivatives of (41) with respect to node locations. For each i ∈ I A , we have ∂ D ∂p ∗ i = 2 η i ( p ∗ i − c ∗ i ) R b v ∗ i + 2 λ N + M X j =1 β i,j ( p ∗ i − p ∗ j ) F ∗ i,j + 2 λ N X j =1 β j,i ( p ∗ i − p ∗ j ) F ∗ j,i = 0 , (42)and for each i ∈ I F , we have ∂ D ∂p ∗ i = 2 λ N X j =1 β j,i ( p ∗ i − p ∗ j ) F ∗ j,i = 0 . (43)By solving Eqs. (42) and (43), we obtain Eq. (18) and the proof is complete. (cid:4) A PPENDIX
DProof of Proposition 3: Note that RL Algorithm iterates between three steps. In what follows,we show that none of these steps will increase the objective function D ( P , W , S ) . For a fixednode deployment P and normalized flow matrix S , the cell partitioning W is updated accordingto Eq. (19) which was shown to be optimal for a given P and S in Proposition 1. Therefore,the first step of RL Algorithm does not increase the objective function. Next, since R ( P , W ) isthe optimal normalized flow matrix for a given node deployment P and cell partitioning W , thesecond step of RL Algorithm does not increase the objective function either. Finally, note thatwhen W , S and { p j } j = i are fixed, the objective function D ( P , W , S ) in Eq. (4) is a convexfunction of the node position p i ; hence, by solving the zero-gradient equations and updating thenode locations according to the Eq. (18), the objective function does not increase. Therefore, theobjective function of RL Algorithm is nonincreasing. In addition, the objective function is lowerbounded by , i.e., D ( P , W , S ) ≥ . As a result, RL Algorithm is an iterative improvementalgorithm and it converges. (cid:4) A PPENDIX
EProof of Lemma 2: Before going through the proof, we state the following lemma:
Lemma 3:
Given a set of points q i ∈ R and non-negative scalar weights a i for i ∈ { , · · · , K } ,and a scalar m , the geometric locus of the point p ∈ R such that the equality K X i =1 a i k p − q i k = m (44)holds, is either an empty set, a single point, or a circle centered at the point c = P Ki =1 a i q i P Ki =1 a i .Proof: Let p = ( p x , p y ) and q i = ( q i,x , q i,y ) . Then, we can rewrite Eq. (44) as K X i =1 a i ! (cid:0) p x + p y (cid:1) − K X i =1 a i q i,x ! p x − K X i =1 a i q i,y ! p y = m − K X i =1 a i k q i k . (45)By manipulating both sides, we can rewrite Eq. (45) as follows: p x − K P i =1 a i q i,xK P i =1 a i + p y − K P i =1 a i q i,yK P i =1 a i = m − K P i =1 a i k q i k K P i =1 a i + (cid:18) K P i =1 a i q i,x (cid:19) + (cid:18) K P i =1 a i q i,y (cid:19) (cid:18) K P i =1 a i (cid:19) . (46)Hence, the geometric locus of the point p = ( p x , p y ) is an empty set or a single point if the right-hand-side of Eq. (46) is negative or zero, respectively; otherwise, the geometric locus is a circlecentered at the point c = P Ki =1 a i q i P Ki =1 a i with the radius r = s m − P Ki =1 a i k q i k P Ki =1 a i + ( P Ki =1 a i q i,x ) + ( P Ki =1 a i q i,y ) ( P Ki =1 a i ) ,and Lemma 3 is proved. Corollary 1:
If the geometric locus in Lemma 3 is a circle centered at c with radius r , thenfor any point p within this circle we have P Ki =1 a i k p − q i k < m , i.e. moving the point p insidethis circle reduces the weighted squared sum in Eq. (44).Now, assume that there exists at least one node, say n , for which Eq. (27) in Lemma 2 doesnot hold for an optimal node deployment P ∗ , cell partitioning W ∗ and normalized flow matrix S ∗ , i.e. p ∗ n does not lie on the segment z ∗ n ˜ p n . We aim to find another deployment such as P ′ , W ′ and S ′ so that E ( P ′ ) ≤ γ and D ( P ′ , W ′ , S ′ ) < D ( P ∗ , W ∗ , S ∗ ) ; hence, contradicting theoptimality assumption of P ∗ , W ∗ and S ∗ , and concluding that Eq. (27) holds for all nodes.For this purpose, let W ′ = W ∗ , S ′ = S ∗ and p ′ i = p ∗ i for all i ∈ I A S I F \{ n } . We aim todetermine the node location p ′ n accordingly. Using the parallel axis theorem [49], we can rewrite D ( P ∗ , W ∗ , S ∗ ) as: D ( P ∗ , W ∗ , S ∗ ) = N X i =1 Z W ∗ i η i k c ∗ i − ω k R b f ( ω ) dω + N X i =1 η i R b v ∗ i k p ∗ i − c ∗ i k + λ N X i =1 N + M X j =1 β i,j k p ∗ i − p ∗ j k F i,j ( W ∗ , S ∗ ) + λ P R A ( W ∗ , S ∗ ) , (47)where v ∗ i and c ∗ i are the volume and centroid of the region W ∗ i , respectively. In what follows,we assume that n ∈ I A , i.e. node n is an AP. Similar proof can be carried out for n ∈ I F . Notethat Eq. (47) can be split as D ( P ∗ , W ∗ , S ∗ ) = D ( P ∗ , W ∗ , S ∗ ) + D ( P ∗ , W ∗ , S ∗ ) , where D ( P ∗ , W ∗ , S ∗ ) = η n R b v ∗ n k p ∗ n − c ∗ n k + N + M X j =1 λβ n,j F ∗ n,j k p ∗ n − p ∗ j k + N X j =1 λβ j,n F ∗ j,n k p ∗ n − p ∗ j k , (48)i.e. D includes those terms in Eq. (47) that involve p ∗ n . In particular, regardless of the node n ’sposition, we have D ( P ∗ , W ∗ , S ∗ ) = D ( P ′ , W ′ , S ′ ) . According to Lemma 3, the geometriclocus of points such as p ∗ n for which the value of D ( P ∗ , W ∗ , S ∗ ) in Eq. (48) remains thesame is a circle Φ ∗ n centered at the point z ∗ n = z n ( P ∗ , W ∗ , S ∗ ) defined in Eq. (16), with radius r ∗ n = k z ∗ n − p ∗ n k . Note that if k z ∗ n − ˜ p n k < k z ∗ n − p ∗ n k , then setting p ′ n = ˜ p n not only leads to themovement energy E ( P ′ ) < E ( P ∗ ) , but also results in D ( P ′ , W ′ , S ′ ) < D ( P ∗ , W ∗ , S ∗ ) since p ′ n lies inside Φ ∗ n . Therefore, we have D ( P ′ , W ′ , S ′ ) < D ( P ∗ , W ∗ , S ∗ ) which is in contradictionwith the optimality of P ∗ , W ∗ and S ∗ ; hence, we have k z ∗ n − ˜ p n k ≥ k z ∗ n − p ∗ n k . Let ˆ p n be theintersection point of the circle Φ ∗ n and segment z ∗ n ˜ p n . Since k ˜ p n − ˆ p n k < k ˜ p n − p ∗ n k , there existsan ǫ n ∈ R + such that k ˜ p n − ˆ p n k + ǫ n < k ˜ p n − p ∗ n k . If p ′ n = ˆ p n + ǫ n × z ∗ n − ˆ p n k z ∗ n − ˆ p n k , then not only we have E ( P ′ ) < E ( P ∗ ) since E ( P ∗ ) − E ( P ′ ) > ζ n ǫ n > , but also D ( P ′ , W ′ , S ′ ) < D ( P ∗ , W ∗ , S ∗ ) since p ′ n lies inside the circle Φ ∗ n . Therefore, we have D ( P ′ , W ′ , S ′ ) < D ( P ∗ , W ∗ , S ∗ ) whichcontradicts the optimality of P ∗ , W ∗ and S ∗ and concludes the proof. (cid:4) A PPENDIX
FProof of Proposition 4: If p ∗ i = z ∗ i for all i ∈ I d , then Eq. (26) implies that E ( P ∗ ) = P i ∈I d ζ i k Γ ∗ i k ≤ γ ; hence, Eq. (29) reduces to the trivial statement p ∗ n = ˜ p n + Γ ∗ n and the proofis complete. Therefore, we assume that there exists at least one node, say n , for which p ∗ n = z ∗ n .Note that if any residual movement energy is left in the optimal deployment, i.e. E ( P ∗ ) < γ ,then there exists an ǫ ∈ R + such that E ( P ∗ ) + ǫ < γ and p n = p ∗ n + ǫ × z ∗ n − p ∗ n k z ∗ n − p ∗ n k lies inside thecircle centered at z ∗ n and radius k z ∗ n − p ∗ n k . Then, according to Lemma 3 and Corollary 1, by fixing the cell partitioning, normalized flow matrix and the location of all nodes except Node n ,and placing Node n at p n we can achieve a lower total multi-hop communication power withoutexhausting the available movement energy, which contradicts the optimality of P ∗ , W ∗ and S ∗ .Therefore, p ∗ n = z ∗ n implies that E ( P ∗ ) = γ . Now, given the optimal node deployment P ∗ , W ∗ and S ∗ , we construct the node deployment P ′ , W ′ and S ′ as follows. Let W ′ = W ∗ , S ′ = S ∗ and p ′ i = p ∗ i for all i ∈ I A S I F \{ m, n } . Let ǫ m , ǫ n ∈ R + be small values and define p ′ m = p ∗ m − ǫ m × z ∗ m − ˜ p m k z ∗ m − ˜ p m k , p ′ n = p ∗ n + ǫ n × z ∗ n − ˜ p n k z ∗ n − ˜ p n k . (49)To satisfy the equality E ( P ′ ) = γ , we have ζ n ǫ n = ζ m ǫ m . Now, we calculate the change in themulti-hop communication power, i.e. D ( P ′ , W ′ , S ′ ) − D ( P ∗ , W ∗ , S ∗ ) . Assume that Node m isfixed at p ∗ m and we move Node n from p ∗ n to p ′ n . Note that this movement only changes the term D defined in Eq. (48); thus, according to Lemma 3 and Eq. (46), this change is proportionalto the difference between the squared radii, i.e. ∆ = (cid:2) k p ′ n − z ∗ n k − k p ∗ n − z ∗ n k (cid:3) × ψ ∗ n , (50)where ψ ∗ n is defined in Eq. (30). Now, with Node n placed at p ′ n , we move Node m from p ∗ m to p ′ m . Similar to the above argument, the term ∆ defined as ∆ = (cid:2) k p ′ m − z ∗ m k − k p ∗ m − z ∗ m k (cid:3) × ψ ∗ m (51)captures the change in D with the assumption that Node n was located at p ∗ n . Now, we take intoaccount that Node n was located at p ′ n instead of p ∗ n during Node m ’s movement. ∆ = λβ n,m F ∗ n,m × (cid:2)(cid:0) k p ′ n − p ′ m k − k p ′ n − p ∗ m k (cid:1) − (cid:0) k p ∗ n − p ′ m k − k p ∗ n − p ∗ m k (cid:1)(cid:3) (52) = λβ n,m F ∗ n,m × (cid:2) (cid:0) k p ′ n − p ∗ m k + ǫ m − ǫ m k p ′ n − p ∗ m k cos ∡ p ′ n p ∗ m p ′ m − k p ′ n − p ∗ m k (cid:1) − (cid:0) k p ∗ n − p ′ m k − k p ∗ n − p ′ m k − ǫ m − ǫ m k p ∗ n − p ′ m k cos ∡ p ∗ n p ′ m ˜ p m (cid:1) (cid:3) (53) = λβ n,m F ∗ n,m × (cid:2) ǫ m − ǫ m ( k p ′ n − p ∗ m k cos ∡ p ′ n p ∗ m p ′ m − k p ∗ n − p ′ m k cos ∡ p ∗ n p ′ m ˜ p m ) (cid:3) (54) = λβ n,m F ∗ n,m × (cid:2) ǫ m − ǫ m ( ǫ m − ǫ n cos θ ) (cid:3) (55) = λβ n,m F ∗ n,m × (cid:20) ζ m ζ n ǫ m cos θ (cid:21) , (56) where s and θ = ∡ z ∗ n sz ∗ m are the intersection point and the angle between the lines z ∗ n ˜ p n and z ∗ m ˜ p m , respectively. Note that in Eq. (52), without any loss of generality, we have assumed thatthe direction of the flow of data, if any, is from Node n to Node m . Moreover, Eq. (53) followsfrom the law of cosines and Eq. (56) follows from the equation ζ n ǫ n = ζ m ǫ m . Hence, we have: D ( P ′ , W ′ , S ′ ) − D ( P ∗ , W ∗ , S ∗ ) = ∆ + ∆ + ∆ (57) = (cid:20) ζ m ζ n ǫ m − ζ m ζ n ǫ m k p ∗ n − z ∗ n k (cid:21) × ψ ∗ n + (cid:2) ǫ m + 2 ǫ m k p ∗ m − z ∗ m k (cid:3) × ψ ∗ m + 2 λβ n,m F ∗ n,m ζ m ζ n ǫ m cos θ. Due to the optimality of P ∗ , W ∗ and S ∗ , Eq. (57) should be non-negative, or equivalently: ǫ m (cid:18) ζ m ζ n ψ ∗ n + ψ ∗ m + 2 λβ n,m F ∗ n,m ζ m ζ n cos θ (cid:19) ≥ (cid:18) ζ m ζ n ψ ∗ n k p ∗ n − z ∗ n k − ψ ∗ m k p ∗ m − z ∗ m k (cid:19) . (58)According to Eq. (30), the term λβ n,m F ∗ n,m is included in both ψ ∗ n and ψ ∗ m , i.e. ψ ∗ n ≥ λβ n,m F ∗ n,m and ψ ∗ m ≥ λβ n,m F ∗ n,m ; therefore, we have: ζ m ζ n ψ ∗ n + ψ ∗ m + 2 λβ n,m F ∗ n,m ζ m ζ n cos θ ≥ ζ m ζ n λβ n,m F ∗ n,m + λβ n,m F ∗ n,m + 2 λβ n,m F ∗ n,m ζ m ζ n cos θ (59) ≥ λβ n,m F ∗ n,m (cid:18) ζ m ζ n − (cid:19) ≥ , (60)thus, the term inside the parentheses on the left hand side of Eq. (58) is always non-negative.Note that if the right hand side of Eq. (58) is strictly positive, then we can choose a smallenough ǫ m such that the inequality in Eq. (58) is contradicted. Hence, we have: ζ m ψ ∗ n k p ∗ n − z ∗ n k ≤ ζ n ψ ∗ m k p ∗ m − z ∗ m k . (61)By swapping the indices m and n in Eq. (49) and repeating the same argument, we have: ζ m ψ ∗ n k p ∗ n − z ∗ n k ≥ ζ n ψ ∗ m k p ∗ m − z ∗ m k . (62)Eqs. (61) and (62) imply that: ζ m ψ ∗ n k p ∗ n − z ∗ n k = ζ n ψ ∗ m k p ∗ m − z ∗ m k . (63)Note that Eq. (49) indicates that Eq. (61) holds for any n but only for a dynamic index m ∈ I d ,and similarly Eq. (62) holds for any m but only for a dynamic index n ∈ I d . Hence, Eqs. (61)and (63) imply that χ ∗ m ≥ χ ∗ n if n ∈ I s , m ∈ I d and χ ∗ m = χ ∗ n if n, m ∈ I d , and Eq. (28) is proved. Now, by using Eq. (63) and the equality E ( P ∗ ) = γ , we can write: X i ∈I d ζ i k Γ ∗ i k − γ = X i ∈I d ζ i k p ∗ i − z ∗ i k = X i ∈I d ζ i ψ ∗ n ζ n ψ ∗ i k p ∗ n − z ∗ n k = ψ ∗ n ζ n k p ∗ n − z ∗ n k X i ∈I d ζ i ψ ∗ i , (64)or equivalently: k p ∗ n − z ∗ n k = P i ∈I d ζ i k Γ ∗ i k − γ ψ ∗ n ζ n P i ∈I d ζ i ψ ∗ i . (65)Hence, we have: p ∗ n = ˜ p n + Γ ∗ n k Γ ∗ n k ( k Γ ∗ n k − k p ∗ n − z ∗ n k ) = ˜ p n + Γ ∗ n − P i ∈I d ζ i k Γ ∗ i k − γ k Γ ∗ n k × ψ ∗ n ζ n × P i ∈I d ζ i ψ ∗ i , (66)and the proof is complete. (cid:4) A PPENDIX
GProof of Proposition 5: We show that none of the steps in MERL Algorithm increases themulti-hop communication power D ( P , W , S ) . Since the movement energy constraint in Eq.(26) does not depend on the cell partitioning and normalized flow matrix, same reasoning as inAppendix D shows that updating W and S according to the generalized Voronoi diagram andBellman-Ford Algorithm, respectively, does not increase D ( P , W , S ) . In what follows, we showthat updating the node deployment according to steps 4 and 5 in Algorithm 2 will not increasethe objective function as well. To show this, we first need the following concepts:Let P k = (cid:0) p k , · · · , p kN , p kN +1 , · · · , p kN + M (cid:1) denote the node deployment after the k -th iteration.In particular, P = ˜ P is the initial deployment. We define the energy allocation after the k -thiteration as E k = (cid:0) e k , · · · , e kN , e kN +1 , · · · , e kN + M (cid:1) where e kn = ζ n k p kn − ˜ p n k is node n ’s movementenergy consumption. Note that after the cell partitioning using the generalized Voronoi diagram,the partitions are fixed as V (cid:0) P k − , S k − (cid:1) . Moreover, let v kn and c kn denote the volume and centroidof V n (cid:0) P k , S k (cid:1) , respectively, and define Γ kn = z kn − ˜ p n where z kn is expressed as in Eqs. (16)and (17). We denote the energy consumed by moving node n from its initial location to z kn by τ kn = ζ n k Γ kn k , and define κ kn = κ n (cid:0) P k , S k (cid:1) = ζ n ψ kn where ψ kn is given by Eq. (30). Finally, wedefine an auxiliary function ˆ χ kn : R N + M −→ R to be ˆ χ kn ( E ) = τ kn − e n κ kn . Note that ˆ χ kn differs from χ n defined in Eq. (31) in the sense that it depends on the energy allocation E rather than thenode deployment and data routing. Lemma 4:
Let I kd and I ks denote the set of dynamic and static nodes after the k -th iterationof the MERL algorithm, respectively. Then, we have: ˆ χ k − i (cid:0) E k (cid:1) = ˆ χ k − j (cid:0) E k (cid:1) , ∀ i, j ∈ I kd (67) ˆ χ k − i (cid:0) E k (cid:1) ≥ ˆ χ k − j (cid:0) E k (cid:1) , ∀ i ∈ I kd , j ∈ I ks (68)Proof: At the end of the deployment step, dynamic node n ’s location in the k -th iteration is: p kn = ˜ p n + Γ k − n − P i ∈I kd ζ i k Γ k − i k − γ k Γ k − n k × ψ k − n ζ n × P i ∈I kd ζ i ψ k − i , (69)thus, its movement energy consumption is: e kn = ζ n k p kn − ˜ p n k = ζ n k Γ k − n k × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − P i ∈I kd ζ i k Γ k − i k − γ k Γ k − n k × ψ k − n ζ n × P i ∈I kd ζ i ψ k − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (70) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ k − n − κ k − n (cid:16)P i ∈I kd τ k − i − γ (cid:17)P i ∈I kd κ k − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ∀ n ∈ I kd (71)where I kd is the set of dynamic nodes in the k -th iteration, determined by the inner loop in steps3 and 4 of the MERL algorithm. According to this inner loop, the term inside the vertical barsin Eq. (71) is positive; hence, we have: e kn = τ k − n − κ k − n (cid:16)P i ∈I kd τ k − i − γ (cid:17)P i ∈I kd κ k − i , ∀ n ∈ I kd . (72)Now, by substituting Eq. (72) into the definition of ˆ χ kn , we have: ˆ χ k − n (cid:0) E k (cid:1) = τ k − n − e kn κ k − n = hP i ∈I kd τ k − i i − γ P i ∈I kd κ k − i , ∀ n ∈ I kd . (73)Therefore, all ˆ χ k − n (cid:0) E k (cid:1) for dynamic nodes are the same and Eq. (67) is proved.In order to prove Eq. (68), we assume that L k inner iterations are performed in steps 3 and 4of the MERL algorithm to determine the dynamic node set in the k -th iteration of the algorithm.For l ∈ { , · · · , L k } , let J kl be the dynamic node set after the l -th inner iteration, where k isthe iteration index of the MERL algorithm. In particular, we have J k = I A S I F and: I kd = J kL k ( J kL k − ( · · · ( J k (74) In other words, in the l -th inner iteration, nodes within the set J kl − − J kl are removed from J kl − due to their non-positive energy allocation, i.e., we have: e kj = τ k − j − κ k − j (cid:16)P i ∈J kl − τ k − i − γ (cid:17)P i ∈J kl − κ k − i ≤ , ∀ j ∈ J kl − − J kl (75)hence, by rearranging the terms in Eq. (75), and summation over all j ∈ J kl − − J kl , we have: X j ∈J kl − −J kl τ k − j X i ∈J kl − κ k − i ≤ X j ∈J kl − −J kl κ k − j X i ∈J kl − τ k − i − γ . (76)Let the auxiliary function ˜ χ k ( J ) = ( P i ∈J τ ki ) − γ P i ∈J κ ki be a mapping from the node set J to the realnumbers. For an inner iteration index l ∈ { , · · · , L k } , we have: ˜ χ k − (cid:0) J kl (cid:1) − ˜ χ k − (cid:0) J kl − (cid:1) (77) = (cid:16)P i ∈J kl τ k − i (cid:17) − γ P i ∈J kl κ k − i − (cid:16)P i ∈J kl − τ k − i (cid:17) − γ P i ∈J kl − κ k − i (78) = (cid:16)P i ∈J kl − κ k − i (cid:17) h(cid:16)P i ∈J kl τ k − i (cid:17) − γ i − (cid:16)P i ∈J kl κ k − i (cid:17) h(cid:16)P i ∈J kl − τ k − i (cid:17) − γ i(cid:16)P i ∈J kl κ k − i (cid:17) (cid:16)P i ∈J kl − κ k − i (cid:17) (79) = (cid:16)P i ∈J kl − κ k − i (cid:17) h(cid:16)P i ∈J kl − τ k − i (cid:17) − (cid:16)P i ∈J kl − −J kl τ k − i (cid:17) − γ i(cid:16)P i ∈J kl κ k − i (cid:17) (cid:16)P i ∈J kl − κ k − i (cid:17) (80) − h(cid:16)P i ∈J kl − κ k − i (cid:17) − (cid:16)P i ∈J kl − −J kl κ k − i (cid:17)i h(cid:16)P i ∈J kl − τ k − i (cid:17) − γ i(cid:16)P i ∈J kl κ k − i (cid:17) (cid:16)P i ∈J kl − κ k − i (cid:17) (81) = P i ∈J kl − −J kl κ k − i ! " P i ∈J kl − τ k − i ! − γ − P i ∈J kl − −J kl τ k − i ! P i ∈J kl − κ k − i !(cid:16)P i ∈J kl κ k − i (cid:17) (cid:16)P i ∈J kl − κ k − i (cid:17) ≥ , (82)where the last inequality follows from Eq. (76). Thus, we have the following ordered sequence: ˜ χ k − (cid:0) J k (cid:1) ≤ ˜ χ k − (cid:0) J k (cid:1) ≤ · · · ≤ ˜ χ k − (cid:0) J kL k (cid:1) = ˜ χ k − (cid:0) I kd (cid:1) = ˆ χ k − n (cid:0) E k (cid:1) , ∀ n ∈ I kd . (83)Let the tentative energy allocation in the l -th inner iteration be ˜ E k ( l ) = (cid:0) ˜ e k ( l ) , · · · , ˜ e kN + M ( l ) (cid:1) . The tentative movement energy consumption of node n in the l -th inner iteration is given by: ˜ e kn ( l ) = τ k − n − κ k − n h(cid:16)P i ∈J kl τ k − i (cid:17) − γ iP i ∈J kl κ k − i , ∀ n ∈ J kl (84)hence, we can rewrite ˜ χ k − (cid:0) J kl (cid:1) as: ˜ χ k − (cid:0) J kl (cid:1) = h(cid:16)P i ∈J kl τ k − i (cid:17) − γ iP i ∈J kl κ k − i = τ k − n − ˜ e kn ( l ) κ k − n , ∀ n ∈ J kl . (85)Note that each node j ∈ J kl − − J kl is removed from the dynamic node set in the l -th inneriteration of the MERL algorithm due to its non-positive tentative energy ˜ e kj ( l ) ≤ ; therefore,we have j ∈ I ks and its allocated movement energy consumption is e kj = 0 . Then, we have: ˆ χ k − j (cid:0) E k (cid:1) = τ k − j − e kj κ k − j = τ k − j κ k − j ≤ τ k − j − ˜ e kj ( l ) κ k − j = ˜ χ k − (cid:0) J kl (cid:1) , ∀ j ∈ J kl − − J kl . (86)Using Eqs. (83) and (86), we have: ˆ χ k − j (cid:0) E k (cid:1) ≤ ˆ χ k − i (cid:0) E k (cid:1) , ∀ i ∈ I kd , j ∈ J kl − − J kl , l ∈ { , · · · , L k } . (87)Note that the static node set I ks consists of all nodes that are removed in the inner loop, i.e. I s = S l ∈{ , ··· ,L k } (cid:0) J kl − − J kl (cid:1) ; hence, Eq. (68) follows from Eq. (87) and the proof is finished. Lemma 5:
For a fixed cell partitioning and normalized flow matrix, the node deployment P k given by the k -th iteration of MERL Algorithm is the unique minimizer to the objective functionin Eqs. (25) and (26).Proof: Using parallel axis theorem [49], the objective function in the k -th iteration is: D = N X i =1 Z V k − i η i k c k − i − ω k R b f ( ω ) dω + N X i η i R b v k − i k p i − c k − i k + λ N X i =1 N + M X j =1 β i,j k p i − p j k F k − i,j + λ P R A (cid:0) W k − , S k − (cid:1) . (88)For a fixed partitioning and routing, a similar reasoning as in the proof of Lemma 2 shows thatnode n ’s optimal location at the end of k -th iteration should be placed on the segment connectingits initial location ˜ p n to the point z k − n given in Eqs. (16) and (17), i.e., if we denote the node n ’s movement energy by e n , we have: p n ( e n ) = ˜ p n + e n ζ n × Γ k − n k Γ k − n k , ∀ n ∈ I A [ I F . (89)By substituting the Eq. (89) into Eq. (88), we can rewrite the objective function as: minimize E D ( E ) s.t. N + M X n =1 e n ! ≤ γ, ≤ e n ≤ ζ n k Γ k − n k , ∀ n ∈ I A [ I F . (90)where: D ( E ) = N X i =1 Z V k − i η i k c k − i − ω k R b f ( ω ) dω + N X i η i R b v k − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ p i + e i ζ i × Γ k − i k Γ k − i k − c k − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + λ N X i =1 N + M X j =1 β i,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ p i + e i ζ i × Γ k − i k Γ k − i k − ˜ p j − e j ζ j × Γ k − j k Γ k − j k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F k − i,j + λ P R A (cid:0) W k − , S k − (cid:1) , (91)Note that the objective function in Eq. (90) and its constraints are convex; hence, it has a uniqueminimizer for a fixed partitioning and routing. If (cid:16)P N + Mn =1 ζ n k Γ k − n k (cid:17) ≤ γ , then the MERLalgorithm moves each node n to z k − n without violating the total energy constraint, indicatingan optimal deployment. On the other hand, if (cid:16)P N + Mn =1 ζ n k Γ k − n k (cid:17) > γ , then nodes will run outof movement energy before they can reach to their corresponding z k − n , and the same reasoningas in Appendix F shows that (cid:16)P N + Mn =1 e n (cid:17) = γ . For the fixed partitioning and routing, let E ∗ = (cid:0) e ∗ , · · · , e ∗ N + M (cid:1) be the optimal energy allocation for the constrained objective functionin Eq. (90), and let P ∗ = (cid:0) p ∗ , · · · , p ∗ N + M (cid:1) be the corresponding optimal deployment. Assumethat the movement energy allocation E k in the k -th iteration is different from the optimal one,i.e., E ∗ = E k . Since (cid:16)P N + Mn =1 e ∗ n (cid:17) = (cid:16)P N + Mn =1 e kn (cid:17) = γ , there exist two distinct indices i and j such that ≤ e ki < e ∗ i and ≤ e ∗ j < e kj . Note that e kj > indicates that j ∈ I kd , i.e., node j is adynamic node in the k -th iteration. Therefore, using Lemma 4 we have: ζ i k Γ k − i k − e ∗ iζ i ψ k − i < ζ i k Γ k − i k − e kiζ i ψ k − i ≤ ζ j k Γ k − j k − e kjζ j ψ k − j < ζ j k Γ k − j k − e ∗ jζ j ψ k − j . (92)Now, we consider a new energy allocation E = ( e , · · · , e N + M ) , where e i = e ∗ i − ǫ , e j = e ∗ j + ǫ and e t = e ∗ t for all t ∈ I A S I F \{ i, j } . Note that (cid:16)P N + Mn =1 e n (cid:17) = γ , and for a sufficiently smallpositive value of ǫ , we have ≤ e ∗ i − ǫ = e i < e ∗ i ≤ ζ i k Γ k − i k and ≤ e ∗ j < e j = e ∗ j + ǫ ≤ e kj ≤ ζ j k Γ k − j k , i.e., E satisfies the constraints in Eq. (90) and it is a valid energy allocation. Similarargument as in Appendix F, that led to the Eq. (61), shows that in order for the energy allocation E not to achieve a lower objective function value in Eq. (90) than D ( E ∗ ) , which contradictsthe optimality of the movement energy allocation E ∗ , we should have: ζ i ψ k − j k p ∗ j − z k − j k ≤ ζ j ψ k − i k p ∗ i − z k − i k , (93)or equivalently: ζ j k p ∗ j − z k − j k ζ j ψ k − j ≤ ζ i k p ∗ i − z k − i k ζ i ψ k − i . (94)According to Eq. (89), each node n ∈ I A S I F is located on the segment connecting ˜ p n to z k − n ;hence: we can rewrite the Eq. (94) as: ζ j k Γ k − j k − e ∗ jζ j ψ k − j ≤ ζ i k Γ k − i k − e ∗ iζ i ψ k − i . (95)But Eq. (95) is in contradiction with Eq. (92); thus, the assumption E ∗ = E k is wrong and wehave E ∗ = E k , i.e. the deployment given by the MERL algorithm is the unique minimizer ofthe constrained objective function and the proof is complete.Now, we have enough materials to prove the convergence of the MERL algorithm. As mentionedin the beginning of the Appendix G, updating the partitioning and normalized flow matrix usingthe generalized Voronoi diagram and Bellman-Ford Algorithm, respectively, does not increasethe objective function. Now, for a fixed partitioning and routing, Lemma 5 indicates that thedeployment given by the MERL algorithm is the unique minimizer of the constrained objectivefunction, i.e., the deployment step in the MERL algorithm does not increase the objective functioneither. Hence, the MERL algorithm generates a sequence of positive non-increasing values forthe objective function D ; thus, it converges. (cid:4) A PPENDIX
HProof of Proposition 6: If p ∗ n = z ∗ n is an optimal deployment P ∗ , W ∗ and S ∗ , then Eq.(34) implies that E n ( P ∗ ) = ζ n k Γ ∗ n k ≤ γ n . Therefore, Eq. (35) reduces to the trivial statement p ∗ n = ˜ p n + Γ ∗ n and the proof is complete. Hence, we assume that p ∗ n = z ∗ n . Now, if any residualmovement energy is left in Node n , i.e. if E n ( P ∗ ) < γ n , then there exists an ǫ n ∈ R + such that E n ( P ∗ ) + ǫ n < γ n and the point p n = p ∗ n + ǫ n × z ∗ n − p ∗ n k z ∗ n − p ∗ n k lies inside the circle centered at z ∗ n with radius k z ∗ n − p ∗ n k . Then, according to Lemma 3, by fixing the cell partitioning, normalized flowmatrix and the location of all nodes except Node n , and placing Node n at p n , we can achieve alower total multi-hop communication power without exhausting the available movement energyin Node n , which contradicts the optimality of P ∗ , W ∗ and S ∗ . Therefore, p ∗ n = z ∗ n implies that E n ( P ∗ ) = γ n , that is ζ n k p ∗ n − ˜ p n k = γ n . (96)According to Lemma 2, we have p ∗ n = δ n ˜ p n + (1 − δ n ) z ∗ n , (97)where δ n ∈ [0 , , which indicates that k p ∗ n − ˜ p n k = (1 − δ n ) k z ∗ n − ˜ p n k . (98)Eqs. (96) and (98) imply that δ n = 1 − γ n ζ n k z ∗ n − ˜ p n k . Therefore, Eq. (97) can be written as: p ∗ n = (cid:18) − γ n ζ n k z ∗ n − ˜ p n k (cid:19) ˜ p n + (cid:18) γ n ζ n k z ∗ n − ˜ p n k (cid:19) z ∗ n (99) = ˜ p n + (cid:18) γ n ζ n k z ∗ n − ˜ p n k (cid:19) ( z ∗ n − ˜ p n ) (100) = ˜ p n + γ n ζ n k Γ ∗ n k Γ ∗ n . (101)Eqs. (96) and (97) imply that γ n = ζ n k p ∗ n − ˜ p n k ≤ ζ n k z ∗ n − ˜ p n k = ζ n k Γ ∗ n k , i.e. γ n ζ n k Γ ∗ n k ≤ . Thus,Eq. (101) can be rewritten as p ∗ n = ˜ p n + min (cid:16) , γ n ζ n k Γ ∗ n k (cid:17) Γ ∗ n which concludes the proof. (cid:4) A PPENDIX
IProof of Proposition 7: In what follows, we show that none of the three steps in LORLAlgorithm will increase the communication power D ( P , W , S ) . Note that the movement energyconstraint in Eq. (34) does not depend on the cell partitioning and normalized flow matrix. Hence,it can be shown via the same argument as in Appendix D that updating W and S according tothe generalized Voronoi diagrams and Bellman-Ford Algorithm, respectively, does not increase D ( P , W , S ) . Note that for a fixed W , S and { p i } i = n , according to Lemma 3, the geometriclocus of node n for which the objective function D ( P , W , S ) remains the same is a circle Φ n centered at z n with radius k z n − p n k . Note that the update rule in Eq. (35) always keeps node n in its valid region determined by its limited movement energy, which is a circle centered at ˜ p n and radius γ n ζ n . A simple geometric reasoning indicates that by updating the position of node n according to Eq. (35), node n will either remain the same or move to the point inside itsvalid region that is closest to the point z n , i.e., node n will either remain on the circle Φ n ormove inside it, and the objective function D ( P , W , S ) does not increase. Since the objectivefunction has a lower bounded, i.e. D ( P , W , S ) ≥ , and it is nonincreasing, LORL Algorithmis in iterative improvement algorithm and it converges. (cid:4) R EFERENCES [1] J. Guo, S. Karimi-Bidhendi, and H. Jafarkhani, “Energy-efficient node deployment in wireless ad-hoc sensor networks,”in
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