Routing Algorithm for Maximizing Lifetime of Wireless Sensor Network for Broadcast Transmission
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Abstract
In the article we discuss solutions of the maximum lifetime broadcasting problem in wireless sensor networks. Due to limited energy resources of the network nodes to find an optimal transmission route of the broadcasted data we minimize the maximum energy consumed by the nodes. We give an analytical solution of the problem in one dimensional regular sensor network for the pointtopoint and pointtomultipoint data transmission scheme. We show that in such a network, when the cost of data transmission is a polynomial function of distance between transmitter and receiver, there exist solutions with an equal energy, i.e., all nodes of the network consume the same amount of energy. We assume that in the networks with sensors evenly distributed over some ddimensional area there always exists an equal energy solution of the problem. To solve the maximum lifetime broadcasting problem in such networks we propose two algorithms. By means of the first algorithm the set of minimum node weight spanning trees in a given network is determined. The second algorithm allows to balance the energy consumption of the sensors for data transmitted along given minimum node weight spanning tree. We show, that the proposed algorithms find an approximate solution of the discussed problem in polynomial time.
Keywords
Wireless communication Broadcast transmission Sensor network lifetime Energy efficiency1 Introduction
Sensor network is a type of wireless adhoc network with the nodes of limited hardware, system and power resources. Usually, sensors collect data in order to perform simple computational tasks and to transmit the resulting data to a set of data collectors. Another type of sensors behavior, important in network management, diagnostics, routing is retransmission of data broadcasted over the network. Collecting, processing and transmitting data is energy absorbing and therefore it requires a sensor cooperation, which relies on shearing the energy between the sensors. Most of the sensor’s energy is utilized during the data transmission, this energy requirement grows with the size of the network and the amount of data transmitted over the network. Efficient use of energy by the network nodes becomes crucial to extend their operating time and the lifetime of the whole network. By a sensor network lifetime we mean the period of time until the first node runs out of energy [1, 2, 3, 4, 5]. The network lifetime can be defined as the number of cycles the network can perform its functions. Namely, if each node of the network has a battery with the initial energy \(E_0\), then by finding the optimal energy utilization of each node in one cycle of network lifetime \(E_i^{\mathrm{opt}}\) we can determine the number of cycles \(N_{\mathrm{cycles}}=[\frac{E_0 }{ E_{i^{\prime }}^{\mathrm{opt}} }]\) the network can perform its functions until the most overloaded node \(i^{\prime }\) runs out of its energy. Various aspects of the lifetime problem in wireless sensor networks were discussed in the literature [6, 7, 8, 9, 10, 11, 12]. In [6] the maximum lifetime broadcasting problem in sensor networks with mobile nodes was analyzed. For several mobility models of the nodes distributed algorithms for energyefficient broadcasting was defined and evaluated. Two algorithms, designed based on trees with minimum highest cost edges and the minimum cost paths trees were compared against the algorithm for broadcast data transmission which utilizes the methods for determining the connected dominating set [7]. In [8] where analyzed polynomial time approximation algorithms for solving the set of rooted maximum network lifetime problems, defined by Nutov and Segal [9]. The objective of the problems is to find the maximumsize collection of routing trees rooted at a given node. The analysis of Nutov and Segal’s algorithms allows the authors to improve the approximations ratios for these problems. In [10] the broadcast tree lifetime problem was presented for two models of receiver energy utilization, the model in which the receiver consumes a fixed amount of energy per unit of received data, and the model in which the energy consumed by a receiver is a function of the transmitter signal power. The lifetime of a broadcast tree is the duration until the first node in the tree fails due to battery energy exhaustion. The authors proposed two algorithms which solve the problem and the solutions were proved to be optimal.
In this article we investigate sensor networks in which the solution of the maximum lifetime broadcasting problem is an equal energy, i.e., the energy consumed by the network is evenly distributed over the sensors. We assume that the sensors are static, they utilize the energy only for sending the data and use the pointtopoint or pointtomultipoint data transmission. For the pointtopoint data transmission scheme the sender transmits the data to a unique receiver. For the pointtomultipoint transmission the transmitter sends in parallel the same data to some set of receivers. Important future of the pointtomultipoint data transmission is the wireless multicast advantage property (the WMA property) [13, 14]. For such transmission the nodes, which are in the range of the transmitting node, can receive the data without additional costs of the transmitter. In the paper we give the solution of the maximum lifetime problem for the pointtopoint and pointtomultipoint broadcast data transmission in one dimensional regular sensor networks \(L_N\). To solve the problem in networks for arbitrary dimension we propose two algorithms which allow to construct the set of trees for data broadcasting and balance the energy consumed by the network nodes. The first algorithm determines the minimum node weight spanning trees in a given network. The second algorithm allows construct the set of trees to balance the energy consumed by the network nodes transmitting data along given minimum node weight spanning tree. Similar algorithms, to discussed in this article, are proposed by Kang and Poovendran [15] and by Wieselthier et al. [13, 14]. The Kang and Poovendran algorithm is based on the transmission along the minimum spanning tree (MST) in equally distributed energy network (EDEN). The Wieselthier et al. broadcast incremental power (BIP) algorithm is similar in principle to the Prim’s algorithm for searching the MSTs, but contrary to our algorithm which uses the pointtopoint transmission, it exploits the wireless multicast advantage in the construction of the broadcast tree [13]. In some special cases of graphs the minimum node weight spanning trees and the MSTs coincide, which means that to a certain extent our algorithms are a generalization of the Kang and Wieselthier algorithms.
The original contribution of this article is the application of the minimum node weight spanning trees to solve the network lifetime problems for broadcast transmission. To find such trees in a complete graph is an NPhard problem [16]. Two polynomial time algorithms for searching the minimum node weight spanning trees in a network graph, based on the function generating trees, are defined in [16]. In this article we propose the new one which is a modification of the minimum degree spanning tree algorithm proposed by Fürer and Raghavachari [17]. In the following chapters we show that the minimum node weight spanning tree searching algorithm inherits many features of the Fürer and Raghavachari algorithm and together with the load balancing broadcast algorithm it allows efficiently find the approximation of an equal energy solutions of the maximum network lifetime broadcasting problem.
2 Definition of the Problem
Lemma 1
Any minimum of the objective function (4) is a solution of the maximum lifetime broadcasting problem.
Proof
If we assume that \(Q_{k}, \; q_{i,j} \in Z_{+}^{}\) in (11), then we get the mixed integer linear programming problem for the network lifetime. The following lemma describes the complexity of the problem.
Lemma 2
The maximum lifetime broadcasting problem is NPhard.
Proof
We reduce the partition problem to the MLB problem (11). To solve the MLB problem in \(Z_{0}^{+}\) we must find the set of integer numbers \(\{q^{k}_{r} \}_{r \in [1,N^{N2}]}\) for all trees from \(G_{k}^{T}\), satisfying the constraint \(\sum _{r} q^{k}_{r} = Q_k\) and which minimizes the objective function (4). Finding such set of numbers \(q^{k}_{r}\in Z_{0}^{+}\) is equivalent to solving the partition problem [19]. \(\square\)
3 Solution of the MLB Problem in One Dimensional Network
Lemma 3
The solution of the maximum lifetime broadcasting problem (11) with \(E_{i,j}\) satisfying (1), for the first and last node of the \(L_N\) network is given by the transmission graphs \((t_{1,2},\ldots , t_{N1,N})\) and \((t_{N,N1},\ldots , t_{2,1})\) with the data flow \(Q_1\) and \(Q_N\) respectively.
Proof
From the property (1) of the data transmission cost energy matrix \(E_{i,j}\) it follows that, the nodes consume minimal energy when they send all of their data to the nearest node. Because the distance between the neighboring nodes of the regular network \(L_N\) is the same, then the energy spent by each node to transmit a given amount of data to its neighbor is also the same. From this it follows that, when the broadcasted data \(Q_1\) is transmitted along the edges \(t_{i,i+1}\), \(i\in [1,N1]\), then each node \(i\in [1,N1]\) spends the same, minimal amount of energy. This means that the optimal transmission graph, the solution to the MLB problem, for the first node is given by the set of graph edges \((t_{1,2} ,\ldots , t_{i,i+1},\ldots , t_{N1,N})\) with the data flow \(Q_1\). In a similar way we can find the solution of (11) for the last node of the \(L_N\) network. It is given by the transmission graph \((t_{N,N1} ,\ldots , t_{i,i1},\ldots , t_{2,1})\) and the data flow \(Q_N\). \(\quad \square\)
The next theorem describes the solution of (11) for the one dimensional, regular sensor network \(L_N\), when the data is broadcasted by the internal nodes, \(k\in [2,N1]\) of the network.
Theorem 1
Proof
The next lemma shows that for a sufficiently large network \(L_N\), when the number of nodes grows, the energy \(E_i^{k}\), \(k\in [2,N1]\) of each node decreases.
Lemma 4
4 Broadcast Transmission with Wireless Multicast Advantage
The numbers on the edges denote the sequence the data is transmitted along the tree and the doted lines indicate the costless transmission.
Lemma 5
Proof
Since the distance between the neighboring nodes in the \(L_N\) network is equal to one, then for a bidirectional antenna the data transmitted by the kth node, \(k\in [2,N1]\) to the \((k1)\)th node is delivered to the \((k+1)\)th node without costs. In other words, the amount \(Q_k\) of data transmitted along the edges \((t_{k,k1}^{k}, t_{k,k+1}^{k})\) are delivered to the both nodes \((k\pm 1) \in L_N\) with the minimal costs \(E_{k}^{k}=Q_k\) of the kth node. From Lemma 3 we know that, the optimal tree to transmit the amount \(Q_k\) of data from the \((k1)\)th node to the nodes \(i\in [1,k2]\) of the \(L_N\) network is given by the sequence of edges \((t_{k1,k2}^{k},\ldots ,t_{2,1}^{k})\) and the energy consumed by each node is qual \(E_{i}^{k}=Q_k\), \(i\in [2,k2]\) and \(E_{1}^{k}=0\). Similarly, the optimal tree to transmit the amount \(Q_k\) of data from the \((k+1)\)th node to the nodes \(i\in [k+2,N]\) of the \(L_N\) network is given by the sequence of edges \((t_{k+1,k+2}^{k},\ldots , t_{N1,N}^{k})\) and the energy utilized by each node is qual \(E_{i}^{k}=Q_k\), \(i\in [k+2,N1]\) and \(E_{N}^{k}=0\). This optimal transmission tree up to an ordering is given by (24). \(\quad \square\)
The next theorem states that for the directional antennas and for the internal nodes of the network \(L_N\), i.e. \(k\in [2,N1]\), the solution of the MLB problem with WMA property coincides with the solution given in (13)–(15).
Theorem 2
For a directional antenna the solution of the MLB problem with the wireless multicast advantage property and \(E_{i,j}\) satisfying (1), for \(k\in [2,N1]\), \(N\ge 3\) is an equal energy solution and it is given by (13)–(15).
Proof
The proof is basically the as the proof of Theorem 1. \(\quad \square\)
The solutions of MLB problem with the wireless multicast advantage property for the first and the last node of the \(L_N\) network and for both types of antennas, the bidirectional and directional is given by Lemma 3.
5 The Maximum Lifetime Broadcasting Algorithms
In sensor networks in which the nodes are evenly distributed on some area of the plane, we can expect that the solution of the MLB problem is an equal energy, i.e. \(\forall _{i\in S_N} E^{\mathrm{Sol}}_{i}=E_{0}\). For such problem we propose two algorithms which allow to find a set of trees to broadcast the data with balancing the energy consumed in the network nodes. The purpose of the first algorithm is to construct in \(S_N\) the set of minimum node weight spanning trees \(G^{\mathrm{M_{{ nw}}ST}}_{k}\) rooted at the broadcasting node. The second algorithm allows to construct, for a given tree T from \(G^{\mathrm{M_{{ nw}}ST}}_{k}\), the set of trees \(G^{{ LBB}}_{k}(T)\) which is used to balance the energy consumed by nodes of the network \(S_N\). To find an approximate solution of the MLB problem we define the objective function (4) over the set \(G^{{ LBB}}_{k}=\bigcup _{T\in G^{\mathrm{M_{{ nw}}ST}}_{k} } G^{{ LBB}}_{k}(T)\) and find its minimum with respect to the variables \(q^{}_{}(T^r)\), \(T^r \in G^{{ LBB}}_{k}\).

The minimum node weight spanning tree algorithm.

Step 1 For a directed, weighted graph \(G_N=\{S_N,V,E\}\) determine the set \(G^{\mathrm{MST}}_{k}\) of minimum outgoing spanning trees rooted at the broadcasting node.
 Step 2 Select a tree T from the set \(G^{\mathrm{MST}}_{k}\) and determine in T the node with the maximum outgoing weight. To decrease the weight of this node select a local subtree \(T^{\prime }\) in T rooted at that node and replace it by another subtree \(T^{\prime \prime }\) attached to some node with lower weight in \(S_N\), such that the inequalityis satisfied. The leaf nodes of the subtree \(T^{\prime \prime }\) attach to the nodes of \(T^{\prime }\).$$\begin{aligned} \max \{ E(T)  E(T^{\prime }) + E(T^{\prime \prime }) \}< \max \{E(T) \} \end{aligned}$$(25)
The requirement (27) allows not only to remove edges from the node but also to add to the ith node a set of edges \(\{t_{i,s}\}_{s}\) satisfying the inequality \(\sum _{s} t_{i,s} E_{i,s} < t_{i,r} E_{i,r}\). If the constructed tree \(T^{\prime }\) is not unique, we can select one tree from the equivalent trees or add all equivalent trees to the objective function of the MLB problem.Procedure 1. Spanning tree graph construction for decreasing energy consumed in the node.
To decrease the energy consumed by the ith node remove one outgoing edge \(t_{i,r}\) from the set \(V^{e}_{i}(T)\) and construct the minimum node weight spanning tree \(T^{\prime }\) for the graph \((S_N, V^{\prime }, E)\), where \(V^{\prime } = V \setminus \{ t_{i,r} \}\), such that the inequalityis satisfied. Repeat the process of constructing the minimum node weight spanning tree in \(S_N\) with removed one, two and finally all the edges from the given nonbroadcasting node. For the broadcasting node we leave at least one outgoing edge. \(\quad \square\)$$\begin{aligned} E_{i}(T^{\prime }) < E_{i}(T) \end{aligned}$$(27)
The requirement (28) means, that we construct the tree \(T^{\prime }\) with a fixed set of outgoing edges of a given node. The weight of the ith node in \(T^{\prime }\) is given by the formula \(E_{i}(T^{\prime }) = E_{i}(T) + Q_k t_{i,r} E_{i,r}\). The two procedures we use to define the LBB algorithm.Procedure 2. Spanning tree graph construction for increasing energy consumed in the node.
To increase the energy consumed by the ith node add to the set of outgoing edges \(V^{e}_{i}(T)\) a new edge \(t_{i,r}\) and construct the minimum node weight spanning tree \(T^{\prime }\) in \(S_N\), such thatRepeat the process of constructing the minimum node weight spanning trees \(T^{\prime }\) with added one, two and finally all \((N2)\) edges to the nonbroadcasting node and \((N1)\) edges to the broadcasting node. \(\quad \square\)$$\begin{aligned} V^{e}_{i}(T^{\prime }) = V^{e}_{i}(T) \bigcup \{ t_{i,r} \}. \end{aligned}$$(28)

The load balancing broadcast algorithm.

Step 1 Select a minimum node weight spanning tree T from the set \(G^{\mathrm{M_{{ nw}}ST}}_{k}\). The formula (26) is used to determine the average energy \(\bar{E}(T)\) utilized by nodes of the network \(S_N\) for the tree T.

Step 2 For the nodes which energy is greater than the average \(\bar{E}(T) < E_i(T)\) apply the Procedure 1, i.e. construct the set of trees for decreasing the energy consumed by that nodes \(G^{LBB}_{k}(T)\). For the nodes which the consumed energy along the tree T is less than the average \(E_i(T) < \bar{E}(T)\) apply the Procedure 2, construct the set of trees for increasing the energy consumed by that nodes. \(\quad \square\)
6 Applications of the Maximum Lifetime Broadcasting Algorithms
In this section we apply the proposed algorithms to find an approximate solution of the MLB problem in sensor networks \(S_5\) build of five nodes. We assume, that the nodes of network \(S_5\) are located at the points \(x_1=(0,2)\), \(x_2=(2,2)\), \(x_3=(3,0)\), \(x_4=(4,3)\), \(x_5=(5,2)\) of the plane, where \(x_i=(x_i^1,x_i^2)\in E^2\) and the second node broadcasts the amount \(Q_{2}=1\) of data. The cost of transmission of one unit of data in the graph \(S_5\) is given by the function \(E_{i,j}=d(x_i,x_j)^4\), where \(d(x_i,x_j)\) is Euclidean distance in \(E^2\).
We construct the set \(G^{{ LBB}}_2(T^i)\), \(i=1,2\), to balance the energy consumed by the nodes for the data transmitted along the two minimum node weight spanning tree \(T^1\) and \(T^2\) The average energy utilized by the nodes of the network \(S_5\) transmitting broadcasted data along the tree \(T^1\) is equal to \(\bar{E}(T^1)=36.33\). Because the weights of the nodes for \(T^{1}\) are \(E(T^{1})=(0,41,0,4,64)\) and \(\bar{E}(T^1) < E_{i}(T^{1})\) for \(i=2,5\), we decrease the energy consumed by the \(x_2\) and \(x_5\) nodes and increase the energy consumed by the \(x_1\), \(x_3\), \(x_4\) nodes of the network.
To decrease the energy consumed by the \(x_5\) node we remove the edge \(t_{5,3}\) from the set \(V(S_5)\) and construct the M\(_{\mathrm{NW}}\)ST tree \(T^{\prime }\) in \(S_5\), such that the inequality \(E_{i}(T^{\prime }) < E_{i}(T^{1})\) is satisfied. In this way we obtain \(T^{\prime }=T^{2}\), the second minimum node weight spanning tree in \(S_5\). We can decrease the energy consumed by the \(x_2\) node by removing from \(V(S_5)\) the edge \(t_{2,1}\) or \(t_{2,4}\). If we remove the edge \(t_{2,4}\), then the minimum node weight spanning tree constructed in \(S_5\) with the smaller set of edges \(V(S_5) \setminus \{ t_{2,4} \}\) is \(T^{3}=(t_{2,1}, t_{1,3}, t_{3,5},t_{5,4})\) with weights of the nodes \(E(T^{3})=(169,16,64,0,4)\). The M\(_{\mathrm{nw}}\)ST tree constructed in \(S_5\) with removed edge \(t_{2,1}\) give us the tree \(T^{4}=(t_{2,4}, t_{4,5}, t_{5,3},t_{3,1})\) with weights of the nodes \(E(T^{4}) = (0, 25, 169, 4, 64)\).
To increase the energy consumed by the first node in \(S_5\) we attach to \(x_1\) the edge \(t_{1,3}\) and fix it in the new tree \(T^{\prime }\), i.e. \(V^{e}_{1}(T^{\prime })=\{ t_{1,3} \}\). Construction of the M\(_{\mathrm{nw}}\)ST tree \(T^{\prime }\) in \(S_5\), with fixed set \(V^{e}_{1}(T^{\prime })\), gives us the tree \(T^{\prime }=T^{5}=(t_{2,1}, t_{1,3}, t_{2,4},t_{4,5})\) with the weights of the nodes \(E(T^{5}) = ( 169, 41, 0, 4,0)\). Applying the LBB algorithm to increase the energy consumed by the \(x_3\) node we obtain \(T^{2}\), the second M\(_{\mathrm{nw}}\)ST in \(S_5\). To increase the energy consumed by the fourth node we attach to \(x_4\) the edge \(t_{4,3}\) and fix the node outgoing edges in \(T^{\prime }\), i.e. \(V^{e}_{4}(T^{\prime })=\{t_{4,5}, t_{4,3} \}\). For a fixed set \(V^{e}_{4}(T^{\prime })\) constructed M\(_{\mathrm{NW}}\)ST tree \(T^{\prime }\) has the form \(T^{\prime }=T^{6}=(t_{2,1}, t_{2,4}, t_{4,5}, t_{4,3})\) with the nodes weights \(E(T^{6})=(0,41,0,104,0)\).
For the set of trees \(G^{\mathrm{LBB}}_{2}(T^1)=\{T^{r}\}_{r\in [1,6]}\) we define the objective function (4) and calculate its minimum with respect to the six variables \(q=(q_1, \ldots , q_6)\). The solution of the MLB problem in \(S_5\), determined by the maximum lifetime broadcasting algorithms, is given by the set of trees \(\{ T^{1}, T^{3}, T^{4}, T^{6}\}\) and the amount of data \(q_{1}=0.393397\), \(q_{3}= 0.201039\), \(q_{4}= 0.124906\), \(q_{6}=0.280658\). The energy consumed by the nodes \(x_1\), \(x_2\), \(x_3\), \(x_5\), \(x_4\) is equal to \(E^{\mathrm{alg}}=33.9755\) and \(E^{\mathrm{alg}}_{4}=31.2616\) respectively. The solution found by means of the algorithms, i.e., \(E^{\mathrm{alg}}=33.9755\), is \(\frac{E^{\mathrm{alg}}}{E^{\mathrm{sol}}}=1.0032\) times worse than the exact solution. When we start constructing the set \(G^{\mathrm{LBB}}_{2}(T)\) from \(T^{2}\), the second M\(_{\mathrm{nw}}\)ST tree in \(S_5\), then we obtain the approximate solution \(E^{\mathrm{alg}}=34.4366\), which is \(\frac{E^{\mathrm{alg}}}{E^{\mathrm{sol }}}=1.01682\) times worse that the exact solution. The set \(G^{\mathrm{LBB}}_{2}(T^2)\) for the solution \(E^{\mathrm{alg}}=34.4366\) consists of the following trees \(T^1\), \(T^2\), \(T^3\), \(T^6\) from \(G^{\mathrm{LBB}}_{2}(T^1)\) and two new trees \(T^7=(t_{2,3}, t_{3,1},t_{3,5},t_{5,4})\) and \(T^8=(t_{2,1}, t_{1,3},t_{2,4},t_{4,5})\).
We would like to apply the maximum lifetime broadcasting algorithms to solve the MLB problem in \(L_N\) network. We show, that the algorithms can find all the trees from the exact solution of the MLB problem (11) in \(O(N^2)\) time.
Lemma 6
The minimum node weight spanning tree and the load balancing broadcast algorithms find the solution of the maximum lifetime broadcasting problem in the \(L_N\) network in \(O(N^2)\) time.
Proof
The minimum node weight spanning trees for the weighted graph \((L_N, V, E)\) are the minimum spanning trees. They can be found in \(O(N^2)\) time by the standard algorithms. For the border nodes of the \(L_N\) network the minimum spanning trees rooted at these nodes form the solution of the MLB problem, see Lemma 3. When the data is broadcasted by the internal node of the network, then the M\(_{\mathrm{NW}}\)ST tree in \(L_N\) is the \(T^k\) tree, \(k\in [2,N1]\), defined in (13). The weights of the nodes in the \(T^k\) tree are \(E_{k}=2\), \(E_{1}=E_{N}=0\) and \(\forall _{i\ne 1,N,k} E_{i}=1\). The average energy consumed by the nodes of the network \(L_N\) transmitting the amount \(Q_k=1\) of data along the tree \(T^{k}\) is equal to \(\bar{E}(T^{k})=\frac{N1}{N2}\). Because of the inequalities \(\bar{E}(T^{k})< E_{k}(T^{k})\) and \(\forall _{i\ne k} \; E_{i}(T^{k}) < \bar{E}(T^{k})\) we should decrease the energy consumed by the broadcasted node and increase the energy consumed by all other nodes. Applying the LBB algorithm to decrease the load of the broadcasting node we obtain two trees, the \(T^{k1}\) and \(T^{k+1}\) tree from (13). In the first iteration, when the LBB algorithm is applied to increase the energy consumed by the nonbroadcasting nodes of the \(L_N\) network, \(i\in [1,N]\), \(i\ne k\), we add to each node one additional outgoing edge and construct the M\(_{\mathrm{NW}}\)ST tree. For the ith node, \(i\in [2,k1]\), we fix the existing outgoing edge \(t_{i,i1}\) and add the \(t_{i,j}\) edge, \(j\ne k,i,i+1\). For \(i\in [k+1,N1]\) we fix the edge \(t_{i,i+1}\), add the new edge \(t_{i,j}\), \(j\ne k,i,i+1\), \(j\ne k,i,i1\) and construct the M\(_{\mathrm{NW}}\)ST tree in \(L_N\). In general, for each node \(i\in [2,N1]\), \(i\ne k\), there are \((N3)\) ways to add the additional edge. Among \((N3)\) trees rooted the ith node, \(i\in [2,N1]\), \(i\ne k\) there is always one tree from the solution (13). Because the border nodes of the \(L_N\) network in the tree \(T^k\) are the leaf nodes we can attach to them a one edge in \((N2)\) ways. Also in this case among constructed \((N2)\) trees rooted at the border nodes there is one tree from the solution (13). Constructed by the LBB algorithm set \(G^{{ LBB}}_k(T^k)\) is composed of \((N3)^2+2(N2)+1\) trees. Because in this set there are all trees from the solution of the MLB problem in \(L_N\) then the algorithm finds them in \(O(N^2)\) time. \(\quad \square\)
In the last step of solving the MLB problem in \(L_N\) by means of the maximum lifetime broadcasting algorithms we have to build the objective function (4) over the set \(G^{{ LBB}}_k(T^k)\) with \(q^r\), \(r\in [1,(N3)^2+2(N2)+1]\) variables and calculate its minimum. Because the MLB problem (11) is equivalent to the linear programming problem to find a minimum of the objective function we can apply the Dantzig simplex algorithm or the Khachian ellipsoid algorithm [21]. In the worst case the complexity of the Dantzig algorithm is exponential and the Khachian algorithm has polynomial time complexity with the bound \(O(N^6 L^2)\), where N is the number of variables and L is the number of bits required to represent the input data.
7 Conclusions
In the paper we investigated the equal energy solutions of the maximum lifetime broadcasting problem in sensor networks. We solved the maximum lifetime problem for the pointtopoint and pointtomultipoint broadcast data transmission in one dimensional regular sensor network \(L_N\). Based on the analytical solution of the problem in one dimension we proposed two algorithms which allow to find an approximate solution of the problem in sensor networks in any dimension. The first algorithm determines the minimum node weight spanning trees in a given network. The second algorithm, called the load balancing broadcast algorithm, allows construct the set of trees to balance the energy consumed by the network nodes transmitting data along given minimum node weight spanning tree. As a sample application of the proposed algorithms we solved the maximum lifetime broadcasting problem in sensor network build of five nodes. Two solutions found by means of these algorithms were \(\frac{E^{\mathrm{alg}}(T^1)}{E^{\mathrm{sol}} }=1.0032\) and \(\frac{E^{\mathrm{alg}}(T^2)}{E^{\mathrm{sol }}}=1.01682\) times worse than the exact solution. We also showed, that by means of these algorithms we can find all trees from the exact solution of the MLB problem in \(L_N\) network in \(O(N^2)\) time.
There are several unsolved problems related to the proposed algorithms. For example, it would be interesting to modify the load balancing broadcast algorithm in such a way that the unequal energy solutions of the maximum lifetime broadcasting problem can be efficiently found. Knowledge about the number of minimum node weight spanning trees in given network, the cardinality of the set \(G^{\mathrm{M_{{ nw}}ST}}_{k}(S_N)\), could help to find better approximation of the solution for the MLB problem. Also the minimum number of the trees necessary to solve the MLB problem for arbitrary network and broadcasting node is not known. It seems, that the solution of the MLB problem in network build of N nodes requires no more than N routing trees.
Notes
Author's contribution
The author declares full contribution to this work.
Compliance with Ethical Standards
Conflict of interest
The author declares that he has no competing interests.
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