Topology Control of wireless sensor network using Quantum Inspired Genetic algorithm

In this work, an evolving Linked Quantum register has been introduced, which are group vector of binary pair of genes, which in its local proximity represent those nodes that will have high connectivity and keep the energy consumption at low, and which are taken into account for topology control. The register works in higher dimension. Here order-2 Quantum inspired genetic algorithm has been used and also higher order can be used to achieve greater versatility in topology control of nodes. Numerical result has been obtained, analysis is done as how the result has previously been obtained with Quantum genetic algorithm and results are compared too. For future work, factor is hinted which would exploit the algorithm to work in more computational intensive problem.


I. INTRODUCTION
The two algorithms, evolutionary and quantum computation has made a remarkable success by solving some problem with relatively less computational time and memory. From it a new mataheuristics field has been made which combined the power of two. The work, Quantum inspired genetic algorithm (QIGA) was put forward by Narayan in 1996 [1].Although evolutionary techniques combined with quantum computing has been for years but the first work on practical problem to work out was used for combinatory optimization [2]. Quantum inspired genetic algorithm used probability based optimization which include algorithms from evolutionary and quantum computing with the likes of parallel computing, superposition and genetic variation [1].QIGA does not require quantum computer for its implementation, but there are metrics when reached certain level, then can quantum computer be required. Those computing techniques combine with probabilities achieve such great result which converge for global optimum. QIGA has been used for combinatory optimization problem, and solved several computational intensive problem. QIGA has been used in Power system optimization and localization of mobile robots [3], Image processing [4], flow shop Scheduling [5], Optimization of Hot Extrusion Process [6].Several algorithm are used for topology control in wireless network (WSN). LEACH [12], SPAN [7], ASCENT [8], and STEM [9]. The first use of evolutionary and Quantum computing used for topology control was in the work [10], which show better result than simple genetic algorithm. The topology control was achieved in fewer generation in Quantum genetic algorithm (QGA).QGA proves superior to simple genetic algorithm. Higher order QIGA was presented in work [11].Order-2 QIGA does need a functional quantum computer for its implementation as it uses random features from quantum mechanical system such as qubits, superposition, and parallel computing. Quantum factor defines when algorithm have to be implemented on quantum computer, and is elaborated in [11] The optimal point of topology control are standardized on connectivity of nodes, number of node exceeding the threshold, energy of nodes, and interference between the nodes. The interference is one of key issue in wireless sensor network (WSN). An optimal point is selected which minimize the interference in register and the intra node between the registers. Interference has not been account in topology control here. Higher order QIGA have terminology which does not exist in QGA. Quantum and relative quantum order, and quantum factor are defined here and for detailed work, reader should understand the full algorithm in [11]. This paper is structured as: The section 1 will describe the various design metrics that influence the topology of nodes. Section 2 discussed the order -2 QIGA, various definition of the algorithm are introduced. In section 3 the linked quantum register has been introduced. In section 4 the algorithm has been implemented for topology control. In Last section, Numerical result has been obtained for topological control of nodes.

CONSTRUCTION AND MAINTENANCE
The sensor network is initialize with n number of sensor node at t=0, all the node have equal resources. So at the start of the process the nodes have identical probability amplitudes which implies from the above statement, initially the nodes are randomly selected in topological space in which they do sensing, surveillance or other data acquisition. The second necessary step in topological control is its construction for which we present a novel higher dimensional register LQR, which self-varies itself across search space and Migration status is updated on each iteration. The Topology maintenance is what the algorithm does in more optimal way than QIGA, On each successive iteration the topological point in space between nodes are varied as to keep the nodes net energy at minimal level, maintain the connectivity of node strong enough so that don't interfere in one another wireless information signal. In wireless communication control decision are most energy consuming, which are kept to minimum number. Suppose there are n number of nodes in a network, 1≤ i ≤ n, which in coordinate space are located at (xᵢ, yᵢ) at the radius of rᵢ. The power consume by each for any node is Pᵢ. Rmax -the maximum a node can transmit Aᵢ -show the connectivity of network, considering it, the network has to ensure it don't interfere in adjacent node' transmission region. Rfthe nodes that exceeds the threshold point Rt are bk in number. The topological factors mathematically are ∀Cij in C (n) , n defines the dimensionality of node The element of adjacency matrix is one if they are connected, which should be in cluster for node connected in a register. For topology control WSN attain these points.
 The total energy of the node should be keep at minimum  The number of node which exceeds the threshold are to be sparse.  The interference between the elements of adjacent register in topological space should be lowered.

III. ORDER-2 QUANTUM INSPIRED GENETIC ALGORITHM
The QIGA uses individual independent qubits to represent binary genes, so they are effectively of order 1. Higher order QIGA was presented in [11]. We here present introductory definition from their work. For full understanding of their work reader should study [11]. QIGA-2 symbolizes classical and quantum individual population by P and Q. order r is defined as the largest register used in algorithms.

IV. LINKED QUANTUM REGISTER (LQR)
In this work we propose a model which takes the adjacent 2-qubit quantum register into account to form a binary paired register. The pair of binary genes in a register represent a node, while in register have a pair of nodes. The information between node in a register is transferred which achieve the topology optimal point in space. As a result the LQR has four points in probability in distribution.|α₀|², |α₁|², |α₂|², and |α₃|². These are few key points about LQR.
 There can be arbitrary number of Quantum binary genes in register .It can varies across different register depending upon the computational complexity a classical machine can cope with. The network can have different number of chromosome in register, and still can achieved the work of having same number of chromosome (nodes) with turning off those nodes for the iterative procedure, but survival of the fittest results from evolutionary computing will keep the nodes ahead in the run.
 Protocol used are application specific in a register. The control decision information in the elements of a register are specific to the individuals only  LQR have memory because in each update step ,which is the distance of nodes in each Register, is to be stored.

a. Algorithm for Topology Control
The QIGA-2 starts by searching the adjacent nodes for which the register identify (RI) is identical. After selecting the nodes of same RI. The algorithm checks for the node connectivity in adjacency matrix which is update by algorithm presented in [11]. The connectivity of nodes is range of space between the nodes or discrete set of point in space. A prior optimal point is set for WSN which is application dependent. A loop is run over the node and the distance is incremented or decremented between the node is achieved which assign node to the same RI. The algorithms then compute the distance between the nodes and set it to be connective. The position of each node is varied by choosing the number of step in each iterative process, which results in step size of the increment or decrement in topology control. The algorithm converges when the given node in WSN are in connectivity range set prior in the algorithm. The update of 2x2 dimensional of LQR depends by choosing one or both of node. the algorithm has select method that choose one of the node in QLR. Several algorithm has been implemented for topology control [7], [8], [9], [10], and [12].
The algorithm starts with an arbitrary number of wireless sensor network (1 ≤ i ≤ n). The initial population will start with equal resources in term of energy, processing Capability and Transceiver power.
In order-2 QIGA the population are represented by vector of chromosome (register).All amplitude are initialized with magnitude of √ Q (t) = {q1 t , q2 t ... qn t }

Solution of individual Q-bits in register:
In this step, binary string are produced from qubits string, this is called observation of states. Q (t) with n bits represent a superposition of 2 n binary states of gene. The states of individual in register is observed For any Register, Rid P (t) = {p1 t , p2 t ,…, pn t } Where P t j represent observational result of j th individual, Rid will have N/r number of P (t) in each iterative step In evaluation step, LQR have the individual nodes, and they are evaluated on basis of how strongly they are connected. The individual use minimum energy and minimum number of nodes that exceeds the power transmission threshold radius, Rf.. The Binary solution is stored. B (t) = { b1 t ,b2 t ,…,bn t } Then the observation of state of Q (t-1) produce binary solution in P (t). The variation operator is model specific, and the user can pick any of the operator that suits the algorithm the best. In order-2, genetic operator are taken 4x4 quantum gates. The observation function from step 2 of algorithm returns strings of binary genes 00, 01, 10 and 11 with a probability of |α0|,| α1|,| α2 |, and |α3| respectively.
The distance update can be done in unidirectional or bi-directional. In unidirectional the position of one of the node is varied keeping the other in its current status, while in bi-directional the position of both node is updated according to the algorithm 1.

VI. NUMERICAL RESULTS
The algorithm has been implemented for 16 nodes. The WSN took 79 generations to accomplish the condition specified in algorithm to converge. Roulette wheel was used for evaluation function of fitness. The QIGA-2 has been compared with Simple Quantum Genetic (SGA) algorithm. Solution found in QIGA-2 earlier, which was obtained because there were two pair of LQR which were in transmission range of one another.so as a result it takes fewer generation for finding optimal topology control. LQR modeling in QIGA-2 results in less generation for topology control than QGA. The solution obtained earlier because the two pair of LQR which have transmission region exceeded than standardized earlier. The power consumption due to fewer generation was beneficial in QIGA-2, and results in less net power than QGA. Goal attained VII. CONCLUSION In this work, a novel register is introduced based on QIGA-2 which uses evolutionary computing in addition to quantum computing to find the optimal solution for topology control. The attributes inherited from quantum computing in LQR are the way of representation of solution of nodes. The order 2 register uses the nodes in register to cooperate for the information necessary to increase the connectivity to an optimal level. At the start of algorithm nodes are paired to make LQR, which transfer the information in each iteration of algorithm for the topology control. Future perspective of the QIGA lies in using more than two number of node in LQR which seem to obtaine better result. Modeling of LQR for more than two node and its implementation on Quantum computer would be challenging.