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- 1. A Genetic Algorithm for the Minimum Cost Localization Problem in Wireless Sensor Networks Angelo F. Assis, Luiz Filipe M. Vieira, Marco T´ulio R. Rodrigues, Gisele L. Pappa Computer Science Department Universidade Federal de Minas Gerais Belo Horizonte - MG - Brasil Abstract—Localization is a paramount concern in wireless sensor networks. Beacon nodes, which have their position deﬁned a priori, might be used in the process, serving as references to ﬁnd the position of other nodes. Many studies focused on ﬁnding the location of as many nodes as possible, given a set of beacons and distance measurements. In this work, we determine the set of beacon nodes in order to localize all nodes in the network. This can reduce the overall cost involved in the network localization process, i.e., reducing the number of nodes in a WSN with GSP. We present a new approach to this problem using Genetic Algorithms. Our simulations results show the efﬁciency of the proposed approach, which has results up to 50% better than the best greedy algorithm found in the literature. I. INTRODUCTION Wireless sensor networks (WSNs) have received a notable investment by the academic community in the last years. Even with the memory and computing limitation of the sensor nodes, WSNs can be employed in different application areas such as medicine, industry, environmental sciences and military. In the ﬁeld of medicine, for example, WSNs can be used to monitor the behavior of the human heart or to detect hazardous substances present in the organism. In the environmental area, WSNs are important to prevent natural disasters like earth- quakes, tsunamis, hurricanes and ﬁres. Sensors can also assist in weather forecasts [3]. Furthermore, WSNs can guarantee the control of data in areas with difﬁcult access or in dangerous regions. Localization is one of the main concerns in WSNs, as information about the sensors’ positions is helpful in many contexts. In traditional applications, sensors generate lots of information which are only relevant when followed by the position of the respective sensing node. Moreover, sensor location supports the performance of network protocols, as, for example, in geographic algorithms. Thus, it is primordial for WSNs to know each individual sensor localization. The main goal in a localization problem is to determine the exact position of each sensor in a bidimensional (2D) region. A way of determining the location of all sensors in a network is to manually set the position of each node. However, in large scale implementations or in scenarios where sensors have moving capabilities, this method may be highly infeasible. An alternative way is equipping each sensor with a Global Positioning System (GPS) [4]. Nevertheless, this approach may also not be practicable, due to the high cost and complexity of embedding the equipment in the sensor, as well as great energy consumption and increase of the sensor size. In this direction, many works aim at determining the location of all nodes in the network, where some special nodes, denominated beacon nodes, are aware of their own position [5], [6], [7]. The remaining sensors will determine their localization via distance measurements to their neighbors, using methods such as the intensity of signal of communication [8], among others. However, in some cases just locating all nodes is not enough, and it is also necessary to do that with the minimum possible cost. Thinking about this last scenario, [10] deﬁned the Minimum Cost Localization Problem (MCLP). In this case, the aim is yet to locate all nodes but with the minimum number of beacon nodes. In [10] the authors show that the MCLP is NP-complete, and deﬁne four greedy methods to deal with the problem. Here, in contrast, we take advantage of the global search properties of genetic algorithms to improve the results obtained in [10]. The use of genetic algorithms and other techniques in WSN problems has been successfully explored, as reported in [9]. However, most of these approaches do not consider minimizing the number of beacon nodes when solving a WSN localization problem. It is important, though, to make the number of beacon nodes as minimum as possible, due to their high ﬁnancial costs. In this work, we present an efﬁcient algorithm that aims at minimizing the number of beacon nodes in a WSN, without jeopardizing the task of locating all remaining nodes in the network. This method makes use of trilateration to calculate the position of non-beacon nodes, determinig the position of all nodes in the network. It is noteworthy that this work does not take into account the location precision of the nodes. The accuracy depends on the method used to deﬁne the position of each node. In this work, the methods considered are trilateration and greedy sweep. In scenarios where the signal propagation models ﬁnds no obstacles, these methods guarantee the desirable precision. The main contributions of this work are: • We provide an innovative formulation for the mini- mum cost localization problem using a genetic algo- rithm; • Our approach improves the results in the literature, in some cases, in more than 50%. The remainder of this paper is organized as follows. Section 2 reviews related work on localization in WSN, while Section 3 deﬁnes the minimum cost localization problem. Section 4 introduces the proposed method based on genetic algorithm, and Section 5 presents the results of computational experi-
- 2. ments. Finally, Section 6 draws some conclusions and points out future research directions. II. RELATED WORK In recent years, the localization problem in WSNs has received a considerable attention by the academic commu- nity [9], [10], [5], [7], [6], [11], [12]. Studies resulted in several techniques that aim at localizing the maximum number of sensors in a network. Recent studies address problems that intend to minimize the number of beacon nodes, in order to localize every node in the network. In this section we present some relevant works in localization in WSN. In [11], theoretical ideas to the localization problem are exposed, in which some nodes have their positions deﬁned a priori and the rest of nodes determine their locations by distance measurements from neighbor nodes. Nodes ﬁxed as beacons are static in the 2D region, but the process of localization can be performed by mobile nodes. In [13], it is presented a range-free localization scheme that employs mobile beacon nodes. The sensors move around the network broadcasting their current locations periodically in such a way that unlocated nodes can locate themselves. Given that, no special hardware or communication technology is necessary. Also, obstacles on the way of the beacon nodes are considered. In a similar way, [12] utilizes mobile beacon nodes to locate the network. Virtual nodes with known location are added during the execution of the algorithm. In this case, a virtual node represents the instantaneous location of a beacon node during its movement. Considering a set of beacons with ﬁxed positions and a WSN, [14] intends to determine the locations of the unlocated nodes. The proposed algorithm estimates the location of the sensors based on the distance measurements among their neighbor nodes. A cross-entropy-based locali- zation algorithm is presented, which aims at improving the accuracy of location estimative of the sensors. In addition, [11] deﬁnes some concepts and conditions to the localizability of a network and the computational com- plexity of the localization method: (i) A network has a unique localization if and only if the graph G that represents the WSN is globally rigid; (ii) Each vertex in G represents a sensor in the network and two nodes are connected if and only if the distance between them is known. The distance measurement is obtained by any method or when the two nodes are beacons; (iii) The computational complexity shows that, to a globally rigid graph, the problem is NP-hard. The Minimum Cost Localization Problem (MCLP), pre- sented in [10], is an optimization problem that aims to locate all the nodes in a WSN using the minimum number of beacon nodes. Localizing all the nodes with minimum cost is NP- hard [10]. Hence, four different greedy algorithms based on trilateration were proposed, and follow two steps: (1) the nodes with less than three neighbors are marked as beacons – because these nodes cannot have their positions deﬁned by other nodes; (2) at each iteration, the unlocated node for which its localization gives the best conﬁguration to the network is selected and deﬁned as a beacon. This best conﬁguration is the one that allows the most number of nodes to be localized. Here we focus on one of the four versions of the algorithms proposed in [10], namely Greedy-Sweep-2. The method served as inspiration and baseline for the algorithm proposed here, and will be described in detail in Section III-B. A similar problem is deﬁned in [9]. Consider a WSN with sensors spread across a 2D region in undeﬁned positions. The goal of this paper is to deﬁne the set B that localizes every other node in the network. This approach solves the localization problem but it disregards minimizing the cost of the set B. We differ from the previous work by formulating the ﬁrst genetic algorithm to solve the MCLP. Our results show improvement of over 50%. We tested the solutions using a well-known network simulator. III. THE MINIMUM COST LOCALIZATION PROBLEM This section presents the problem deﬁnition for the MCLP and describes Greedy-Sweep-2, one of the greedy algorithms proposed so far to solve the problem. A. Problem Deﬁnition Consider a WSN represented by a graph G = (V, E), where V is the set of n sensors v1, . . . , vn and E is the set of edges. The edge vivj ∈ E if and only if the distance between the vertices vi and vj is known. The signal range of each sensor in the network may vary. It is assumed that, for each subset V ⊂ V , with |V | = 3, the sensors in V are not collinear. The subset B ⊂ V is the set of beacon nodes in which each vi ∈ B has its global position deﬁned in the beginning of the process of localization. During the process, the position of the remaining sensors will be determined by the positions of nodes in B. Given the set of sensors and the distance measurements among them, the objective of MCLP is to ﬁnd the set B of beacon nodes with minimum size. The MCLP is formally deﬁned as follows: Minimum Cost Localization Problem: Given a graph G = (V, E) that represents the WSN, determine the subset B of sensors to be beacons such that the remaining nodes can be localized and the number of beacons vi ∈ B is minimized. Figure 1(a) illustrates a network used as an input to the problem, while Figure 1(b) shows the result with beacon nodes marked as black, as shown in Figure 1(b). Green nodes correspond to nodes that are located using the position of the beacons. The MCLP always return a feasible solution since, in the worst case, each sensor in the network is marked as beacon (B = V ). It is known that all sensors with degree less than three must be included in B and |B| ≥ 3 . Let V<3 the set of sensors with degree less than 3 and Bopt the optimal solution of MCLP. It is easily discernible that |V<3| is a lower bound of |Bopt|. In the scenario where all node degrees are larger or equal to three, the size of the minimum 3-dominating set (M3DS) of V gives an upper bound of |Bopt|. The minimum k-dominating set is a subset MkDS of nodes so that ni must have k neighbors included in MkDS, ∀ni /∈ MkDS. Furthermore, the size of MkDS is minimized. Thus, max{3, V<3} ≤ |Bopt| ≤ |M3DS|. Determining the minimum k-dominating set is NP-hard [15].
- 3. (a) Network before the execution (b) Network after the execution Fig. 1: Simulation in a network with 100 nodes B. Greedy-Sweep-2: A Greedy Algorithm for the MCLP As already mentioned, so far four different method have been proposed to solve the MCLP. Here we describe in detail Greedy-Sweep-2, which will be used as a baseline here and solves the MCLP as follows. It ﬁrst deﬁnes the status of the nodes by assigning colors to them. A white node represents a point which position has not been discovered yet. A black node deﬁnes a beacon node, which has the position deﬁned beforehand. Finally, a green node is a non-beacon node, but its position can be obtained by the position of three localized nodes using both trilateration (when three nodes are available) and local sweep (when two nodes with two other neighbors each are known). The process of trilateration in a 2D plane works as follows. Given three points r, s and t in the space with their positions deﬁned as well as the distance measurements among them and the point v to be localized. Three circles are deﬁned centered in r, s and t. The distance between each of the three points and v means the value of the radius. The intersection point of these three circles is the position of the point v. Figure 2 illustrates this scenario. We can observe that, if only the points r and s were deﬁned, the intersection between the circles deﬁned by them would return two points. The third point t gives the reference to deﬁne the correct position of the point v. Fig. 2: Trilateration In the case of local sweep, the process works as follows. Let r(u) be the number of neighbors of the node u with known locations. The idea is to identify two nodes v and w such that r(v) = r(w) = 2, i.e., two nodes with exactly two located neighbors. Neither v nor w can be located through trilateration, but there are only two possible positions for each of them. The distance between v and w may be used in some cases, though, to eliminate one of the two potential positions, identifying the remaining position as the true position of the node. We say that, when the method succeeds, a unique match has been found. Otherwise, it is said that the nodes have no unique match. More details on this method can be found in [10]. In the case of the algorithms proposed here, when using trilateration, the position of a node can be determined if the number of neighbors marked as either black or green is larger than or equal to 3. For each white node v it is maintained a rank r(v) to store the number of located neighbors of v. Once the algorithm marks the node v as black or green, the ranks of v’s neighbors are updated. If trilateration does not assign green to a node, the method based on local sweep is employed. Whenever r(v) = 3 or it is guaranteed the unique match, the color of v (and w, when using local sweep) is changed to green. The procedure is done recursively, as shown in Algorithm 1. The algorithm aims to deﬁne the minimum number of black nodes and, through their position, color the remaining nodes as green calculating, thus, their localization. The algorithm works as follows: (1) every node with degree less than three is marked as black, since they cannot be located by the position of other nodes (trilateration is based upon the position of three nodes). (2) In each step of the algorithm, the best white node is selected and marked as black. Here, we understand best node as the node that can beneﬁt most the
- 4. Algorithm 1: MARK(u, color) s(u) = color;1 for all u’s neighbor v and s(v) = white do2 r(v) = r(v) + 13 if any u’s white neighbor v and r(v) ≥ 3 then4 if color = black or green then5 MARK(v, green);6 if color = blue then7 MARK(v, blue);8 if any u’s white neighbors v and any v’s white neighbor9 w satisfying r(v) = r(w) = 2 and they are neighbor to each other then if both v and w have unique positions to guarantee10 the consistence of distance measurement then if color = black ou green then11 MARK(v, green);12 MARK(w, green);13 if color = blue then14 MARK(v, blue);15 MARK(w, blue);16 localization procedure in next step if marked as black. The procedure is shown in Algorithm 2. The algorithm terminates when the set of white nodes is empty. Algorithm 2: Greedy Localization Algorithm for each v ∈ V do1 s(v) = white and r(v) = 0;2 for each v ∈ V do3 if the degree of v ≤ 2 then4 MARK(v, black);5 while ∃ v such that s(v) = white do6 u = GREEDY-SELECTION;7 MARK(u, black);8 In order to pick the next white node to be colored, it is necessary to ﬁnd the one that will contribute most to the localization of the network. The algorithm runs a false MARK (Algorithm 1) in each white node v, assigning the blue color to it and, recursively, coloring other nodes as blue through trilateration. The number of blue nodes marked by the node v is checked and stored in c(v). The node with greater c(v) is selected as the next beacon. This procedure is shown in Algorithm 3. IV. A GENETIC ALGORITHM FOR THE MCLP This section presents a metaheuristic based on genetic algorithms to solve the MCLP. We believe the global search and noisy tolerance provided by the GA will improve search conditions of the beacons. We use a binary coded GA, where each individual represents a list of all sensors in the network, where one indicates the sensor is a beacon node and zero represents nodes that will be localized using trilateration and local sweep. Algorithm 3: GREEDY-SELECTION for all v and s(v) = white do1 MARK(v, blue);2 Let c(v) be the number of blue nodes;3 for all v and s(v) = blue do4 s(v) = white;5 Let r(v) be the number of its black and green6 neighbors; Return v with the maximum c(v) (tie is broken by ID).7 Algorithm 4: GENETIC-ALGORITHM Initialize the population1 while currentGeneration < limGenerations do2 Evaluate each individual using the number of3 beacon nodes Save the n individuals with best ﬁtness4 elitism5 Select the best individuals using a tournament6 Perform crossover with probability pc7 Perform mutation with probability pm8 currentGeneration++9 Output the best individual10 Having deﬁned the representation, the algorithm follows the steps described in Algorithm 4. The population is ran- domly initialized, and the ﬁtness calculated using the a WSN simulator. The simulator checks the feasibility of the solutions, and returns the ﬁnal ﬁtness of the individuals. The ﬁtness evaluation is followed by a tournament selection [16], where k individuals compete to undergo crossover and mutation operations, subject to probabilities pc and pm, respectively. The process goes on until a maximum number of generations is reached. As previously mentioned, the ﬁtness is calculated counting the number of beacon nodes in the individual, as deﬁned in Equation 1, and the minimum ﬁtness is the best solution to the problem. In Equation 1, nij always has one of the values 0 or 1 and represents the node j of the individual i. The variable nNodes represents the number of nodes in the network and j ranges from 1 to nNodes. During the evaluation process, the algorithm checks the feasibility of the solutions by determining the location of all nodes. This is done by executing Algorithm 1 for all selected beacon nodes. If, at the end of the process, there are still white nodes in the network, these nodes are automatically converted to beacon nodes. In this way, all individuals are valid (can localize all sensor nodes) after the evaluation process. fitnessi = nNodes j=1 nij (1) Concerning the genetic operators, nothing sophisticated is proposed in this ﬁrst version of the system. As observed in the results, a simple version of the method can outperform the greedy search without increasing computational time with more complex operators. Hence, a uniform crossover is used
- 5. Fig. 3: Example of crossover. in order to avoid the position bias of one-point crossover, as illustrated in Figure 3. The mutation process is also based on individual gene probabilities of swap. V. COMPUTATIONAL RESULTS In order to test the efﬁcacy of the proposed GA, extensive simulations were performed in different network sizes. All results are compared with Greedy-Sweep-2 [10], described in Section III-B. In order to test the algorithm, we used the simulator Sinalgo [17]. Sinalgo is a framework to test and validate algorithms in networks. It features conﬁgurable network conditions, such as the range of the nodes and their distribution on the plane. The project is written is Java, it is free and is published under the BSD license. Figure 4 shows the simulation environment of Sinalgo presenting a network with 10 nodes. After the execution of the program, the nodes 1, 2 and 9 were deﬁned as beacons. The remaining nodes have their locations found using the position of the beacon nodes. Fig. 4: Sinalgo with a simulation result In the simulations performed, random wireless sensor net- works were generated by Sinalgo. The number of nodes in the network varied from 500 to 3000. The nodes were uniformly distributed in a rectangular plane with dimensions of 1200 and 1000. The communication range of the nodes is deﬁned as 80. This feature means that, if the distance between two nodes is less or equal to 80, each one can communicate with the other. The evaluation of each test is based on the number of TABLE I: GA Parameters Parameter Value Number of generations 50 Number of individuals 100 Crossover rate 60% Mutation rate 30% Elitism 10% Tournament size 5 Fig. 5: Best solutions obtained by the greedy and genetic algorithms in networks with different number of nodes beacon nodes found in the solution of each algorithm. Each experiment performed with the GA was executed 30 times. The results show the mean and the conﬁdence interval out of all executions. The GA parameters were deﬁned using the smallest net- work, and preliminary tests executed in the bigger instances. The values are not optimized, and this task is left for future work. Their values are listed in Table I. We can observe that the performance of the genetic algo- rithm in the MCLP is always equal or better than the greedy algorithm. Figure 5 summarizes the results of the simulations performed in a network with different number of nodes. The conﬁdence intervals presented in Figure 5 show that results given by the GA are reliable, since the smaller the interval the more certain is the solution. For networks with more than 1500 nodes the solution of the algorithm is always similar and very close to the optimal solution. Figure 6 shows the results obtained in a different simulation in which only the range of the nodes would vary from 80 to 135, with a step of 5. We tested the greedy and genetic algorithms in all scenarios. It is clear that the results obtained by the genetic algorithm are, at least, equal to the solutions of the greedy algorithm. This happens because, in this case, the results given by the latter are inputted to the GA, serving as an individual in the initial population and reducing the time of convergence. It can be also observed that the GA always yields better results than the greedy algorithm, when the range is at least 95. This might be due to the fact that, the smaller
- 6. Fig. 6: Simulation results in a ﬁxed network with 500 nodes, varying the range of the nodes the range is, the greater it tends to be the number of connected components of the network, resulting in small subproblems that can be easily resolved by the greedy algorithm. With a greater range value, though, the graph tends to be more connected, making the problem too complex for the greedy algorithm to perform as well as the GA. Furthermore, the GA solution reaches a stable value when the communication range is greater than 110. Again, the connectivity of the nodes justiﬁes the fact because with a large communication range, the network becomes highly connected and helps the GA to ﬁnd the best solution in most of their performances. Figure 7 shows results of simulations in networks with number of nodes varying from 500 to 3000, with a step of 500. For each network, the results of the greedy algorithm are compared to the results of the best individual produced by the GA and the mean ﬁtness values of all individuals in the population. In Figure 7(a), where the number of nodes is the smallest, the genetic algorithm gave just slightly better results in terms of number of beacon nodes selected (y axis in the graph). In the remaining simulations, represented by Figu- res 7(b)-7(f), the GA generated a substantial difference in the number of beacon nodes. The mean of all solutions generated by the genetic algorithm converges to a ﬁnal value during the execution. This behavior shows that the population is evolving in each generation, i.e., the GA ﬁnds better solutions in next generations and the ﬁnal population has better individuals than the initial population. It is clear that, in the populations of GA, there exist many different individuals since the mean is worse than the greedy solution and the best solution found in GA is better than solutions of the greedy one. The curve representing the best solution of the GA shows that results of GA are always better than the greedy algorithm. In this simulation, the genetic algorithm reached a solution 50% better than the greedy algorithm (see for example, the network with 1500 nodes). VI. CONCLUSION In this paper we investigated the Minimum Cost Locali- zation Problem for WSNs. We introduced in the literature an innovative formulation for this problem. We also elaborated a genetic algorithm to identify the minimum set of beacon nodes enough to locate every sensor belonging to a network. The experiments conducted verify the efﬁciency of the method, as it overcame the greedy algorithms presented in the literature in the tested scenarios. In some cases, the genetic algorithm reached a solution more than 50% better. A possible optimization might be to use different methods other than trilateration and local sweep to calculate the ﬁtness function. Other related problems remain as future work. A problem of interest might be, for example, to minimize not the number of beacon nodes itself, but the size of the minimum closed path that traverses every beacon node, maintaining the restriction of localizing every sensor in the network. Another important issue is to consider how precise the location given by the GA is. In this case, a multi-objective algorithm could be considered to take both measures into account. REFERENCES [1] A. A. F. Loureiro, J. M. S. Nogueira, L. B. Ruiz, R. A. Mini, E. F. Nakamura, and C. M. S. Figueiredo, “Wireless sensors networks (in portuguese),” in Proceedings of the 21st Brazilian Symposium on Computer Networks (SBRC’03), Natal, RN, Brazil, May 2003, pp. 179– 226, tutorial. [2] L. B. Ruiz, L. H. A. Correia, L. F. M. Vieira, D. F. Macedo, E. F. Nakamura, C. M. S. Figueiredo, M. A. M. Vieira, E. H. B. Maia, D. Cˆamara, A. A. F. Loureiro, J. M. S. Nogueira, D. C. da Silva Jr., and A. O. Fernandes, “Arquiteturas para redes de sensores sem ﬁo,” in Proceedings of the 22nd Brazilian Symposium on Computer Networks (SBRC’04), Gramado, RS, Brazil, May 2004, pp. 167–218, tutorial. ISBN: 85-88442-82-5. [3] C. Yawut and S. Kilaso, “A wireless sensor network for weather and disaster alarm systems,” in Proceedings 2011 International Conference on Information and Electronics Engineering (ICIEE 2011), vol. 6, 2011, pp. 155–160. [4] B. Hofmann-Wellenhof, H. Lichtenegger, and J. Collins, Global Positioning System: Theory and Practice, 5th ed. Springer, Feb. 2001. [Online]. Available: http://www.worldcat.org/isbn/3211835342 [5] T. He, C. Huang, B. M. Blum, J. A. Stankovic, and T. Abdelzaher, “Range-free localization schemes for large scale sensor networks,” in Proceedings of the 9th annual international conference on Mobile computing and networking, ser. MobiCom ’03. New York, NY, USA: ACM, 2003, pp. 81–95. [Online]. Available: http://doi.acm.org/10.1145/938985.938995 [6] A. Savvides, C.-C. Han, and M. B. Strivastava, “Dynamic ﬁne-grained localization in ad-hoc networks of sensors,” in Proceedings of the 7th annual international conference on Mobile computing and networking, ser. MobiCom ’01. New York, NY, USA: ACM, 2001, pp. 166–179. [Online]. Available: http://doi.acm.org/10.1145/381677.381693 [7] D. Moore, J. Leonard, D. Rus, and S. Teller, “Robust distributed network localization with noisy range measurements,” in Proceedings of the 2nd international conference on Embedded networked sensor systems, ser. SenSys ’04. New York, NY, USA: ACM, 2004, pp. 50–61. [Online]. Available: http://doi.acm.org/10.1145/1031495.1031502 [8] M. Rodrigues, L. Vieira, and M. Campos, “Fingerprinting-based radio localization in indoor environments using multiple wireless technolo- gies,” in Personal Indoor and Mobile Radio Communications (PIMRC), 2011 IEEE 22nd International Symposium on, sept. 2011, pp. 1203 – 1207. [9] Y. Gao, Y. Zhuang, T. Ni, K. Yin, and T. Xue, “An improved genetic algorithm for wireless sensor networks localization.” in BIC-TA. IEEE, 2010, pp. 439–443.
- 7. [10] M. Huang, S. Chen, and Y. Wang, “Minimum cost localization problem in wireless sensor networks,” Ad Hoc Networks, vol. 9, pp. 387–399, May 2011. [Online]. Available: http://dx.doi.org/10.1016/j.adhoc.2010. 08.013 [11] J. Aspnes, T. Eren, D. K. Goldenberg, A. S. Morse, W. Whiteley, Y. R. Yang, B. D. O. Anderson, and P. N. Belhumeur, “A theory of network localization,” IEEE Transactions on Mobile Computing, vol. 5, no. 12, pp. 1663–1678, Dec. 2006. [12] C.-H. Wu, W. Sheng, and Y. Zhang, “Mobile sensor networks self localization based on multi-dimensional scaling,” in Robotics and Automation, 2007 IEEE International Conference on, april 2007, pp. 4038 –4043. [13] K. feng Ssu, C. ho Ou, and H. C. Jiau, “Localization with mobile anchor points in wireless sensor networks,” IEEE Transactions on Vehicular Technology, pp. 1187–1197, 2005. [14] M. Azim, Z. Aung, W. Xiao, and V. Khadkikar, “Localization in wireless sensor networks by cross entropy method,” 2012. [15] W. Shang, F. F. Yao, P.-J. Wan, and X. Hu, “On minimum m -connected k -dominating set problem in unit disc graphs,” J. Comb. Optim., vol. 16, no. 2, pp. 99–106, 2008. [16] T. Blickle and L. Thiele, “A mathematical analysis of tournament selection,” in Proceedings of the Sixth International Conference on Genetic Algorithms. Morgan Kaufmann, 1995, pp. 9–16. [17] D. C. G. A. E. Zurich, “Sinalgo - simulator for network algorithms,” 2013, http://dcg.ethz.ch/projects/sinalgo/, accessed 30-January-2013.
- 8. (a) Network with 500 nodes (b) Network with 1000 nodes (c) Network with 1500 nodes (d) Network with 2000 nodes (e) Network with 2500 nodes (f) Network with 3000 nodes Fig. 7: Simulations in networks with 500, 1000, 1500, 2000, 2500 and 3000 nodes. Figure 7(a) shows that GA gave slightly better results than the greedy algorithm. In Figures 7(b)-7(f) is shown that the performance of the GA is better than performance of the greedy algorithm, ﬁnding a reduced number of beacon nodes.

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