cec2013

A Genetic Algorithm for the Minimum Cost
Localization Problem in Wireless Sensor Networks
Angelo F. Assis, Luiz Filipe M. Vieira, Marco Túlio 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-

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].

(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

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

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

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. Câmara, 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.

[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.

(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.

cec2013

More Related Content

What's hot

Viewers also liked

Similar to cec2013

cec2013