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A study on the performance of local search versus population-based methods for mesh router nodes placement problem

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J Intell Manuf (2012) 23:2057–2067DOI 10.1007/s10845-011-0507-7
A study on the performance of local search versuspopulation-based methods for mesh router nodes placementproblem
Admir Barolli
·
Fatos Xhafa
·
Christian Sánchez
·
Makoto Takizawa
Received: 4 October 2010 / Accepted: 24 January 2011 / Published online: 5 February 2011© Springer Science+Business Media, LLC 2011
Abstract
Node placement problems have been long inves-tigatedintheoptimizationﬁeldduetonumerousapplicationsinfacilitylocation,logistics,services,etc.Suchproblemsareattractingagaintheattentionofresearchersnowfromthenet-working domain, and more especially from Wireless MeshNetworks(WMNs)ﬁeld.Indeed,theplacementofmeshrou-ters nodes appears to be crucial for the performance andoperability of WMNs, in terms of network connectivity andstability. However, node placement problems are known fortheir hardness in solving them to optimality, and thereforeheuristics methods are approached to near-optimally solvesuch problems. In this work we evaluate the performance of differentheuristicmethodsinordertojudgeontheirsuitabil-ity of solving mesh router nodes problem. We have selectedmethods from two different families, namely, local searchmethods (Hill Climbing and Simulated Annealing) and pop-ulation-basedmethods(GeneticAlgorithms).Theformerareknown for their capability to exploit the solution space byconstructing a path of visited solutions, while the later meth-ods use a population of individuals aiming to largely explorethe solution space. In both cases, a bi-objective optimiza-
A. Barolli
·
M. TakizawaDepartment of Computers and Information Science,Seikei University, 3-3-1 Kichijoji-Kitamachi, Musashino-Shi,Tokyo 1808633, Japane-mail: admir.barolli@gmail.comM. Takizawae-mail: makoto.takizawa@computer.orgF. Xhafa (
B
)
·
C. SánchezDepartment of Languages and Informatics Systems,Technical University of Catalonia, C/Jordi Girona, 1-3,08034 Barcelona, Spaine-mail: fatos@lsi.upc.eduC. Sáncheze-mail: csanchez@lsi.upc.edu
tion consisting in the maximization of the size of the giantcomponent in the mesh routers network (for measuring net-work connectivity) and that of user coverage are considered.In the experimental evaluation, we have used a benchmark of instances—varying from small to large size—generatedusing different distributions of mesh node clients (Uniform,Normal, Exponential and Weibull).
Keywords
Wireless mesh networks
·
Local search
·
Genetic algorithms
·
Size of giant component
·
User coverage
Introduction
Node placement problems have been long investigated in theoptimization ﬁeld due to numerous applications in locationscience (facility location, logistics, services, etc) and classi-ﬁcation (clustering). In such problems, we are given a num-ber of potential facilities to serve to costumers connected tofacilities aiming to ﬁnd locations such that the cost of serv-ing all customers is minimized. In traditional versions of theproblem, facilities could be hospitals, polling centers, ﬁrestations serving to a number of stationary clients and aim-ing to minimize some distance function in a metric spacebetween stationary clients and such facilities. Recently, suchproblemsareshowingtheirusefulnesstocommunicationnet-works,wherefacilitiescouldbeservers,routers,etc.offeringconnectivity services to clients.Facility location problems are showing their usefulness tocommunication networks, and more especially from Wire-less Mesh Networks (WMNs) ﬁeld. Wireless Mesh Net-works (Akyildiz et al. 2005; Nandiraju et al. 2007) are
currently attracting a lot of attention from wireless researchand technology community due to their importance as a
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2058 J Intell Manuf (2012) 23:2057–2067
means for providing cost-efﬁcient broadband wireless con-nectivity. WMNs infrastructures are currently used in devel-oping and deploying medical, transport and surveillanceapplications in urban areas, metropolitan, neighboring com-munities and municipal area networks (Chen and Chekuri2007). WMNs are based on mesh topology, in which everynode (representing a server) is connected to one or morenodes, enabling thus the information transmission in morethan one path. The path redundancy is a robust feature of this kind of topology. Compared to other topologies, Meshtopology needs not a central node, allowing networks basedon such topology to be self-healing. These characteristicsof networks with mesh topology, make them very reliableand robust networks to potential server node failures. InWMNs mesh routers provide network connectivity servicesto mesh client nodes. The good performance and operabil-ity of WMNs largely depends on placement of mesh routersnodes in the geographical area to achieve network connec-tivity, stability and user coverage. The objective is to ﬁnd anoptimal and robust topology of the mesh network to supportconnectivity services to stationary clients.For most formulations, node placement problems areshown to be computationally hard to solve to optimality(Garey and Johnson 1979; Lim et al. 2005; Amaldi et al.
2008; Wanget al. 2007), and therefore heuristic and meta-
heuristic approaches are the de facto approach to solve theproblem for practical purposes. Node placement problemsare also closely related to facility layout problems (Chungand Chung 1999). Several heuristic approaches are foundin the literature for node placement problems in WMNs(Muthaiah and Rosenberg 2008; Zhou et al. 2007; Tang
2009; Franklin and Murthy 2007; Vanhatupa et al. 2007).
In this work, we evaluate the performance of different heu-ristic methods in order to judge on their suitability of solvingmesh router nodes problem. We have selected methods fromtwo different families, namely, local search methods (HillClimbing and Simulated Annealing) and population-basedmethods (Genetic Algorithms). The former are known fortheir capability to exploit the solution space by constructingapathofvisitedsolutions,whilethelatermethodsuseapop-ulation of individuals aiming to largely explore the solutionspace. In both cases, a bi-objective optimization, namely, themaximization of the size of the giant component in the net-work (for measuring network connectivity) and that of usercoverage are considered. In the experimental evaluation, wehave used a benchmark of instances—varying from small tolarge size—generated using different distributions of meshnode clients (Uniform, Normal, Exponential and Weibull).The rest of the paper is organized as follows. In section“Problem definition” we present the definition of the meshrouter nodes placement problem in WMNs. Local searchmethods are presented in section “Local search algorithms”andGAsformeshrouternodesplacementinsection“Geneticalgorithms”. The experimental evaluation of local searchmethods and GAs is given in section “Experimental study”.Weendthepaperinsection“Conclusion”withsomeconclu-sions.
Problem definition
In a general setting, location models in the literature havebeendeﬁnedasfollows.Wearegiven:(a)auniverse
U
,fromwhichaset
C
ofclientinputpositionsisselected;(b)aninte-ger,
N
≥
1, denoting the number of facilitiesto be deployed;(c) one or more metrics of the type
d
:
U
×
U
→
R
+
,which measure the quality of the location; and, (d) an opti-mization model. Then, the optimization model takes in inputthe universe where facilities are to be deployed, a set of cli-ent positions and returns a set of positions for facilities thatoptimize the considered metrics.Itshouldbenotedthatdifferentmodelscanbeestablisheddepending on whether the universe is considered: (a)
contin-uous
(universe is a region, where clients and facilities maybe placed anywhere within the
continuum
leading to an un-countably inﬁnite number of possible locations); (b)
discrete
(universe is a discrete set of predeﬁned positions); and, (c)
network
(universe is given by an undirected weighted graph;in the graph, client positions are given by the vertices andfacilities may be located anywhere on the graph).Weconsidertheversionofthemeshnodeplacementprob-lem corresponding to the network space model above. Thus,in this version, we are given a 2D area where to distributea number of mesh router nodes and a number of mesh cli-ent nodes of ﬁxed positions (of an arbitrary distribution) andﬁnds a location assignment for the mesh routers that maxi-mizesthenetworkconnectivity(sizeofthegiantcomponent)andclientcoverage.Aninstanceoftheproblemconsiststhusof: (a)
N
mesh router nodes, each having its own radio cov-erage, deﬁning thus a vector of routers; (b) an area
W
×
H
wheretodistribute
N
meshrouters.Positionsofmeshroutersare not pre-determined. The area is divided in square cellsof an a priori ﬁxed length and mesh router nodes are to bedeployed in the cells of the grid area; and, (c)
M
client meshnodes located in arbitrary cells of the considered grid area,deﬁning a matrix of clients.An instance of the problem can be formalized by an adja-cency matrix of the WMN graph, whose nodes are of twotypes: router nodes and client nodes and whose edges arelinks in the mesh network (there is a link between a meshrouter and mesh client if the client is within radio cover-age of the router). Each mesh node in the graph is a triple
v
=
x
,
y
,
r
representing the 2D location point and
r
is theradius of the transmission range. There is an arc betweentwo nodes
u
and
v
, if
v
is within the transmission circulararea of
u
. The deployment area is partitioned by grid cells,
1 3
J Intell Manuf (2012) 23:2057–2067 2059
representing graph nodes, where we can locate mesh routernodes. In fact, in a cell, both a mesh and a client node can beplaced.The objective is to place mesh router nodes in cells of considered area to maximize network connectivity and usercoverage.Networkconnectivityandusercoverageareamongmost important metrics in WMNs. The former measures thedegreeofconnectivityofthemeshnodeswhilethelaterrefersto the number of mesh client nodes connected to the WMN.Both objectives areimportantand directlyaffectthenetwork performance; nonetheless, network connectivity is consid-ered as more important than user coverage. It should also benoted that in general optimizing one objective could affecttheotherobjectivealthoughthereisnodirectrelationamongthese objectives nor are they contradicting.Optimization settingFor optimization problems having two or more objectivefunctions, two settings are usually considered: the hierarchi-cal and simultaneous optimization. In the former, the objec-tives are classiﬁed (sorted) according to their priority. Thus,for the two objective case, one of the objectives, say
f
1
, isconsidered as primary objective and the other, say
f
2
, as sec-ondary one. The meaning is that the optimization is carriedout in two steps: in the ﬁrst we try to optimize
f
1
, and then,wetrytooptimize
f
2
withoutworseningthebestvalueof
f
1
.In the later approach, both objectives are optimized simulta-neously.Inthisworkwehaveconsideredthehierarchicalapproachin which the size of the giant component is a primary objec-tive and the user coverage is a secondary one. The network connectivity(throughthemaximizationofsizeofgiantcom-ponent) is in fact crucial to WMNs.Client mesh nodes distributionsIt should be noticed from the above problem description thatmesh client nodes can be arbitrarily situated in the givenarea. For evaluation purposes, it is interesting, however, toconsider concrete distributions of clients. For instance, it hasbeen shown from studies in real urban areas or universitycampuses that mobile users tend to cluster to hotshots. Wehave considered Uniform, Normal, Exponential and Weibulldistributions for client mesh nodes (see section “Experimen-tal study”).
Local search algorithms
Forthisstudy,wehaveselectedHillClimbing(HC)andSim-ulated Annealing (SA) local search methods.Hill climbing for mesh router node placementWe present here the particularization of the Hill Climbingalgorithm (see Algorithm 1) for the mesh router node place-ment problem in WMNs.
Algorithm 1
Hill Climbing algorithm for maximization.
f
is the ﬁtness function
1:
Start
: Generate an initial solution
s
0
;2:
s
=
s
0
;
s
∗
=
s
0
;
f
∗
=
f
(
s
0
)
;3:
repeat
4:
Movement Selection
: Choose a movement
m
=
select
_
mo
v
ement
(
s
)
;5:
Evaluate & Apply Movement
:6:
if
δ(
s
,
m
)
≥
0
then
7:
s
′ =
appply
(
m
,
s
)
;8:
s
=
s
′
;9:
end if
10:
Update Best Solution
:11:
if
f
(
s
′
) >
f
(
s
∗
)
then
12:
f
∗
=
f
(
s
′
)
;13:
s
∗
=
s
′
;14:
end if
15:
Return
s
∗
,
f
∗
;16:
until
(stopping condition is met)
Initial solution.
The algorithms starts by generating an ini-tialsolutioneitherrandomorbyadhocmethods(Xhafaetal.2009).
Evaluation of ﬁtness function
An important aspect is thedetermination of an appropriate objective function and itsencoding. In our case, the ﬁtness function follows a hierar-chical approach in which the main objective is to maximizethe size of giant component in WMN.
Neighborselectionandmovementtypes
Theneighborhood
N
(
s
)
of a solution
s
consists of all solutions that are accessi-ble by a local move from
s
. We have considered three differ-enttypesofmovements.Theﬁrst,called
Random
,consistsinchoosing a router at random in the grid area and placing it ina new position at random. The second move, called
Radius
,choosestherouterofthelargestradioandplacesitatthecen-ter of the most densely populated area of client mesh nodes(see Algorithm 2). Finally, the third move, called
Swap
, con-sists in swapping two routers: the one of the smallest radiosituated in the most densely populated area of client meshnodes with that of largest radio situated in the least denselypopulated area of client mesh nodes. The aim is that largestradio routers should serve to more clients by placing them inmore dense areas.We also considered the possibility to combine the abovemovements in sequences of movements. The idea is to see if
1 3
2060 J Intell Manuf (2012) 23:2057–2067
Algorithm 2
Radius movement
1:
Input
: Values
H
g
and
W
g
for height and width of a small grid area.2:
Output
: New conﬁguration of mesh nodes network.3: Compute the most dense
H
g
×
W
g
area and
(
x
dense
,
y
dense
)
itscentral cell point.4: Compute the position of the router of largest radio coverage
(
x
largest
_
co
v
,
y
largest
_
co
v
)
.5: Move router at
(
x
largest
_
co
v
,
y
largest
_
co
v
)
to new position
(
x
dense
,
y
dense
)
.6: Re-establish mesh nodes network connections.
the combination of these movements offers some improve-ment over the best of them alone. We called this type of movement
Combination
:
<
Rand
1
,...,
Rand
k
;
Radius
1
,...,
Radius
k
;
S
w
ap
1
,...,
S
w
ap
k
>
,where
k
is a user speciﬁed parameter.
Acceptability criteria
The acceptability criteria for newlygenerated solution can be done in different ways (simpleascent, steepest ascent, or stochastic). In our case, we haveadopted the simple ascent, that is, if
s
is current solutionand
m
is a movement, the resulting solution
s
′
obtained byapplying
m
to
s
will be accepted, and hence become currentsolution, iff the ﬁtness of
s
′
is at least as good as ﬁtness of solution
s
. In terms of
δ
function,
s
′
is accepted and becomescurrent solution if
δ(
s
,
m
)
≥
0. It should be noted that in thisdefinition we are also accepting solutions that have the sameﬁtness as previous solution. The aim is to give chances to thesearch to move towards better solutions in solution space.A more strict version would be to accept only solutions thatstrictly improve the ﬁtness function (
δ(
s
,
m
) >
0).Simulated annealingSA algorithm is a generalization of the Metropolis heuristic.Indeed, SA consists of a sequence of executions of Metrop-olis with a progressive decrement of the temperature startingfrom a rather high temperature, where almost any move isaccepted, to a low temperature, where the search resemblesHillClimbing.Infact,itcanbeseenasahill-climberwithaninternal mechanism to escape local optima (see pseudo-codein Algorithm 3). In SA, the solution
s
′
is accepted as the newcurrent solution if
δ
≤
0 holds, where
δ
=
f
(
s
′
)
−
f
(
s
).
Toallow escaping from a local optimum, moves that increasethe energy function are accepted with a decreasing probabil-ity exp
(
−
δ/
T
)
if
δ >
0, where
T
is a parameter called the“temperature”. The decreasing values of T are controlled bya
cooling schedule
, which speciﬁes the temperature valuesat each stage of the algorithm, what represents an impor-tant decision for its application (a typical option is to use aproportional method, like
T
k
=
α
·
T
k
−
1
). SA usually givesbetter results in practice, but uses to be very slow. The moststriking difﬁculty in applying SA is to choose and tune itsparameters such as initial and ﬁnal temperature, decrementofthetemperature(coolingschedule),equilibriumdetection,etc.
Algorithm 3
: Pseudo-code of simulated annealing (SA)
t
:= 0Initialize
T s
0 := Initial_Solution()
v
0 := Evaluate(
s
0)
while
(stopping condition not met)
dowhile
t
mod MarkovChainLen = 0
do
t
:=
t
+1
s
1 := Generate(
s
0
,
T
) //
Move
v
1 := Evaluate(
s
1)
if
Accept(
v
0
,v
1
,
T
)
then
s
0 :=
s
1
v
0 :=
v
1
end if end while
T
:= Update(
T
)
end while
return
s
0
The initial solution, ﬁtness evaluation and movementtypes speciﬁed in subsection “Hill climbing for mesh routernode placement” are also valid to SA (refer to Xhafa et al.2010c for further details.) However, the acceptability crite-ria of neighboring solutions is now different, as explainednext.
Acceptability criteria
The acceptability criteria for newlygenerated solution is based on the definition of a thresholdvalue (accepting threshold) as follows. We consider a suc-cession
t
k
such that
t
k
>
t
k
+
1
,
t
k
>
0 and
t
k
tends to 0 as
k
tends to inﬁnity. Then, for any two solutions
s
i
and
s
j
, if
f itness
(
s
j
)
−
f itness
(
s
i
) <
t
k
, then accept solution
s
j
.For the SA,
t
k
values are taken as accepting threshold butthe criterion for acceptance is probabilistic:– If
f itness
(
s
j
)
−
f itness
(
s
i
)
≤
0 then
s
j
is accepted.– If
f itness
(
s
j
)
−
f itness
(
s
i
) >
0 then
s
j
is acceptedwithprobabilityexp
[
(
f itness
(
s
j
)
−
f itness
(
s
i
))/
t
k
]
(atiteration
k
thealgorithmgeneratesarandomnumber
R
∈
(
0
,
1
)
and
s
j
is accepted if
R
<
exp
[
(
f itness
(
s
j
)
−
f itness
(
s
i
))/
t
k
]
)
.In this case, each neighbor of a solution has a positiveprobability of replacing the current solution. The
t
k
valuesare chosen in way that solutions with large increase in thecost of the solutions are less likely to be accepted (but thereis still a positive probability of accepting them).
1 3
J Intell Manuf (2012) 23:2057–2067 2061
Genetic algorithms
Genetic Algorithms (GAs) (Holland 1975) have shown theirusefulness for the resolution of many computationally com-binatorial optimization problems. For the purpose of thiswork we have used the
template
given in Algorithm 4.
Algorithm 4
Genetic Algorithm template
Generate the initial population
P
0
of size
µ
;Evaluate
P
0
;
while
not termination-condition
do
Select the parental pool
T
t
of size
λ
;
T
t
:=
Select
(
P
t
)
;Perform crossover procedure on pairs of individuals in
T
t
withprobability
p
c
;
P
t c
:=
Cross
(
T
t
)
;Perform mutation procedure on individuals in
P
t c
with probability
p
m
;
P
t m
:=
Mutate
(
P
t c
)
;Evaluate
P
t m
;Create a new population
P
t
+
1
of size
µ
from individuals in
P
t
and/or
P
t m
;
P
t
+
1
:=
Replace
(
P
t
;
P
t m
)
t
:=
t
+
1;
end whilereturn
Best found individual as solution;
We present next the particularization of GAs for the meshrouternodesplacementinWMNs(seeXhafaetal.2010bformore details).EncodingThe encoding of individuals (also known as chromosomeencoding) is fundamental to the implementation of GAs inordertoefﬁcientlytransmitthegeneticinformationfrompar-ents to offsprings.In the case of the mesh router nodes placement problem,a solution (individual of the population) contains the infor-mation on the current location of routers in the grid area aswell as information on links to other mesh router nodes andmeshclientnodes.Thisinformationiskeptindatastructures,namely,
pos_routers
for positions of mesh router nodes,
routers_links
for link information among routers and
client_router_link
for link information among rou-ters and clients (matrices of the same size as the grid area areused.) Based on these data structures, the size of the giantcomponent and the number of users covered are computedfor the solution.It should be also noted that routers are assumed to havedifferent radio coverage, therefore to any router could belinked to a number of clients and other routers. Obviously,whenever a router is moved to another cell of the grid area,theinformationonlinkstobothotherroutersandclientsmustbe computed again and links are re-established.Population and ﬁtness evaluationThe ﬁrst population of individuals is done generating a fewindividuals using ad hoc methods in Xhafa et al. (2010a) and
the rest of individuals by large perturbations of the ad hocindividuals. Regarding the ﬁtness evaluation, ss stated ear-lier, we have adopted the bi-objective case, in which the sizeof the giant component is considered primary and the usercoverage secondary (see subsection “Optimization setting”)Selection operatorsThe selection operators considered in this work are thosebased on
implicit ﬁtness re-mapping
technique. They are:(a)
Linear Ranking Selection
This operator selects the indi-viduals in the population with a probability directly propor-tional to their ﬁtness value; (b)
Best Selection
This operatorselects higher ﬁtness individuals in the population. Clearly,by always choosing the best ﬁtted individuals of the popu-lation, the GA converges prematurely; and, (c)
Tournament Selection
This operator selects the individuals based on theresult of a tournament among individuals.Crossover operatorsThe crossover operators are the most important ingredientof GAs. Indeed, by selecting individuals from the parentalgeneration and interchanging their
genes
, new individuals(descendants) are obtained. It is very important to stress thatthecrossoveroperatorsdependonthechromosomerepresen-tation. The crossover operator should thus take into accountthe specifics of mesh router nodes encoding. We have con-sidered the crossover operator, called
intersection operator
(see Algorithm 5).
Algorithm 5
Crossover operator
1:
Input
: Two parent individuals
P
1
and
P
2
; values
H
g
and
W
g
forheight and width of a small grid area;2:
Output
: Two offsprings
O
1
and
O
2
;3: Select at random a
H
g
×
W
g
rectangle
RP
1
in parent
P
1
. Let
RP
2
be the same rectangle in parent
P
2
;4: Select at random a
H
g
×
W
g
rectangle
RO
1
in offspring
O
1
. Let
RO
2
be the same rectangle in offspring
O
2
;5: Interchange the mesh router nodes: Move the mesh router nodes of
RP
1
to
RO
2
and those of
RP
2
to
RO
1
;6: Re-establish mesh nodes network connections in
O
1
and
O
2
(linksbetweenmeshrouternodesandlinksbetweenclientmeshnodesandmesh router nodes are computed again);7:
return
O
1
and
O
2
;
1 3

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