2012 International Conference on Information and Computer Networks (ICICN 2012)
IPCSIT vol. 27 (2012) © (2012) IACSIT Press, Singapore
A Shared Incumbent Environment for the Minimum Power
Broadcasting Problem in Wireless Networks
N.E. Toklu, R. Montemanni*, G. Di Caro and L.M. Gambardella
Istituto Dalle Molle di Studi sull'Intelligenza Artificiale (IDSIA)
Galleria 2, CH-6928 Manno, Switzerland
Abstract. In this study, we consider the minimum power broadcasting problem in wireless actuator
networks. We attack the problem with a method - the shared incumbent environment - that executes two
algorithms in parallel: a mathematical programming approach and a simulated annealing approach.
According to the shared incumbent environment paradigm, when an incumbent solution is found by one
method, the other method is notified and profits from the information received. Experimental results show
that the shared incumbent environment lead to results which are better than those of the two algorithms
combined in it taken singularly.
Keywords: Shared incumbent environment, mixed integer linear programming, simulated annealing,
matheuristics, minimum power broadcast, wireless networks.
1. Introduction
Wireless actuator networks establish communication by using devices called terminals that use omnidirectional antennea to send and receive radio signals. The same data can be sent to multiple terminals at the
same time, as long as they are within the coverage area of the sender terminal (this is called wireless
multicast advantage [1]). Terminals which are outside the coverage area of the sender terminal can still
receive the data with the help of intermediate terminals acting as routers [1, 2].
Among the applications of wireless networks, an interesting one is that consisting in commanding from
remote the actuators, which are at locations difficult to be reached by people [3]. In such applications, the
command is generated in a source terminal and sent to the wireless terminals attached to the actuators.
The Minimum Power Broadcast Problem (MPBP) is faced because of the fact that the terminals usually
depend on small mobile batteries. This means, there is limited power available for the network. As the
coverage area of a terminal increases, the power usage also increases. Therefore, finding a topology where
the coverage areas are minimized would decrease the power usage and ensure a longer life span for the
network. Thus, MPBP is the problem of finding a topology in which all terminals can receive data from the
source terminal, with the total transmission power is minimized [1].
Many different approaches were previously proposed for solving MPBP. Among them are a polynomialtime heuristic called broadcast incremental power algorithm [1] and different approaches based on mixed
integer linear programming (MILP) [4, 5]. A matheuristic [6] approach for MPBP was also presented, where
the algorithm was a combination of linear programming and a quantum inspired evolutionary algorithm [2].
Recently, a matheuristic called shared incumbent environment, which executes a linear programming
solver and an ant system meta-heuristic search in parallel, was proposed for sequential ordering problems [7].
According to the shared incumbent environment, the MILP and the meta-heuristic search inform each other
*
Corresponding author. Tel.: +41 58 666 666 7; fax: +41 58 666 666 1.
E-mail address: roberto@idsia.ch.
158
when they improve the best solutioon retrieved since the beeginning of the
t computattion. The pu
urpose of thee
shared incumbent enviroonment is to quickly findd solutions with
w the help of the meta--heuristic and inform thee
er,
so
that,
k
knowing
that
t
better
solut
tions
already
y exist, the solver
s
does nnot explore unpromising
u
g
MILP solve
areas of the solution space
s
and reeaches to thhe useful sollutions moree quickly. O
On the otherr hand, new
w
improving solutions
s
rettrieved by thhe MILP solvver are used
d by the metaa-heuristic too exit from situations off
premature convergence/
c
/stagnation.
In this study,
s
we aim
m to obtain improved
i
low
wer and uppeer bound for large MPBP
P instances under
u
limitedd
computationn times, by implementinng a shared incumbent environment,
e
, where the solver work
ks in parallell
with a simuulated annealiing meta-heuuristic algorithm.
This paaper is organnized as folloows. In sectiion 2 the pro
oblem definiition is proviided via a MILP
M
model..
Section 3 exxplains a sim
mulated anneealing approaach for the problem. In section 4 intrroduces the details
d
of thee
shared incuumbent envirronment for this problem
m. Experimen
ntal results are
a shown inn section 5. Finally, thee
conclusionss are drawn inn section 6.
2. A mixxed integeer linear programm
p
ming mod
del
The MP
PBP consistss in setting thhe transmissiion power off each terminnal to create such a topollogy that thee
source term
minal will bee able to trannsmit to all other termin
nals directly or with the help of oth
her terminalss
acting as rouuters, with thhe total transsmission pow
wer minimizeed.
To form
mulate the MPBP,
M
let us have graph G=(V,A), where
w
V is thee set of term
minals and A is the set off
arcs which represent coonnections beetween pairs of terminalss. An arc connsisting of thhe two termiinals i, j ∈ V
is denoted as
a (i,j). The source
s
terminnal is represeented by s ∈ V. The transsmission pow
wer required for terminall
i to includee terminal j within its coverage areaa is denoted
d by rij. We also set thee power requ
uirement forr
terminal i to reach itsellf as rii =0. Now,
N
let us have an arraay vi for eacch terminal ii. The array vi stores thee
indices of all
a terminals, sorted in a non-decreasi
n
ing order acccording to the power requuirement theey impose onn
terminal i. We
W call vik ass the k-th desstination of teerminal i.
The prooblem can bee expressed as a flow-baased MILP formulation
fo
[ In the foormulation, xik representss
[2].
the binary decision
d
wheether terminaal i should have
h
its k-th destination as the fartheest reaching point for itss
coverage arrea or not, annd yij represents the flow in arc (i,j). The
T assumpttion behind tthe formulatiion is that ann
arborescencce is containeed in each feasible conneected topolog
gy.
Accordiing to the coonstraints (2), only one faarthest reachiing point cann be selected for each term
minal i. Thiss
means, one terminal caan have only one coveragge area. Con
nstraints (3) connect x vaariables and y variables..
t (4), the network
n
flow
w representedd by yij mustt form an arbborescence. The domain
ns of x and y
According to
variables arre specified inn constraintss (5) and (6), respectively
y.
3. A sim
mulated an
nnealing approach
a
159
We now introduce a simulated annealing approach which was proposed for finding suboptimal solutions
for MPBP quickly [8]. Simulated annealing [9] (SA) is a meta-heuristic approach which simulates the
process of annealing in metallurgy. This process involves heating, which triggers the atoms to change their
initial positions and slow cooling, which slowly decreases the chances for the atoms to change to a worse
configuration, turning the process into a controlled local search. Into the simulated annealing algorithm, the
temperature is reflected as a variable which effects the chance of accepting a candidate solution vector with a
worse quality than the current solution vector.
The simulated annealing approach we use here starts by applying the broadcast incremental power and
the sweep algorithms [1]. The solution of the sweep becomes both the initial current solution and the initial
best solution. Then the simulated annealing process starts with temperature t=tinit, where tinit is a parameter
regulating the initial temperature. At each iteration, a candidate solution, which is obtained by modifying the
current solution, is generated. The modification is done as follows: A terminal i is randomly selected and its
coverage area is decreased in such a way that if terminal i reaches its k-th destination before the modification,
it reaches its (k-1)-th destination after this step; The modified solution is checked. If the network is still
connected, the modification is finished. Otherwise, the modification goes on by selecting a terminal j≤i,
increasing its coverage area and fixing the disconnectivity. There are two alternative behaviors for the
selection of terminal j: with a probability pr, terminal j is selected in such a way that fixing the
disconnectivity will have the least addition of transmission power; with a probability 1- pr, terminal j is
selected randomly.
If the candidate solution has a lesser cost (total transmission power), it is accepted, which the new
current solution becomes the candidate solution. Otherwise, the acceptance occurs with probability
calculated as e(cost(candidate)-cost(current))/t. During the execution, the temperature t is decreased at every Ct
iterations where the best solution was not improved. The decreasing of the temperature is done by
multiplying it by a factor α < 1. The process of modifying the current solution and probabilistically accepting
the modified candidate as the new current solution is repeated until t ≥ tmin, where t_{min} is the temperature
threshold. As a result of previous experiments [8], the following parameter values were found to be effective
for this algorithm pr = 0.2, tinit = 0.2, Ct = 30000, α = 0.9, tmin =0.1.
4. A shared incumbent environment
A MILP solver represents the problem space as a tree, where each node is a solution subspace. When an
integer variable is found to be fractional after a linear programming optimization procedure, the current node
is divided into branches, where the fractional variable is forced to be rounded up and down to become an
integer. Such expansion of the tree takes exponential time. To decrease the time requirement for solving the
problem, the solver applies pruning: stopping the expansion of the nodes that are worst in quality than the
current known best solution.
The benefit of a shared incumbent environment is that it incorporates a meta-heuristic algorithm which
finds useful heuristic solutions very quickly and informs the solver, so that the solver is aware of good
feasible solutions in very early stages and applies pruning more efficiently. This leads the solver to
promising solution subspaces more quickly. In addition to this, MILP solver can also improve the shared best
solution and push the meta-heuristic out of premature convergence situations by informing it.
The implementation of this approach was done in a multi-threaded fashion: a MILP solver thread and a
simulated annealing thread. The simulated annealing thread is run repeatedly until a given maximum
computation time is reached, or the MILP thread has found a proven optimal solution. When the minimum
temperature is reached, the simulated annealing restarts the search with the initial temperature and a different
random seed.
5. Experimental results
To evaluate the performance of the shared incumbent environment, three alternative methods were
considered for solving some random problem instances: MILP, a MILP solver, allowed to use two threads in
parallel; SA, two simulated annealing threads working in parallel, sharing the best solutions; SIE, one thread
160
dedicated too the MILP solver
s
and once
o
thread too the simulaated annealinng algorithm,, working in parallel andd
sharing the best solutionns.
mented in C+
++ and IBM ILOG CPLE
EX 12.3 (httpp://www.cpleex.com) wass
The alggorithms werre all implem
used as the MILP solverr, and a maxiimum compuutation time of 3000 secoonds is allow
wed for each combinationn
instance/meethod. All teests were runn on a compputer with In
ntel Core 2 Duo
D 2.66 G
GHz processo
or and 4 GB
B
RAM.
Each prroblem instance was gennerated by laaying terminaals randomlyy on a two ddimensional area. On thee
two dimenssional area, dij being the euclidean diistance between the term
minals i and jj, the power requirementt
2
value rij was calculated as dij , for eaach arc (i,j). A random teerminal was selected
s
as thhe source s.
Let us start the evaaluation by analysing
a
thee results for a single ranndomly geneerated instan
nce with 1800
terminals, which
w
well reepresents thee usual behavvior of the methods
m
consiidered. Figurre 1 shows th
he upper andd
lower bounds obtained over time byy MILP, SA and SIE. It can be seen that the uppper bound (so
olution cost))
found by MILP
M
is over 20 000 000; SA was ablee to find a much
m
more usseful solutionn very quicklly, but couldd
not further improve the solution, beeing stuck alrready in the first minute over solutioon cost 310 000.
0
SIE wass
finally able to find a feaasible solutioon very quickkly, like SA, but also oveer time it keppt improving the solutionn
M
and SA
A, with a so
olution cost near 300 0000 at the en
nd. It is alsoo
and provideed better ressults than MILP
interesting to
t compare the
t lower boounds providded by MILP
P and SIE ovver time. It ccan be seen that a lowerr
bound wass provided earlier by SIE and itt kept increeasing, whicch means tthe branch and boundd
implementaation of the solver can appproximate too desired solu
utions faster..
Figure 1. Upper
U
bounds (solution costts, UB) and loower bounds (LB)
(
found byy MILP, SA annd SIE on a 18
80-terminal
example insstance, drawn versus time inn seconds. Noote that the y-aaxis contains gaps
g
to be ablle to cover all the progress
made by
b the three methods
m
To makke a general evaluation
e
off SIE, 10 insstances for eaach number of
o terminals were generaated, numberr
of terminalss being 130, 140, 145, 150,
1
160, 1700, 180, 190 and 200. Thhen, each insstance was so
olved by thee
approaches MILP, SA annd SIE. The results are shhown in tablle 1. In the taable, each row
w representss the averagee
S was able to improve is given in parentheses),
p
,
of 10 instannces (and oveer how manyy of these 100 instances SIE
the column group “Impprovement ovver MILP” represents
r
th
he improvem
ments made bby SIE in co
omparison too
MILP, and the columnn group “Im
mprovement over SA” represents the
t improvem
ments madee by SIE inn
b
are calculated as 1-a/b, wheree a is the upp
per bound off
comparisonn to SA. Imprrovements foor the upper bounds
the solutionn of SIE andd b is the uppper bound of
o the solutio
on of the othher approachh. Improvem
ments for thee
lower bounds are calculated as a/b--1, where a is
i the lower bound of thhe solution of SIE and b is the lowerr
bound of thhe solution of the otherr approach. The
T row “ov
verall” show
ws the averaage and totall number off
instances where
w
SIE was able to impprove calculaated over all instances.
161
Taable 1. Resultts generated by
y each approaach
In tablee 1, it can bee seen that, for
f both uppper bounds (ii.e. solution costs) and loower boundss, the sharedd
incumbent environmentt SIE providdes consideraable improveements comppared to MIL
LP, except for
f instancess
with the num
mber of term
minals 130 annd 140. This shows that the
t shared inncumbent envvironment is effective onn
large instannces to provvide non-trivvial lower and
a
upper bounds.
b
On top of thatt, the shared
d incumbentt
environmennt also provvides improvvements oveer the heuristic solutionn costs of tthe simulateed annealingg
approach most
m of the tim
me, which shhows that thee shared incu
umbent envirronment alsoo results in a competitivee
heuristic meethod.
6. Concllusions
A shareed incumbennt environm
ment, which executes a mathematica
m
al programm
ming solver and
a a meta-heuristic seearch in paraallel and maakes them shhare their beest solutionss, was consiidered for th
he minimum
m
power broaadcasting problem in wireless
w
actuator network
ks. Experim
mental results show thatt the sharedd
incumbent environment
e
t finds betterr solutions inn limited time for the minnimum poweer broadcastiing problem..
Therefore, this
t approachh can be seeen as an effecctive heuristtic for the larrge instancess of this prob
blem, and att
the same tim
me as a tool to
t provide beetter lower bounds than a standard MILP
M
solver.
7. Acknoowledgem
ments
N.E. Tooklu was suppported by the Swiss Natiional Sciencee Foundationn through prooject 200021-119720.
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