HYBRID POPULATION-BASED INCREMENTAL LEARNING
TO ASSIGN TERMINALS TO CONCENTRATORS
Eugénia Moreira Bernardino, Anabela Moreira Bernardino
Research Center for Informatics and Communications, Dep. of Computer Science, School of Technology and Management
Polytechnic Institute of Leiria, Leiria, Portugal
Juan Manuel Sánchez-Pérez, Juan Antonio Gómez-Pulido, Miguel Angel Vega-Rodríguez
Dep. of Technologies of Computers and Communications, Polytechnic School, University of Extremadura, Cáceres, Spain
Keywords: Communication networks, Terminal assignment problem, Optimisation algorithms, Population-based
incremental learning.
Abstract: In the last decade, we have seen a significant growth in communication networks. In centralised
communication networks, a central computer serves several terminals or workstations. In large networks,
some concentrators are used to increase the network efficiency. A collection of terminals is connected to a
concentrator and each concentrator is connected to the central computer. In this paper we propose a Hybrid
Population-based Incremental Learning (HPBIL) to assign terminals to concentrators. We use this algorithm
to determine the minimum cost to form a network by connecting a given collection of terminals to a given
collection of concentrators. We show that HPBIL is able to achieve good solutions, improving the results
obtained by previous approaches.
1 INTRODUCTION
In last years, we have observed tremendous research
activities in optimisation methods for
communication networks. This is mainly due to the
dramatic growth in the use of the Internet (Salcedo-
Sanz and Yao, 2004; Yao et al. 2005). The
assignment of terminals to concentrators is an
important issue in communication networks'
optimisation. The number of concentrators and
terminals and their locations are known. Each
concentrator is limited in the amount of traffic that it
can accommodate. For that reason, each terminal
must be assigned to one node of the set of
concentrators in a way that no concentrator
oversteps its capacity (Khuri and Chiu, 1997;
Salcedo-Sanz and Yao, 2004; Xu et al. 2004). This
problem is known as Terminal Assignment Problem
(TAP). Our purpose is to minimise the cost to form a
network between a specified set of terminals and
concentrators (Khuri and Chiu, 1997). The objective
is to assign terminals to concentrators under three
constraints (Bernardino et al. 2009b): (1) each
terminal is assigned to one (and only one)
concentrator; (2) the total number of terminals
assigned to any concentrator does not overload that
concentrator, i.e. is within the concentrators'
capacity and (3) balanced distribution of terminals
among concentrators. Under these constraints, an
assignment with the lowest possible cost is sought.
The TAP is a NP-Hard combinatorial
optimisation problem (Salcedo-Sanz and Yao,
2004). This means that we cannot guarantee to find
the best solution in a reasonable amount of time. The
intractability of this problem is a motivation for the
pursuits of an algorithm to produce approximate
solutions.
Estimation of distribution algorithms (EDAs) are
a class of Evolutionary Algorithms (EAs). EDAs use
sampling with probabilities instead of traditional
crossover and mutation operators.
The Population-Based Incremental Learning
(PBIL) algorithm is an EDA, proposed by Baluja
(1994). The PBIL uses a stochastic guide search
process to obtain new solutions based on the
directional information from the previous best
solution. The PBIL maintains statistics about the
search space (learning probabilities) and uses them
182
Moreira Bernardino E., Moreira Bernardino A., Sánchez-Pérez J., Gómez-Pulido J. and Vega-Rodríguez M..
HYBRID POPULATION-BASED INCREMENTAL LEARNING TO ASSIGN TERMINALS TO CONCENTRATORS .
DOI: 10.5220/0003076301820189
In Proceedings of the International Conference on Evolutionary Computation (ICEC-2010), pages 182-189
ISBN: 978-989-8425-31-7
Copyright
c
2010 SCITEPRESS (Science and Technology Publications, Lda.)
to direct its exploration (Baluja, 1994, 1996, 1997).
The algorithm produces new solutions according
to the learning probabilities. Based on analysing the
principle of PBIL algorithm, in this paper we present
an improvement version of the PBIL algorithm. We
extend the standard PBIL algorithm to work with an
integer representation (terminal-based
representation). The standard PBIL algorithm uses
learning probabilities to build complete solutions. In
our Hybrid PBIL (HPBIL) algorithm, we use the
learning probabilities to perform modifications on
TAP solutions. We incorporate the HPBIL with an
intensification mechanism to allow returning to
previous best solutions. The HPBIL also uses a
diversification mechanism that periodically
reinitialises all the learning probabilities.
We compare the performance of HPBIL with
five algorithms: Local Search Genetic Algorithm
(LSGA), Tabu Search (TS), Hybrid Ant Colony
Optimisation (HACO), Hybrid Differential
Evolution with a Multiple strategy (MHDE), and
Hybrid Scatter Search (HSS), used in literature.
The paper is structured as follows. In Section 2
we describe the TAP; in Section 3 we describe the
implemented HPBIL algorithm; in Section 4 we
discuss the computational results obtained and,
finally, in Section 5 we report about the conclusions.
2 TAP
In the TAP, a communication network will connect
N terminals and each with T
i
demand (weight) via M
concentrators and each with C
j
capacity. No
terminal’s demand exceeds the capacity of any
concentrator. A terminal site has a fixed and known
location CT
i
(x,y). A concentrator site has also a
fixed and known location CP
j
(x,y).
Problem Instance:
Terminals - a set N of n distinct terminals;
Weights - a vector T, with the capacity required
for each terminal;
Terminals' Location - a vector CT, with the
location (x,y) of each terminal;
Concentrators - a set M of m distinct
concentrators;
Capacities - a vector C, with the capacity
required for each concentrator;
Concentrators' Location - a vector CP, with the
concentrators' location (x,y).
Each terminal must be assigned to one node of
the set of concentrators, in a way that no
concentrator oversteps its capacity. To minimise the
cost, the distances between concentrators and
terminals assigned to them must be minimised.
Other objective is to ensure a balanced distribution
of terminals among concentrators.
Figure 1 illustrates an assignment to a problem
with N=10 terminal sites, and M=3 concentrator
sites. The figure shows the coordinates for the
concentrators and terminals and also their capacities.
Figure 1: TAP - Example.
In this work, the solutions are represented using
integer vectors. We use the terminal-based
representation (Figure 1). Each position in the
vector corresponds to a terminal. The value carried
by position i of the chromosome specifies the
concentrator that terminal i is to be assigned to.
3 PROPOSED HPBIL
PBIL is a population-based optimisation method to
solve hard combinatorial optimisation problems.
PBIL combines characteristics from EAs and
reinforcement learning (Baluja, 1994).
The PBIL creates a real-valued probability vector
P, which is used to generate new solutions. The
coding method of Baluja’s PBIL uses binary
representation (Baluja, 1994; He et al. 1999).
Initially, the values of the probability vector are
initialised to 0.5. As the search proceeds, the values
in the probability vector gradually shift to reflect the
search experience of the best solutions found in
previous generations.
The probability vector is used to create a number
of ns solutions. The probability vector is updated,
HYBRID POPULATION-BASED INCREMENTAL LEARNING TO ASSIGN TERMINALS TO CONCENTRATORS
183
taking the best solution B of the previous generation.
After the probability vector is updated, the vector is
mutated and a new set of solutions is created by
sampling from the updated probability vector. The
algorithm continues until the stop criterion is
reached (He et al. 1994).
The main steps of the standard PBIL are the
following:
Initialise parameters
Initialise the probability vector P
i
Update the probability vector: P
i
=0.5(1<=i<=n)
While stop criterion is not reached:
Generate ns sample solutions according to
the probability vector P.
Evaluate ns solutions.
Find best solution B.
Update probability vector towards best
solution B:
P
i
=P
i
*(1.0-lr)+B
i
*lr(1<=i<=n)
Mutate probability vector:
if (random(0,1)<mp)
P
i
=P
i
*(1.0-mutShift)+
random(0,1)*(mutShift)
(1<=i<=n)
ns – number of solutions in the population
n - length of encoded solution
lr - learning rate
mp - mutation probability
mutShift - amount for mutation to affect the
probability vector
The probability vector in PBIL serves as
distributed, numerical information which the
algorithm uses to probabilistically construct
solutions to the problem being solved and which the
algorithm adapts during the algorithm execution, to
reflect the search experience. The probabilities
induce a probability distribution over the search
space and determine which parts of the search space
are effectively sampled.
To extend the binary PBIL algorithm for an
integer representation, the probabilities need to be
maintaned in a matrix. For the TAP, the set of
probabilities is maintained in a matrix T of size N*M,
where the entry T
ij
measures the desirability of
assigning terminal i to concentrator j.
The standard PBIL algorithm uses the probability
vector to construct complete solutions. Our HPBIL
algorithm uses the probabilities to perform
modifications on TAP solutions.
In HPBIL, we incorporate a parameter q to
control the exploration and exploitation processes.
We use these two processes to modify the solutions.
The management of the probabilities is the most
important component of PBIL. Exploration is a
stochastic process, in which the choice of the
component used to modify a solution to the problem
is made in a probabilistic way. Exploitation chooses
the component that maximises a blend of probability
values and partial objective function evaluations.
In this paper, we also explore one of the most
successful emerging ideas combining Local Search
(LS) with a population-based search algorithm.
HPBIL uses a modified PBIL to explore several
regions of the search space and simultaneously
integrates a LS algorithm to intensify the search
around some selected regions.
For the best solution B in some generation, the
corresponding learning formula to update the
probabilities is: T
iBi
= T
iBi
+ lr.
HPBIL uses an intensification mechanism. This
mechanism allows returning to previous best
solutions. The algorithm also uses a diversification
mechanism after a predefined number of nid
iterations, without improving the best solution found
so far. When combined with appropriate choices for
the probabilities update, the diversification
mechanism can be very useful to refocus the search
on a different search space region and to avoid the
early convergence of the algorithm.
The main steps of HPBIL are the following:
Initialise parameters
iteration = 0
Generate initial population S
for s=1 to ns do
Apply Local Search to S
s
Evaluate Solutions in S
Find best global solution G in S
Initialise the probability matrix T
ij
:
T
ij
=1/M, (1<=i<=N), (1<=j<=M)
intensification=true
While stop criterion is not reached:
iteration = iteration+1
for s=1 to ns do
Modify s solution according to the
probability matrix T:
S’
s
=ModifyProcess(S
s
)
Apply Local Search to S’
s
cond=true
if (intensification=true)
if(fitness(S’
s
)>
fitness(S
s
))
S’
s=
S
s
else
cond=false
if (cond=true) //no solution improved
ICEC 2010 - International Conference on Evolutionary Computation
184
intensification=false
Find best solution B in S’
if(fitness(B)<fitness(G))
G=B
Intensification=true
Update probability matrix towards best
solution B:
T
iBi
= T
iBi
+ lr
Mutate probability matrix:
if (random(0,1)<mp)
T
ij
=T
ij
*(1.0-mutShift)+
random(0 or 1)*(mutShift)
(1<=i<=N), (1<=j<=M)
if (iteration % nid =0)
Apply Diversification Mechanism
S=S’
S - population of solutions
lr - learning constant
The next subsections describe each step of the
algorithm in detail.
3.1 Initialisation of Parameters
The following parameters, must be defined by the
user: (1) ms – number of seconds; (2) ns – number
of solutions in the population; (3) mp – mutation
probability; (4) mutShift – amount for mutation
to affect the probability matrix; (5) nm – number of
modifications; (6) q – exploitation/exploration
probability and (7) nid – number of iterations
without improvement (used for diversification).
3.2 Create Initial Population
The solutions are created using a deterministic form.
The deterministic form is based in the Greedy
algorithm proposed by Abuali et al. (1994). The
Greedy algorithm randomly assigns terminals to the
closest feasible concentrators.
3.3 Local Search Procedure
HPBIL uses the LS algorithm proposed by
Bernardino et al. (2008b). The evaluation process is
the most time-consuming step of the algorithm,
which is usually the case in many real-life problems.
We improve the LS proposed by Bernardino et al.
(2008b). After creating a neighbour, the algorithm
does not perform a full examination to calculate the
new fitness value; it only updates the fitness value
based on the modifications made to create the
neighbour. The running time is considerably
reduced. We observe 80% of improvement in terms
of execution time.
3.4 Evaluation of Solutions
The fitness function is the same used in Bernardino
et al. (2008a, 2008b, 2009a, 2009b, 2010a, 2010b).
onPenalisati
dist
balfitness
N
t
tct
M
c
c
1
0
)(,
1
0
*1,0
*9,0
c(t)= concentrator of terminal t
t = terminal, c = concentrator
The fitness function is based on:
(1) the total number of terminals connected to each
concentrator (the purpose is to guarantee a balanced
distribution of terminals among concentrators);

1
0
)(1
0
1
10
1*20
N
t
ctcif
c
M
N
roundtotalif
total
M
N
roundabs
c
total
bal
c
c
(2) the distances between concentrators and
terminals assigned to them (the goal is to minimise
the distances);
(3) the penalisation if a solution is not feasible (the
objective is to penalise the solutions when the total
capacity of one or more concentrators is
overloaded).

Feasibleif
onPenalisati
0
500




22
)(,
..)(..)( ytCTytcCPxtCTxtcCPdist
tct
The main objective is to minimise the fitness
function.
3.5 Initialisation of Probability Matrix
All the values in the probability matrix are initialised
with the same probability:
T
ij
=1/M, (1<=i<=N), (1<=j<=M)
For example, if we have 10 terminals and 4
concentrators, all the values are initialised with the
same probability ¼=0.25. All the concentrators
have the same probability of being selected.
(
1
)
(
2
)
(
3
)
HYBRID POPULATION-BASED INCREMENTAL LEARNING TO ASSIGN TERMINALS TO CONCENTRATORS
185
3.6 Modification of Solutions
It consists of repeating nm modifications. The
modification is done assigning a terminal t to a
concentrator c. First, a terminal t is randomly
chosen (between 1 and N) and then a concentrator c
is selected. Then, a random number x is generated
between 0 and 1. If x is smaller than q (parameter),
the best feasible concentrator c is chosen in a way
that T
tc
will be maximum. If x is higher than q,
the feasible concentrator c is chosen with a
probability proportional to the values contained in
the pheromone trail. We only consider feasible
concentrators. This means that we only consider the
concentrators that have a free capacity equal or
higher than the demand of terminal t.
3.7 Intensification Mechanism
The intensification mechanism allows a more
complete exploration of the neighbourhood and
allows returning to previous best solutions. The
objective is to return towards attractive regions to
search them thoroughly. If the intensification is
active and the solution S
i
in the beginning of the
iteration is better, the new solution S’
i
returns to the
initial solution S
i
. The intensification is activated
when the best solution found so far has been
improved and remains active while at least one
solution succeeds on improving its solution during
the iteration. In the end of each iteration, if no
solution improves its last solution, the intensification
is deactivated. The objective is to explore other
regions, avoiding the algorithm to become trapped in
a local minimum.
3.8 Probability Matrix Update
The probability matrix is updated by taking into
account only the best solution of the previous
generation. The probability matrix is updated by
setting: T
iBi
= T
iBi
+ lr
Based on preliminary observations, we consider
the value 0.5 for lr (learning constant).
3.9 Probability Matrix Mutation
PBIL does not use a crossover operator and a
selection mechanism like the most EAs. Instead, the
values in T are mutated once per iteration. During
this step, a random number between 0 and 1 is
generated. If this random value is smaller than mp
(mutation probability), the probability is mutated by
setting:
T
ij
=T
ij
*(1.0-mutShift)+
random(0 or 1)*(mutShift)
3.10 Diversification Mechanism
The diversification mechanism restarts the
probability matrix and creates new solutions. For the
following iteration, we kept the best solution found
so far (G).
3.11 Termination Criterion
The algorithm stops when a maximum number of
seconds (ms) is reached.
More information on PBIL can be found in
(Baluja, 1994, 1996, 1997; Baluja and Caruana,
1995; He et al. 1999).
4 RESULTS
In order to test the performance of our approach, we
use a collection of TAP instances of different sizes.
We take 9 problems from literature (Bernardino et
al. 2008a).
To compare our results we consider the results
produced with LSGA, TS, HACO, MHDE and HSS.
We compare our algorithm with the algorithms
proposed by Bernardino et al. (2008a, 2008b, 2009b,
2010a, 2010b), because they (1) used the same test
instances; (2) adopted the same fitness function; (3)
implemented the algorithms using the same
language (C++), and; (4) adopted the same
representation (terminal-based).
Table 1 presents the best-obtained results with
HPBIL, LSGA, TS, MHDE, HACO and HSS. The
first column represents the number of the problem
(Prob) and the remaining columns show the results
obtained (BestF – Best Fitness, Ts – Run Times).
The initial solutions for all algorithms were created
using the Greedy algorithm. The algorithms have
been executed using a processor Intel Core Duo
T2300. The Ts (Run Time) corresponds to the
execution time that each algorithm needs to obtain
the best feasible solution.
The HPBIL algorithm can reach the best-known
solutions for all instances. MHDE, HACO, HSS and
LSGA can also find the best–known solutions, but in
a higher execution time. Since we are not trying to
dynamically assign terminals to concentrators, the
running time is not enough to determine the quality
of the algorithms. The best-known solutions are
ICEC 2010 - International Conference on Evolutionary Computation
186
Table 1: Results.
Prob LSGA TS MHDE HACO HSS HPBIL
BestF Ts BestF Ts BestF Ts BestF Ts BestF Ts BestF Ts
1 65.63 <1s 65.63 <1s 65.63 <1s 65.63 <1s 65.63
<1s
65.63 <1s
2 134.65 <1s 134.65 <1s 134.65 <1s 134.65 <1s 134.65 <1s 134.65 <1s
3 270.26 <1s 270.26 <1s 270.26 <1s 270.26 <1s 270.26 <1s 270.26 <1s
4 286.89 <1s 286.89 <1s 286.89 <1s 286.89 <1s 286.89 <1s 286.89 <1s
5 335.09 <1s 335.09 <1s 335.09 <1s 335.09 2s 335.09 <1s 335.09 <1s
6 371.12 1s 371.12 <1s 371.12 <1s 371.12 3s 371.12 1s 371.12 <1s
7 401.21 1s 401.49 1s 401.21 2s 401.21 4s 401.21 1s 401.21
<1s
8 563.19 7s 563.34 1s 563.19 10s 563.19 14s 563.19 4s 563.19
3s
9 642.83 7s 642.86 2s 642.83 15s 642.83 25s 642.83 6s 642.83
5s
Table 2: Results – fitnesses and standard deviations.
Prob LSGA TS MHDE HACO HSS HPBIL
AvgF Std AvgF Std AvgF Std AvgF Std AvgF Std AvgF Std
1 65.63 0.00 65.63 0.00 65.63 0.00 65.63 0.00 65.63 0.00 65.63 0.00
2 134.65 0.00 134.65 0.00 134.65 0.00 134.65 0.00 134.65 0.00 134.65 0.00
3 270.69 0.23 270.76 0.30 270.75 0.15 270.42 0.08 270.35 0.06 270.26 0.01
4 286.99 0.13 287.93 0.75 287.17 0.14 286.96 0.08 286.90 0.01 286.89 0.00
5 335.99 0.60 335.99 0.59 336.55 0.39 335.79 0.34 335.20 0.14 335.09 0.00
6 371.68 0.24 372.44 0.45 373.19 0.42 372.45 0.39 371.70 0.24 371.41 0.13
7 402.41 0.50 403.25 0.73 403.61 0.33 402.28 0.40 401.82 0.34 401.61 0.15
8 564.94 0.52 564.50 0.54 572.04 0.76 565.64 0.84 563.87 0.37 563.76 0.18
9 646.52 0.84 644.18 0.48 648.46 0.48 644.82 0.58 643.94 0.51 643.35 0.19
reached with almost all algorithms (except TS). For
that reason, to establish which is the best algorithm,
we must observe the average quality of the produced
solutions and the standard deviations.
Table 2 presents the average fitnesses and
standard deviations. The first column represents the
number of the problem (Prob) and the remaining
columns show the results obtained (AvgF – Average
Fitness, Std – Standard Deviation). To compute the
results in table 2, we use 0.5 second for instances
1-3, 1 second for instances 4-5, 2 seconds for
instance 6, 5 seconds for instance 7, 10 seconds for
instance 8 and 15 seconds for instance 9.
The suggestions from literature helped us to
guide our choice of parameter values for TS
(Bernardino et al. 2008a), LSGA (Bernardino et al.
2008b), MHDE (Bernardino et al. 2010a), HACO
(Bernardino et al. 2009b) and HSS (Bernardino et al.
2010b). For the TS, we consider a number of
elements in the tabu list between 5 and 20. The
parameters of LSGA are set to crossover probability
between 0.3 and 0.4, selection operator=
“tournament”, mutation probability between 0.6
and 0.8, crossover operator=“one-point” and
mutation operator= “multiple”. The parameters of
the MHDE algorithm are set to crossover probability
between 0.3 and 0.4, factor F between 0.9 and
1.6 and strategy=“Best1Exp”. The parameters of
the HACO algorithm are set to the number of
iterations used for diversification between 200
and 400, Q=100, q=0.9, pheromone
influence=0.8, pheromone evaporation=0.8 and
number of modifications between 2 and 10. The
parameters of the HSS algorithm are set to ni=100,
b1=8, b2=8, and nid between N/15 and N/2.
The parameters of HPBIL are set to mp=0.3,
mutShift=0.1, nid<N/20 and q=0.6. The
MHDE and LSGA were applied to populations of
200 individuals, HSS to populations of 100
individuals and HPBIL and HACO to populations of
30 individuals.
The values presented in table 2 have been
computed based on 50 different executions (50 best
executions out of 100 executions) for each test
instance.
As it can be seen in table 2, for larger instances
the standard deviations and the average fitnesses for
the HPBIL algorithm are smaller. It means that the
HPBIL algorithm is slightly more robust than
LSGA, TS, HACO, MHDE and HSS.
The best results obtained with HPBIL use
nm<N/20, mp between 0.2 and 0.7 (Figure 2)
and mutShift>=0.1 (Figure 2), q between 0.3
and 0.6 (Figure 2), number of solutions between
HYBRID POPULATION-BASED INCREMENTAL LEARNING TO ASSIGN TERMINALS TO CONCENTRATORS
187
30 and 100 and nid in the range [N*3 and
N*4]. These parameters were experimentally found
to be good and robust for the problems tested.
We perform comparisons between all parameters
(using the 9 instances) in order to establish the
correct parameter setting for the HPBIL algorithm.
We consider the same instance – 7 (a problem with
average difficulty) to show the comparisons between
parameters. To compute the results we use 1000
iterations.
Figure 2: Influence of parameters – Problem 7.
In our experiments we use a growing number of
solutions. The number of solutions (ns) was set to
{10, 20, 30, 40, …, 200}. We studied the
impact on the execution time and the average fitness
(Figures 3 and 4). A high number of solutions
significantly increases the algorithm execution time
(Figure 3). The results show that the best values are
in the range [30 and 100]. With these values, the
algorithm can reach, in a reasonable amount of time,
a reasonable number of good solutions. With a
higher number of solutions, the algorithm can reach
a better average fitness (Figure 4), but it is more
time consuming. We also observe that a small
number of solutions allows an initial faster
convergence, but a worse final result, following to
an increased amount of suboptima values. This can
be explained, because the quality of the initial best-
located solution previous to the first restart depends
highly on the population size: they need more
population diversity – it depends on the population
size – to avoid premature stagnation.
Figure 3: ns – Execution Time – Problem 7.
For nid<N*3 and nid>N*4 we observed
phenomena of stagnation and insufficient
intensification.
Figure 4: ns – Average Fitness – Problem 7.
For parameter nm, the number of modifications,
nm<N/20 has been shown experimentally more
efficient (Figure 5). A high nm has a significant
impact on the execution time (Figure 6). A small nm
did not allow the system to escape from local
minima, because after the LS, the resulting solution
was, in most cases, the same as the starting solution.
Figure 5: nm – Average Fitness – Problem 7.
Figure 6: nm – Execution Time – Problem 7.
In general, experiments have shown that the
proposed parameter setting is very robust to small
modifications.
5 CONCLUSIONS
In this paper we present a HPBIL algorithm to solve
the TAP. The performance of' our algorithm is
compared with five algorithms: a LSGA, a TS
algorithm, a HACO algorithm, a MHDE algorithm
ICEC 2010 - International Conference on Evolutionary Computation
188
and a HSS algorithm. All algorithms were applied to
TAP by the same authors.
HPBIL presents better results for TAP. The
experimental results show that the proposed
algorithm is an effective and competitive approach
in composing satisfactory results with respect to
solution quality and execution time for TAP.
Moreover, in terms of standard deviation, the
algorithm also proved to be more stable and robust
than the other algorithms.
For future work we suggest the implementation
of Evolutionary Swarm Intelligence algorithms. The
combination of EAs and SI algorithms can unify the
fast speed of EAs for global solutions and good
precision of SI algorithms for good solutions by the
feedback information.
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