Modiﬁed Differential Evolution in the Load Balancing Problem for the

iFDAQ of the COMPASS Experiment at CERN

Ond

rej

Subrt

, Martin Bodl

, Matou

s Jandek

, Vladim

ır Jar

, Anton

ın Kv

eto

, Josef Nov

Jan Tomsa

and Miroslav Virius

Czech Technical University in Prague, Prague, Czech Republic

Charles University, Prague, Czech Republic

Keywords:

Data Acquisition System, Differential Evolution, Genetic Algorithm, Load Balancing, Optimization.

Abstract:

In general, state-of-the-art data acquisition systems in high energy physics experiments must satisfy high re-

quirements in terms of reliability, efﬁciency and data rate capability. The paper introduces the Load Balancing

(LB) problem of the intelligent, FPGA-based Data Acquisition System (iFDAQ) of the COMPASS experi-

ment at CERN and proposes a solution based on genetic algorithms. Since the LB problem is N P -complete,

it challenges analytical and heuristic methods in ﬁnding optimal solutions in reasonable time. Differential

Evolution (DE) is a type of evolutionary algorithms, which has been used in many optimization problems due

to its simplicity and efﬁciency. Therefore, the Modiﬁed Differential Evolution (MDE) is inspired by DE and

is presented in more detail. The MDE algorithm has newly-designed crossover and mutation operator and its

selection mechanism is inspired by Simulated Annealing (SA). Moreover, the proposal uses an adaptive scal-

ing factor and recombination rate affecting the exploration and exploitation of the MDE algorithm. Thus, the

MDE represents a new efﬁcient stochastic search technique for the LB problem. The proposed MDE algorithm

is examined on two LB test cases and compared with other LB solution methods.

1 INTRODUCTION

The intelligent, FPGA-based Data Acquisition Sys-

tem (iFDAQ) (Bodlak et al., 2016; Bodlak et al.,

2014) reads out data from the detectors of the COM-

PASS (COmmon Muon Proton Apparatus for Struc-

ture and Spectroscopy) experiment (Alexakhin et al.,

2010) being a high-energy particle physics experi-

ment with ﬁxed-target situated on the M2 beamline of

the Super Proton Synchrotron (SPS) particle acceler-

ator at the CERN laboratory in Geneva, Switzerland.

In complex readout data systems, such as the iF-

DAQ, the data streams must be properly allocated in

order for the load to be well-balanced in the system

(Kameda et al., 1997). Thus, the necessity of solving

the Load Balancing (LB) problem arises.

The paper is organized as follows. Firstly, the LB

problem is introduced in Section 2. Subsection 2.1

gives the proper deﬁnition of the LB problem. Com-

plexity of the LB problem is proved in Subsection 2.2.

Secondly, Section 3 describes the proposal of the

Modiﬁed Differential Evolution (MDE) algorithm be-

ing applicable to the LB problem. All parts of the

MDE algorithm are discussed in an extensive way.

Finally, numerical results are stated in Section 4

to demonstrate how the MDE approach is successful

and efﬁcient in solving the LB problem. Then, the

results acquired by the MDE are compared with other

LB solution methods.

2 LOAD BALANCING PROBLEM

For the iFDAQ, the most challenging task from the

LB point of view is load balancing at the multiplexer

(MUX) level. The optimization criterion is mini-

mization of the difference between the output ﬂows

of the individual multiplexers. This minimization is

achieved by remapping the connection of inputs to

input ports of the multiplexers. Each input port es-

tablishes a connection between a data source (a de-

tector or a data concentrator) and the MUX level. For

the COMPASS experiment, it is necessary to consider

ﬂows varying from 0 B to 10 kB for each input port.

In Figure 1, a visualization of LB at the MUX

level is given. There are m MUXes with p ingoing

ports each. Moreover, n ∈ N ﬂows f

, f

,..., f

∈

Šubrt, O., Bodlák, M., Jandek, M., Jarý, V., Kv

eto

n, A., Nový, J., Tomsa, J. and Virius, M.

Modiﬁed Differential Evolution in the Load Balancing Problem for the iFDAQ of the COMPASS Experiment at CERN.

DOI: 10.5220/0008319102130220

In Proceedings of the 11th International Joint Conference on Computational Intelligence (IJCCI 2019), pages 213-220

ISBN: 978-989-758-384-1

213

p + 1

p + 2

MUX

(m − 1)p + 1

p+1

p+2

(m−1)p+1

i=1

i=p+1

i=(m−1)p+1

Figure 1: Visualization of LB at the MUX level.

, where n = m · p, are shown in the ﬁgure with in-

dices k

,. . .,k

∈ {i | 1 ≤ i ≤ n} ∧ ∀i, j : k

6= k

Despite the fact that each ﬂow varies from 0 B

to 10 kB in the COMPASS experiment, the domain

is used. The motivation comes from a general ap-

proach to LB. Moreover, a ﬂow with 0 B can be either

a physical connected input port sending no data or an

empty input port where no data source is connected

to. In brief, there are always n = m · p ﬂows regard-

less whether all ports are used or not.

2.1 Problem Formulation

This subsection deals with a proper deﬁnition of the

LB problem and preparation for discussion of the

complexity of the LB problem. The Multiple Knap-

sack (MK) problem (Kellerer et al., 2004) is useful

for the examination of the LB problem complexity as

it can be shown that there exists a polynomial reduc-

tion from the MK problem to the LB problem. As MK

problem is N P -complete, this implies the LB prob-

lem is N P -complete.

Deﬁnition 1. Let m ∈ N denote the number of MUXes

with p ∈ N ingoing ports each, i.e., n = m · p ∈ N

ingoing ports in total and ﬂows f

, f

,. . ., f

∈ N

Let S

,. . .,S

be subsets of indices and F =

∑

i=1

be a theoretical average ﬂow for one

MUX. The Load Balancing (LB) problem is an op-

timization problem such that:

To minimize

∑

i=1

F −

∑

j∈S

, (1)

subject to the constraints

• each ﬂow must be allocated

[

i=1

= {i | i ∈ 1,... ,n} (2)

• each ﬂow must be allocated at most once

∩ S

0 ∀i, j = 1,..., m ∧ i 6= j (3)

• each MUX has p ports

| = p ∀i = 1, . .., m (4)

Being a generalization of the well-known Knap-

sack problem (Kellerer et al., 2004), the MK problem

represents an extension to m knapsacks.

Deﬁnition 2. Given n ∈ N items with weights

,. . .,w

∈ N and values v

,. . .,v

∈ R

, and

m ∈ N knapsacks with capacities W

,. . .,W

∈ N.

The Multiple Knapsack (MK) problem is an optimiza-

tion problem such that:

To maximize

∑

i=1

∑

j=1

i j

, (5)

subject to the constraints

• maximum knapsack capacity

∑

j=1

i j

≤ W

∀i = 1,... ,m (6)

• each item must be allocated at most once

∑

i=1

i j

≤ 1 ∀ j = 1,... ,n (7)

• assignment of item

i j

∈ {0,1} ∀i = 1, . .., m, ∀ j = 1,..., n (8)

2.2 Problem Complexity

The Knapsack problem is N P -complete (Kellerer

et al., 2004). Being a generalization of the Knapsack

problem, the MK problem is N P -complete.

In order to determine the complexity of the LB

problem, the decision version of the LB problem must

be deﬁned at ﬁrst: Are there subsets S

,. . .,S

such that

[

i=1

= {i | i ∈ 1,. . .,n} and S

∩ S

0,∀i, j = 1,. . .,m ∧ i 6= j and |S

| = p, ∀i = 1,... ,m

and

∑

j∈S

≤ F,∀i = 1, ..., m, where F =

∑

i=1

Theorem 1. The Load Balancing (LB) problem is

N P -complete.

Proof. First, the LB problem is a N P problem. The

proof are the subsets S

,. . .,S

of ﬂow indices that

are chosen and the veriﬁcation process is to compute

| = p,∀i = 1,. ..,m and

∑

j∈S

≤ F,∀i = 1,... ,m,

which takes polynomial time in the size of input.

ECTA 2019 - 11th International Conference on Evolutionary Computation Theory and Applications

214

Second, it will be shown there is a polynomial re-

duction from the MK problem to the LB problem. It

sufﬁces to show that there exists a polynomial time

reduction Q(·) such that Q(x) is a YES instance to the

LB problem iff x is a YES instance to the MK prob-

lem. Suppose there are given f

, f

,. . ., f

, for the LB

problem, consider the following MK problem: W

F,V

= p,∀i = 1, . .., m and w

= f

= 1 + f

/h,∀i =

1,. . .,n, where h >

∑

i=1

and K =

[

i=1

⊆ {1,..., n}

and K

∩K

0,∀i, j = 1,. ..,m ∧i 6= j, where K

rep-

resents indices of items assigned to the i-th knapsack.

Here, Q(·) is the process converting the MK problem

to the LB problem. It is clear that this process is poly-

nomial in the input size.

If x is a YES instance for the MK prob-

lem, with the chosen sets K

,. . .,K

, let R =

{1,. . .,n}\

[

i=1

. It follows that

∑

j∈K

∑

j∈K

≤

= F,∀i = 1,... ,m and it remains to prove there are

p items in each knapsack. It follows that

∑

j∈K

∑

j∈K

(1 + f

/h) ≥ V

= p,∀i = 1,. .., m and thus, there

must be at least p items in each knapsack to satisfy

the inequality. Moreover, n = m · p implies there must

be exactly p items in each knapsack and thus, R =

Therefore, the sets K

,. . .,K

correspond to sets

,. . .,S

, respectively, and x is a YES instance

for the LB problem.

Conversely, if Q(x) is a YES instance for the

LB problem, there exists S

,. . .,S

such that S

∩

0,∀i, j = 1, . .., m ∧ i 6= j and

[

i=1

= {i | i ∈

1,. . .,n} and |S

| = p,∀i = 1, ..., m and

∑

j∈S

≤

F,∀i = 1, . .., m. Let the MK problem consist of m

knapsacks and let the i-th knapsack contain the items

corresponding to indices in S

, and it follows that

∑

j∈K

∑

j∈K

≤ W

= F, ∀i = 1,. ..,m and

∑

j∈K

∑

j∈K

(1 + f

/h) ≥ V

= p, ∀i = 1,... ,m. Therefore,

Q(x) is a YES instance for the MK problem.

This proves the N P -completeness of the LB

problem.

3 MODIFIED DIFFERENTIAL

EVOLUTION

Belonging to the Evolutionary Algorithms (EA) class

(Eiben and Smith, 2015), a Genetic Algorithm (GA)

(Affenzeller et al., 2018) is a heuristic technique in-

spired by the process of natural selection and evolu-

tionary biology. It attempts to simulate evolutionary

principles in order to ﬁnd estimate solutions of opti-

mization problems. These algorithms use techniques

and strategies simulating processes well-known from

nature – heredity, mutation, natural selection and

crossover.

The main principle of the GA process is grad-

ual production of stronger generations containing in-

dividuals representing different solutions of a prob-

lem. An individual is represented by the vector x =

,. . .,x

)

. In the optimization process, a new

population is created in each generation and each in-

dividual in a population represents just one solution

of a problem. As the population evolves, solutions

improve.

Differential Evolution (DE) (Storn and Price,

1995; Das et al., 2016) is a stochastic, population-

based search strategy developed using the same prin-

ciples as GA. However, it differs signiﬁcantly in the

mutation step, crossover operator and the following

selection mechanism. Unlike GA, a mutation of DE

is applied ﬁrst to generate a trial vector, which is then

used within the crossover operator to produce one

offspring, and the mutation step sizes are inﬂuenced

by differences between the individuals of the current

population.

The Modiﬁed Differential Evolution (MDE) is

a heuristic algorithm based on DE and has a new

mutation operation, crossover operator and selection

mechanism. In this section, the newly proposed op-

erators are described in more detail at ﬁrst. Conse-

quently, the complete MDE algorithm is presented.

3.1 Mutation Operator

The mutation operator produces a trial vector for each

individual of the current population. This trial vector

will then be used by the crossover operator to produce

offspring.

Let the ﬁrst (i − 1) MUXes be already allocated.

In general, the mutation operator tries to mutate the

i-th MUX only. The trial vector u

(s) is created based

on a random individual x

(s) selected from the cur-

rent population for a parent x

(s) for the i-th MUX in

iteration s. Firstly, all n elements from the random in-

dividual x

(s) are copied to the trial vector u

(s). Ac-

tually, the ﬂows represented by elements with indices

1,. . .,(i − 1)p have already been allocated to the ﬁrst

(i−1) MUXes. Therefore, the mutation operator does

not consider elements with indices 1,... ,(i − 1)p and

leaves them as they are copied from the random indi-

vidual x

(s). Otherwise, the so far achieved solution

Modiﬁed Differential Evolution in the Load Balancing Problem for the iFDAQ of the COMPASS Experiment at CERN

215

⌈βp⌉–times

j,1

(s)

j,(i−1)p

(s)

j,(i−1)p+1

(s)

j,ip

(s)

j,ip+1

(s)

j,mp

(s)

i-th MUX

Flows already allocated

to (i − 1) MUXes

Remaining (m − i)

(s)

all n

elements

j,1

(s)

j,(i−1)p

(s)

j,(i−1)p+1

(s)

j,ip

(s)

j,ip+1

(s)

j,mp

(s)

i-th MUX

Flows already allocated

to (i − 1) MUXes

Remaining (m − i)

copy of

MUXes

Figure 2: The mutation operator used to produce a trial vec-

tor u

(s) from a random individual x

(s) selected from cur-

rent population for a parent x

(s) for the i-th MUX in itera-

tion s using dβpe swaps.

would have been damaged or completely lost.

Thus, the mutation operator considers only ele-

ments with indices (i − 1)p + 1,... ,mp. It performs

dβpe swaps. In more detail, it randomly selects one

element from the elements with indices (i − 1)p +

1,. . .,ip and one element from elements with indices

ip + 1,. . .,mp in each swap from dβpe swaps and

swaps them. To sum it up, a detailed diagram of the

mutation operator is shown in Figure 2.

β is the scaling factor, controlling the ampliﬁca-

tion of the differential variation. Theoretically β ∈

(0,∞), but it is usually taken from the range [0.1, 1].

3.2 Crossover Operator

The MDE crossover operator implements a discrete

recombination of the trial vector u

(s) and the par-

ent vector x

(s) to produce offspring x

(s). x

i, j

(s)

refers to the j-th element of the vector x

(s). Elements

i, j

(s) and x

i, j

(s) are deﬁned in the same way and re-

fer to the j-th element of vectors x

(s) and u

(s), re-

spectively. CR is the crossover or recombination rate

in the range [0,1].

The approach is similar to the mutation operator.

Let the ﬁrst (i − 1) MUXes be already allocated. In

general, the crossover operator tries to cross the i-th

MUX only. Firstly, all n elements from the parent

(s) are copied to the offspring x

(s). Actually, ﬂows

represented by elements with indices 1, ... , (i − 1)p

have already been allocated to the ﬁrst (i−1) MUXes.

Therefore, the crossover operator does not consider

elements with indices 1,.. . ,(i − 1)p and leaves them

as they are copied from the parent x

(s). By analogy

with the mutation operator, the so far achieved solu-

tion would be damaged or completely lost.

j,1

(s)

j,(i−1)p

(s)

j,(i−1)p+1

(s)

j,ip

(s)

j,ip+1

(s)

j,mp

(s)

i-th MUX

Flows already allocated

to (i − 1) MUXes

Remaining (m − i)

(s)

′

(s)

all n

elements

if rand(k) ≤ C R k ∈ {(i − 1)p + 1, . . . , ip}

then ﬁnd element

with value x

′

j,k

(s) in x

′

(s)

in x

′

(s) with value u

j,k

(s) and swap it

′

j,1

(s)

′

j,(i−1)p

(s)

′

j,(i−1)p+1

(s)

′

j,ip

(s)

′

j,ip+1

(s)

′

j,mp

(s)

i-th MUX

Flows already allocated

to (i − 1) MUXes

j,1

(s)

j,(i−1)p

(s)

j,(i−1)p+1

(s)

j,ip

(s)

j,ip+1

(s)

j,mp

(s)

i-th MUX

Flows already allocated

to (i − 1) MUXes

copy of

′

j,l

(s) l ∈ {(i − 1)p + 1, . . . , mp}

MUXes

Remaining (m − i)

MUXes

Remaining (m − i)

MUXes

Figure 3: The crossover operator used to produce an off-

spring x

(s) from a parent x

(s) and a trial vector u

(s) for

the i-th MUX in iteration s.

Thus, the crossover operator considers only the el-

ements with indices (i − 1)p + 1,.. .,mp. Then, for

each element

j,(i−1)p+1

(s),. ..,x

j,ip

(s), (9)

rand(k) ≤ CR k ∈ {(i − 1)p + 1,. . .,ip}, (10)

then ﬁnd element

j,l

(s) l ∈ {(i − 1)p + 1, ... , mp} (11)

in x

(s) with value u

j,k

(s) and swap the values of ele-

ment x

j,k

(s) and element x

j,l

(s) in x

(s). If the condi-

tion in Equation 10 is not satisﬁed, then do nothing.

In Figure 3, a diagram of the crossover operator

is given. The dashed arrows represent actions being

subjected to the condition in Equation 10. Once the

condition is not satisﬁed for a given element, it does

nothing and continues to the next element.

3.3 Adaptive Parameters

The scaling factor β and recombination rate CR af-

fect the exploration and exploitation of the algorithm

(Das et al., 2005). Exploration is the algorithm abil-

ity to cover and explore different areas in the feasible

search space while exploitation is the ability to con-

centrate only on promising areas in the search space

and to enhance the quality of the potential solution in

the promising region. The scaling factor β controls

the ampliﬁcation of the differential variations. The

smaller the value of β, the smaller the mutation step

ECTA 2019 - 11th International Conference on Evolutionary Computation Theory and Applications

216

sizes, and the longer it will take for the algorithm to

converge. Larger values of β facilitate exploration,

but may cause the algorithm to overshoot good op-

tima. The value of β should be small enough to allow

differentials to explore tight valleys, and large enough

to maintain diversity. In this paper, an adaptive scal-

ing factor is adopted to achieve a favorable compro-

mise between exploration and exploitation. For in-

creasing exploration, a large initial value of β is cho-

sen. Then, it is reduced linearly along the iterations

for good exploitation:

β(s) = c

− c

max

, (12)

where s

max

is the maximum number of iterations and

are constants. In this way, the mutation operator

performs a wider search in the solution space at the

early stages of the evolution and at the later stages,

the search is restricted around the local area.

The probability of recombination, CR, has a di-

rect inﬂuence on a diversity of the MDE. The higher

the probability of recombination, the more variation

is introduced in a new population, thereby increasing

diversity and exploration. Increasing CR often results

in faster convergence, while decreasing CR increases

search robustness. In the MDE, CR is changed along

the evolution process like β as follows:

CR(s) = k

− k

max

, (13)

where s

max

is the maximum number of iterations and

are constants.

3.4 Selection Mechanism

In the MDE, a probabilistic selection mechanism (Das

et al., 2007) is used instead of the deterministic selec-

tion of the original DE. The selection mechanism has

been inspired by Simulated Annealing (SA) (Dela-

haye et al., 2019). SA uses a random search strategy,

which not only accepts new solutions that decrease

the objective function value (assuming a minimiza-

tion problem), but may also accept new solutions that

rather increase the objective function value based on

a predetermined probability distribution function. Ex-

ponential probability distribution function is normally

used for this purpose. Based on this idea, the selection

mechanism of the MDE can be described as follows:

(s + 1) =











(s) if f (x

(s)) ≤ f (x

(s))

(s) if f (x

(s)) > f (x

(s)) ∧

h(x

(s),x

(s)) > rand()

(s) otherwise,

(14)

h(x

(s),x

(s)) = exp

f (x

(s)) − f (x

(s))

f (x

(s))T

, (15)

where T is the temperature, as deﬁned in the SA tech-

nique. Here, the temperature T is adaptively changed

in the evolution process as follows:

T (s + 1) = αT (s)

T (0) = T

(16)

The parameter α is the rate of reducing the tem-

perature (α < 1). T

is the initial temperature. A nor-

malized difference between the parent and offspring

objective functions has been considered in Equation

15 to eliminate the effect of different ranges of objec-

tive functions. The selection mechanism begins with

a large value for T

and thus, many new worse solu-

tions x

(s) have a chance to be selected to increase the

exploration of the MDE. However, the temperature T

decreases along the iterations and so the probability

of selecting worse solutions is decreased.

3.5 The MDE Algorithm

The MDE algorithm consists of the following steps,

the objectives of which are described below:

Step 1 – Parameter Setup

To determine the size of the population N

∈ N,

the maximum number of iterations s

max

, constants

,α and the initial temperature T

Step 2 – Initialization of Population

To initialize the individuals of the population ran-

domly by the assignment of the input values.

Step 3 – Evaluation of Population

To evaluate the f itness of each individual accord-

ing to the objective function in Equation 1.

Step 4 – Mutation Operation (see Subsection 3.1)

The MDE mutation operator produces a trial vec-

tor u

(s) for each individual (parent) x

(s) of the

current population. This trial vector will then be

used by the crossover operator to produce off-

spring.

Step 5 – Crossover (see Subsection 3.2)

The MDE crossover operator implements a dis-

crete recombination of the trial vector u

(s) and

the parent vector x

(s) to produce offspring x

(s).

Step 6 – Selection (see Subsection 3.4)

Either the parent x

(s) or the produced offspring

(s) survives and enters the next generation. To

construct the population of the next generation,

the MDE selection mechanism based on probabil-

ity is used.

Step 7 – Stopping Criterion

If the stopping criterion is not satisﬁed, go to Step

3, else return the individual with the best f itness

as the solution. Here, the maximum number of

iterations s

max

is selected as the stopping criterion.

Modiﬁed Differential Evolution in the Load Balancing Problem for the iFDAQ of the COMPASS Experiment at CERN

217

Table 1: Parameters used for the MDE.

max

50 20,000 0.6 0.4 0.3 0.1 1 0.7

Table 2: The TC1 results using the MDE in each execution.

Ex.

C++ Matlab

Error t [ms] Error t [ms]

1 2.24 1,601 2.24 71,413

2 2.24 1,652 2.24 25,558

3 2.24 1,997 2.24 23,688

4 2.24 2,220 2.24 25,378

5 2.24 1,382 2.24 43,370

6 2.24 2,099 2.24 90,298

7 2.24 862 2.24 78,666

8 2.24 1,901 2.24 71,448

9 2.24 2,379 2.24 57,260

10 2.24 1,803 2.24 28,210

4 NUMERICAL RESULTS

The MDE has been implemented in C++ and Mat-

lab (R2018a, 64-bit) on a personal computer equipped

with Intel(R) Core(TM) i7-8750H CPU (@2.20 GHz,

6 Cores, 12 Threads, 9M Cache, Turbo Boost up to

4.10 GHz) and 16 GB RAM (DDR4, 2 666 MHz)

memory. The MDE is examined on two test cases and

numerical results are compared with other methods of

solving the LB problem.

The results are investigated with respect to the er-

ror and computation time. The error is deﬁned as the

objective function of the LB problem, see Equation 1.

4.1 Test Case 1

The Test Case 1 (TC1) consists of m = 6 MUXes with

p = 15 ingoing ports each and with respect to the

number of MUXes, it corresponds to the iFDAQ setup

used in the COMPASS Run 2016, 2017 and 2018. It

considers n = m · p = 6 · 15 = 90 ﬂows with values

randomly generated in the range from 0 B to 10 kB.

The proposed MDE algorithm is partially stochas-

tic and hence, it might produce different solutions in

every execution. The parameters used for the MDE

algorithm to solve the TC1 are given in Table 1.

In Table 2, the results produced in C++ and Mat-

lab for the TC1 using the MDE in each execution

are stated. The error is equal approximately to 2.24

giving a global optimum in each execution, however,

the solutions might be different – multiple global op-

tima are possible. The best TC1 ﬂow allocation based

on the MDE produced in C++ corresponding to Ex-

82,494

82,493 82,493 82,493 82,493 82,493

MUX

10,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

90,000

Flow [B]

Port 15

Port 14

Port 13

Port 12

Port 11

Port 10

Port 9

Port 8

Port 7

Port 6

Port 5

Port 4

Port 3

Port 2

Port 1

Figure 4: The best TC1 ﬂow allocation based on the MDE

produced in C++ corresponding to Execution 7.

82,494

82,493 82,493 82,493 82,493 82,493

MUX

10,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

90,000

Flow [B]

Port 15

Port 14

Port 13

Port 12

Port 11

Port 10

Port 9

Port 8

Port 7

Port 6

Port 5

Port 4

Port 3

Port 2

Port 1

Figure 5: The best TC1 ﬂow allocation based on the MDE

produced in Matlab corresponding to Execution 3.

ecution 7 is given in Figure 4 and produced in Mat-

lab corresponding to Execution 3 is given in Figure

5. In both ﬁgures, the total ﬂow allocated to the ﬁrst

MUX is 82,494 B and to the remaining ﬁve MUXes is

82,493 B each. Thus, the mutual comparison reaches

82,493/82, 494 ≈ 100.00%. The computational time

is very low in C++ due to the fast memory access pro-

vided by C++ and the usage of pointers trying to avoid

any copying of memory. Hence, the MDE algorithm

represents a good candidate for a real-time LB solver.

To demonstrate the behaviour of the MDE algo-

rithm, the evolution process of the best TC1 ﬂow al-

location produced in Matlab corresponding to Execu-

tion 3 is shown in Figure 6. At the beginning of the

evolution process, it shows a fast convergence of the

proposed MDE algorithm. It reached a solution close

to a global optimum after just 150 iterations. Never-

theless, there are several short-term deteriorations of

the error in evolution, which are caused by selection

mechanism based on SA. The selection mechanism

sometimes selects individuals with a worse f itness

value. They have a chance to show their potential

to produce a new population. However, this feature

weakens at the end of the evolution process.

ECTA 2019 - 11th International Conference on Evolutionary Computation Theory and Applications

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0 500 1,000 1,500 2,000 2,500 3,000

Iteration

Error

Figure 6: The evolution process of the best TC1 ﬂow alloca-

tion based on the MDE produced in Matlab corresponding

to Execution 3.

Table 3: Comparison of LB solution methods based on the

best ﬂow allocation for the TC1.

Met.

C++ Matlab

Error t [ms] Error t [ms]

DP 2.24 17,529 2.24 16,521

GH 243.89 47 243.89 1

ILP 21.19 17,127 23.81 1,465

MDE 2.24 862 2.24 23,688

RL 10.44 32,404 10.44 117,106

Finally, a global optimum is reached after 3,400

iterations and the error is equal approximately to 2.24.

The TC1 results produced by the MDE are com-

pared with other LB solution methods – Dynamic

Programming (DP), Greedy Heuristic (GH), Integer

Linear Programming (ILP) and Reinforcement Learn-

ing (RL) – in Table 3. In sum, the MDE and GH, both

reaching a global optimum, can be used for real-time

LB due to the low computational time.

Table 4: The TC2 results using the MDE in each execution.

Ex.

C++ Matlab

Error t [ms] Error t [ms]

1 15.30 6,050 2.00 87,064

2 2.00 2,507 2.00 67,601

3 2.00 1,911 2.00 104,993

4 2.00 3,266 2.00 61,987

5 2.00 2,358 2.00 94,706

6 2.00 2,659 2.00 80,280

7 2.00 5,398 4.24 151,224

8 2.00 3,153 2.00 108,325

9 2.00 2,718 2.00 37,530

10 2.00 5,100 2.00 82,067

68,401 68,401 68,401 68,401

68,400 68,400 68,400 68,400

MUX

10,000

20,000

30,000

40,000

50,000

60,000

70,000

Flow [B]

Port 15

Port 14

Port 13

Port 12

Port 11

Port 10

Port 9

Port 8

Port 7

Port 6

Port 5

Port 4

Port 3

Port 2

Port 1

Figure 7: The best TC2 ﬂow allocation based on the MDE

produced in C++ corresponding to Execution 3.

68,401 68,401 68,401 68,401

68,400 68,400 68,400 68,400

MUX

10,000

20,000

30,000

40,000

50,000

60,000

70,000

Flow [B]

Port 15

Port 14

Port 13

Port 12

Port 11

Port 10

Port 9

Port 8

Port 7

Port 6

Port 5

Port 4

Port 3

Port 2

Port 1

Figure 8: The best TC2 ﬂow allocation based on the MDE

produced in Matlab corresponding to Execution 9.

4.2 Test Case 2

The Test Case 2 (TC2) consists of m = 8 MUXes with

p = 15 ingoing ports each and thus, it corresponds

to the iFDAQ full setup. However, the iFDAQ full

setup has never been in operation for the COMPASS

experiment since it was not required by any physics

program. It considers n = m · p = 8 · 15 = 120 ﬂows

with values randomly generated in the range from 0 B

to 10 kB. The parameters used for the MDE algorithm

to solve the TC2 are the same as for TC1, see Table 1.

In Table 4, the results produced in C++ and Mat-

lab for the TC2 using the MDE in each execution

are stated. The error is almost always equal to 2.00

giving a global optimum in each execution. The

best TC2 ﬂow allocation based on the MDE pro-

duced in C++ corresponding to Execution 3 is given

in Figure 7 and produced in Matlab corresponding to

Execution 9 is given in Figure 8. In both ﬁgures,

the total ﬂow allocated to the ﬁrst four MUXes is

68,401 B each and to the remaining four MUXes is

68,400 B each. Thus, the mutual comparison reaches

68,400/68, 401 ≈ 100.00%.

Modiﬁed Differential Evolution in the Load Balancing Problem for the iFDAQ of the COMPASS Experiment at CERN

219

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000

Iteration

Error

Figure 9: The evolution process of the best TC2 ﬂow alloca-

tion based on the MDE produced in Matlab corresponding

to Execution 9.

Table 5: Comparison of LB solution methods based on the

best ﬂow allocation for the TC2.

Met.

C++ Matlab

Error t [ms] Error t [ms]

DP 2.00 24,873 2.00 23,873

GH 224.22 63 224.22 1

ILP 194.43 49,927 134.61 95,251

MDE 2.00 1,911 2.00 37,530

RL 2.83 75,882 2.45 204,824

The evolution process of the best TC2 ﬂow allo-

cation based on the MDE produced in Matlab corre-

sponding to Execution 9 can be seen in Figure 9. A

global optimum is reached after 4,900 iterations and

the error is equal to 2.00. Finally, the TC2 results

produced by the MDE are compared with results ac-

quired by DP, GH, ILP and RL in Table 5.

5 CONCLUSION

The paper has introduced the LB problem of the iF-

DAQ of the COMPASS experiment at CERN. N P -

completeness of the LB problem makes optimiza-

tion more challenging. The proposed MDE has a

new crossover and mutation operator and its selection

mechanism is inspired by SA. Results have shown the

MDE matches requirements in terms of the best error

and ability to ﬁnd a global optimum. Thus, the MDE

represents a solver of the long-term LB setup, where

no frequent changes in the ﬂows are expected.

Since 2019, a crosspoint switch connecting all

involved links in the iFDAQ provides a fully pro-

grammable system topology making the iFDAQ re-

conﬁgurable on-the-ﬂy and replaces the ﬁxed point-

to-point connections. Thus, the crosspoint switch will

analyze ﬂows and automatically assign them to input

ports of MUXes in order to equally distribute load

over all MUXes. The low computational time of the

MDE opens up a perspective for real-time LB.

ACKNOWLEDGEMENTS

This research has been supported by OP VVV, Re-

search Center for Informatics, CZ.02.1.01/0.0/0.0/

16 019/0000765.

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