Adaptive Grouping Quantum Inspired Shuffled Frog Leaping Algorithm

The International Journal Of Engineering And Science (IJES)
|| Volume || 4 || Issue || 2 || Pages || PP.19-30|| 2015 ||
ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805
www.theijes.com The IJES Page 19
Adaptive Grouping Quantum Inspired Shuffled Frog Leaping
Algorithm
Xin Li1
, Xu Huangfu2
, Xuezhong Guan2
, Yu Tong2
,
1
School of Computer and Information Technology, Northeast Petroleum University, Daqing, China
2
School of Electrical and Information Engineering, Northeast Petroleum University, Daqing, China
--------------------------------------------------------ABSTRACT-----------------------------------------------------------
To enhance the optimization ability of classical shuffled frog leaping algorithm, a quantum inspired shuffled
frog leaping algorithm with adaptive grouping is proposed. In this work, the frog swarms are adaptive grouped
according to the average value of the objective function of child frog swarms, the frogs are encoded by
probability amplitudes of Multi-Qubits system. The rotation angles of Multi-Qubits are determined based on the
local optimum frog and the global optimal frog, and the Multi-Qubits rotation gates are employed to update the
worst frog in child frog swarms. The experimental results of some benchmark functions optimization shows that,
although its single step iteration consumes a long time, the optimization ability of the proposed method is
significantly higher than the classical leaping frog algorithm.
KEYWORDS : Quantum computing, Swarm intelligent optimization, Shuffled leaping frog algorithm, Adaptive
grouping
---------------------------------------------------------------------------------------------------------------------------------------
Date of Submission: 12 February 2015 Date of Accepted : 28 February 2015
---------------------------------------------------------------------------------------------------------------------------------------
I. INTRODUCTION
Shuffled frog leaping algorithm (SFLA) is a new heuristic cooperative search algorithm, which
simulates the foraging behavior of a group of frogs jumping in wetlands [1]. As a new bionic intelligent
optimization algorithm, SFLA combines the advantages of memetic evolutionary algorithm and particle swarm
optimization algorithm, with the concept of simple, less adjustment parameters, calculation speed, strong global
search capability, easy implementation, currently used in the rough set attribute reduction [2], fuzzy-means
clustering [3], drawing water bath to control [4], cooperative spectrum sensing cognitive radio [5- 6], speech
recognition [7], etc. In terms of performance improvements SFLA, paper [8] by introducing in the frog
population attraction and repulsion mechanism, effectively avoiding premature convergence; paper [9] by
introducing the extreme dynamic optimization to improve the optimization efficiency of the algorithm; paper
[10] through the integration, simulated annealing, immunizations, Gaussian mutation, chaotic disturbance,
enhanced algorithm optimization capabilities; paper [11] by using a dispersion and fitness, improve the
convergence speed and accuracy. However, these improvements do not consider the effect of the grouping
SFLA for optimal performance. Since SFLA each group to update only a worst frog, in general, when a fixed
number of frog, the fewer packets (the more the number of the group frog), the higher the computational
efficiency, and optimization capability is relatively weak, so frog group grouping has an important influence on
the optimization of the performance of SFLA. Taking into account the optimal number of packets SFLA usually
related to specific issues as well as the optimization phase, this paper presents an Adaptive Grouping Quantum
Inspired Shuffled Frog Leaping Algorithm. Quantum computing is an emerging interdisciplinary; combining
information science and quantum mechanics, and its integration with intelligent optimization algorithms began
in the 1990s. Currently, there are a lot more mature algorithms, such as quantum-behaved particle swarm
optimization algorithm [12], quantum inspired evolutionary algorithm [13], quantum derivative harmony search
algorithm [14], quantum inspired immune algorithm [15], quantum inspired genetic algorithm [16], quantum
inspired differential evolution algorithm [17]. In addition to the literature [12] using the real-coded, the rest are
encoded using a single bit probability amplitude. Single-bit probability amplitude coding disadvantage is that
the adjustment of a qubit can only change one gene locus on the individual, while the probability amplitude for
multi-bit coding, adjust a qubit, can change all of the ground state probability amplitude in multi-bit quantum
superposition states , thereby changing the position of all genes on the individual. Therefore, this paper will also
propose a new multi-bit probability amplitude coding mechanism to further improve the SFLA of optimizing
capacity and optimize efficiency. The results of the optimization criterion function demonstrate the superiority
of the proposed method.

Adaptive Grouping Quantum Inspired…
II.SHUFFLED FROG LEAPING ALGORITHM
First, There is N frogs 1 2
( , , , )i i i in
x x x x
, calculated the target value
( )( 1, 2, , )i
f x i N
of each
frog, sort all Target value from good to bad. Order N = k × m, k is the number of sub-group, m is the number of
subgroups frogs, the N sorted frogs cycle divided into k sub-groups. namely: the first sub-group divided into
optimal frog, The second sub-group divided into Suboptimal frog, and so on, until all the frogs have been
allocated. In each subgroup, the best and worst frogs were recorded as b
x
and w
x
, the whole frog group of the
best frog recorded as g
x
, in each iteration, the following formula is updated sub-group w
x
.
( )b b
D r x w 
(1)
w w
x x D  
(2)
where m ax m axi
D D D  
, i
D
is the ith
value of the vector D , i = 1, 2,…, n, m ax
D
is the biggest leap frog step,
r is a random number between 0 and 1.
If w
x 
is better than w
x
, then w
x 
instead of w
x
; otherwise, b
x
instead of g
x
, repeat the Eq.(1) and (2). If the w
x 
inferior to w
x
,, then randomly generate a solution to replace w
x
. So the cycle until the termination condition is
met.
III. ADAPTIVE GROUPING STRATEGY
In SFLA, by updating the group worst frog to achieve local optimization, by mixing and re-grouped in
each group to achieve global optimization. From the optimization mechanism, the group number reflects the
optimization of the global characteristics, while the number of frogs in the group reflects the optimization of the
locality. So, how to achieve the balance of global and localized search, is worthy of in-depth study of the
problem, and this balance by grouping decision. However, the balance of global and localized search is often
also associated with specific issues and optimization phase, grouping of frog group cannot be fixed pattern.
Therefore, this paper proposes an adaptive grouping strategy.
Let the total number of frogs be N, we rewritten N as following equation
1 1 2 2 s s
N n m n m n m      
(3)
where 1 2
,s
n n n   1 2 s
m m m  
.
Therefore, there are s kinds of groupings. Namely: 1
n
group, each group has 1
m
frogs; 2
n
group, each group
has 2
m
frogs; …; s
n
group, each group has s
m
frogs. As an example of the minimum value optimization,
recorded the average value of the objective function as a vg
f
, Frog group is divided into k
n
groups, each with
k
n
frogs, the average value of the ith
group objective function is
i
avg
f
, the
i
a vg a vg
f f
number is
1
k
n
, the
i
a vg a vg
f f
number is
2
k
n
. Adaptive grouping strategy proposed in this paper can be described as follows.
(1) If
1 2
k k
n n
k s
 

 , than
1
1
k k
k k
n n
m m




 , (2) If
1 2
1
k k
n n
k
 

 , than
1
1
k k
k k
n n
m m




 .
For this strategy, we explain below. 1k
n
≥ 2k
n
represents the sub-groups of
i
avg
f
less than a vg
f
have advantages,
This means that the optimization approach tends more parallel sub-groups, and therefore need to increase the
number of sub-groups; Conversely, 1k
n
< 2k
n
represents the sub-groups of
i
avg
f
less than a vg
f
have
disadvantages, this means that the optimization approach tends to parallel the minority carrier group, hence the
need to reduce the number of sub-groups.
IV. MULTI-BIT QUANTUM SYSTEM AND THE MULTI-BIT QUANTUM ROTATION GATE
4.1. Qubits and single qubit rotation gate
What is a qubit? Just as a classical bit has a state-either 0 or 1- a qubit also has a state. Two possible
states for a qubit are the state | 0
and | 1
, which as you might guess correspond to the states 0 and 1 for a
classical bit.
Notation like
|
is called the Dirac notation, and we will see it often in the following paragraphs, as it is the
standard notation for states in quantum mechanics. The difference between bits and qubits is that a qubit can be

in a state other than 0|
or 1|
. It is also possible to form linear combinations of states, often called
superposition.
T
| co s | 0 sin | 1 [co s sin ] ,         
(4)
where θ is the phase of | 
, cos  and sin  denote the probability amplitude of | 
.
In the quantum computation, the logic function can be realized by applying a series of unitary transform to the
qubit states, which the effect of the unitary transform is equal to that of the logic gate. Therefore, the quantum
services with the logic transformations in a certain interval are called the quantum gates, which are the basis of
performing the quantum computation. A single qubit rotation gate can be defined as
co s sin
( ) .
sin co s
 

 
   
   
  
R
(5)
Let the quantum state
cos
|
sin



 
   
  , and | 
can be transformed by
cos( )
( )
sin( )
 

 
  
   
  
R
. It is obvious
that ( )R
shifts the phase of
| 
4.2. The tensor product of matrix
Let the matrix A has m low and n column, and the matrix B has p low and q column. The tensor product
of A and B is defined as.
1 1 1 2 1
2 1 2 2 2
1 2
,
n
n
m m m n
A A A
A A A
A A A
 
 
  
 
 
 
B B B
B B B
A B
B B B
L
L
M M M M
L
(6)
where ,i j
A
is the element of matrix A.
4.3. Multi-bit quantum system and the multi-bit quantum rotation gate
In general, for an n-qubits system, there are 2n
of the form 1 2
| n
x x x L
ground states, similar to the
single-qubit system, n-qubits system can also be in the a linear superposition state of 2n
ground states, namely
1 2
1 1 1
T
1 2 1 2 1 2 0 0 0 0 0 1 1 1 1
0 0
| | [ ] ,
n
n x x xn n
x x x
a x x x a a a  
 
      L L L L
L L L L
(7)
where 1 2x x xn
a L
is called probability amplitude of the ground state 1 2
| n
x x x L
, and to meet the following
equation.
1 2
1 1 1
2
1 2
0 0
1 .
n
x x xn
x x x
a
 
   L
L
(8)
Let
| co s |0 sin |1i i i
      
, according to the principles of quantum computing, the 1 2
| n
   L
can be written
as
1 2
1 1 2
1 2 1 2
1
1 2
co s co s co s
co sco s co s co s sin
| | | | .
sin sin
sin sin sin
n
n n
n n
n
n
  
   
     
 
  
 
 
  
            
    
 
 
L
L
L L L
M M M M
L
(9)
It is clear from the above equation that, in an n-qubits system, any one of the ground state probability amplitude
is a function of n-qubits phase 1 2
( , , , )n
  L
, in other words, the adjustment of any i

can update all 2n
probability amplitudes.
In our works, the n-qubits rotation gate is employed to update the probability amplitudes. According to the
principles of quantum computing, the tensor product of n single-qubit rotation gate
( )i
R
is n-qubits rotation
gate. Namely

2 2
( ) ( ) ( ) ( ),i n i n
              R R R RL L
(10)
where
cos sin
( ) , 1, 2, , .
sin cos
i i
i
i i
i n
 

 
   
   
  
R L
Taking n=2 as an example, the 1 2
( )  R
can be rewritten as follows.
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
1 2
1 2 1 2 1 2 1 2
1 2 1 2 1
co s co s co s sin sin co s sin sin
co s sin co s co s sin sin sin co s
( )
sin co s sin sin co s co s co s sin
sin sin sin co s co s sin
       
       
 
       
     
         

        
  
          

     
R
2 1 2
.
co s co s 





   (11)
It is clear that
1 2 1 2 1 2
ˆ ˆ ˆ( ) | | | | ,n n n n
                 R L
L L
(12)
where
ˆ| cos( ) | 0 sin( ) | 1 .i i i i i
            
V. HUFFLED FROG LEAPING ALGORITHM ENCODING METHOD BASED ON MULTI-
BITS PROBABILITYAMPLITUDES
In this paper, the frogs group is encoded by multi-qubits probability amplitudes. Let N denote the
number of particles, D denote the dimension of optimization space. Multi-qubits probability amplitudes
encoding method can be described as follows.
5.1. The number of qubits needed to code
For an n-bits quantum system, there are 2n probability amplitudes, which can be used directly as a
result of an individual encoding. In the D-dimensional optimization space, it is clear that 2
n
D  . Due to the
constraint relation between each probability amplitude (see to Eq.(10)), hence 2
n
D  . For the D-dimensional
optimization problem, the required number of qubits can be calculated as follows.
log( ) 1.n D 
(13)
5.2. The encoding method based on multi-qubits probability amplitudes
First, generating randomly N n-dimensional phase vector i
θ
, 1, 2, , ,i N L
as follows
 1 2
, , , ,i i i in
  θ L
(14)
where
2ij
rand  
, rand is a random number uniformly distributed within the (0,1), 1, 2j n L
.
Let
| cos | 0 sin | 1ij ij ij
      
, Using Equation (11), we can obtain following N n-qubits systems
1 1 1 2 1
| n
   L
, 21 22 2
| n
   L
,…, 1 2
| N N N n
   L
. In each of the quantum system, the first D probability
amplitudes can be regarded as a D-dimensional particle code.
VI. THE UPDATE METHOD BASED ON MULTI-QUBITS PROBABILITYAMPLITUDES
In this paper, the multi-bit quantum rotation gates are employed to update particles. Let the phase
vector of the global optimal frog be 1 2
[ , , ]g g g gn
  F L
, the phase vector of the group optimal frog is
1 2
[ , , , ]b b b b n
  F L
, the phase vector of the group worst frog 1 2
[ , , , ]w w w w n
  F L
. Similar to traditional
SFLA, for each subgroup, we only just need to update w
F
.
By formula (9) it is clear that, once w
F
has been updated, all its corresponding probability amplitudes will be
updated. To improve the search capability, in an iteration, all phases w
F
are updated in turn, which allows all
particles are updated n times. Let 0

denote the phase update step size, the specific update can be described as
follows.

Step1. Set j=1, 
T
1 2 1 2
( ) cos cos cos , sin sin sin ]w i i in i i in
      F L L L L，
.
Step2. Set 1 2
0i i in
        L
.
Step3: Determine the value of the rotation angle, where the sgn donates the symbolic function.
If
| |
i
b j ij
   
, then 0
sg n ( )
b b
ij ij ij
      
.
If
| |
i
b j ij
   
, then 0
sg n ( )
b i
ij b j ij
       
.
If
| |g j ij
   
, then 0
sg n ( )
b
ij g j ij
      
.
If
| |g j ij
   
, then 0
sg n ( )
g
ij g j ij
       
.
Step4.
0 .5 0 .5
b g
j j j
      
,
(1)
1 2
( ) ( , , , ) ( )w n n w
R       F FL
.
Step5: Let 0j
rnds    
, rn d s is a random number of -1 to 1.
( 2 ) (1)
1 2
( ) ( , , , ) ( )w n n w
R       F FL
. If
(1)
( )w
F
is better than
( 2 )
( )w
F
, then
(1)
( ) ( )w w
 F F
, otherwise,
( 2 )
( ) ( )w w
 F F
.
Step6: If
j n
, then j = j + 1, back to step2, Otherwise, end.
VII. ADAPTIVE GROUPING QUANTUM INSPIRED SHUFFLED FROG LEAPING
ALGORITHM
Suppose that, N denote the number of frogs, D denote the number of optimization space dimension.
1 1 s s
n l m l m    L
, i
l
and i
m
is positive integers and 1 2 s
l l l  L 1 2 s
m m m  L
. For adaptive
grouping quantum-inspired shuffled frog leaping algorithm, called AGQISFLA, the optimization process can be
described as follows.
(1) Initialize the frog group
According to Eq.(12) to determine the number of qubits n, according to Eq.(13) initialize phase of each
particle, according to Eq.(9) to calculate the probability amplitude of 2n
each particle, where the first D
probability amplitudes are the coding of the particles. Set the jth
probability amplitude of the ith
particle be ij
x
,
coding result can be expressed as the following equation.
T
1 1 1 1 2 1
T
1 2 1 2 2 2
T
1 2
[ , , , ]
[ , , , ]
.
[ , , , ]
D
D
n N N N D
P x x x
P x x x
P x x x
 




 
L
L
L L L L L L L
L
(15)
Initialization phase update step 0

, the limited number of iteration G. Set the current iteration step t=1.
(2) Calculation of the objective function value
Set the j-dimensional variable range be
[ , ]j j
M inX M axX
, because of the probability amplitude ij
x
values in
the interval [0,1], it is need to make the solution space transformation. The transformation equation is below.
1
[ (1 ) (1 )],
2
ij j ij j ij
X M a xX x M in X x   
(16)
where i = 1, 2, ···, N, j = 1, 2, ···, D.
With the above formula, calculate the objective function values of all frogs. And ascending objective function
value, global optimal frog phase be 1 2
ˆ ˆ ˆˆ [ , , , ]g g g g n
  F L
, global optimal objective function value be
ˆ
g
f
,tLS is
the total number of iteration.
(3) Frog Segmentation
The N sorted frogs cycle into L sub-groups, each group have M frogs, namely: Optimal frog divided into the
first subgroup, suboptimal frog divided into the second subgroups, and so on, until all frog allocated.
(4) Update subgroup worst frog
The Each subgroup evolution times is LS, in each evolution, update subgroup worst frog , follow the steps in
the previous section (1) to step (6). As such, the worst frog of each subgroup are updated LS × n times.
(5) Adaptive calculation of the number of subgroups

In each sub-group, the average value of the objective function is
l
avg
f
, where l= 1, 2,.., L. Mixing the
subgroups, and arranged in ascending order according to the value of the objective function, remember the
objective function is the average of the entire frog groups is avg
f
, the number of
i
a vg a vg
f f
is
1
L
n
, the number
of
i
a vg a vg
f f
is
2
L
n
.The following two steps to recalculate the number of sub-groups are L and the number of
subgroups frog are M.
(a)If
1 2
L L
n n
k s
 

 , than
1
1
k
k
L l
M m




 . (b) If
1 2
1
L L
n n
k
 

 ,than
1
1
k
k
l l
M m




 .
(6) Update the global optimal solution
Let the optimal frog phase be 1 2
[ , , , ]g g g gn
  F
, the corresponding objective function value be g
f
. If
ˆ
g g
f f
, then
ˆ
g g
f f
,
ˆ
g g
F F
; otherwise
ˆ
g g
f f
,
ˆ
g g
F F
.
(7) Determine the termination condition
If t <G, t = t + 1, back to (3); otherwise, save the result, end.
VIII. COMPARATIVE EXPERIMENT
In this study, the 25 standard test functions are employed to verify the optimization ability of
AGQISFLA, and compare with the traditional leapfrog algorithm (SFLA), adaptive grouping leapfrog algorithm
(AGSFLA). All test functions are minimal value optimization, All functions belong to minimum optimization,
where D is the number of independent variables, Ω is the solution space,
*
X is the exact minimum point,
*
( )f X
is the corresponding minimum.
8.1. Test function
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
2 * *
1 1
( ) ; [ 100,100] ; [0, 0, , 0]; ( ) 0.
D D
ii
f x f

    X Ω X XL
* *
2 1 1
( ) ; [ 100,100] ; [0, 0, , 0]; ( ) 0.
DD D
i ii i
f x x f
 
      X Ω X XL
2 * *
3 1 1
( ) ( ) ; [ 1 0 0,1 0 0 ] ; [0, 0, , 0 ]; ( ) 0 .
D i D
ji j
f x f
 
     X Ω X XL
* *
4
1
( ) m ax ( ); [ 100,100] ; [0, 0, , 0]; ( ) 0.
D
i
i D
f x f
 
    X Ω X XL
1 2 2 2 * *
5 11
( ) (100( ) ( 1) ); [ 100,100] ; [0, 0, , 0]; ( ) 0.
D D
i i ii
f x x x f


       X Ω X XL
2
6 1
( ) [ 0.5] ; [ 100,100] ; [1,1, ,1]; ( ) 0.
D D
ii
f x X f X
 

      X L
4 * *
7 1
( ) (1 (0,1)); [ 100,100] ; [0, 0, , 0]; ( ) 0.
D D
ii
f ix random f

     X Ω X XL
2 * *
8
( ) [ 1 0 co s(2 ) 1 0 ]; [ 1 0 0,1 0 0 ] ; [0, 0, , 0 ]; ( ) 0 .
D
i i
f x x f      X Ω X XL
2
9 1 1
* *
1 1
( ) 2 0 ex p ( 0 .2 ) ex p ( co s(2 )) 2 0 ;
[ 1 0 0,1 0 0 ] ; [0, 0, , 0 ]; ( ) 0 .
D D
i ii i
D
f x x e
D D
f

 
     
   
 X
Ω X XL
2 * *
10 1 1
1
( ) cos( ) 1; [ 100,100] ; [0, 0, , 0]; ( ) 0.
4000
DD Di
ii i
x
f x f
i
 
       X Ω X XL
4 2 * *
1 1 1
1
( ) ( 1 6 5 ) 7 8 .3 3 2 3 3 1 4; [ 1 0 0,1 0 0 ] ; 2 .9 0 3 5 3 4; ( ) 0 .
D D
i i i ii
f x x x x f
D 
        X Ω X
12 2 2 2
1 2 1 11 1
* *
( ) [1 0 s in ( ) ( 1) (1 1 0 s in ( )) ( 1) ( ,1 0 ,1 0 0 .4 );
( ) ,
1
( , , , ) 0 , 1 ( 1);
4
( ) , ,
[ 1 0 0 ,1 0 0 ] ; [ 1, 1, , 1]; ( ) 0 .
D D
i i D ii i
m
i i
i i i i
m
i i
D
f y y y y u x
D
k x a x a
u x a k m a x a y x
k x a x a
f

 

 
      

  

      

  

      
 X
Ω X XL

(13)
13 2 2
1 3 1 11
2 2
1
*
1
( ) [sin (3 ) ( 1) (1 sin (2 ))
( 1) (1 sin (2 ))] ( , 5,1 0 0, 4 );
( ) ,
( , , , ) 0 ,
( ) , ,
[ 1 0 0,1 0 0 ] ; [ 1, 1, , 1]; ( ) 0 .
D
i ii
D
D D ii
m
i i
i i
m
i i
D
f x x x
D
x x u x
k x a x a
u x a k m a x a
k x a x a
X f X
 





    
  
  

   

  

       


X
L
(14)
(15)
2
1 5 1 1
2 2 2
* *
( ) ( co s( ) 1);
4 0 0 0
1 0 0 ( ) (1 ) ;
[ 1 0 0,1 0 0 ] ; [1,1, ,1]; ( ) 0 .
D D jk
jkK j
jk k j j
D
y
f y
y x x x
f
 
  
   
    
 X
X XL
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
8.2. The experimental scheme and parameter design
The total number of frog group is set to N = 100, in order to achieve adaptive grouping, the N is
decomposed into the following eight situations, as follows.
2 5 0 4 2 5 5 2 0 1 0 1 0 2 0 5 2 5 4 5 0 2 1 0 0 1               N (17)
Among them, each instance is a grouping scheme, the first number is the number of sub-groups, and the second
number is the number of frogs in the group.
SFLA and AGSFLA limit the number of iteration steps is set to G = 100 and G = 1000, Biggest jump step taken
m ax
5D 
; limited iteration number of AGQISFLA is set to G = 100, The subgroups iterations of these three
1 2 2
1 4 1 11
* *
( ) ( 2 0 .3 co s(3 ) co s(4 ) 0 .3);
[ 1 0 0,1 0 0 ] ; [0, 0, , 0 ]; ( ) 0 .
D
i i i ii
D
f x x x x
f
 

 
   
   
X
Ω X XL
/ 4 2 2 4 4
1 6 4 3 4 2 4 1 4 4 2 4 1 4 3 41
* *
( ) [( 1 0 ) 5( ) ( 2 ) 1 0 ( 1 0 ) ];
[ 1 0 0,1 0 0 ] ; [0, 0, , 0 ]; ( ) 0 .
D
i i i i i i i ii
D
f x x x x x x x x
f
     
       

   
X
Ω X XL
1 2 2 0 .2 5 2 2 2 0 .1
1 7 1 11
* *
( ) ( , ) ( , ) ( , ) ( ) [sin (5 0 ( ) ) 1];
[ 1 0 0,1 0 0 ] ; [0, 0, , 0 ]; ( ) 0 .
D
i i Di
D
f g x x g x x g x y x y x y
f


      
   
X
Ω X XL
2
1 8 1
* *
1 / 2
( ) 1 0 ( 1 0 co s(2 )); ;
(2 ) / 2 , 1 / 2
[ 1 0 0,1 0 0 ] ; [0, 0, , 0 ]; ( ) 0 .
D i i
i i ii
i i
D
x x
f D y y y
ro u n d x x
f


 
    

   
X
Ω X XL

m ax m ax
1 9 1 0 0
* *
( ) [ co s(2 ( 0 .5 ))] ( co s( ));
0 .5; 0 .3; m ax 3 0; [ 1 0 0,1 0 0 ] ; [0, 0, 0 ]; ( ) 0 .
D k kk k k k
ii k k
D
f a b x D a b
a b k f
 
  
  
      
  X
Ω X XL
2 2 4 *
2 0
1 1 1
( ) ( 0 .5 ) ( 0 .5 ) ; [ 1 0 0 .1 0 0 ] ; [0, 0, 0 ]; ( ) 0 .
D D D
D
i i i
i i i
f x ix ix X f X

  
        X Ω L
1
2 1 1
( ) ; [ 1 0 0,1 0 0 ] ; [0, 0, , 0 ]; ( ) 0 .
iD D
ii
f x X f X

 

     X L
22 1
( ) sin( ) 0.1 ; [ 100,100] ; [0, 0, , 0]; ( ) 0.
D D
i i ii
f x x x X f X
 

      X L
 
2 21
2 21 1
2 3 1 1
1
* *
( 0 .5 )
( ) ex p ) co s(4 0 .5 ) 1;
8
[ 1 0 0 .1 0 0 ] ; [0, 0, 0 ]; ( ) 0 .
D
i i i i
i i i i
i
D
x x x x
f x x x x D
f

 
 

  
      
   
X
Ω X XL
2
24 1
( ) 1 exp( 0.5 ); [ 1,1] ; [0, 0, , 0]; ( ) 0.
D D
ii
f x X f X
 

       X L
2 2
25 1 1
( ) 1 cos(2 ) 0.1 ; [ 100,100] ; [0, 0, , 0]; ( ) 0.
D D D
i ii i
f x x X f X
 
 
        X L

algorithms is set to LS=15;phase update step is set to 0
0 .0 5  
.
For SFLA, each function are used eight kinds of groupings were optimized 50 times, namely, each function is
independent optimization .400 times. 400 times optimal results averaged and the average value of a single
iteration of the running time as a comparative index; for AGSFLA and AGQISFLA, The initial number of frog
groups is L = 10, the group number of frog is M = 10, Each function independently optimized 50 times, taking
the average of 50 times optimal results, with the average value of a single iteration of the running time as a
comparison index.
8.3. Comparative Experiment Results
Experiments conducted using Matlab R2009a. For comparison, the average time of a single iteration of
the function i
f
is set to i
T
, the average optimal results for i
O
, i = 1, 2,…,25. All test functions, Taking G=50 as
an example ( 1 6
f
for D = 52), the results of such comparison are shown in Table 1, he average optimization
results for D=100, are shown in Table 2.
For the function i
f
, the average time of the single iteration of SFLA, AGSFLA, AGQISFLA is set to
A
i
T
,
F
i
T
,
Q
i
T
.the average optimal results is set to
A
i
Q
,
F
i
Q
,
Q
i
Q
.To facilitate a further comparison, Taking AGSFLA and
SFLA as example, there are the following two formulas. Eq.(18) is the ratio of the average running time. Eq.(19)
is the ratio of the average optimal results
2 0
1
/
2 5
A FA
i ii
F
T TT
T



(18)
2 0
1
/
2 5
A FA
i ii
F
Q QQ
Q



(19)
For these three algorithms, the ratio of average running time and the average optimal results is shown in Table 3.
Table 1 The 50 times contrast of the average results for three algorithms optimization (D = 50)
SFLA AGSFLA AGQISFLA
i
f ( )i
T s
i
O
( )i
T s
i
O
( )i
T s
i
O
2
1 0G 
3
1 0G 
2
1 0G 
3
1 0G 
2
1 0G 
1
f 0.0146 2.03E+03 3.09E+02 0.1452 2.61E+02 2.59E-008 0.1700 3:26E-015
2
f 0.0163 4.23E+08 3.41E+02 0.1103 5.30E+05 5.64E+02 0.2091 5:19E-008
3
f 0.0984 5.16E+04 8.11E+03 0.5064 5.55E+03 3.35E+02 1.2738 1:17E-015
4
f 0.0190 14.9593 10.4918 0.1055 15.3773 12.8575 0.2239 0:514605
5
f 0.0372 7.04E+07 7.73E+05 0.2028 7.42E+05 2.03E+02 0.3585 48:24962
6
f 0.0279 1.58E+03 43.23333 0.1434 85.70000 34.30000 0.2793 0
7
f 0.0341 2.84E+07 3.24E+03 0.1532 4.92E+04 0.001770 0.2827 3:55E-028
8
f 0.0249 2.40E+03 9.78E+02 0.1377 1.09E+03 9.71E+02 0.2221 3:16E-013
9
f 0.0316 17.18656 16.07727 0.1465 14.31819 13.61492 0.4036 0:1850413
1 0
f 0.0308 1.62E+02 3.149318 0.0740 4.434261 0.002976 0.3502 1:16E-015
1 1
f 0.0317 1.35E+04 1.21E+02 0.1606 1.45E+02 12.72304 0.3003 6:1248557
1 2
f 0.0773 8.40E+06 11.55566 0.4046 2.94E+02 17.54648 0.6654 2:99E-005
1 3
f 0.0772 2.59E+07 8.99E+03 0.3996 5.32E+04 1.00E+02 0.7448 0:0322664
1 4
f 0.0366 6.05E+03 9.57E+02 0.1735 7.94E+02 11.75275 0.4062 1:42E-013
1 5
f 0.3756 3.33E+13 2.35E+09 0.7196 1.41E+10 2.52E+03 3.1635 2:90E+002
1 6
f 0.0515 3.76E+07 3.98E+05 0.2787 1.97E+05 2.26E+03 0.6207 9:26E-023
1 7
f 0.0515 1.93E+02 1.62E+02 0.3485 1.77E+02 1.65E+02 0.5558 0:0016304
1 8
f 0.0447 2.42E+03 9.61E+02 0.3161 1.24E+03 9.68E+02 0.3490 1:26E-012
1 9
f 0.4529 58.03894 41.94107 0.8332 60.97842 44.34244 3.7331 6:8459530

2 0
f 0.0202 1.55E+07 1.19E+04 0.1772 3.08E+04 7.66E+03 0.2160 5:43E-015
2 1
f 0.0338 4.17E+65 2.50E+47 0.2809 4.13E+41 1.53E+15 0.3652 1:27E-206
2 2
f 0.0167 1.47E+02 74.83271 0.0969 71.211607 46.331449 0.1750 6:76E-009
2 3
f 0.0348 44.76880 43.062524 0.1367 45.373445 42.675330 0.2528 0:1789964
2 4
f 0.0158 0.073135 0.0317157 0.0996 0.0041286 1.01E-14 0.1133 0
2 5
f 0.0173 1.92E+02 15.005397 0.1037 24.769981 4.357839 0.1543 4:72E-014
Table2 The 50 times contrast of the average results for three algorithms optimization (D = 100)
i
f
SFLA AGSFLA AGQISFLA
( )i
T s
i
O
( )i
T s
i
O
( )i
T s
i
O
2
1 0G 
3
1 0G 
2
1 0G 
3
1 0G 
2
1 0G 
1
f 0.0226 4.99E+03 1.02E+03 0.2278 2.16E+03 0.061430 0.2606 8:91E-015
2
f 0.0246 3.64E+51 6.45E+04 0.1723 1.27E+35 1.15E+03 0.3227 1:15E-007
3
f 0.1544 2.10E+05 3.12E+04 0.7998 1.41E+04 5.87E+03 1.9394 1:61E-013
4
f 0.0292 18.15317 13.80720 0.1683 17.63319 11.72408 0.3488 0:559791
5
f 0.0589 1.48E+08 2.98E+06 0.3147 2.58E+07 4.78E+02 0.5650 69:8203
6
f 0.0442 3.68E+03 3.46E+02 0.2277 8.68E+02 3.80E+02 0.4337 0
7
f 0.0531 1.43E+08 2.38E+05 0.2324 6.12E+06 6.29E+03 0.4421 1:39E-027
8
f 0.0385 5.66E+03 2.59E+03 0.2200 4.01E+03 2.95E+03 0.3377 013
9
f 0.0502 18.44872 17.75022 0.2237 17.56241 12.90171 0.6207 0:404291
1 0
f 0.0487 3.37E+02 6.909176 0.1128 27.02232 0.077702 0.5527 9:11E-014
1 1
f 0.0495 1.62E+04 2.06E+02 0.2549 5.62E+02 13.19321 0.4708 9:089264
1 2
f 0.1222 1.80E+07 17.88713 0.6367 3.51E+05 16.03746 1.0287 1:23E-004
1 3
f 0.1208 5.76E+07 7.37E+04 0.6048 1.03E+07 2.06E+02 1.1594 5:01E-002
1 4
f 0.0561 1.48E+04 3.04E+03 0.2604 5.33E+03 32.15902 0.6415 2:04E-013
1 5
f 0.5742 1.95E+14 2.05E+10 1.1437 6.90E+12 1.47E+04 4.7639 3:99E+002
1 6
f 0.0790 5.23E+07 7.03E+05 0.4235 1.03E+06 1.72E+04 0.9684 4:32E-022
1 7
f 0.0800 4.16E+02 3.54E+02 0.5331 3.73E+02 1.43E+02 0.8364 0:091603
1 8
f 0.0703 5.37E+03 2.70E+03 0.4950 4.30E+03 1.88E+03 0.5379 5:11E-012
1 9
f 0.6933 1.39E+02 1.16E+02 1.2734 1.44E+02 21.29647 5.7135 9:935560
2 0
f 0.0319 1.17E+05 4.09E+04 0.2741 1.15E+05 1.89E+04 0.3428 3:88E-014
2 1
f 0.0526 7.74E+84 1.71E+55 0.4231 1.87E+60 1.63E+17 0.5483 9:36E-187
2 2
f 0.0256 3.56E+02 1.98E+02 0.1549 2.64E+02 1.58E+02 0.2759 8:34E-008
2 3
f 0.0546 93.59604 92.42014 0.2130 94.61277 90.05981 0.4037 0:9899643
2 4
f 0.0245 0.183484 0.096426 0.1536 0.029085 1.12E-08 0.1811 0
2 5
f 0.0267 4.92E+02 91.40583 0.1608 2.06E+02 14.42596 0.2436 2:67E-013
Table 3 the ratio of average running time and the average optimal results
D /
A F
T T
/
A F
i i
Q Q
/
Q A
T T
1 0 0
/
Q A
G
Q Q
/
Q F
T T
1 0 0
/
Q F
G
Q Q
2
10G 
3
10G 
2
10G 
3
10G 
2
10G 
3
10G 
50 5.495 0.2784 0.5133 2.107 0.0819 0.0418 10.06 0.0067 0.0111
100 5.486 0.4143 0.3492 2.117 0.0602 0.0567 10.04 0.0054 0.0081
AVG 5.491 0.3463 0.4312 2.112 0.0710 0.0492 10.05 0.0060 0.0096
From Tab.1-Tab.3, the introduction of adaptive grouping strategy and quantum computing makes AGQISFLA
single-step running time of about 10 times that of traditional SFLA. Therefore, in order to enhance the fairness
of the comparison results, we not only need to look at the same contrast iteration number, and must be further
investigated algorithm to optimize the results of comparison in the same time. This is the fundamental reason for

the SFLA and AGSFLA the iteration number is set to G = 100 and G = 1000. According to the optimization
results of f1 ~ f25, when G = 100, AGSFLA is about 0.35 times of SFLA; when G = 1000, AGSFLA is about
0.43 times of SFLA. This shows that the introduction of adaptive grouping strategy can indeed enhance the
ability to optimize the algorithm. When G = 100, AGQISFLA (G = 100) is about 0.07 times of AGSFLA; when
G = 1000, AGQISFLA (G = 100) of approximately 0.05 times of AGSFLA. This shows that the use of multi-bit
probability amplitude coding and evolutionary mechanisms can indeed improve the algorithm optimization
capabilities. From
Table 3, in the same iteration steps, optimization results of AGQISFLA only 6/1000 of SFLA; at the same time
optimization, optimization results of AGQISFLA only one percent of SFLA. Experimental results show that the
adaptive grouping and multi-bit probability amplitude coding can indeed significantly improve the ability to
optimize the traditional leapfrog algorithm.
In this paper, when D = 50 and D = 100, the average results of the algorithm AGQISFLA is show in Fig.1, the
average results of 50 times each function after optimization, the figure shows that when the dimension of
AGQISFLA increases, there is a good stability.
Figure 1 When D = 50 and D = 100, contrast optimization results of AGQISFLA
8.4 Analysis of experimental results
About adaptive grouping strategy, when f1 ~ f25 is 100-dimensional. Average number of iteration steps
show in different groups AGQISFLA and AGSFLA, as shown in Figure 2 ~ 4.
Figure 2 when G = 100, the average value of the iteration number of different groups of AGSFLA
Figure 3 when G = 1000, the average value of the iteration number of different groups of AGSFLA
1 5 10 15 20 25
0
100
200
300
400
f
1
 f
25
Avg.results
50 dimension
100 dimension
2 4 5 10 20 25 50 100
0
20
40
60
80
100
The number of subgroups
Avg.iterations
2 4 5 10 20 25 50 100
0
200
400
600
800
1000
Avg.iterations

Figure 4 when G = 100, the average value of the iteration number of different groups of AGQISFLA
After adaptive grouping, the percentages of the number iterations for each subgroup in total iterations
are shown in Table 4.
Table 4 the percentages of the number iterations for each subgroup in total iterations (%)
Algorithm Iterations
Number of Subgroups
2 4 5 10 20 25 50 100
AGSFLA 100 0.14 1.22 2.04 3.32 5.08 5.64 5.78 76.78
AGSFLA 1000 0.02 0.08 0.17 0.35 1.87 3.18 6.66 87.67
AGQISFLA 100 0.04 2.22 3.62 2.86 4.10 5.18 6.52 75.46
The experimental results demonstrate the AGQISFLA long run time, high capacity optimization
features, we give the following analysis.
First, as previously described, the more subgroups, the more number of frogs need to update for each iteration,
and so the Longer the time of single iteration. Secondly, the individual coding method based on multi-bit
probability amplitude, if the number of qubits encoded set is n, each iteration subgroup evolutionary times for
LS, then each iteration, each subgroup of the worst frog are updated LS × n times, which in the extended run
time, but also greatly improve the number updates of worst frog; at the same time, this by individually adjusting
the phase of qubits to cycle update individual approach, making individuals more elaborate update, which also
enhances the solution space of ergodic. Third, in the update strategy based on multi-bit revolving door, and
draws on the thinking of particle swarm optimization, taking the leading role of subgroups optimal frogs and
global optimum frog, To some extent, to avoid the tendency to fall into premature convergence; at the same
time, the use of multi-bit quantum revolving door, one operation can be achieved for all the updates on the
individual probability amplitude, the characteristics of quantum rotation gates guaranteed probability amplitude
of the "length" unchanged, effectively avoiding iterative sequence divergence, and thus improve the
convergence capability. In summary, the adaptive grouping and multi-bit encoding probability amplitude of
these two mechanisms, at the expense of time in exchange for the ability to optimize, which is consistent with
the theorem no free lunch.
IX. CONCLUSION
In this paper, a quantum inspired shuffled frog leaping algorithm algorithms is presented which
encoded by adaptive grouping and multi-bit probability amplitude. Frog group individual coding approach is to
use multi-qubits system in the ground state of the probability amplitude, frog group of individuals update
method is a multi-bit quantum revolving door. Function extreme optimization results show that under the same
running time, the optimization ability of proposed algorithm has greatly superior to the conventional leapfrog
algorithm. thus revealing, adaptive grouping and multi-bit encoding probability amplitude of these two
mechanisms is indeed an effective way to greatly improve traditional leapfrog algorithm optimization
capabilities.
X. ACKNOWLEDGEMENTS
This work was supported by the Youth Foundation of Northeast Petroleum University (Grant No.2013NQ119).
XI. REFERENCES
[1] Eusuff M M, Lansey K E. Optimization of water distribution network design using the shuffled frog leaping
algorithm. Journal of Water Resources Planning and Management. 2003, 129(3): 210-225.
[2] Ding Wei-Ping, Wang Jian-Dong, Chen Shen-Bo, Chen Xue-Yun, Sen Xue-Hua. Rough attribute reduction with
cross-entropy based on improved shuffled frog-leaping algorithm. Journal of Nanjing University (Natural Sciences),
2014, 50(2): 159-166.
[3] Cui Wen-Hua, Liu Xiao-Bing, Wang Wei, Wang Jie-Sheng. Product family design method based on fuzzy Cmeans
clustering method optimized by improved shuffled frog leaping algorithm. Journal of Dalian University of
Technology, 2013, 53(5): 760-765.
2 4 5 10 20 25 50 100
0
20
40
60
80
100
Avg.iterations

[4] Xio Chun-Cai, Hao Kuang-Rong, Ding Yong-Sheng. An improved shuffled frog leaping algorithm for solving
controlled optimization problems of water bath stretching slot. Journal of East China University of Science and
Technology (Natural Science Edition), 2014, 40(1): 102-106.
[5] Zheng Shi-Lian, Luo Cai-Yi, Yang Xiao-Niu. Cooperative spectrum sensing for cognitive radios based on a modified
shuffled frog leaping algorithm. Acta Physica Sinica, 2010, 59(5): 3611-3617.
[6] Zheng Shi-Lian, Yang Xiao-Niu. Swarm initialization of shuffled frog leaping algorithm for cooperative spectrum
sensing in cognitive radio. Acta Physica Sinica, 2013, 62(7): 078405.
[7] Zhang Xiao-Dan, Huang Cheng-Wei, Zhao Li, Zou Cai-Rong. Recognition of practical speech emotion using
inproved shuffled frog leaping algorithm. Acta Acustica, 2014, 39(2): 271-280.
[8] Wang Jie-Sheng, Gao Xian-Wen. Design of multivariable PID controller of electroslag remelting process based on
improved shuffled frog leaping algorithm. Control and Decision, 2011, 26(11): 1731-1734.
[9] Luo Jian-Ping, Li Xia, Chen Min-Rong. Improved Shuffled Frog Leaping Algorithm for Solving CVRP. Journal of
Electronics & Information Technology, 2011, 33(2): 429-434.
[10] Zhang Xiao-Dan, Zhao Li, Zou Cai-Rong. An improved shuffled frog leaping algorithm for solving constrained
optimization problems. Journal of Shandong university (Engineering Science), 2013, 43(1): 1-8,21.
[11] Xiao Ying-Ying, Chai Xu-Dong, Li Bo-Hu, Wang Qiu-Sheng. Convergence analysis of shuffled frog leaping
algorithm and its modified algorithm. Journal of Huazhong University of Science & Technology (Natural Science
Edition), 2012, 40(7): 15-18,28.
[12] Sun J,Wu X J, Vasile P, Fang W, Lai C H, XuWB. quantum-behaved particle swarm optimization. Information
Sciences, 2012, 193: 81-103.
[13] Lu T C, Yu G R. An adaptive population multi-objective quantum-inspired evolutionary algorithm for
multiobjective0/1 knapsack problems. Information Sciences, 2013, 243: 39-56.
[14] Abdesslem L. A hybrid quantum inspired harmony search algorithm for 0-1 optimization problems. Journal of
Computational and Applied Mathematics, 2013, 253: 14-25.
[15] Jiaquan G, Jun W. A hybrid quantum-inspired immune algorithm for multi-objective optimization. Applied
Mathematics and Computation, 2011, 217: 4754-4770.
[16] Han Kuk-Hyun and Kim Jong-Hwan. Quantum-inspired evolutionary algorithm for a class of combinatorial
optimization. IEEE Transactions on Evolutionary Computation,2002, 6(6): 580-593.
[17] Liu Xian-De, Li Pan-Chi, Yang Shu-Yun, Pan Jun-Hui, Xiao Hong, Cao Mao-Jun. Design and implementation of
quantum-inspired differential evolution algorithm. Journal of Signal Processing, 2014, 30(6): 623-633.
[18] Giuliano Benenti, Giulio Casati, Giuliano Strini. Principles of Quantum Computation and Information (Column I:
Basic Concepts). Singapore: World Scientific Publishing Company. 2004, 99-187.
[19] Cheng Dai-zhan, Qi hong-sheng. Semi-tensor product of matrix: theory and application (Second Edition).Beijing:
Science Press. 2011: 1-24.

Adaptive Grouping Quantum Inspired Shuffled Frog Leaping Algorithm

More Related Content

What's hot (20)

Viewers also liked (19)

Similar to Adaptive Grouping Quantum Inspired Shuffled Frog Leaping Algorithm (20)

Recently uploaded (20)

Adaptive Grouping Quantum Inspired Shuffled Frog Leaping Algorithm