Distance Maintaining Compact Quantum Crossover Based ... · PDF fileDistance Maintaining Compact Quantum Crossover Based Clonal Selection Algorithm Hongwei Dai, Yu Yang, Cunhua Li

Distance Maintaining Compact Quantum Crossover Based Clonal Selection Algorithm

Hongwei Dai, Yu Yang, Cunhua Li

Distance Maintaining Compact Quantum Crossover Based Clonal

Selection Algorithm

*1Hongwei Dai,

1Yu Yang,

1Cunhua Li

*1,Corresponding Author School of Computer Engineering, Huaihai Institute of Technology,

[email protected]

doi:10.4156/jcit.vol5. issue10.8

Abstract Premature convergence is one of the major difficulties with Clonal Selection Algorithm (CSA). It

has been observed that this problem is closely tied to the problem of losing diversity in the population.

In this paper, we propose a distance maintaining compact quantum crossover based CSA. The compact

quantum crossover is useful for information exchanging between different solutions, whereas uses

fewer antibodies. On the other hand, distance maintaining scheme permits the algorithm to fine tune its

solutions. Simulation results on Traveling Salesman Problems (TSP) show that the novel algorithm is

able to effectively balance exploration and exploitation of the search space and can also present the

optimal or near-optimal solutions.

Keywords: Clonal Selection Algorithm (CSA), Diversity,

Distance Maintaining Compact Quantum Crossover, Optimization Problem

1. Introduction

Over the last few decades, there has been a rapid increasing interest in the area of natural inspired

computing. such as Genetic Algorithm (GA) [21][29], Evolutionary Algorithm (EA) [3][28], Ant

Colony Optimization (ACO) [10][2] and Particle Swarm Optimization (PSO) [12]. Unlike other

traditional optimization algorithms, which emphasize accurate and exact computation at the cost of

spending much more computation time, these nature inspired algorithms have better convergence

speeds to the optimal or near optimal results in reasonable time.

More recently however, an artificial computational systems named artificial immune system (AIS),

exploits the immune system’s characteristics of learning and memory in order to develop adaptive

systems capable of performing a wide range of tasks in various engineering applications, such as patter

recognition [6],intrusion detection [13] [8], computer security [16], optimization problems [11][1], and

fault detection [4]. In particular, Clonal Selection Algorithm (CSA) [9], one of the most famous

artificial immune algorithms which was designed based on the clonal selection principle of adaptive

immunity, has been verified as having a great number of useful mechanisms from the viewpoint of

programming, controlling, information processing and so on [8], [11], [1].

As mentioned above, CSA is very useful for solving some difficult problems, however, suffers from

poor convergence properties and difficulties in reaching high-quality solutions in reasonable time [30].

Before analyzing this problem, the concept of diversity is introduced at first.

Diversity, a measure of a population independent of the evaluation function, typically represents the

relative distance between individuals with respect to their representation [25]. Population based

algorithms such as evolutionary algorithms and genetic algorithms have a tendency to converge to local

optima. CSA, a population based AIS, has been observed that its premature convergence problem is

closely tied to the problem of losing diversity [7].

In traditional CSA, according to the clonal selection theory, low affinity antibodies have to be

deleted and replaced by the new random antibodies during mutation process. On the other hand, initial

antibody repertoire is also generated randomly. These random operations clearly reduce the complexity

of algorithm. However, random antibodies have a bad influence on the diversity of repertoire. Hence,

the search within affinity landscape will become premature and can not produce better solutions.

Moreover, both somatic hypermutation of clonal selection theory and receptor editing generate not only

high affinities, but also low affinity antibodies usually [26]. As a result, the search efficiency is low.

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Journal of Convergence Information Technology

Volume 5, Number 10. December 2010

To solve this problem, a distance maintaining compact quantum crossover is constructed in this

paper.

The classical quantum interference crossover proposed by Narayanan [22] has been proved useful

for solving multicast routing problem [19]. An improved quantum crossover has also proposed for

information exchanging between different antibodies in our previous work [7]. In our paper submitted

to ICICEL, it has been also approved that the compact quantum crossover is able to exchange

information like the improved crossover in [7], whereas uses fewer antibodies. In addition, distance

maintaining scheme is expected to further improve the algorithm performance.

The remainder of this paper is organized as follows: In Section 2, we introduce the clonal selection

theory and quantum interference crossover. We then describe the distance maintaining compact

quantum crossover based clonal selection algorithm in Section 3. In Section 4, a number of benchmark

problems - traveling salesman problems are solved for validating the performance of the proposed

algorithm. Finally, we give some general remarks to conclude this paper in Section 5.

2. Background

This section will describe the clonal selection theory behind CSA. Also, a brief description of the

classical quantum crossover and the improved crossover in [7] will be provided.

2.1. Clonal Selection Theory[5]

Clonal selection theory is a form of natural selection and it explains the essential features which

contain sufficient diversity, discrimination of self and non-self and long-lasting immunologic memory.

This theory interprets the response of lymphocytes in the face of an antigenic stimulus.

When an animal is exposed to an antigen, some subpopulation of its bone marrow derived cells (B

lymphocytes) can recognize the antigen with a certain affinity (degree of match).B lymphocytes will be

stimulated to proliferate and eventually mature into terminal (non-dividing) antibody secreting cells,

called plasma cells. Proliferation of the B lymphocytes is a mitotic process whereby the cells divide

themselves, creating a set of clones identical to the parent cell. The proliferation rate is directly

proportional to the affinity level, i.e. the higher affinity levels of B lymphocytes, the more of them will

be readily selected for cloning and cloned in larger numbers.

According to the clonal selection theory, random point mutation is performed during the maturation

process. However, frequently, a large proportion of the cloned population becomes dysfunctional or

develops into harmful anti-self cells after the mutation. Moreover, the authors [23] interpret that clonal

deletion, previously regarded as the major mechanism of central B cell tolerance, has been shown to

operate secondarily and only when receptor editing is unable to provide a non-autoreactive specificity.

Recent investigations [24], [26], [31] indicate that receptor editing has played a major role in

shaping the lymphocyte repertoire. Both B and T lymphocytes that carry antigen receptors are able to

change specificity through subsequent receptor gene rearrangement.

2.2. Quantum Interference Crossover

Quantum computing, a research area including concepts such as quantum mechanical computers

and quantum algorithms, has gained much attention and wide applications because of its superiority to

classical computer on various problems. Some works on combining quantum computing with

evolutionary algorithms (EA), genetic algorithms (GA) and particle swarm optimization (PSO) for

some combinatorial optimization problems can be found in [14][15][27].

In addition, a quantum interference crossover based on quantum computing was first proposed by

Narayanan [22]. In paper [19], a quantum clonal algorithm (QCA) merging immune clonal selection

algorithm and quantum crossover was presented to deal with multicast routing problem.

The classical quantum crossover occurs as follows: take the 1st city of tour one, 2nd city of tour two,

3rd city of tour three, etc. Note that no duplicates are permitted within the same universes. If a city is

already presented in the visiting tour, choose the next alphabetical city not already contained. Based on

simulation results, it can be found that this classical quantum crossover, we call it position based

quantum crossover, is not suitable for TSPs. In [7], we proposed an improved quantum crossover based

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CSA for solving traveling salesman problem (TSP). This distance based quantum crossover works well

and presents better solutions.

It should be noticed that both the position based classical quantum crossover and the improved

distance based ones have to face the same problem: The number of universes required for crossover is

equal to the number of cities of the TSP instance. In order to overcome this problem, a distance

maintaining compact quantum crossover based CSA is proposed in this paper. The novel crossover is

able to exchange information like the improved crossover in [7] whereas uses fewer antibodies.

Furthermore, distance maintaining scheme is expected to further improve the algorithm performance.

3. Distance Maintaining Compact Quantum Crossover Based Clonal Selection

Algorithm

This section provides the full architecture of the distance maintaining compact quantum crossover

based clonal selection algorithm (DMCQCSA). Firstly, we introduce operation process of the distance

maintain compact quantum crossover. Then, we propose the DMCQCSA and list its procedure.

3.1. Distance Maintaining Compact Quantum Crossover

As discussed in the above section, the number of universes required for crossover is equal to the

number of cities of the TSP instance. What is more, simulation results on TSPs indicate that the low

affinity antibodies in crossover population have less contribution to improve fitness. As a result, a

distance maintaining compact quantum crossover is designed.

Figure 1 illustrates the new crossover. S1, S2, S3, ..., SN−1, SN are N current solutions sorted in a

descending order of affinity. In traditional compact quantum crossover, antibody pairs S1 − S2, S2 −

S3, ..., SN−2 − SN−1, SN−1 − SN undergo crossover. However crossover operation based on these “similar”

antibodies always generates low fitness offspring. To solve this problem, a distance maintaining

mechanism is introduced. Compact quantum crossover occurs between S1−SN, S2−SN−1, S3−SN−2 etc.

Fig. 2 shows detail information of the new crossover.

Figure 1. Distance maintaining compact quantum crossover

Figure 2. Operation process of the new crossover

Fig. 2 (a) exhibits the crossover process and (b) shows the operation results of each step.

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Without loss of generality, assuming pair S1−SN is selected for crossover. The crossover procedure

can be explained as follows:

Step1: Randomly select a city in S1 (For example city A). Then compare two edges: edge A-B in S1

and edge A-C in SN. City C will be selected if the length of A-C is shorter than that of edge A-B. Now,

the offspring sequence includes A and C.

Step2: City C is the current city. Then compare two edges: edge C-E in SN and edge C-D in S1. City

D will be selected if the length of C-D is shorter than that of edge C-E. Now, the offspring sequence

includes A, C and D.

Step3: Now, city D is the current city. Then compare edge D-E in S1 and edge D-B in SN. City B

will be selected if the length of D-B is shorter than that of edge D-E. New city B is added into the

offspring sequence, that is A, C, D and B.

Step4: Now, city B is the current city. Then compare edge B-C in S1 and edge B-F in SN. Because

city C has been already selected, city F will be selected and added to the offspring sequence. Then we

get A, C, D, B, F sequence.

Step5: Finally, the only one remaining city E will be added into the former sequence. Then a close

visiting tour A − C − D − B − F − E − A is constructed.

Figure 3. Flowchart of the proposed algorithm

3.2. DMCQCSA

The new DMCQCSA by merging distance maintaining compact quantum crossover with CSA is

presented in Figure 3. The full produce of the proposed algorithm can be represented as follows:

Step 1 Randomly creating an initial pool of m antibodies (S1, S2, ..., Sm).

Step 2 Compute the affinity of all antibodies (A(S1),A(S2), ...,A(Sm)). Then sort them in a descending

order. Where A(.) is the affinity function.

Step 3 Select n (n ≤ m) best affinity elites from the m original antibodies.

Step 4 Place each of the n selected elites in n separate and distinct pools (EP1,EP2, ...,EPn). They

will be referred to as the elite pools.

Step 5 Clone the elites in each elite pool with a rate proportional to its fitness, i.e., the fitter the

antibody, the more clones it will have. The amount of clone generated for these antibodies is given by

Eq.(1):

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)( Mn

inroundpi

(1)

where i is the ordinal number of the elite pools, M is a multiplying factor which determines the scope

of the clone and round(.) is the operator that rounds its argument towards the closest integer. After this

step, we can obtain ip antibodies just as (EP1,1,EP1,2, ...,EP1,p1; ...;EPn,1,EPn,2, ...,EPn,pn).

Step 6 Subject the clones in each pool through either random point mutation or receptor editing

processes. Some of the clones in each elite pool undergo the random point mutation process and the

remainder of the clones pass the receptor editing process. After this step, we obtain ip mutated cells

just as (EP’1,1,EP’1,2, ...,EP’1,p1; ...;EP’n,1,EP’n,2, ...,EP’n,pn)

Step 7 According to the distance maintaining compact quantum interference crossover described

above, subject the selected n elites through crossover.

Step 8 Reselect the fittest cell from each elite pool and quantum crossover generated cells.

Step 9 Update the parent cells in each elite pool with the fittest cells selected in Step 8.

If the iteration number reaches a pre-specified maximal generation number Gmax, the produce will

be terminated. Otherwise, it returns to Step 3.

4. Simulation on TSPs

In this section we first provide a description of the benchmark problems - TSP used for our

simulation. Then we present the simulation results, as well as comparison between the new method and

other algorithms.

4.1. Traveling Salesman Problem

Through the years the TSP, one of the most famous NP-hard combinatorial optimization problems,

has gathered a lot of attentions from academic researchers and industrial practitioners. The TSP is very

easy to describe, yet very difficult to solve.

Paper [18] introduced the TSP problem whose objective is to find the shortest route for a traveling

salesman who, starting from his home city has to visit every city on a given list precisely once and then

return his home city. The main difficulty of this problem lies in the immense number of possible tours.

No polynomial time algorithm is known with which can be solved.

In simulation, the gene sequence of immune cell can be expressed as a set of coordinates in a N-

dimensional shape-space. That is X = (x1, x2, ..., xN). Naturally, we can use different gene sequences

denote the solutions of a N-city TSP. If d(i, j) denotes the distance between city i and j which is

symmetric and known, the object of TSP is to find a permutation of the set {1, 2, 3, ...,N} that

minimizes the quantity [18]:

1

1

))1(),(())1(),(()(N

i

NdiidC (2)

4.2. Simulation Results

In this section, we present the simulation results from the application of the distance maintaining

compact quantum crossover based CSA and some other traditional algorithms to a number of TSP[17]

problems. All these simulations are implemented in 10 replications of C++ on a Pentium Dual-Core

2.2GHz CPU with 2.0GB memory. Table 1 lists the TSP instances to be tested.

The meaning of the parameters used in the proposed algorithm and their values are illustrated in

Table 2.

Premature convergence is one of the main difficulties with the CSA. It has been observed that this

problem is closely related to the problem of losing diversity in the population. Here, the diversity is

defined as the mean edge-distance between the best tour and all other tours. Edge-distance means

different edges between two tours [20].

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In order to provide insight into the relationship between the diversity and the actual results, we

present the experimental results of population diversity and tour distance for eil51, rd100 and pr124

respectively.

Table 1. Problems to be solved

Table 2. The meaning of the user-defined parameters and their values

Figure 4. (a)Convergence curve of CQCSA and DMCQCSA for eil51 instance (b)Improvement of

DMCQCSA comparing with CQCSA for eil51

Figure 5. Comparison of population diversity of CQCSA and DMCQCSA on eil51 problem

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Figure 4 (a) shows the experimental results of CQCSA and DMCQCSA for applying eil51 instance.

In Figure 4 (b), the improvements of DMCQCSA comparing with CQCSA for eil51 is presented. The

plotted lines represent that the DMCQCSA has outperforms better than CQCSA when line segments

are under the axis, which means, the negative value indicates the DMCQCSA has improvement

(shorter route) comparing with CQCSA.

As we know population diversity is a key issue in the performance of CSA. A common hypothesis

is that high diversity is important for the process of algorithm to avoid premature convergence and

escaping from local optimal solution.

Figure 5 presents the population diversity of CQCSA and DMCQCSA for eil51 during the

maturation process. In this figure, we can easily discover that the CQCSA has much higher population

diversity than DMCQCSA. It seems that the diversity is not the only key. Naturally, the same question

presented in [7] will be asked again: Is it possible for a lower population diversity algorithm to

generate better solutions than higher diversity ones?

Figure 6. (a)Convergence curve of CQCSA and DMCQCSA for rd100 instance (b)Improvement of

DMCQCSA comparing with CQCSA for ed100

Figure 7. Comparison of population diversity of CQCSA and DMCQCSA on rd100 instance

To answer this question, simulations on rd100 instance are also performed. Figure 6 (a) shows the

experimental results of CQCSA and DMCQCSA for applying rd100 instance. In Figure 6 (b), the

improvements of DMCQCSA comparing with CQCSA for rd100 is illustrated.

Figure 7 presents the population diversity of CQCSA and DMCQCSA for rd100 during the

maturation process.

Based on the above simulation results, it is apparent that higher population diversity is useful for a

greater exploration of the search space during the first half process, however a lower diversity which

permits the algorithm to fine-tune its results during the latter stages is also necessary.

In order to confirm the effectiveness and the robustness of the new algorithm, we perform

simulations on TSPs from eil51 to kroA200 and also compare our method with RECSA, TQCSA and

CQCSA. (RECSA: basic clonal selection without crossover; TQCSA: RECSA with traditional

quantum interference crossover; CQCSA: RECSA with compact quantum crossover.)

Table 3 shows the experimental results. Parameters PDM and PDB which indicate the percentage

deviation from the optimal tour length Dopt of the mean distance Dm and the best distance Db by the

following formula.

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100

opt

optm

D

DDPDM

(3)

100

opt

optb

D

DDPDB (4)

Table 3. Experimental results of RECSA, TQCSA, CQCSA and DMCQCSA for TSPs from eil51 to

kroA200

From Table 3, we can observe that the proposed distance maintaining compact quantum crossover

based CSA (DMCQCSA) has a superior ability to search better solutions than that of the RECSA,

TQCSA and CQCSA.

5. Conclusions

In this paper, we have proposed a distance maintaining compact quantum crossover based cloanal

selection algorithm (DMCQCSA) for overcoming premature convergence. The compact quantum

crossover is useful for information exchanging between different solutions, whereas uses fewer

antibodies than the traditional quantum interference crossover. What is more, the distance maintaining

mechanism permits the algorithm to fine tune its solutions. Experimental results of population diversity

and solutions on eil51 and re100 instances confirm that higher population diversity is useful for a

greater exploration of the search space during the first half process, and a lower diversity which

permits the algorithm to further exploit its results during the latter stages is also necessary. Simulation

results on a wide range of TSP instances demonstrate that the performance of the proposed DMQCSA

is superior as compared to other related CSAs.

6. Acknowledgements

This work was supported by the Ministry of Science and Technology of China under Grant No.

2009GJC10043, the Natural Science Foundation of Jiangsu Province under Grant No. BK2008190, the

Natural Science Foundation of Huaihai Institute of Technology under Grant No. KX08013, No.

KQ09014.

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Distance Maintaining Compact Quantum Crossover Based ... · PDF fileDistance Maintaining Compact Quantum Crossover Based Clonal Selection Algorithm Hongwei Dai, Yu Yang, Cunhua Li