issn: 0975-766x - coden: ijptfi

D.Venkatesan* et al. /International Journal of Pharmacy & Technology

IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 22384-22394 Page 22384

ISSN: 0975-766X

CODEN: IJPTFI

Available Online through Research Article

www.ijptonline.com META-HEURISTIC METHOD FOR SOLVING TARRY ESCOTT PROBLEM

D.Venkatesana, S.Raja Balachandar

b, R.Srikanth

b, K.Kannan

b, S.G.Venkatesh

b

aSchool of Computing,

bSchool of Humanities and Sciences, SASTRA University, Thanjavur, Tamilnadu, India.

Email: [email protected]

Received on: 15.10.2016 Accepted on: 12.11.2016

Abstract

The paper introduces Gravitational Search Algorithm (GSA) to find numerical solutions of Diophantine equations,

namely Tarry Escott Problem for which there exists no general method of finding solutions. This algorithm finds upon

introducing randomization concept along with the two of the four primary parameters 'velocity' and 'gravity' in physics.

The performance of this algorithm has been evaluated on a set of random values. Computational results show that the

gravitational search algorithm - based heuristic is capable of producing high quality solutions, can offer many solutions

of such equations.

Keywords: Meta-heuristic; Diophantine; Tarry Escott problem; Gravitational Search Algorithm.

1. Introduction

Diophantine problems, named after Diophantus of Alexandria (c. 250 A.D), are concerned with the integral solutions of

polynomial equations with integer coefficients. An equation which has two or more unknowns is called an indeterminate

equation. More generally, a system of equations is called indeterminate, if the number of equations is less than that of

the unknown. Diophantu proposed many indeterminate problems in his arithmetic and made systematic use of algebraic

symbols. He was the first Mathematician to make such an effort towards developing a symbolism for the powers of

algebraic expressions. He was content with a single numerical rational solution, although, the problems usually had

infinitely many rational solutions and often integral solutions.

One can easily understand that Diophantine problems offer almost unlimited field for research by reason of their variety

by [1-8]. A Diophantine problem is considered as solved if a method is available to decide whether the problem is

solvable or not and, in case of its solvability, to exhibit all integers satisfying the requirements set forth in the problem.

A partial solution of Diophantine problem has only a very limited interest. Although the study of indeterminate equation


IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 22384-22394 2385

and its solution in integers has had a very important place in the development of Number Theory, there is no well-

unified body of knowledge concerning general methods. There are but very few Diophantine problems of a general type

in which the complete solution is known [7]. The successful completion of exhibiting all integers satisfying the

requirements set forth in the problem add to further progress of Number Theory. In particular, Diophantine equations of

the form

11

n

j

s

i

m

i

s

i ba (1)

have been studied by numerous mathematicians for many years and by a variety of methods. Classes of this equation

are the Pythagorean theorem (n=1, m=s=2), Fermat’s Last theorem (n=1, m=2, s>2), Euler’s conjecture (n=1, s>m>2)

and the deficient symmetric equal sum of like powers (n>1, m=n, s>m). Some specific examples of (1) are considered

in [9-15 ], namely the equations which have s, m (=n) = 4,4; 3,2; 5,4; 4,2; 7,4; 5,3; and 6,3.

q

nnn

p BBBAAAr

,...,,,...,, 21

,...,

21

21

(2)

Further, the notation designates a so-called multidegreed equality and means that the sum of the numbers on the left

equals the sum of the numbers on the right for each of the ),...,( 21 rnnnr positive integral powers of the numbers. In

[16], parametric solutions of the two multi-degree equalities

213213321

4,2

321 B ,A ,,,,, BBAABBBAAA

and

721

8,6,4,2,1

721 ,...,,,...,, DDDCCC

are obtained. A special case of (2) is the (k, s) multigrade Diophantine equation of the form

1,2,....k)(j , 11

s

i

j

i

s

i

j

i yx (3)

The above equation, conveniently denoted by the symbol

s

k

s yyyxxx ,...,,,...,, 2121

is known as the Prouhet Tarry Escott problem of degree k and has been analyzed by various authors [5-7,16,17] for its

nontrivial integral solutions with particular reference to certain values of s and k.



In this paper, we consider a general form of non-ideal non-trivial parametric integral solutions of the system (3), with k

= 2, s = 4

4321

2

4321 ,,,,,, yyyyxxxx (4)

and the system

2

4

2

3

2

2

2

1

22

4

2

3

2

2

2

1 ,,,,,, yyyyxxxx (5)

i.e., 2,1 ,)()(4

1

24

1

2

jyxi

j

i

i

j

i

There have been some attempts to apply soft computing techniques to find a numerical solution of some Diophantine

equations. Abraham and Sanglikar [18] tried to find numerical solutions of a1 . x1 p1

+ a2 . x2 p2

+ .. + an . xn pn

= N

type Diophantine equations by applying genetic operators mutation and crossover. Though the methodology could find

solutions, it was not fully random in nature and seemed more like a steepest ascent hill climbing rather than a genetic

algorithm. Hsiung and Mathews [19] tried to illustrate the basic concepts of a genetic algorithm using first-degree linear

Diophantine equation given by a + 2b + 3c+ 4d = 30. Literature also talks about an application of higher order Hopfield

neural network to find solution of Diophantine equation [20]. Abraham and Sanglikar[21] explains the process of

avoiding premature converging points using Host Parasite Co-evolution [22-24 ] in a typical GA.. They also used [25]

simulated annealing as a viable probabilistic search strategy for tackling the problem of finding numerical solution.

These methods, though effective to a certain extent for smaller equations, are not good enough to deal with the

complexities of Diophantine equations.

Hence, we present an algorithm called gravitational search algorithm (GSA) . This algorithm is based on the Newtonian

gravity: “Every particle in the universe attracts every other particle with a force that is directly proportional to the

product of their masses and inversely proportional to the square of the distance between them” [26,27 ] to find an

infinite number of nontrivial integral solutions to a few interesting Tarry Escott problem.

This paper is organized as follows: In section 2, the outline of the proposed algorithm is given. Sections 3 and 4 deal

with experimental and comparative study to show the performance of GSA. Finally, the concluding remarks are given in

section 5.

2. Gravitational Search Algorithm (GSA)



In this section, we employ gravitational search algorithm [28] based on (6).

2

21

R

MGMF (6)

In (6) F, G, M1, M2 and R are representing the magnitude of the gravitational force, gravitational constant, the mass of

the first and second particles, and the distance between the two particles respectively. Newton’s second law says that

when a force, F, is applied to a particle, its acceleration, a, depends only on the force and its mass, M :

M

Fa (7)

From (6) and (7), we conclude that there is an attracting gravity force among all particles of the universe where the effect

of bigger and the closer particle is higher.

The gravitation is the tendency of masses to accelerate towards each other. It is one of the four fundamental interactions

in nature [27] (the others are: the electromagnetic force, the weak nuclear force, and the strong nuclear force).

In Physics, agent’s performance is measured by their masses and they are considered as objects. Based on gravity force

the agents are attracted each other and their tendency is towards heavier masses. As a result, the heavy masses move in a

very slow manner than the lighter ones and this induces the exploitation.

Each and every mass (agent) has four specifications: position, inertial mass, active gravitational mass, and passive

gravitational mass in GSA. The algorithm updates gravitational and inertia masses with the help of heavy masses and

finds the optimum. This artificial world of masses obeys the Newton’s law of gravity and motion [28].

The detailed discussion of the GSA is available in [28] and the outline of the proposed algorithm is given below.

(a) Search space identification.

(b) Randomized initialization.

(c) Fitness evaluation of agents.

(d) Update G(t), best(t), worst(t) and Mi(t) for i = 1,2,. . .,N.

(e) Calculation of the total force in different directions.

(f) Calculation of acceleration and velocity.

(g) Updating agents’ position.

(h) Repeat steps c to h until the stop criteria is reached.

(i) End.



Many applications of this gravitational algorithm are available in the literature such as Prototype classifier; Advanced

reservation and Scheduling in grid computing; Forecasting; Dispatch problem; Hydraulic turbine governing system;

Clustering; Traveling salesman problem; Vertex covering problem; Set covering problem and Filter Modeling[29-40].

The complete review of GSA, application of GSA and other theoretical analysis is presented in [40].

Initial population and Fitness function

The procedure of finding a numerical solution to the assumed Diophantine equation starts with a population of random

agents or solutions of fixed size. The agents are constructed as integer agents based on probable values of variables

appearing in the assumed Diophantine equation. The construction of these agents is facilitated by incorporating

knowledge of the domain and the constraints imposed in the problem. Table 1 show that the possible values of each

unknown variables in the population and size of the population for all the Diophantine equations.

Table 1. Population Size and Range.

Equation Unknown

Variables

Range of

Unknown variables

Population size

(4) x1,x2,x3,x4

y1,y2,y3,y4

1,2500 100

(5) x1,x2,x3,x4

y1,y2,y3,y4

1,110000 1000

The Fitness function value of an agent gives the effectiveness of the agent in the search space. The definition of fitness

function of an agent in GSA is presented in Table 2. Fit is the fitness function to be minimized, max{ abs(f j) } is the

maximum absolute value of individual equations of the system f(x)=0 and the number of equations in the system is 2.

Table 2: Fitness function.

Equation System of Equations Fitness function

(4) )()(

4

1

4

1

1

i

i

i

i yxf

4

1

24

1

2

2 )()(i

i

i

i yxf

Fit = max{ abs(f j) }

j = 1,2

(5) )()(

4

1

24

1

2

1

i

i

i

i yxf

4

1

224

1

22

2 )()(i

i

i

i yxf

Fit = max { abs(f j) }

j = 1,2



The value of the fitness function indicates the distance between the current position of the agent and its solution. If

fitness=0 for a particle, then that position of the agent is taken as a solution. At each iteration, the attempt is to reduce

this distance. Thus, the procedure becomes a minimization process in which each of the agents tries to reduce the

distance between its present status and the solution of the equation whose fitness function value is given to be zero. We

proceed our approach with the steps given in the previous section and the details of the results and comparison report are

given in the next section.

3. Experimental Results

Since heuristics for solving this type of problems are not based on theoretical analysis, the only objective way to

evaluate their performance is by conducting a comparative study based on a set of problems are chosen randomly for

each type of Diophantine Equations.

In this section, we first define the experimental environment and problem instances. Then a thorough performance

evaluation will be conducted to compare both the solution quality and iterations for two different heuristics namely

Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) for solving the same test problems along with our

proposed method. All experiments have been conducted on 30 randomly generated problem instances to test the GSA

performance. The problem instances divided into 2 sets as in Table-3. The ranges of the coefficients are reported in the

table for 2 different equations. Table 4 and Table 5 exhibit the GSA solutions of (4) and (5) for different values of

(x1,x2,x3,x4) and (y1,y2,y3,y4).

Table-3 Random Problem Instances.

Type of

Diophantine

Equation

Coefficients Range of the

Coefficients

Number of

problems

(4) (x1,x2,x3,x4)

(y1,y2,y3,y4)

]999,999[xi

]999,999[yi

20

(5) (x1,x2,x3,x4)

(y1,y2,y3,y4)

]31000,4000[xi

]32000,140[yi

10



Table 4. Solutions of (4).

S.No (x1,x2,x3,x4) (y1,y2,y3,y4) S.no (x1,x2,x3,x4) (y1,y2,y3,y4)

1 13,35,-1,31 31,23,29,-5 11 88,60,-76,-72 104,-84,36,-56

2 -29,88,-5,46 13,70,61,-44 12 68,54,-44,-78 84,-58,36,-62

3 -39,117,-67,17 17,93,21,-103 13 181,147,-157,-171 209,-183,111,-137

4 -125,277,-59,31 205,-35,139,-185 14 288,162,-216,-234 324,-234,108,-198

5 -6,239,316,323 309,64,386,113 15 783,491,-603,-671 939,-727,237,-449

6 529,865,57,593 697,529,785,33 16 310,126,-246,-190 370,-234,-6,-130

7 189,319,-61,479 387,143,445,-49 17 187,153,-163,-177 239,-213,69,-95

8 89,33,-57,-65 91,-51,29,-69 18 561,355,-465,-451 635,-489,249,-395

9 128,72,-96,-104 144,-104,48,-88 19 843,185,-323,-705 843,-305,177,-715

10 54,-10,16,-60 -16,46,30,-60 20 52,4,8,-64 -8,28,44,-64

Table 5. Solutions of (5).

S.No (x1,x2,x3,x4) (y1,y2,y3,y4)

21 32,178,324,-426 146,292,438,144

22 432,648,864,-216 216,432,648,864

23 864,1296,1728,432 432,864,1296,1728

24 576,3204,5832,7668 2628,5256,7884,2592

25 128,712,1296,-1704 584,1168,1752,576

26 972,1458,1944,-486 486,972,1458,1944

27 3456,5184,6912,1728 1728,3456,5184,6912

28 2304,12816,23328,30672 10512,21024,31536,10368

29 288,1602,2916,-3834 1314,2628,3942,1296

30 1728,2592,3456,-864 864,1728,2592,3456

4. Comparative Study

To compare the performance of the algorithms we have used the parameter, namely the average fitness value of all the

test problems with GA and PSO. In GA, one-point crossover, uniform mutation and roulette wheel selection were used.

The crossover probability and mutation probability were set to 0.9 and 0.005, respectively. The parameters of PSO are

set as in [ 25]. The parameters of our GSA are given in [35-38].



0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

P roblem Number

Av

era

ge

Fit

ne

ss

va

lue

s

P S O

GA

GS A

Fig 1: Comparison of Average fitness values.

Figure 1 shows that the average fitness value for all the test problems with GSA , GA and PSO. From the figure, it can

be seen that GSA performance is better than the other algorithms for all the test problems.

5. Conclusion

In this paper, Gravitational Search Algorithm (GSA) is applied to find the numerical solution of the Tarry Escott

Problem, for which there exists no general method of finding solutions. This algorithm is found upon introducing

randomization concept along with the two of the four primary parameters 'velocity' and 'gravity' in physics. The

performance of this algorithm has been evaluated on a set of random values. Computational results showed that the

gravitational search algorithm - based heuristic is capable of producing high quality solutions, and can offer many

solutions of such equations.

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