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COMPUTATIONAL COMPLEXITY OF PARAMETRIC GAME MODEL SOLVING ALGORITHMS
Baghinyan M.K.
PhD graduate, NATIONAL POLYTECHNIC UNIVERSITY
OF ARMENIA
ABSTRACT
In this paper the computational complexity of parametric game model solving algorithms [1-3] is measured. In the first part of the analysis we present the count of all the possible actions of the algorithms, and then the big O notation: the worst case scenario. The final results are presented in the form of tables.
Keywords: parametric game model, big O notation, complexity analysis, parametric simplex method.
plied software package we apply computational complexity analysis [4-6]. We perform such analysis for one-parametric, two-parametric game model solving algorithms taking in account the following parameters: The m,n sizes of the input game model payoff matrix and the K number of discrete.
The computational complexity for one-parametric game model solving algorithms is presented in table 1.
Table 1
One-parametric algorithm analysis
Image Count of operations
) (k +1)2 (n + m)
0<f* K ) (K +1)2 (n + m\n -1)
pj) K) (K +1)2 (n + m\n -1)
^(K) (K +1)2 (n -1)
vif~l) (K ) (K +1)2 (n + m)
Rbq\k) (k+1)2
Aq) (K) 2 (n + m)
In general, computational complexity is the sum of the basic operations of the algorithm. Big O notation is an important characteristic to describe the complexity of the algorithm for the worst possible performance cases. In order to measure performance of the algorithms used in [3] parametric game model solver ap-
To evaluate two-parametric game models solving algorithms, first let's obtain it from the multi-parametric game model solution method [2] having parameters d,t.
p^d t)=t)d, t), j-m+m,
^(dAP^/p^A i*k, j-u+m,
d,t)-R^1 /pj\d,t),
R&\d,t)-R^-Rb^'p£%d,t)/pj>(d,t), i * ,0,
A(q(d,t)- A(q-1) - p(qTl)(d, t) • A(q-l) / p(q-)(d, t), j - 1,n + m, Whose image equations after applying differential transforms method can be described as:
pWKx, K 2 ) = (pj\Kx, K 2 )
*oJ V 1 2 x ^ iJ
-Z2: pjK, - /„ K2 - /2). p^^d,, /2 x/pj')(o,o).
/,=o /2 =o
j = 1n + m ,K = (kx,K2) = 0»,/x * /2 * o, We make the following assignments,
4-,}(d,t)=^(d,,).p<q-i)M,, *io, j=mrm,
p(q)(d,t)=4'%,)-c<'-1)(d,,)/p(q-1>(d,,), i * io, j=in
With the corresponding images:
K, K2
(k,,K 2 )=2 .pj1 (K, - /,,K 2 - /2). p^ (/,,/ 2),
/, =o /2=o
i * ^, j = 1, n + m, K = o, * /2 * o,
pj) (K,, K 2 )=piq-1) (K,, K 2)- (j-1) (K,, K 2)-
K, K2
-2 2: (k, - /,,K2 - /2 )■ pr1 (/,,/2)) /p'0 (o,o)
/,=o /2=o
, i * j =,,n + m, K = (Ki,K2) = o,№,/x * /2 * o, R^ (K,,K 2 )= (Rb^ (K,,K 2 )-
K, K2
-5: Rb') (k, - /,,k2 - /2 )■ pfq,^1) (/,,/2 );/piq-1) (o,o;,
/, =o /2 =o
h * ^ * o /
And for the following assignments,
w(q-1)(d,t)=p<q-1)M/pj(d,t), i * io,
v^ (d,, )=p^ (d,,) /pj M, i * io,
We get these images:
wi'-1 (K, ,K 2 ) = j (K,,K 2)-
io
K K2
2 2 Wi(q-l| (k, - /,,K2 - /2 )■ pfe1* (/,,/ 2 );/p(r) (o,o),
/, =o /2=o
i * i0, ^ * /2 * o,
v(q-l) (k,, ,^2 )=fp^ (K,,K 2)-
K1 K2
"j (Ki - li,K2 -/2 ). p,, V1, / 2
/, =o /2 =o
/j * /2 * o,
22vi^(Ki -/i,K2 -/2)■ pfe1*(/,,/2))/p^l}(o,o),
Rbq](k,,K2)=Rir1}-Rir1}- Wi(q-1)(K,,K2), i * io, A(q) (k ) = A(q-1) - ¿(q-l|v(q-1) (k , K),
i * i'o, j = 1, n + m, K = o,»:
The count of the operations performed for the equations for solving two-parametric game models presented
in table 2.
Table 1.
Two-parametric algorithm analysis
image Count of operations
piqj (Ki,K 2 ) K2 K2 2 (n + m)
j1 (k ,K 2 ) Kx 2 K2 2 (n + m)n -1)
pj) (K ,K 2 ) Kj2 K 22 (n + m)n -1)
M 2 ) Ki2 K 22 (n -1)
viq"l) (K1;K 2 ) K2 K2 2 (n + m)
rti] (ki, k 2 ; Kj2 K 22
4}(Ki, K2 ) 2 (n + m)
Now, let's present computational complexity according to big O notation for the fixed size payoff matrix (for example n=3, m=3) presented in table 3.
Table 3
Big O notation_
image Big O notation
one-parametric game models
pW K), (K), p« K), Wq-i) (k ), v^k ), Rq ](K) O( (K + i)2)
A(q ) (K ) O(i)
Two-parametric game models
p(j (Ki ,K 2 ), c<f-i) (Ki,K 2 ), p(q) (Ki,K 2 ), w(q-1) Ki ,K 2 ), v(q-i) (Ki,K 2 ), Rbql ] (Ki,K 2 ) O(Ki2 K222)
A(q} (Ki,K 2 ) O (i)
Conclusion: Computational complexity analysis was performed for one-parametric and two-parametric game models solving algorithms. The analysis describes the performance of the algorithms according to all the counts of the single operations and big O notation.
References
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2. Baghinyan M. K. A solution method for none cooperative multi-parametric game models based on differential transforms // Bulletin NPUA, Series of Technical Sciences.- Yerevan.- 2017.-Volume 1.-p.
226-232:
3. Baghinyan M. K. AN APPLIED SOFTWARE PACKAGE FOR SOLVING PARAMETRIC GAME MODELS // PROCEEDINGS OF ENGINEERING ACADEMY OF ARMENIA, SCIENTIFIC AND TECHNOLOGICAL COLLECTED ARTICLES.- Yerevan.- 2017.-Volume 14, N 2.-p. 303-308:
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