1. Optimal Paralleling for Solving Combinatorial Modelling Problems Volodymyr Stepashko, and Serhiy Yefimenko International Research and Training Centre of Information Technologies and Systems of the National Academy of Sciences and Ministry of Education and Sciences of Ukraine, astrid@irtc.org.ua, syefim@ukr.net

2. GMDH combinatorial algorithm Matrix Х [n×m] vector y[n×1] y = f (θ, x)= θ 1 x 1 + θ 2 x 2 + … + θ m x m amount of models – Σ C m i = 2 m -1

3. Variants of structures generation 1. Binary numbers generator 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 2. Successive complication of structures 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1

4. Paralleling on 2 processors Binary numbers generator 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 I processor: 8 models, 13 arguments II processor: 7 models, 19 arguments

5. Successive complication of structures (paralleling on 2 processors) 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1

6. Algorithm of determination of the initial state of binary structural vector by position at successive complication Step 1.Calculation of amount of combinations – 2.Determination of the initial state of binary vector b for every processor as a decimal number – 3.Conversion from the decimal number to appropriate binary number for every processor: position = ; u=i-1, d=m-1, Cycle on if position<=C, then b[l]=1, u= u -1, d=d -1, else b[i]=0, position = position -С, u= u -1,

7. Application of scheme of paralleling with successive complication for solving high dimensional problems Amount of arguments Amount of modelsAmount of processors 1 1000.0001 2 49500.005 3 1617000.16 4 39212253.9 5 7528752075 6 11920524001192 7 1600756080016008... 99 1000.0001 100 10.000001 Amount of arguments – 100 For one processor – 1.048.575 models (20 arguments) ~ 1 min (2 GFLOPS)

8. Time plot for 8 processes on one processor amount of arguments – 22, amount of models – 2 22 -1 Efficiency binary numbers generator successive complication Efficiency80,7%99,8%

9. Run-time of combinatorial algorithm Time, s Processors Arguments

10. Efficiency of schemes of paralleling of combinatorial algorithm Binary counter Successive complication

11. Conclusion  The scheme of operations paralleling in a combinatorial algorithm on principle of binary counter is explored. It is shown that it does not provide the uniform loading on all processors of the cluster system. With the increase of the number of processors of cluster system efficiency of paralleling decreases considerably.  The new method of paralleling is developed on the basis of algorithm of generation of the successively complicated structures of models.  By the tests experiments it is shown that the use of the offered scheme provides the equal total amount of models and estimated parameters on every processor.

12. Optimal Paralleling for Solving Combinatorial Modelling Problems Volodymyr Stepashko, and Serhiy Yefimenko International Research and Training Centre of Information Technologies and Systems of the National Academy of Sciences and Ministry of Education and Sciences of Ukraine, astrid@irtc.org.ua, syefim@ukr.net