GMDH combinatorial algorithm
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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, [email protected], [email protected]
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Optimal Paralleling for Solving Combinatorial Modelling Problems
Volodymyr Stepashko, and Serhiy YefimenkoInternational Research and Training Centre of
Information Technologies and Systems of the National Academy of Sciences and Ministry of Education and
Sciences of Ukraine, [email protected], [email protected]
GMDH combinatorial algorithm
Matrix Х [n×m]
vector y[n×1]
y = f (θ, x)= θ1 x1+ θ2 x2 + … + θm xm
amount of models – Σ Cmi = 2m -1
Variants of structures generation
1. Binary numbers generator
0 0 0 10 0 1 00 0 1 10 1 0 00 1 0 10 1 1 00 1 1 11 0 0 01 0 0 11 0 1 01 0 1 11 1 0 01 1 0 11 1 1 01 1 1 1
2. Successive complication of structures
0 0 0 10 0 1 00 1 0 01 0 0 00 0 1 10 1 0 11 0 0 10 1 1 01 0 1 01 1 0 00 1 1 11 0 1 11 1 0 11 1 1 01 1 1 1
Paralleling on 2 processorsBinary numbers
generator
0 0 0 10 0 1 00 0 1 10 1 0 00 1 0 10 1 1 00 1 1 11 0 0 01 0 0 11 0 1 01 0 1 11 1 0 01 1 0 11 1 1 01 1 1 1
I processor:8 models,13 arguments
II processor:7 models,19 arguments
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
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,
mii ,1, 1i
mC
kjj ,1, 1)1(1
j
k
C im
1)1(1
j
k
C im
mll ,1, udCC udCC
udCC
Application of scheme of paralleling with successive complication for solving high dimensional problems
Amount of arguments Amount of models Amount of processors
1 100 0.0001
2 4950 0.005
3 161700 0.16
4 3921225 3.9
5 75287520 75
6 1192052400 1192
7 16007560800 16008
... ... ...
99 100 0.0001
100 1 0.000001
Amount of arguments – 100
For one processor – 1.048.575 models (20 arguments) ~ 1 min (2 GFLOPS)
Time plot for 8 processes on one processoramount of arguments – 22, amount of models – 222-1
0
20
40
60
80
100Time, s
1 2 3 4 5 6 7 8
№ of processbinary counter successive complication
%,Tk
TE
kk 1001
Efficiency
binary numbers generator successive complication
Efficiency 80,7% 99,8%
Run-time of combinatorial algorithm
1 2 4 8 16
20
21
22
2324
0
200
400
600
800
1000
1200
1400
1600Binarycounter
Succesivecomplication
Time, s
Processors
Arguments
Efficiency of schemes of paralleling of combinatorial algorithm
ArgumentsProcessors 20 21 22 23 24
1 100% 100% 100% 100% 100%2 92% 93% 93% 93% 93%4 86% 86% 87% 87% 87%8 79% 80% 80% 81% 82%16 74% 75% 70% 77% 77%
Binary counter
Processors 20 21 22 23 241 100% 100% 100% 100% 100%2 100% 99% 100% 100% 99%4 100% 100% 100% 98% 99%8 100% 99% 100% 98% 99%16 99% 98% 98% 100% 100%
Successive complication
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.
Optimal Paralleling for Solving Combinatorial Modelling Problems
Volodymyr Stepashko, and Serhiy YefimenkoInternational Research and Training Centre of
Information Technologies and Systems of the National Academy of Sciences and Ministry of Education and
Sciences of Ukraine, [email protected], [email protected]
