CS6035 Parallel/Distributed Processing II:

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CS6035 Parallel/Distributed Processing II
An Efficient Static Assignment Parallelization
Scheme for Algebraic Fractals
By Chris MacPhee Supervisor Dr. Bhavsar
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Outline

• Introduction
• Computational Characteristics
• Serial Program
• Parallelization
• Experimental Results
• IBM SP
• SGI Onyx
• Conclusion

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Introduction
What are fractals?

• Possess non-Euclidian geometry (formless)
• Self-similar (same type of structure at all
  scales)
• Clouds are not spheres, mountains are not cones,
  coastlines are not circles, and bark is not
  smooth, nor does lightning travel in a straight
  line.
• - Benoit Mandelbrot, 1983

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Introduction
Examples of fractals images
From Fractal Gallery http//projekt.pinknet.cz/f
ractal/
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Computational Characteristics
The Mandelbrot set

• z ? z2 c , where z, c ? ?
• z0 is a constant
• c varies
• z is iterated until either
• z diverges beyond a preset limit
• the maximum number of iterations
• is reached

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Computational Characteristics
The Mandelbrot set (cont)
z ? z2 c
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Computational Characteristics
The generalized function
z ? z2 c , where z, c ? ? z ? z? c ,
where z, c ? ? ? ? ?
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Computational Characteristics
The generalized function (cont)
z ? z10 c
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Computational Characteristics
The generalized function (cont)
z ? z100000 c
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Computational Characteristics
Distribution of iterations
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Parallelization
Two architectures

• Shared memory programming
• Run on SMP machines (e.g. Sun SGI)
• Uses OpenMP
• Message passing programming
• Run on distributed memory machines
• (e.g. Compaq IBM)
• Uses Message Passing Interface (MPI)

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Parallelization
Three work assignments
Static work assignment Dynamic work
assignment New static work assignment
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Parallelization
Static work assignment
Divide column groups evenly between processors
Master
257-512
1-256
513-768
769-1024
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Parallelization
Dynamic work assignment
Farm work to the slaves in work sizes of 64
columns
Master (in queue 513-1024)
449-512
321-384
385-448
257-320
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Parallelization
New static work assignment
Divide workload evenly over processors
Master
370-513
1-339
514-657
658-1024
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Experimental Results
Two machines

• Symphony (University of New Brunswick)
• IBM SP
• 16 ? 375 MHz processors
• 4 GB of RAM
• Distributed memory architecture
• Herzberg (Memorial University of Newfoundland)
• SGI Onyx
• 28 ? 400 MHz processors
• 14 GB of RAM
• Shared memory architecture

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Experimental Results
IBM SP Timings
Computing time for each slave processor for ? 2
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Experimental Results
IBM SP Timings
Computing time for each slave processor for ?
10
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Experimental Results
IBM SP Timings
Computing time for each slave processor for ?
100000
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Experimental Results
SGI Onyx Timings
Computing time for each slave processor for ? 2
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Experimental Results
SGI Onyx Timings
Computing time for each slave processor for ?
10
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Experimental Results
SGI Onyx Timings
Computing time for each slave processor for ?
100000
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Summary
Summary

• The computational characteristics of fractal
  images have been analyzed.
• A static assignment method for efficient parallel
  processing has been developed.
• The static assignment method becomes more
  efficient as ? increases.

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References
1 H. O. Peitgen and P. Richter, The Beauty of
Fractals, Springer-Verlag, Berlin,
1996.   2 U. G. Gujar and V. C. Bhavsar,
"Fractals from z z a c in the Complex
z- plane", Comp. and Graph., 16(1), pp. 45-49,
1992.   3 S. V. Dhurandhar, V. C. Bhavsar, and
U. G. Gujar, "Analysis of z-plane fractal images
from z z a c for a lt 0", Comp. and Graph.,
17(1), pp. 89-94, 1993.   4 V. C. Bhavsar, U.
G. Gujar, N. Vangala, "Vectorization of
generation of fractals from z z a c on IBM
3090 / 180VF", Comp. and Graph., 17(2), pp.
169-174, 1993.   5 E. Aubanel, "Parallel
Programming with Generalized Fractals," Faculty
of Computer Science, University of New
Brunswick, February 2002, http//www.cs.unb.ca/p
rofs/aubanel/aubanel_fractals.html.   6 B.
Wilkinson and M. Allen, Parallel Programming,
Prentice Hall, Upper Saddle River, 1999.