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CS6035 Parallel/Distributed Processing II:

CS6035 Parallel/Distributed Processing II:. An Efficient Static Assignment Parallelization Scheme for Algebraic Fractals. By: Chris MacPhee Supervisor: Dr. Bhavsar. Outline:. Introduction Computational Characteristics Serial Program Parallelization Experimental Results IBM SP

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CS6035 Parallel/Distributed Processing II:

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

  2. Outline: • Introduction • Computational Characteristics • Serial Program • Parallelization • Experimental Results • IBM SP • SGI Onyx • Conclusion

  3. 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

  4. Introduction Examples of fractals images From: Fractal Gallery http://projekt.pinknet.cz/fractal/

  5. 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

  6. Computational Characteristics The Mandelbrot set (cont) zz2 + c

  7. Computational Characteristics The generalized function zz2 + c, where z, c  zz + c, where z, c    

  8. Computational Characteristics The generalized function (cont) zz10 + c

  9. Computational Characteristics The generalized function (cont) zz100000 + c

  10. Computational Characteristics Distribution of iterations

  11. 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)

  12. Parallelization Three work assignments Static work assignment Dynamic work assignment New static work assignment

  13. Parallelization Static work assignment Divide column groups evenly between processors Master 1-256 257-512 513-768 769-1024

  14. 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

  15. Parallelization New static work assignment Divide workload evenly over processors Master 1-339 370-513 514-657 658-1024

  16. 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

  17. Experimental Results IBM SP Timings Computing time for each slave processor for  = 2

  18. Experimental Results IBM SP Timings Computing time for each slave processor for  = 10

  19. Experimental Results IBM SP Timings Computing time for each slave processor for  = 100000

  20. Experimental Results SGI Onyx Timings Computing time for each slave processor for  = 2

  21. Experimental Results SGI Onyx Timings Computing time for each slave processor for  = 10

  22. Experimental Results SGI Onyx Timings Computing time for each slave processor for  = 100000

  23. 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.

  24. 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 < 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/profs/aubanel/aubanel_fractals.html. [6] B. Wilkinson and M. Allen, Parallel Programming, Prentice Hall, Upper Saddle River, 1999.

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