An Efficient Static Assignment Parallelization Scheme for

Algebraic Fractals

By: Chris MacPheeSupervisor: Dr. Bhavsar

CS6035 Parallel/Distributed Processing II:

Outline:

IntroductionComputational CharacteristicsSerial Program ParallelizationExperimental Results

• IBM SP• SGI Onyx

Conclusion

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

Introduction

Examples of fractals images

From: Fractal Gallery http://projekt.pinknet.cz/fractal/

Computational Characteristics

The Mandelbrot set

z z2 + c , where z, c

z0 is a constantc varies

z is iterated until either: • z diverges beyond a preset limit• the maximum number of iterations

is reached

Computational Characteristics

The Mandelbrot set(cont)

z z2 + c

Computational Characteristics

The generalized function

z z2 + c , where z, c

z z + c , where z, c

Computational Characteristics

The generalized function(cont)

z z10 + c

Computational Characteristics

The generalized function(cont)

z z100000 + c

Computational Characteristics

Distribution of iterations

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)

Parallelization

Three work assignments

Static work assignmentDynamic work assignmentNew static work assignment

Parallelization

Static work assignment

257-5121-256 513-768 769-1024

Divide column groups evenly between processors

Master

Parallelization

Dynamic work assignment

449-512257-320 321-384 385-448

Farm work to the slaves in work sizes of 64 columns

Master (in queue: 513-1024)

Parallelization

New static work assignment

370-5131-339 514-657 658-1024

Divide workload evenly over processors

Master

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

Experimental Results

IBM SP Timings

Computing time for each slave processor for = 2

Experimental Results

IBM SP Timings

Computing time for each slave processor for = 10

Experimental Results

IBM SP Timings

Computing time for each slave processor for = 100000

Experimental Results

SGI Onyx Timings

Computing time for each slave processor for = 2

Experimental Results

SGI Onyx Timings

Computing time for each slave processor for = 10

Experimental Results

SGI Onyx Timings

Computing time for each slave processor for = 100000

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.

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.