Mandelbrot set computation in Java

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Consider the well known Mandelbrot set [Dou86] We discuss this problem and its parallelization as a task parallel problem in the Task Parallelism and SPMD patterns Each pixel is colored based on the behavior of the quadratic recurrence relation in Eq 56 Equation 56
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C and Z are complex numbers and the recurrence is started with Z0 = C The image plots the
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imaginary part of C on the vertical axis ( 15 to 15) and the real part on the horizontal axis ( 1 to 2) The color of each pixel is black if the recurrence relation converges to a stable value or is colored depending on how rapidly the relation diverges Pseudocode for the sequential version of this code is shown in Fig 526 The interesting part of the problem is hidden inside the routine compute_Row() The details of this routine are not important for understanding the parallel algorithm, however, so we will not show them here At a high level, the following happens for each point in the row
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Each pixel corresponds to a value of C in the quadratic recurrence We compute this value based on the input range and the pixel indices
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Figure 526 Pseudocode for a sequential version of the Mandelbrot set generation program
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Int const Nrows // number of rows in the image Int const RowSize // number of pixels in a row Int const M // number of colors in color map Real :: conv // divergence rate for a pixel Array of Int :: color_map (M) // pixel color based on conv rate Array of Int :: row (RowSize) // Pixels to draw Array of Real :: ranges(2) // ranges in X and Y dimensions manage_user_input(ranges, color_map) // input ranges, //color map initialize_graphics(RowSize, Nrows, M, ranges, color_map) for (int i = 0; i<Nrows; i++){ compute_Row (RowSize, ranges, row) graph(i, RowSize, M, color-map, ranges, row) } // end loop [i] over rows
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We then compute the terms in the recurrence and set the value of the pixel based on whether it converges to a fixed value or diverges If it diverges, we set the pixel value based on the rate of divergence
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Once computed, the rows are plotted to make the well known Mandelbrot set images The colors used for the pixels are determined by mapping divergence rates onto a color map Creating a parallel version of this program using the Loop Parallelism pattern is trivial The iterations of the loop over rows are independent All we need to do is make sure each thread has its own row to work on We do this with the single pragma:
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#pragma omp parallel for private(row)
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The scheduling can be a bit tricky because work associated with each row will vary considerably
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depending on how many points diverge The programmer should try several different schedules, but a cyclic distribution is likely to provide an effective load balance In this schedule, the loop iterations are dealt out like a deck of cards By interleaving the iterations among a set of threads, we are likely to get a balanced load Because the scheduling decisions are static, the overhead incurred by this approach is small
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#pragma omp parallel for private(Row) schedule(static, 1)
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For more information about the schedule clause and the different options available to the parallel programmer, see the OpenMP appendix, Appendix A Notice that we have assumed that the graphics package is thread safe This means that multiple threads can simultaneously call the library without causing any problems The OpenMP specifications require this for the standard I/O library, but not for any other libraries Therefore, it may be necessary to protect the call to the graph function by placing it inside a critical section:
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#pragma critical graph(i, RowSize, M, color_map, ranges, row)
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We describe this construct in detail in the Implementation Mechanisms design space and in the OpenMP appendix, Appendix A This approach would work well, but it could have serious performance implications if the rows took the same time to compute and the threads all tried to graph their rows at the same time
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