// skip if already solved // done

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// try all expansions

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for (int q = 0; q < boardSize; ++q) { row // Check if queen can be placed in column q of next boolean attacked = false; for (int i = 0; i < row; ++i) { int p = sofar[i]; if (q == p || q == p - (row-i) || q == p + (row-i)) attacked = true; break;

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// If so, fork to explore moves from new configuration if (!attacked) { // build extended board representation int[] next = new int[row+1];

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} } } }

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for (int k = 0; k < row; ++k) next[k] = sofar[k]; next[row] = q; new NQueens(next)fork();

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442 Computation Trees

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A number of computationally intensive algorithms involve tasks of the form:

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For a fixed number of steps, or until convergence, do { Update one section of a problem; Wait for other tasks to finish updating their sections; }

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Most often, such algorithms perform update operations on partitioned arrays, matrices, or image representations For example, many physical dynamics problems involve repeated local updates to the cells of a matrix Jacobi algorithms and related relaxation techniques repeatedly recalculate estimated values across neighboring cells, typically using an averaging formula such as:

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void oneStep(double[][] oldMatrix, double[][] newMatrix, int i, int j) { newMatrix[i][j] = 025 * (oldMatrix[i-1][j] + oldMatrix[i][j-1] + oldMatrix[i+1][j] + oldMatrix[i][j+1]); }

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Normally, to save space, two different matrices are swapped as newMatrix and oldMatrix across successive steps Algorithms requiring that all tasks periodically wait for all others to complete do not always scale quite as well as more loosely coupled fork/join designs Even so, these algorithms are common, efficient, and amenable to significant parallel speedups 4421 Building and using trees It would be inefficient to repeatedly apply fork/join decomposition in iterative designs in order to update sections in parallel Because the sections are the same across iterations, they can be constructed just once and then repeatedly invoked so that on each iteration, the corresponding updates execute in the same order as would be produced by a recursive solution Computation trees are explicit representations of the tree-structured computations implicitly arising in fork/join recursion These trees have two kinds of nodes, internal nodes and leaf nodes, corresponding to the recursive and base cases of a recursive solution They can be constructed and used for iterative update problems via the following steps: 1 Create a tree of task objects representing the recursive partitions, where: o Each internal node contains references to subpartitions, and has an update method that performs fork/join processing of each of them o Each leaf node represents a finest-granularity partition, and has an update method that operates directly on it 2 For a fixed number of steps, or until convergence, do: o Execute the task performing the root partition's update method For example, the following code illustrates the highlights of a set of classes that perform Jacobi iteration using the averaging formula shown above In addition to updating, this version also keeps track of the differences among computed cell values across iterations, and stops when the maximum difference is within a constant EPSILON Also, like many programs of this form, this code assumes that the matrices have been set up with extra edge cells that are never updated, so boundary conditions never need to be checked (Alternatives include recomputing edge values using special edge formulas after each pass, and treating edges as toroidally wrapping around the mesh) The recursive decomposition strategy used here is to divide the mesh into quadrants, stopping when the number of cells is at most leafCells, which serves as the granularity threshold This strategy works well so long as the numbers of rows and columns in the matrix are approximately equal If they are not, additional classes and methods could be defined to divide across only one dimension at a time The approach here assumes that the matrix as a whole already exists, so rather than actually dividing up cells, task nodes just keep track of the row and column offsets of this matrix that each partition is working on The subclass-based design used here reflects the different structure and behavior of internal versus leaf nodes Both are subclasses of abstract base JTree:

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abstract class JTree extends FJTask { volatile double maxDiff; // for convergence check } class Interior extends JTree {

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private final JTree[] quads; Interior(JTree q1, JTree q2, JTree q3, JTree q4) { quads = new JTree[] { q1, q2, q3, q4 }; } public void run() { coInvoke(quads); double md = 00; for (int i = 0; i < quadslength; ++i) { md = Mathmax(md,quads[i]maxDiff); quads[i]reset(); } maxDiff = md; } } class Leaf extends JTree { private final double[][] A; private final double[][] B; private final int loRow; private final int hiRow; private final int loCol; private final int hiCol; private int steps = 0; Leaf(double[][] A, double[][] B, int loRow, int hiRow, int loCol, int hiCol) { thisA = A; thisB = B; thisloRow = loRow; thishiRow = hiRow; thisloCol = loCol; thishiCol = hiCol; } public synchronized void run() { boolean AtoB = (steps++ % 2) == 0; double[][] a = (AtoB) A : B; double[][] b = (AtoB) B : A; double md = 00; for (int i = loRow; i <= hiRow; ++i) { for (int j = loCol; j <= hiCol; ++j) { b[i][j] = 025 * (a[i-1][j] + a[i][j-1] + a[i+1][j] + a[i][j+1]); md = Mathmax(md, Mathabs(b[i][j] - a[i][j])); } } maxDiff = md; } }

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The driver class first builds a tree that represents the partitioning of its argument matrix The build method could itself be parallelized But because the base actions are just node constructions, the

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granularity threshold would be so high that parallelization would be worthwhile only for huge problem sizes The run method repeatedly sets the root task in motion and waits out completion For simplicity of illustration, it continues until convergence Among other changes necessary to turn this into a realistic program, you would need to initialize the matrices and deal with possible lack of convergence within a bounded number of iterations Because each iteration entails a full synchronization point waiting for the root task to finish, it is relatively simple to insert additional operations that maintain or report global status between iterations

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class Jacobi extends FJTask { static final double EPSILON = 0001; // convergence criterion final JTree root; final int maxSteps; Jacobi(double[][] A, double[][] B, int firstRow, int lastRow, int firstCol, int lastCol, int maxSteps, int leafCells) { thismaxSteps = maxSteps; root = build(A, B, firstRow, lastRow, firstCol, lastCol, leafCells); } public void run() { for (int i = 0; i < maxSteps; ++i) { invoke(root); if (rootmaxDiff < EPSILON) { Systemoutprintln("Converged"); return; } else rootreset(); } } static JTree build(double[][] a, double[][] b, int lr, int hr, int lc, int hc, int size) { if ((hr - lr + 1) * (hc - lc + 1) <= size) return new Leaf(a, b, lr, hr, lc, hc); int mr = (lr + hr) / 2; // midpoints int mc = (lc + hc) / 2; return new Interior(build(a, b, lr, mr, lc, mc, size), build(a, b, lr, mr, mc+1, hc, size), build(a, b, mr+1, hr, lc, mc, size), build(a, b, mr+1, hr, mc+1, hc, size)); } }

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