MatrixMultiply

Background

Matrix multiplication is a simple mathematical operation which we will replicate in this studio. For our purposes, we will only deal with square matrices (same number of rows and columns). However, we will approach this problem with several different parallel constructs and approaches.

For those unfamiliar on how to multiply two matrices, take a look at this link for a quick overview.

Math Definition

source: Matrix Multiplication on Wikipedia

If $\mathbf {A}$ is an $n\times m$ matrix and $\mathbf {B}$ is an $m\times p$ matrix

for each i=[0..n) and for each j=[0..p)

$(\mathbf {A} \mathbf {B} )_{ij}=\sum _{k=1}^{m}A_{ik}B_{kj}$

Sequential Java Implementation

The point of the required portion of this studio is not to struggle with matrix multiplication, but rather to get some experience with the parallel for loop constructs in Habanero.

Feel free to use the provided sequential implementation in SequentialMatrixMultiplier as a reference:

	@Override
	public double[][] multiply(double[][] a, double[][] b) {
		double[][] result = MatrixUtils.createMultiplyResultBufferInitializedToZeros(a, b);
		int n = a.length;
		int m = a[0].length;
		int p = b[0].length;
		for (int i = 0; i < n; i++) {
			for (int j = 0; j < p; j++) {
				// NOTE: result is already initialized to 0.0
				// result[i][j] = 0.0;
				for (int k = 0; k < m; k++) {
					result[i][j] += a[i][k] * b[k][j];
				}
			}
		}
		return result;
	}

Where to Start

MatrixMultiplyTestSuite

In src/test/java in package matrixmultiply.studio you will find MatrixMultiplyTestSuite.

From there you can tunnel into the three MatrixMultipliers you will need to implement.

Required Studio Solutions

There are three methods you will need to implement, all of which are different ways to use parallel for loops to solve the problem. To assist you, the sequential implementation has already been completed for you. We recommend starting from the top and working your way down. There is also an optional recursive implementation and a manual grouping implementation which has been done for you (this is just to demonstrate how chunking works behind the scenes).

ForallForallMatrixMultiplier

In this implementation, you will simply convert the sequential solution into a parallel one using two forall loops.

Forall2dMatrixMultiplier

In this implementation, we will cut down the syntax of the two forall implementation with the use of HJ’s forall2d method. Functionally, this method serves the purpose of using two forall loops. The input parameters should equal the input parameters of the two separate forall loops. For example,

forall(min, max, (i) -> {
	forall(min, max, (j) -> doSomething());
});

Would appear as forall2d(min, max, min, max, (i, j) -> doSomething()); in the forall2d syntax.

Forall2dChunkedMatrixMultiplier

In this implementation, we will add a minor performance boost to the process by using the forall-chunked construct. Although similar to the traditional forall loop, it increases speedup using iteration grouping/chunking. This topic is discussed in detail in this Rice video and explained in the HJ documentation. There is no need to specify anything, allow the runtime to determine the chunking.

In this approach, we will show you how you can specify the grouping yourself and what forAllChunked is doing behind the scenes. There is no need to implement this method, this is just here to serve as a reference for you. However, you should understand the concept of iteration grouping before moving forward.

Optional Fun Divide and Conquer Solutions

In this implementation, you will solve the matrix multiply problem sequentially and in parallel using recursion. Although this class should be able to take in a matrix of any size, try to imagine this as a 2x2 matrix in order to make it easier to solve. Once you solve the sequential method, the parallel method should look very similar with exception of an async/finish block.

In order to obtain the desired result matrix, you will need to recursively call the correct submatrices for each of the four result submatrices. Imagining this as a 2x2 matrix, remember that the dot products of the rows of the first matrix and the columns of the second matrix create the result matrix.

Hint: Each result submatrix should have two recursive calls, for a total of eight recursive calls.

SequentialDivideAndConquerMatrixMultiplier

In class SubMatrix, method sequentialDivideAndConquerMultiplyKernel you will find your base case and the sub matrices prepared for you.

	private static void sequentialDivideAndConquerMultiplyKernel(SubMatrix a, SubMatrix b, SubMatrix result) {
		if( result.size == 1 ) {
			result.values[result.row][result.col] += a.values[a.row][a.col] * b.values[b.row][b.col];
		} else {
			SubMatrix a11 = a.newSub11();
			SubMatrix a12 = a.newSub12();
			SubMatrix a21 = a.newSub21();
			SubMatrix a22 = a.newSub22();
			
			SubMatrix b11 = b.newSub11();
			SubMatrix b12 = b.newSub12();
			SubMatrix b21 = b.newSub21();
			SubMatrix b22 = b.newSub22();
			
			SubMatrix result11 = result.newSub11();
			SubMatrix result12 = result.newSub12();
			SubMatrix result21 = result.newSub21();
			SubMatrix result22 = result.newSub22();

			throw new NotYetImplementedException();
		}
	}

You simply need to make the appropriate recursive calls to compute the result on the right:

${\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}\\\mathbf {A} _{21}&\mathbf {A} _{22}\\\end{pmatrix}}{\begin{pmatrix}\mathbf {B} _{11}&\mathbf {B} _{12}\\\mathbf {B} _{21}&\mathbf {B} _{22}\\\end{pmatrix}}={\begin{pmatrix}\mathbf {A} _{11}\mathbf {B} _{11}+\mathbf {A} _{12}\mathbf {B} _{21}&\mathbf {A} _{11}\mathbf {B} _{12}+\mathbf {A} _{12}\mathbf {B} _{22}\\\mathbf {A} _{21}\mathbf {B} _{11}+\mathbf {A} _{22}\mathbf {B} _{21}&\mathbf {A} _{21}\mathbf {B} _{12}+\mathbf {A} _{22}\mathbf {B} _{22}\\\end{pmatrix}}$

source: Wikipedia Parallel Matrix Multiplication

ParallelDivideAndConquerMatrixMultiplier

Again, given the following:

${\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}\\\mathbf {A} _{21}&\mathbf {A} _{22}\\\end{pmatrix}}{\begin{pmatrix}\mathbf {B} _{11}&\mathbf {B} _{12}\\\mathbf {B} _{21}&\mathbf {B} _{22}\\\end{pmatrix}}={\begin{pmatrix}\mathbf {A} _{11}\mathbf {B} _{11}+\mathbf {A} _{12}\mathbf {B} _{21}&\mathbf {A} _{11}\mathbf {B} _{12}+\mathbf {A} _{12}\mathbf {B} _{22}\\\mathbf {A} _{21}\mathbf {B} _{11}+\mathbf {A} _{22}\mathbf {B} _{21}&\mathbf {A} _{21}\mathbf {B} _{12}+\mathbf {A} _{22}\mathbf {B} _{22}\\\end{pmatrix}}$

source: Wikipedia Parallel Matrix Multiplication

What computation can be done in parallel? What computation must be performed sequentially?