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.

Where to Start

You can find all of the relevant files for the assignment under the matrix-multiply directory. From there, you will need to implement the Matrix and SubMatrix classes.

Matrix

There are seven 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.

parallelForAllIMultiply

In this implementation, you will simply convert the sequential solution into a parallel one using a forall loop. This should be the easiest parallel implementation and should only require minor changes to work properly.

parallelChunkedForAllIMultiply

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.

parallelGroupedForAllTasksForSeqIMultiply

In this implementation, we will make use of HJ’s iteration regions in order to manually group them for the sake of performance. You only need to use a 1D region for this method, but you will need to make notable changes from your previous work. For optimal performance, we will break down the parallel tasks so that they equal the exact number of working threads. You can access this information using HJ’s numWorkerThreads() method. After launching a number of tasks equal to the number of worker threads, use HJ’s myGroup() method to create a new 1D region from the index of the thread, a 1D region specified to the size of the matrix, and the number of threads. After that, launch the code as you did previously and multiply the two matrices.

Hint: use forseq to create a sequential for loop that can access the indices of a HJ iteration region.

parallelForAllIForAllJMultiply

In this implementation, we will increase the parallelism of the previous implementations using two forall loops instead of one. There is no need to chunk or group the iterations. This should look very similar to the original forall implementation.

parallelForAll2dIJMultiply

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.

parallelChunkedForAll2dIJMultiply

This method should build on the previous 2d method and optimize its performance using chunked(). This should look very familiar from the previous implementations. Again, there is no need to specify how to chunk, leave it all to the runtime.

parallelGroupedForAllTasksForSeq2dIJMultiply (need help explaining this)

This is the most difficult implementation out of all of the previous methods. However, if you did everything beforehand, a lot of the steps are a synthesis of the forall2d implementations and the 1D grouping implementation.

UNDER CONSTRUCTION

SubMatrix

In this implementation, you will solve the matrix multiply problem 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.