Question

Write a CRCW PRAM algorithm that uses \(n^{2}\) processors to multiply two \(n \times n\) matrices. Your algorithm should perform better than the standard \(\Theta\left(n^{3}\right)\) -time serial algorithm.

Short answer

The algorithm divides the task among different processors to each multiply corresponding elements in the row of the first matrix and column of the second matrix. Multiplications are done in parallel, which is far quicker than the serial algorithm. The time complexity of this algorithm is \(\Theta\left(n^{2}\right)\), assuming add and multiply operations can be done in constant time.

Step-by-step solution

Initialization

Firstly, initialize \(n^{2}\) processors and two matrices A and B of size \(n \times n\) for multiplication.

Element Assignment

Assign each element of matrices A and B to a single processor. Given there are \(n^{2}\) processors, each cell in the \(n \times n\) matrix will be assigned to one processor, resulting in distributing the matrices among the processors.

Parallel Processing

In parallel, each processor performs the multiplication of the corresponding cell values of matrices A and B. The calculation will be the sum of multiplication of corresponding elements in a row of the first matrix and a column of the second matrix. All processors perform this step simultaneously.

Calculating Result

Finally, the partial results calculated by each processor are summed up to compute the value each cell in the resultant matrix.

Key concepts

Parallel Processing in CRCW PRAM

Understanding parallel processing is key when working with algorithms designed for optimal performance. The CRCW (Concurrent Read, Concurrent Write) PRAM model enables processors to perform tasks in parallel, meaning they can execute multiple operations simultaneously. This is crucial in reducing the computation time of complex tasks such as matrix multiplication.

In the context of multiplying two matrices, using parallel processing involves multiple processors working at the same time to achieve a result that would traditionally take longer with a single processor. By assigning each matrix element to a different processor, the algorithm divides the work into smaller, manageable tasks, significantly speeding up the overall process.

This is particularly effective in CRCW PRAM where each processor can read from and write to shared memory simultaneously. In situations where both speed and efficiency are essential, utilizing parallel processing can lead to more effective and quicker computing solutions.

Matrix Multiplication Made Easy

Matrix multiplication is a fundamental operation in many algorithms and applications, including those in computer graphics, scientific computing, and more. When multiplying two matrices, the result is another matrix whose elements are the dot products of the rows from the first matrix with the columns of the second.

In the traditional sense, matrix multiplication involves three nested loops to calculate the products and sums of matrix elements. This results in a time complexity of \(\Theta(n^3)\), where \(n\) is the size of the matrices involved. However, the use of parallel computing and models like CRCW PRAM can drastically enhance performance by coordinating multiple processors to work on different parts of the calculation simultaneously.

• First, each processor assigned to an element in matrix A reads its corresponding row.
• Simultaneously, each processor assigned to an element in matrix B reads its corresponding column.
• The processors then perform the multiplication tasks independently and concurrently, reducing the operation time.

By breaking the problem into concurrent tasks, matrix multiplication becomes far more efficient, making it a highly applicable technique in parallel computing tasks.

Understanding Computational Complexity

Computational complexity refers to the amount of resources required to solve a problem, specifically in terms of time and space. For algorithms, understanding and improving computational complexity is essential for optimizing performance, especially regarding execution time and memory usage.

In the traditional matrix multiplication, the computational complexity is \(\Theta(n^3)\), which can be limiting as the size of matrices increases. This level of complexity can lead to significant time consumption, which becomes impractical for very large matrices.

Parallel processing approaches, like CRCW PRAM, allow for the distribution of computational tasks over multiple processors, thus reducing the time taken for calculations without increasing resource consumption beyond the number of processors. While the time complexity in such parallel approaches significantly decreases, careful planning is necessary to manage the concurrent operations and ensure proper summation of results without errors.
Understanding computational complexity is thus pivotal in developing efficient algorithms and leveraging parallel processing capabilities to solve large-scale problems effectively.