Problem Solving using Multithreaded Algorithms:

Multithreading involves concurrent execution of multiple threads within a process. In problem-solving, this concurrency allows for parallel execution of tasks, leading to potential performance improvements.

 

Matrix multiplication:

Matrix multiplication is a mathematical operation that takes two matrices, say A and B, and produces another matrix C. The element C[i][j] of the resulting matrix is obtained by multiplying the elements of the ith row of matrix A by the corresponding elements of the jth column of matrix B and summing up the results.

Multithreading can be used for the computation of matrix multiplication, where each thread is responsible for computing a subset of the resulting matrix. The strategy involves dividing the workload among threads, each handling a portion of the rows in the resulting matrix.

 

Steps for Multithreaded Matrix Multiplication:

 

Check Compatibility: Ensure that the number of columns in the first matrix is equal to the number of rows in the second matrix.

Initialize Result Matrix: Create a result matrix C with dimensions (number of rows in A) x (number of columns in B).

Determine Thread Workload: Divide the rows of the resulting matrix into equal portions, assigning each portion to a separate thread.

Create Threads: Spawn threads, each responsible for computing the matrix multiplication for its assigned portion of the result matrix.

Thread Execution: Within each thread, perform the matrix multiplication for the assigned rows, calculating the dot product for each element.

Synchronize Threads: Ensure proper synchronization to avoid race conditions, especially when updating shared data (the result matrix).

Wait for Threads to Complete: Wait for all threads to finish their computations before proceeding.

Example:

Matrix A:   | 1  2 |                            Matrix B:    | 5  6 |                 

| 3  4 |                                                | 7  8 |

Step-by-Step Multithreaded Matrix Multiplication:

1. Check Compatibility: Both matrices have 2 columns, so they are compatible for multiplication.

2. Initialize Result Matrix: Create a result matrix C with dimensions 2x2 (matching the rows of A and columns of B).

3. Determine Thread Workload: Divide the rows of C among two threads. Thread 1 will handle the first row of C, and Thread 2 will handle the second row.

4. Create Threads: Spawn two threads: Thread 1 and Thread 2.

5. Thread Execution (Thread 1): Thread 1 calculates the first row of the resulting matrix C.

     - Element C[0][0] = (1*5 + 2*7) = 19

     - Element C[0][1] = (1*6 + 2*8) = 22

6. Thread Execution (Thread 2): Thread 2 calculates the second row of the resulting matrix C.

     - Element C[1][0] = (3*5 + 4*7) = 43

     - Element C[1][1] = (3*6 + 4*8) = 50

 

7. Synchronize Threads: Ensure proper synchronization, especially when updating shared data (result matrix C).

8. Resulting Matrix C:  | 19  22 |

| 43  50 |

 

In this example, we've divided the workload between two threads, each responsible for calculating a portion of the resulting matrix. The final result is the product of matrices A and B, calculated concurrently using multithreading.

 

Multithreaded Merge Sort:

Merge Sort is a classic divide-and-conquer sorting algorithm. When implemented in a multithreaded fashion, the sorting process can be parallelized, potentially improving overall performance. Here's a step-by-step explanation of multithreaded Merge Sort:

Algorithm Overview:

Base Case: If the array has one element or is empty, it is already sorted.

Divide: Split the array into two halves.

Multithreaded Recursion: Recursively apply Merge Sort on each half concurrently using multiple threads.

Merge: Combine the sorted halves into a single sorted array.

Example with Two Threads:

Consider the array: [38, 27, 43, 3, 9, 82, 10]

 

 

Step-by-Step Execution:

Initial Array: [38, 27, 43, 3, 9, 82, 10]

STEP 1(Divide): Split the array into two halves: [38, 27, 43] and [3, 9, 82, 10]

STEP 2(Multithreaded Recursion): Two threads are created, each sorting one of the divided halves.

-Thread 1 (Sorting [38, 27, 43]):

[27, 38, 43]

-Thread 2 (Sorting [3, 9, 82, 10]):

[3, 9, 10, 82]

STEP 3(Merge): Combine the sorted halves into a single sorted array.

[3, 9, 10, 27, 38, 43, 82]

The final sorted array is achieved by merging the results from the two threads.