In many coding problems, one often needs to manipulate a matrix based on certain conditions. A classic problem in algorithm design is the Set Matrix Zeroes problem. The task involves modifying a given matrix such that if an element in the matrix is zero, the entire row and column containing that element are set to zero.

In this article, we will explore both the brute force and optimal approaches to solving this problem. Each method has its own trade-offs in terms of time complexity and space usage, so understanding both approaches will help you decide the best solution based on your specific use case.

Brute Force Approach for Set Matrix Zeroes

The brute force approach for this problem is simple but inefficient. The idea is to iterate through each element of the matrix and if the element is zero, set all elements in that row and column to zero.

Algorithm:

1. Traverse the matrix to find all the positions of zeros.
2. For each zero element, set all elements in its row and column to zero.
3. Return the updated matrix.

Code Example:

def setZeroes(matrix):
    rows = len(matrix)
    cols = len(matrix[0])

    # Create a copy of the matrix
    matrix_copy = [row[:] for row in matrix]

    for i in range(rows):
        for j in range(cols):
            if matrix[i][j] == 0:
                # Set all elements in the row and column to zero in the copy
                for k in range(cols):
                    matrix_copy[i][k] = 0
                for k in range(rows):
                    matrix_copy[k][j] = 0

    # Update original matrix
    for i in range(rows):
        for j in range(cols):
            matrix[i][j] = matrix_copy[i][j]

Time Complexity: O(m * n)

Space Complexity: O(m * n)

This approach requires extra space to store the updated matrix and processes each element multiple times, making it inefficient for larger matrices.

Optimal Approach for Set Matrix Zeroes

The optimal approach reduces the space complexity and avoids the need for a second matrix. Instead, it uses the original matrix itself to mark which rows and columns need to be set to zero.

Algorithm:

1. Use the first row and the first column of the matrix to store whether a particular row or column should be set to zero.
2. Traverse the rest of the matrix:
   • If an element is zero, mark the first row and first column at the respective positions.
3. After marking, update the matrix by setting the marked rows and columns to zero.
4. Finally, handle the first row and first column separately.

Code Example:

def setZeroes(matrix):
    rows = len(matrix)
    cols = len(matrix[0])

    first_row_zero = False
    first_col_zero = False

    # Check if the first row and first column need to be set to zero
    for i in range(rows):
        if matrix[i][0] == 0:
            first_col_zero = True
            break

    for j in range(cols):
        if matrix[0][j] == 0:
            first_row_zero = True
            break

    # Mark zeros in the first row and column for the rest of the matrix
    for i in range(1, rows):
        for j in range(1, cols):
            if matrix[i][j] == 0:
                matrix[i][0] = 0
                matrix[0][j] = 0

    # Set zeros for the inner matrix based on marks
    for i in range(1, rows):
        for j in range(1, cols):
            if matrix[i][0] == 0 or matrix[0][j] == 0:
                matrix[i][j] = 0

    # Set zeros for the first row and first column if needed
    if first_row_zero:
        for j in range(cols):
            matrix[0][j] = 0

    if first_col_zero:
        for i in range(rows):
            matrix[i][0] = 0

Time Complexity: O(m * n)

Space Complexity: O(1)

This optimal solution runs in linear time but reduces space complexity to O(1) by modifying the matrix in place.

Conclusion:

The Set Matrix Zeroes problem can be solved in multiple ways, but the brute force approach often isn’t the best choice for larger matrices due to its high time and space complexity. The optimal approach, which uses the matrix itself to store information, is more efficient and better suited for handling larger datasets.

By choosing the optimal method, you reduce the space complexity to O(1), which can significantly improve performance, especially in resource-constrained environments. Always consider both time and space complexity when deciding which approach to use for a given problem.

This article has covered the key insights into solving the Set Matrix Zeroes problem using both brute force and optimal strategies. If you’re working on a similar problem, you can apply the same techniques to optimize your solutions.