MedianInARowWiseSortedMatrix

The MedianInARowWiseSortedMatrix is a utility function designed for finding the median value in a row-wise sorted matrix. It takes an n x m matrix as input, where each row is sorted in ascending order, and returns the median value of all the elements in the matrix.

Usage

def median_in_row_wise_sorted_matrix(matrix: List[List[int]]) -> int:
    """
    Finds the median value in a row-wise sorted matrix.
  
    Args:
    - matrix: An n x m matrix (List[List[int]]) where each row is sorted.

    Returns:
    - The median value (int) of all the elements in the matrix.
    """
    # Implementation goes here

Example

Consider the following row-wise sorted matrix:

matrix = [
    [1, 3, 5],
    [2, 6, 9],
    [3, 6, 9]
]

Calling the median_in_row_wise_sorted_matrix(matrix) function would return 5, which is the median value of all the elements in the matrix.

Approach

The function utilizes the property of a row-wise sorted matrix to find the median efficiently. The approach involves flattening the matrix into a sorted list and then finding the median element. Here are the steps:

1. Flatten the matrix by concatenating all rows into a single list.
2. Sort the flattened list in ascending order.
3. Determine the median index based on the total number of elements. If the number of elements is odd, the median index is (total_elements - 1) // 2. If the number of elements is even, the median index is total_elements // 2.
4. Return the element at the median index from the sorted list.

Complexity Analysis

The time complexity of the median_in_row_wise_sorted_matrix function is O(n * m log(n * m)), where n and m are the dimensions of the input matrix. This complexity arises from flattening and sorting the matrix.

The space complexity is O(n * m) since the function creates a flattened list to store all the matrix elements.

Conclusion

The MedianInARowWiseSortedMatrix utility function provides an efficient way to find the median value in a row-wise sorted matrix. By following the documented usage and understanding the underlying approach, programmers can easily leverage this function in their code to solve similar problems efficiently.