Minimum Operations to Make Columns Strictly Increasing

Problem Description

You are given a m x n matrix grid consisting of non-negative integers. In one operation, you can increment the value of any grid[i][j] by 1. Return the minimum number of operations needed to make all columns of grid strictly increasing.

Key Insights

• Each column in the matrix must be strictly increasing.
• For each column, we need to ensure that each element is greater than the one directly above it.
• We can calculate the number of operations needed by comparing each element with its predecessor in the same column.
• The total operations can be aggregated across all columns.

Space and Time Complexity

Time Complexity: O(m * n)
Space Complexity: O(1)

Solution

To solve this problem, we will use a greedy approach. We will iterate through each column of the matrix and for each element in the column (starting from the second row), we will check if it is greater than the element directly above it. If it is not, we will calculate how many increments are needed to make it strictly greater. We will keep a running total of these increments for all elements in all columns.

1. Iterate through each column from left to right.
2. For each column, iterate through its rows from the second row to the last row.
3. For each element, check if it is less than or equal to the element above it.
4. If it is, calculate the difference needed to make it strictly greater and increment the element.
5. Accumulate the total number of increments required.

Code Solutions