Problem Description

In this problem, we have a matrix of size m x n, which is initially filled with zeros. We are also given a list of operations, where each operation is represented by a two-element array. The first element, a_i, signifies the number of rows (from the top), and the second element, b_i, signifies the number of columns (from the left), which will be affected by the operation. An operation increments by one each element in the submatrix defined by the given number of rows and columns. After applying all the operations, some elements in the matrix might be incremented multiple times.

Our task is to count the number of times the maximum value in the matrix occurs after carrying out all the given operations.

Intuition

To find the solution to this problem effectively, we notice that any overlap between operations will result in those specific cells being incremented multiple times. Therefore, the cells that are incremented the most are the ones that fall within the area of overlap of all the operations.

Instead of executing all operations which can lead to a time-consuming process for large matrices, we can find the intersection of all operation ranges. This intersection will give us the smallest submatrix that each operation influences.

We can find the minimum a_i and b_i across all operations since the operations are inclusive from index 0 up to but not including a_i for rows and b_i for columns. The intersection of these minimum values gives us the range of rows and columns that every operation will affect, and hence, these cells will have the maximum value.

Finally, we calculate the size of the range by multiplying the minimum a_i by the minimum b_i, and this gives us the count of maximum integers in the matrix after performing all operations. This is the key insight that allows us to arrive at an efficient solution.

Learn more about Math patterns.

Solution Approach

The solution approach for this problem is straightforward once the intuition is understood. We use a simple greedy algorithm to find the minimum values of a_i and b_i. A greedy algorithm is an approach for solving problems by choosing the best option available at the moment, without considering the bigger picture. In this case, we are selecting the minimum a_i and b_i because we are certain that the best option to find the maximum overlapped area is to find the smallest range that is affected by all operations.

Here are the steps in the algorithm using Python code as an example:

1. Initialize two variables min_row and min_col with the maximum possible values of m and n respectively, which represent the dimensions of the matrix.

2. Iterate over each operation (ops) in the list of operations:

   • For each operation, take the pair [a_i, b_i].
   • Update min_row to be the minimum of the current min_row and a_i.
   • Update min_col to be the minimum of the current min_col and b_i. By doing this, we find the intersection of all the operations which is the smallest submatrix affected by all operations.

3. After finding the minimum values for the rows (min_row) and columns (min_col), the maximum number of times an element has been incremented is exactly the region defined by min_row x min_col.

We do not use any complex data structures for this problem as the Python solution demonstrates efficiency through avoiding explicit matrix construction and operation simulation, which could be costly in terms of both time and space for large matrices. The problem is reduced to simple value comparisons and multiplication which makes it an O(k) solution, where k is the number of operations, irrespective of the actual dimensions of the matrix.

The Python solution uses this approach to provide a concise and efficient solution. Here's a brief walk-through of the Python code:

class Solution:
    def maxCount(self, m: int, n: int, ops: List[List[int]]) -> int:
        # Initialize `m` and `n` to represent the max possible numbers of rows and columns.
        for a, b in ops:  # Loop through each operation.
            m = min(m, a)  # Find the smallest value of `a_i` from all operations.
            n = min(n, b)  # Find the smallest value of `b_i` from all operations.
        return m * n  # The size of the overlapped area is the result we want.

The min functions are used to find the smallest values from the provided a and b because every a affects rows 0 to a-1 and every b affects columns 0 to b-1. By multiplying m and n, which are now the sizes of the maximum overlapped area, we get the count of elements in the matrix with the maximum values after all operations.

Example Walkthrough

Let's consider a small example to illustrate the solution approach.

Suppose we have a matrix of size 4 x 5 (m = 4, n = 5), and we are given a list of three operations: ops = [[2,3], [3,3], [3,2]]. To visualize the initial state of the matrix filled with zeros, it would look something like this:

0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0

Now, let's perform these operations according to the described algorithm without actually incrementing each cell:

1. The first operation is [2, 3], which means we would increment the submatrix from rows 0 to 1 and columns 0 to 2. The affected area, for now, is 2 x 3.

2. The second operation is [3, 3], which would cover rows 0 to 2 and columns 0 to 2, but since we are looking for the overlap, our affected area is still 2 x 3 (as the first operation only covers two rows).

3. The third operation is [3, 2], which has a coverage from rows 0 to 2 and columns 0 to 1. When looking for the overlapping area, we observe that the rows overlap with the first and second operations, but the columns now only overlap within the first two columns. Therefore, the final smallest submatrix that all operations influence is now 2 x 2.

Given that we want the count of maximum values after all these operations, we don't need to perform the increment. We simply need the size of the range defined by the minimum a_i and b_i for rows and columns respectively.

For our example, the smallest a_i from the operations is 2 (from the first operation), and the smallest b_i is 2 as well (from the third operation).

Now we multiply these values: min_row = 2 and min_col = 2, therefore, the count is min_row * min_col = 2 * 2 = 4.

So, after all the operations, there are four elements in the matrix that have the maximum value, and that's the result. No element will be incremented more times than these four elements that lie in the intersection of all operation ranges.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of this function is O(k), where k is the number of operations in the ops list. The reason for this is that the function loops through each operation once and performs a constant-time operation of updating m and n to the minimum of their current values and the values in the operation. Since looping through the list of operations is the only step that depends on the size of the input, and each iteration involves constant time work, the overall time complexity is linear with respect to the number of operations in ops.

Space Complexity

The space complexity of this function is O(1) because it only uses a fixed amount of extra space. No additional space is allocated that is dependent on the input size of the problem; the variables m and n are updated in place. As the amount of space used does not change with varying input sizes (as ops are merely iterated over and not stored), the space complexity is constant.

Learn more about how to find time and space complexity quickly using problem constraints.