Two Pointers on matrix

Question

Given a 2D matrix of integers, implement a solution using the two pointers technique to find the maximum sum of a subarray of a specified size. The input will consist of the matrix dimensions and the size of the subarray, and the output should be the maximum sum found.

Codemia · Accepted Answer

The problem requires us to find the maximum sum of a subarray of a specified size in a 2D matrix. Given that we're looking for sums within a defined boundary (the size of the subarray), the sliding window or two pointers technique is quite applicable here. This technique allows us to efficiently compute sums without needing to repeatedly iterate over the same elements. By maintaining two pointers that define the boundaries of our current subarray, we can dynamically adjust our sum and keep track of the maximum found.

1. **Input Parsing**: Start by reading the dimensions of the matrix and the size of the subarray. We'll denote the matrix as `matrix` of size `m x n` and the subarray size as `k`.
2. **Initialize Variables**: Create a variable `max_sum` to store the maximum sum found, initially set to a very small number. Also, set up a variable `current_sum` to accumulate the sum of the current subarray.
3. **Iterate Over the Matrix**: Use two nested loops to generate the top-left corner of each possible subarray of size `k`. The outer loop iterates over possible starting rows, and the inner loop iterates over possible starting columns.
4. **Calculate Subarray Sum**: For each top-left corner identified, calculate the sum of the subarray by using two additional loops to cover the `k x k` area. Update `current_sum` and check if it exceeds `max_sum`.
5. **Return Result**: After iterating through all possible subarrays, return `max_sum`.

Here is a concrete Python implementation of the described approach:

```python
def max_subarray_sum(matrix, k):
    if not matrix or not matrix[0]:
        return 0
    m, n = len(matrix), len(matrix[0])
    max_sum = float('-inf')

for i in range(m - k + 1):
        for j in range(n - k + 1):
            current_sum = 0
            for x in range(k):
                for y in range(k):
                    current_sum += matrix[i + x][j + y]
            max_sum = max(max_sum, current_sum)

return max_sum
```

The time complexity of this approach is O((m - k + 1) * (n - k + 1) * k^2), where `m` and `n` are the dimensions of the matrix. This is because for every starting position of the top-left corner of the subarray, we need to iterate through `k^2` elements to calculate the sum. The space complexity is O(1) since we are using only a fixed amount of extra space regardless of the input size.

1. **Empty Matrix**: If the matrix is empty (i.e., `matrix = []`), we should return 0 as there are no elements to sum.
2. **Subarray Size Larger Than Matrix**: If `k` is larger than either dimension of the matrix, the function should handle this gracefully, likely returning 0 since a valid subarray cannot be formed.
3. **Negative Numbers**: If the matrix contains all negative numbers, the algorithm should still correctly identify the maximum sum, which would be the least negative number or the sum of the least negative subarray.

Two Pointers on matrix

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