Maximum Sum of an Hourglass

Problem Description

You are given an m x n integer matrix grid. We define an hourglass as a part of the matrix with the following form:

Return the maximum sum of the elements of an hourglass. Note that an hourglass cannot be rotated and must be entirely contained within the matrix.

Key Insights

• An hourglass is defined by a specific pattern of elements in the matrix.
• The hourglass sums can be calculated by iterating through possible starting points in the matrix.
• We need to ensure that the hourglass fits within the bounds of the matrix (i.e., starts no later than the second-to-last row and second-to-last column).

Space and Time Complexity

Time Complexity: O(m * n)
Space Complexity: O(1) (since we're using a constant amount of extra space)

Solution

To solve the problem, we will iterate through each possible hourglass starting position in the matrix. For each starting position, we will calculate the sum of the hourglass elements and keep track of the maximum sum found. This approach utilizes a nested loop to access the necessary elements for the hourglass shape.

The algorithm can be summarized as follows:

1. Loop through each element in the matrix that can serve as the top-left corner of an hourglass.
2. For each position, compute the sum of the hourglass elements.
3. Update the maximum sum if the current hourglass sum exceeds the previously recorded maximum.

Code Solutions