Find the Grid of Region Average - Solution & Explanation

Problem Overview: You receive an m x n image grid and a threshold. For every possible 3 x 3 region, check whether all adjacent cells inside the region differ by at most the threshold. If the region is valid, compute the floor of the average of its 9 values and contribute that value to every cell in the region. A cell may belong to multiple valid regions, so the final grid value becomes the average of all region averages covering that cell. If no valid region covers a cell, keep the original value.

Approach 1: Brute-Force Region Detection (O(m*n) time, O(m*n) space)

Iterate through the matrix and treat every cell (i, j) as the top-left corner of a potential 3 x 3 region. For each candidate region, verify the constraint by checking all horizontal and vertical adjacent pairs inside the region. There are only a fixed number of comparisons (12 adjacency checks), so validation is constant time. If the region satisfies the threshold condition, compute the sum of the 9 cells and derive the region average using integer division.

Maintain two helper matrices: one for accumulating the sum of region averages affecting each cell and another for counting how many valid regions contributed to that cell. When a valid region is found, add its computed average to all 9 cells and increment their counters. After scanning the entire grid, compute the final result cell-by-cell: if the counter is greater than zero, divide the accumulated sum by the count; otherwise copy the original pixel value.

This approach works well because the region size is fixed. Even though every cell may trigger a region check, each validation and averaging step touches only 9 cells, making the total runtime proportional to the grid size. The technique relies heavily on systematic iteration over a matrix and careful boundary handling.

Approach 2: Prefix Sum Assisted Averaging (O(m*n) time, O(m*n) space)

You can slightly streamline the average calculation by precomputing a 2D prefix sum of the grid. This allows constant-time retrieval of any 3 x 3 region sum instead of summing the nine elements directly. The adjacency validation still requires checking neighbors within the region, but the averaging step becomes a single prefix-sum query.

In practice the improvement is minor because the region size is small, but prefix sums can help if the region size becomes variable in a follow-up question. This pattern appears frequently in array and matrix range-sum problems.

Recommended for interviews: The brute-force region detection approach is typically expected. It demonstrates that you can systematically scan a grid, validate local constraints, and correctly handle overlapping contributions. Prefix sums are a reasonable optimization discussion but rarely necessary given the fixed 3x3 window.