Problem Description

You are given a 2D integer matrix grid with dimensions m x n and an integer k. Your task is to find the minimum absolute difference between any two distinct values within every possible k x k square submatrix of the grid.

For each k x k submatrix in the grid, you need to:

1. Find all pairs of distinct values within that submatrix
2. Calculate the absolute difference for each pair
3. Determine the minimum absolute difference among all pairs

The result should be a 2D array ans with dimensions (m - k + 1) x (n - k + 1), where each element ans[i][j] represents the minimum absolute difference for the k x k submatrix that starts at position (i, j) in the original grid (with (i, j) being the top-left corner of the submatrix).

Special case: If all elements in a submatrix are identical (no distinct pairs exist), the minimum absolute difference for that submatrix is 0.

For example, if you have a 4x4 grid and k=2, you would examine all possible 2x2 submatrices:

• The submatrix starting at (0,0) covers positions (0,0), (0,1), (1,0), (1,1)
• The submatrix starting at (0,1) covers positions (0,1), (0,2), (1,1), (1,2)
• And so on...

Each of these submatrices would contribute one value to the result array based on the minimum absolute difference between its distinct elements.

Intuition

The key insight is that to find the minimum absolute difference between any two distinct values, we need to compare all possible pairs within each submatrix. However, comparing all pairs directly would be inefficient.

Think about what happens when we have a list of numbers and want to find the smallest difference between any two of them. If we sort the numbers first, the minimum difference must occur between two adjacent elements in the sorted list. Why? Because for any two non-adjacent elements in a sorted array, there's always an element between them that would create a smaller difference with one of them.

For example, if we have sorted values [1, 3, 7, 9], the minimum difference could be between (1,3), (3,7), or (7,9). We don't need to check (1,7) or (1,9) because 3 is closer to both 1 and 7 than they are to each other.

This leads us to the approach:

1. For each k x k submatrix, extract all its elements
2. Sort these elements
3. Check only consecutive pairs in the sorted list (using pairwise)
4. Skip pairs where both elements are the same (we need distinct values)
5. Find the minimum among these differences

The time complexity for each submatrix becomes O(k² log k²) for sorting plus O(k²) for finding the minimum difference, which is much better than checking all O(k⁴) possible pairs without sorting.

The edge case where all elements are identical is handled naturally - when we filter for distinct pairs (if a != b), no pairs remain, so we default to 0.

Learn more about Sorting patterns.

Solution Approach

The solution follows a straightforward enumeration approach where we systematically examine each possible k x k submatrix in the grid.

Step 1: Initialize the result array

ans = [[0] * (n - k + 1) for _ in range(m - k + 1)]

We create a 2D array ans with dimensions (m - k + 1) x (n - k + 1) to store the results. This size corresponds to the number of possible top-left positions for a k x k submatrix in an m x n grid.

Step 2: Enumerate all possible submatrices

for i in range(m - k + 1):
    for j in range(n - k + 1):

We iterate through all valid top-left corner positions (i, j) where a k x k submatrix can fit within the grid boundaries.

Step 3: Extract elements from each submatrix

nums = []
for x in range(i, i + k):
    for y in range(j, j + k):
        nums.append(grid[x][y])

For each submatrix starting at (i, j), we collect all k² elements into a list nums. The nested loops iterate from row i to i + k - 1 and column j to j + k - 1.

Step 4: Sort and find minimum difference

nums.sort()
d = min((abs(a - b) for a, b in pairwise(nums) if a != b), default=0)

After sorting the elements, we use pairwise(nums) to generate consecutive pairs from the sorted list. The pairwise function creates pairs like (nums[0], nums[1]), (nums[1], nums[2]), etc. We calculate the absolute difference for each pair where the elements are distinct (a != b), and find the minimum among these differences. If no distinct pairs exist (all elements are the same), the default=0 parameter ensures we return 0.

Step 5: Store the result

ans[i][j] = d

We store the minimum absolute difference for the current submatrix at the corresponding position in our result array.

The algorithm processes a total of (m - k + 1) × (n - k + 1) submatrices, and for each submatrix, it performs sorting on k² elements, resulting in an overall time complexity of O((m - k + 1) × (n - k + 1) × k² log k²).

Example Walkthrough

Let's walk through a concrete example to illustrate the solution approach.

Consider a 4×4 grid with k = 3:

grid = [[5, 2, 9],
        [3, 8, 1],
        [7, 4, 6]]

We need to find the minimum absolute difference for each possible 3×3 submatrix. With a 4×4 grid and k=3, we have (4-3+1) × (4-3+1) = 2×2 = 4 possible submatrices.

Wait, let me correct that. For a 3×3 grid with k=3, we actually have (3-3+1) × (3-3+1) = 1×1 = 1 possible submatrix (the entire grid itself).

Let me use a better example with a 4×4 grid and k=2:

grid = [[5, 2, 9, 3],
        [3, 8, 1, 4],
        [7, 4, 6, 2],
        [1, 5, 3, 8]]

With k=2, we'll have (4-2+1) × (4-2+1) = 3×3 = 9 possible 2×2 submatrices.

Submatrix at position (0,0):

• Top-left corner: (0,0)
• Elements: [5, 2, 3, 8]
• After sorting: [2, 3, 5, 8]
• Consecutive pairs: (2,3), (3,5), (5,8)
• Differences: |3-2|=1, |5-3|=2, |8-5|=3
• Minimum difference: 1

Submatrix at position (0,1):

• Top-left corner: (0,1)
• Elements: [2, 9, 8, 1]
• After sorting: [1, 2, 8, 9]
• Consecutive pairs: (1,2), (2,8), (8,9)
• Differences: |2-1|=1, |8-2|=6, |9-8|=1
• Minimum difference: 1

Submatrix at position (1,2):

• Top-left corner: (1,2)
• Elements: [1, 4, 6, 2]
• After sorting: [1, 2, 4, 6]
• Consecutive pairs: (1,2), (2,4), (4,6)
• Differences: |2-1|=1, |4-2|=2, |6-4|=2
• Minimum difference: 1

Following this process for all 9 submatrices, our result array would be:

ans = [[1, 1, 1],
       [1, 2, 1],
       [2, 2, 3]]

Special case example: If we had a submatrix with all identical elements like [5, 5, 5, 5]:

• After sorting: [5, 5, 5, 5]
• Consecutive pairs: (5,5), (5,5), (5,5)
• Since we filter out pairs where a == b, no valid pairs remain
• The default=0 parameter gives us 0 as the result

This walkthrough demonstrates how the algorithm systematically processes each submatrix by extracting elements, sorting them, and efficiently finding the minimum difference between distinct values using consecutive pairs in the sorted list.

Solution Implementation

Time and Space Complexity

Time Complexity: O((m - k + 1) × (n - k + 1) × k² × log(k))

The analysis breaks down as follows:

• The outer two loops iterate through all possible k × k submatrices, giving us (m - k + 1) × (n - k + 1) iterations
• For each submatrix position, we extract k² elements using two nested loops (lines 8-10)
• Sorting these k² elements takes O(k² × log(k²)) = O(k² × log(k)) time
• The pairwise comparison to find minimum absolute difference iterates through O(k²) consecutive pairs in the sorted array
• Overall: O((m - k + 1) × (n - k + 1) × (k² + k² × log(k) + k²)) = O((m - k + 1) × (n - k + 1) × k² × log(k))

Space Complexity: O(k²)

The space analysis:

• The ans matrix uses O((m - k + 1) × (n - k + 1)) space, but this is the output and typically not counted in auxiliary space
• The nums list stores k² elements for each submatrix
• The sorting operation may use additional O(k²) space depending on the implementation
• The dominant auxiliary space is O(k²) for storing the submatrix elements

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

Common Pitfalls

Pitfall 1: Misunderstanding the "Distinct Values" Requirement

The Problem: A common mistake is calculating the minimum difference between ALL pairs of elements in the submatrix, including pairs with identical values. This would always return 0 when duplicate values exist in the submatrix.

Incorrect Implementation:

# WRONG: This includes differences between identical values
min_difference = min(abs(a - b) for a, b in pairwise(subgrid_values))
# This would always be 0 if any consecutive elements after sorting are the same

Correct Implementation:

# CORRECT: Only consider pairs with distinct values
min_difference = min(
    (abs(a - b) for a, b in pairwise(subgrid_values) if a != b), 
    default=0
)

Pitfall 2: Inefficient Duplicate Pair Checking

The Problem: Some might try to check all possible pairs (not just consecutive ones after sorting) or use nested loops to find distinct pairs, leading to unnecessary complexity.

Inefficient Approach:

# INEFFICIENT: Checking all O(k^4) pairs
min_diff = float('inf')
for i in range(len(subgrid_values)):
    for j in range(i + 1, len(subgrid_values)):
        if subgrid_values[i] != subgrid_values[j]:
            min_diff = min(min_diff, abs(subgrid_values[i] - subgrid_values[j]))

Optimal Solution: After sorting, the minimum difference between distinct values must occur between consecutive elements in the sorted array. This reduces the problem from O(k^4) comparisons to O(k^2) comparisons.

Pitfall 3: Incorrect Handling of All-Identical Submatrices

The Problem: When all elements in a submatrix are identical, there are no distinct pairs. The code might crash or return an incorrect value if this edge case isn't handled properly.

Problematic Code:

# WRONG: This would raise ValueError if no distinct pairs exist
min_difference = min(abs(a - b) for a, b in pairwise(subgrid_values) if a != b)

Correct Handling:

# CORRECT: Use default=0 to handle the case when all elements are identical
min_difference = min(
    (abs(a - b) for a, b in pairwise(subgrid_values) if a != b), 
    default=0
)

Pitfall 4: Off-by-One Errors in Submatrix Boundaries

The Problem: Incorrectly calculating the range of valid starting positions or the boundaries of each submatrix.

Common Mistakes:

# WRONG: This would go out of bounds
for i in range(m - k + 2):  # Should be m - k + 1
    for j in range(n - k + 2):  # Should be n - k + 1

# WRONG: Incorrect submatrix extraction
for x in range(i, i + k - 1):  # Should be i + k
    for y in range(j, j + k - 1):  # Should be j + k

Correct Implementation:

# Valid starting positions
for i in range(m - k + 1):
    for j in range(n - k + 1):
        # Extract k×k submatrix
        for x in range(i, i + k):
            for y in range(j, j + k):
                subgrid_values.append(grid[x][y])