#3567

Medium

LeetCode Problem Workspace

Minimum Absolute Difference in Sliding Submatrix

Find the minimum absolute difference in each k x k submatrix within a given 2D grid using array and sorting techniques.

Problem Statement

You are given a 2D grid with integer values and an integer k. For every contiguous k x k submatrix of the grid, compute the minimum absolute difference between any two distinct values within that submatrix.

The result should be a 2D array, where each element in the array represents the minimum absolute difference for the corresponding k x k submatrix. The size of the output should be (m - k + 1) x (n - k + 1).

Examples

Example 1

Input: grid = [[1,8],[3,-2]], k = 2

Output: [[2]]

Example 2

Input: grid = [[3,-1]], k = 1

Output: [[0,0]]

Example 3

Input: grid = [[1,-2,3],[2,3,5]], k = 2

Output: [[1,2]]

Constraints

1 <= m == grid.length <= 30
1 <= n == grid[i].length <= 30
-105 <= grid[i][j] <= 105
1 <= k <= min(m, n)

Solution Approach

Brute Force with Sorting

For each k x k submatrix, extract all its values, sort them, and find the minimum absolute difference between adjacent values. This approach guarantees that you can calculate the differences efficiently by leveraging sorting.

Sliding Window Optimization

Use a sliding window technique to iterate over the grid, ensuring that you avoid redundant recalculations of values in overlapping submatrices. This allows for optimized performance while maintaining simplicity.

Efficient Data Structures

Use heaps or balanced trees to track the minimum absolute difference dynamically as the window slides across the grid. This reduces the need for sorting every time.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the method used. A brute force approach with sorting for each submatrix results in O(k^2 log(k)) for each submatrix, multiplied by the number of submatrices. Optimized methods using sliding windows or dynamic data structures can lower the complexity.

What Interviewers Usually Probe

Look for a deep understanding of sorting and array manipulation.
Watch for clear communication on trade-offs between brute-force and optimized solutions.
Ensure the candidate can discuss sliding window techniques and their efficiency.

Common Pitfalls or Variants

Common pitfalls

Overlooking the overlap of values in sliding windows and recalculating from scratch.
Not properly handling edge cases such as submatrices that span the entire grid.
Misunderstanding the relationship between grid size and the required output size.

Follow-up variants

Consider variations with larger k values or non-square matrices.
What if the grid contains duplicate values? How does that affect the calculation?
Explore different data structures for optimizing the search for minimum differences.

FAQ

What is the minimum absolute difference in sliding submatrices?

It is the smallest absolute difference between any two distinct values in each k x k submatrix of a given grid.

How does the sliding window technique optimize the problem?

By reusing values from previously calculated submatrices, the sliding window technique minimizes redundant calculations, improving efficiency.

What is the primary pattern for solving the Minimum Absolute Difference in Sliding Submatrix problem?

The primary pattern involves using sorting and array manipulation to calculate the minimum absolute difference in submatrices.

How do I handle overlapping submatrices in this problem?

Use a sliding window approach to handle overlaps efficiently without recalculating values from scratch in overlapping submatrices.

What are common pitfalls to avoid when solving this problem?

Be careful with inefficient recalculation for overlapping submatrices, and ensure correct handling of grid boundaries and edge cases.

terminal

Solution

Solution 1: Enumeration

We can enumerate all possible $k \times k$ submatrices by their top-left coordinates $(i, j)$. For each submatrix, we extract all its elements into a list $\textit{nums}$. Then, we sort $\textit{nums}$ and compute the absolute differences between adjacent distinct elements to find the minimum absolute difference. Finally, we store the result in a 2D array.

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class Solution:
    def minAbsDiff(self, grid: List[List[int]], k: int) -> List[List[int]]:
        m, n = len(grid), len(grid[0])
        ans = [[0] * (n - k + 1) for _ in range(m - k + 1)]
        for i in range(m - k + 1):
            for j in range(n - k + 1):
                nums = []
                for x in range(i, i + k):
                    for y in range(j, j + k):
                        nums.append(grid[x][y])
                nums.sort()
                d = min((abs(a - b) for a, b in pairwise(nums) if a != b), default=0)
                ans[i][j] = d
        return ans

Continue Practicing

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