Maximum Sum Submatrix III in JavaScript (Time Complexity: O(m^2 * n^2))

Given a m x n grid filled with integers, find the submatrix with maximum sum among all submatrices.

Example:

Input:
[
  [ 1,  -9, -10,  1],
  [-1,  10,  10,  1],
  [ 0,   9,   9, -9],
  [-1,  -1,  -1, -1]
]
Output: 38
Explanation: Submatrix [[10, 10], [9, 9]]

Understanding the Problem

The core challenge of this problem is to find the submatrix within a given m x n grid that has the maximum sum. This problem is significant in various applications such as image processing, financial analysis, and more. A common pitfall is to assume that the submatrix must be of a fixed size, but it can be of any size within the given matrix.

Approach

To solve this problem, we can use a dynamic programming approach combined with the concept of prefix sums. Here's a step-by-step breakdown:

Naive Solution

The naive solution involves checking all possible submatrices and calculating their sums. This approach is not optimal due to its high time complexity of O(m^3 * n^3).

Optimized Solution

We can optimize the solution by using prefix sums to reduce the time complexity to O(m^2 * n^2). The idea is to precompute the sum of elements in submatrices starting from the top-left corner to any point (i, j) in the matrix. This allows us to quickly calculate the sum of any submatrix using these precomputed sums.

Algorithm

1. Compute the prefix sum matrix.

2. Iterate over all possible pairs of rows.

3. For each pair of rows, use a sliding window technique to find the maximum sum subarray for the columns between these rows.

Code Implementation

// Function to find the maximum sum submatrix
function maxSumSubmatrix(matrix) {
    const m = matrix.length;
    const n = matrix[0].length;
    
    // Step 1: Compute the prefix sum matrix
    const prefixSum = Array.from({ length: m + 1 }, () => Array(n + 1).fill(0));
    for (let i = 1; i <= m; i++) {
        for (let j = 1; j <= n; j++) {
            prefixSum[i][j] = matrix[i - 1][j - 1] + prefixSum[i - 1][j] + prefixSum[i][j - 1] - prefixSum[i - 1][j - 1];
        }
    }
    
    let maxSum = -Infinity;
    
    // Step 2: Iterate over all pairs of rows
    for (let row1 = 1; row1 <= m; row2++) {
        for (let row2 = row1; row2 <= m; row2++) {
            let currentSum = 0;
            let maxSubarraySum = -Infinity;
            
            // Step 3: Use sliding window to find the maximum sum subarray for columns between row1 and row2
            for (let col = 1; col <= n; col++) {
                const submatrixSum = prefixSum[row2][col] - prefixSum[row1 - 1][col];
                currentSum = Math.max(submatrixSum, currentSum + submatrixSum);
                maxSubarraySum = Math.max(maxSubarraySum, currentSum);
            }
            
            maxSum = Math.max(maxSum, maxSubarraySum);
        }
    }
    
    return maxSum;
}

// Example usage
const matrix = [
    [1, -9, -10, 1],
    [-1, 10, 10, 1],
    [0, 9, 9, -9],
    [-1, -1, -1, -1]
];

console.log(maxSumSubmatrix(matrix)); // Output: 38

Complexity Analysis

The time complexity of this approach is O(m^2 * n^2) due to the nested loops iterating over pairs of rows and columns. The space complexity is O(m * n) for the prefix sum matrix.

Edge Cases

Potential edge cases include:

Matrix with all negative numbers.
Matrix with a single row or column.
Matrix with all zeros.

Each of these cases should be tested to ensure the algorithm handles them correctly.

Testing

To test the solution comprehensively, consider the following test cases:

Simple cases with small matrices.
Large matrices with mixed positive and negative numbers.
Edge cases as mentioned above.

Using a testing framework like Jest can help automate and manage these tests effectively.

Thinking and Problem-Solving Tips

When approaching such problems, it's essential to:

Break down the problem into smaller, manageable parts.
Consider different approaches and their trade-offs.
Practice similar problems to build familiarity with common patterns and techniques.

Conclusion

Finding the maximum sum submatrix is a classic problem that can be efficiently solved using dynamic programming and prefix sums. Understanding and implementing this solution helps build a strong foundation in algorithmic problem-solving.

Additional Resources

For further reading and practice, consider the following resources: