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Maximum Sum Subsquare in JavaScript (Time Complexity: O(n^3 * m^3)) /homework


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

Example:

Input:
[
  [ 1,  -9, -10,  1],
  [-1,  10,  10,  1],
  [ 0,   9,   9, -9],
  [-1,   3,  -1, -1]
]
Output: 38
Explanation: 2x2 Submatrix [[10, 10], [9, 9]]

Note: A subsquare is a submatrix having the same number of columns and rows


Understanding the Problem

The core challenge of this problem is to identify the subsquare (a submatrix with equal number of rows and columns) within a given n x m grid that has the maximum sum of its elements. This problem is significant in various applications such as image processing, data analysis, and financial modeling where identifying regions of interest with maximum values is crucial.

Potential pitfalls include misunderstanding the definition of a subsquare and not considering all possible subsquares within the grid.

Approach

To solve this problem, we need to consider all possible subsquares within the grid and calculate their sums. A naive approach would involve iterating through all possible subsquares and calculating their sums, but this can be highly inefficient for larger grids.

We can optimize this by using a prefix sum array to quickly calculate the sum of any subsquare. The prefix sum array allows us to compute the sum of any submatrix in constant time after an initial preprocessing step.

Naive Solution

The naive solution involves iterating through all possible subsquares and calculating their sums directly. This approach has a time complexity of O(n^3 * m^3) and is not feasible for large grids.

Optimized Solution

The optimized solution involves using a prefix sum array to calculate the sum of any subsquare in constant time. This reduces the time complexity to O(n^2 * m^2) for preprocessing and O(n^2 * m^2) for querying, resulting in an overall time complexity of O(n^2 * m^2).

Algorithm

1. Create a prefix sum array where each element at (i, j) contains the sum of all elements in the submatrix from (0, 0) to (i, j).

2. Iterate through all possible subsquares and use the prefix sum array to calculate their sums efficiently.

3. Keep track of the maximum sum encountered and the corresponding subsquare.

Code Implementation

// Function to calculate the prefix sum array
function calculatePrefixSum(grid) {
    const n = grid.length;
    const m = grid[0].length;
    const prefixSum = Array.from({ length: n }, () => Array(m).fill(0));

    for (let i = 0; i < n; i++) {
        for (let j = 0; j < m; j++) {
            prefixSum[i][j] = grid[i][j];
            if (i > 0) prefixSum[i][j] += prefixSum[i - 1][j];
            if (j > 0) prefixSum[i][j] += prefixSum[i][j - 1];
            if (i > 0 && j > 0) prefixSum[i][j] -= prefixSum[i - 1][j - 1];
        }
    }
    return prefixSum;
}

// Function to find the maximum sum subsquare
function maxSumSubsquare(grid) {
    const n = grid.length;
    const m = grid[0].length;
    const prefixSum = calculatePrefixSum(grid);
    let maxSum = -Infinity;

    for (let size = 1; size <= Math.min(n, m); size++) {
        for (let i = size - 1; i < n; i++) {
            for (let j = size - 1; j < m; j++) {
                let total = prefixSum[i][j];
                if (i >= size) total -= prefixSum[i - size][j];
                if (j >= size) total -= prefixSum[i][j - size];
                if (i >= size && j >= size) total += prefixSum[i - size][j - size];
                maxSum = Math.max(maxSum, total);
            }
        }
    }
    return maxSum;
}

// Example usage
const grid = [
    [1, -9, -10, 1],
    [-1, 10, 10, 1],
    [0, 9, 9, -9],
    [-1, 3, -1, -1]
];

console.log(maxSumSubsquare(grid)); // Output: 38

Complexity Analysis

The time complexity of the optimized solution is O(n^2 * m^2) due to the preprocessing step and the querying step. The space complexity is O(n * m) for storing the prefix sum array.

Edge Cases

Potential edge cases include:

Each algorithm handles these edge cases by ensuring that the prefix sum array is correctly calculated and that all possible subsquares are considered.

Testing

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

Use testing frameworks like Jest or Mocha for automated testing.

Thinking and Problem-Solving Tips

When approaching such problems, consider breaking down the problem into smaller parts and using auxiliary data structures like prefix sums to optimize calculations. Practice similar problems to develop a deeper understanding of dynamic programming and matrix manipulation.

Conclusion

Understanding and solving the maximum sum subsquare problem involves recognizing the need for efficient calculations and leveraging data structures like prefix sums. Practice and exploration of similar problems can enhance problem-solving skills and algorithmic thinking.

Additional Resources