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]]
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 consider only submatrices of fixed sizes or to overlook negative values that might affect the sum.
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:
A naive solution would involve checking all possible submatrices and calculating their sums. This approach is not optimal due to its high time complexity of O(m2 * n2 * m * n), which is impractical for large grids.
We can optimize the solution by using prefix sums to reduce the time complexity. The idea is to fix the top and bottom rows and then use a 1D array to store the sum of elements between these rows for each column. This reduces the problem to finding the maximum sum subarray, which can be solved using Kadane's algorithm.
Here is a step-by-step breakdown of the optimized algorithm:
// Function to find the maximum sum submatrix
function maxSumSubmatrix(matrix) {
const rows = matrix.length;
const cols = matrix[0].length;
let maxSum = -Infinity;
// Iterate over all pairs of top and bottom rows
for (let top = 0; top < rows; top++) {
const temp = new Array(cols).fill(0);
for (let bottom = top; bottom < rows; bottom++) {
// Calculate the sum of elements between the top and bottom rows for each column
for (let col = 0; col < cols; col++) {
temp[col] += matrix[bottom][col];
}
// Find the maximum sum subarray in the temp array using Kadane's algorithm
let currentMax = temp[0];
let currentSum = temp[0];
for (let i = 1; i < temp.length; i++) {
currentSum = Math.max(temp[i], currentSum + temp[i]);
currentMax = Math.max(currentMax, currentSum);
}
// Update the maximum sum found
maxSum = Math.max(maxSum, currentMax);
}
}
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
The time complexity of the optimized solution is O(m2 * n), where m is the number of rows and n is the number of columns. This is a significant improvement over the naive approach. The space complexity is O(n) due to the temporary array used to store column sums.
Potential edge cases include:
To test these edge cases, you can use the following examples:
// Edge case: All negative elements
const matrix1 = [
[-1, -2, -3],
[-4, -5, -6],
[-7, -8, -9]
];
console.log(maxSumSubmatrix(matrix1)); // Output: -1
// Edge case: Single row
const matrix2 = [
[1, 2, 3, 4]
];
console.log(maxSumSubmatrix(matrix2)); // Output: 10
// Edge case: Single column
const matrix3 = [
[1],
[2],
[3],
[4]
];
console.log(maxSumSubmatrix(matrix3)); // Output: 10
To test the solution comprehensively, include a variety of test cases, from simple to complex. You can use testing frameworks like Jest or Mocha for automated testing.
When approaching such problems, it's essential to break down the problem into smaller parts and think about how to optimize each part. Practice solving similar problems and study different algorithms to improve your problem-solving skills.
In this blog post, we discussed how to find the maximum sum submatrix in a given grid. We explored a naive solution and an optimized solution using prefix sums and Kadane's algorithm. Understanding and solving such problems is crucial for improving algorithmic thinking and problem-solving skills.
For further reading and practice, consider the following resources: