Maximum Sum of Three Non-Overlapping Subarrays II - Java Solution and Time Complexity Analysis

Input: [2, 3, -8, 7, -2, 9, -9, 7, -2, 4]
Output: 28
Explanation: Subarrays [2, 3], [7, -2, 9] and [7, -2, 4]
             have the maximum sum of 28

Understanding the Problem

The core challenge of this problem is to find three non-overlapping subarrays that yield the maximum possible sum. This problem is significant in scenarios where we need to maximize profit or benefits from non-overlapping intervals, such as in financial analysis or resource allocation.

Potential pitfalls include misunderstanding the requirement for non-overlapping subarrays and not considering all possible combinations of subarrays.

Approach

To solve this problem, we can break it down into manageable steps:

1. Calculate the sum of all possible subarrays of a given length.
2. Use dynamic programming to keep track of the maximum sums of subarrays ending at or before each index.
3. Combine these results to find the maximum sum of three non-overlapping subarrays.

Let's start with a naive solution and then optimize it.

Naive Solution

A naive solution would involve iterating through all possible combinations of three subarrays and calculating their sums. This approach is not optimal due to its high time complexity of O(n^3).

Optimized Solution

We can optimize the solution using dynamic programming. The idea is to precompute the sums of subarrays and use auxiliary arrays to store the maximum sums of subarrays up to each index.

Algorithm

Here is a step-by-step breakdown of the optimized algorithm:

1. Compute the sum of all subarrays of length k.
2. Use two auxiliary arrays to store the maximum sum of subarrays up to each index from the left and from the right.
3. Iterate through the array to find the maximum sum of three non-overlapping subarrays using the precomputed sums.

Code Implementation

public class MaxSumOfThreeSubarrays {
    public int maxSumOfThreeSubarrays(int[] nums, int k) {
        int n = nums.length;
        int[] sum = new int[n + 1];
        for (int i = 0; i < n; i++) {
            sum[i + 1] = sum[i] + nums[i];
        }

        int[] left = new int[n];
        int[] right = new int[n];
        int maxSum = 0;

        // Calculate max sum of subarray from the left
        for (int i = k - 1; i < n; i++) {
            int currentSum = sum[i + 1] - sum[i + 1 - k];
            if (currentSum > maxSum) {
                maxSum = currentSum;
                left[i] = i + 1 - k;
            } else {
                left[i] = left[i - 1];
            }
        }

        // Reset maxSum for right calculation
        maxSum = 0;

        // Calculate max sum of subarray from the right
        for (int i = n - k; i >= 0; i--) {
            int currentSum = sum[i + k] - sum[i];
            if (currentSum >= maxSum) {
                maxSum = currentSum;
                right[i] = i;
            } else {
                right[i] = right[i + 1];
            }
        }

        // Find the maximum sum by combining left, middle, and right subarrays
        int result = 0;
        for (int i = k; i <= n - 2 * k; i++) {
            int l = left[i - 1];
            int r = right[i + k];
            int total = (sum[l + k] - sum[l]) + (sum[i + k] - sum[i]) + (sum[r + k] - sum[r]);
            result = Math.max(result, total);
        }

        return result;
    }
}

Complexity Analysis

The time complexity of the optimized solution is O(n), where n is the length of the input array. This is because we make a constant number of passes through the array. The space complexity is also O(n) due to the auxiliary arrays used for storing sums and indices.

Edge Cases

Potential edge cases include:

• Arrays with all positive or all negative numbers.
• Arrays with the minimum length of three.

Each algorithm handles these cases by ensuring that subarrays are non-empty and by correctly computing sums and indices.

Testing

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

• Simple cases with small arrays.
• Cases with mixed positive and negative numbers.
• Edge cases with the minimum number of elements.

Use a testing framework like JUnit to automate and validate these test cases.

Thinking and Problem-Solving Tips

When approaching such problems:

• Break down the problem into smaller, manageable parts.
• Consider both naive and optimized solutions.
• Use dynamic programming to store intermediate results and avoid redundant calculations.

Practice by solving similar problems and studying different algorithms to improve problem-solving skills.

Conclusion

Understanding and solving problems like the maximum sum of three non-overlapping subarrays is crucial for developing strong algorithmic thinking. Practice and exploration of different approaches are key to mastering such problems.

Additional Resources

For further reading and practice, consider the following resources:

• LeetCode Problem
• GeeksforGeeks Dynamic Programming
• Coursera Algorithms Course