Maximum Sum of Three Non-Overlapping Subarrays II in C++ (Time Complexity: O(n))

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 together have the maximum possible sum. This problem is significant in scenarios where we need to maximize the sum of multiple segments of data, such as in financial analysis or signal processing.

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

Approach

To solve this problem, we need to consider the following 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 starting from 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

The naive solution involves 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 maximum sums of subarrays ending at or starting from each index and then combine these results efficiently.

Algorithm

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

1. Compute the prefix sums of the array to quickly calculate the sum of any subarray.
2. Use dynamic programming to find the maximum sum of subarrays ending at each index.
3. Use dynamic programming to find the maximum sum of subarrays starting from each index.
4. Combine the results to find the maximum sum of three non-overlapping subarrays.

Code Implementation

#include <vector>
#include <algorithm>
#include <iostream>

using namespace std;

// Function to find the maximum sum of three non-overlapping subarrays
int maxSumOfThreeSubarrays(vector<int>& nums, int k) {
    int n = nums.size();
    vector<int> sum(n + 1, 0);
    for (int i = 0; i < n; ++i) {
        sum[i + 1] = sum[i] + nums[i];
    }

    vector<int> left(n, 0), right(n, 0);
    int maxSum = sum[k] - sum[0];
    for (int i = k; i < n; ++i) {
        if (sum[i + 1] - sum[i + 1 - k] > maxSum) {
            maxSum = sum[i + 1] - sum[i + 1 - k];
            left[i] = i + 1 - k;
        } else {
            left[i] = left[i - 1];
        }
    }

    maxSum = sum[n] - sum[n - k];
    right[n - k] = n - k;
    for (int i = n - k - 1; i >= 0; --i) {
        if (sum[i + k] - sum[i] >= maxSum) {
            maxSum = sum[i + k] - sum[i];
            right[i] = i;
        } else {
            right[i] = right[i + 1];
        }
    }

    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[i + k] - sum[i]) + (sum[l + k] - sum[l]) + (sum[r + k] - sum[r]);
        result = max(result, total);
    }

    return result;
}

int main() {
    vector<int> nums = {2, 3, -8, 7, -2, 9, -9, 7, -2, 4};
    int k = 2; // Length of each subarray
    cout << "Maximum sum of three non-overlapping subarrays: " << maxSumOfThreeSubarrays(nums, k) << endl;
    return 0;
}

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 additional arrays used for storing prefix sums and dynamic programming results.

Edge Cases

Potential edge cases include:

• Arrays with all negative numbers.
• Arrays with all positive numbers.
• Arrays with a mix of positive and negative numbers.

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 arrays.
• Cases with large arrays to test performance.
• Edge cases as mentioned above.

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

Thinking and Problem-Solving Tips

When approaching such problems, consider breaking them down into smaller subproblems and using dynamic programming to solve them efficiently. Practice solving similar problems and studying different algorithms to improve your problem-solving skills.

Conclusion

In this blog post, we discussed how to solve the problem of finding the maximum sum of three non-overlapping subarrays. We explored a naive solution and then optimized it using dynamic programming. Understanding and solving such problems is crucial for developing strong algorithmic skills.

Additional Resources

For further reading and practice, consider the following resources:

• LeetCode Problem
• GeeksforGeeks Dynamic Programming
• Coursera Algorithms Specialization