Maximum Sum of Three Non-Overlapping Subarrays 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 overlapping subarrays and not considering all possible subarray combinations.

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 best subarray sums up to each point in the array.
3. Combine these results to find the maximum sum of three non-overlapping subarrays.

Naive Solution

A naive solution would involve checking all possible combinations of three subarrays, which would be computationally expensive and inefficient (O(n^3) time complexity). This is not optimal for large arrays.

Optimized Solution

We can optimize the solution using dynamic programming and prefix sums. The idea is to precompute the sums of all subarrays of a given length and then use these precomputed sums to find the maximum sum of three non-overlapping subarrays efficiently.

Algorithm

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

1. Compute the prefix sums of the array to quickly calculate subarray sums.
2. Use three arrays to store the maximum subarray sums up to each point in the array for the first, second, and third subarrays.
3. Iterate through the array to update these arrays with the maximum sums found so far.
4. Combine the results to find the maximum sum of three non-overlapping subarrays.

Code Implementation

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

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); // Prefix sums
    for (int i = 0; i < n; ++i) {
        sum[i + 1] = sum[i] + nums[i];
    }

    vector<int> left(n, 0), right(n, 0);
    int maxSum = 0;

    // Calculate the best subarray sum for the left part
    for (int i = k, total = sum[k] - sum[0]; i < n; ++i) {
        if (sum[i + 1] - sum[i + 1 - k] > total) {
            total = sum[i + 1] - sum[i + 1 - k];
            left[i] = i + 1 - k;
        } else {
            left[i] = left[i - 1];
        }
    }

    // Calculate the best subarray sum for the right part
    right[n - k] = n - k;
    for (int i = n - k - 1, total = sum[n] - sum[n - k]; i >= 0; --i) {
        if (sum[i + k] - sum[i] >= total) {
            total = sum[i + k] - sum[i];
            right[i] = i;
        } else {
            right[i] = right[i + 1];
        }
    }

    // Calculate the maximum sum by combining the results
    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]);
        maxSum = max(maxSum, total);
    }

    return maxSum;
}

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 prefix sums and tracking the best subarray sums.

Edge Cases

Potential edge cases include:

• Arrays with negative numbers.
• Arrays where the best subarrays are at the beginning or end of the array.
• Arrays with all positive or all negative numbers.

Each of these cases is handled by the algorithm as it considers all possible subarrays and uses prefix sums to efficiently calculate their sums.

Testing

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

• Simple cases with small arrays.
• Cases with mixed positive and negative numbers.
• Cases with all positive or all negative numbers.
• Edge cases with the smallest possible arrays (length 3).

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 store intermediate results. Practice solving similar problems and study common algorithms to improve 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 using an optimized approach with dynamic programming and prefix sums. Understanding and solving such problems is crucial for developing strong algorithmic thinking and problem-solving skills.

Additional Resources

For further reading and practice, consider the following resources:

• LeetCode Problem
• GeeksforGeeks Dynamic Programming
• Coursera Algorithms Course