Maximum Sum of Three Non-Overlapping Subarrays III in Java (Time Complexity: O(n))

Given an array nums of integers, find three non-overlapping subarrays with maximum sum.

Return the total sum of the three subarrays

Example:

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

Note:

Subarrays must be non-empty
nums contains at least three numbers

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:

Calculate the sum of all possible subarrays of a given length.
Use dynamic programming to keep track of the maximum sums of subarrays ending at or starting from each index.
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:

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

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 ending at each index
        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];
            }
        }

        maxSum = 0;
        // Calculate max sum of subarray starting at each index
        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];
            }
        }

        int result = 0;
        // Find 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[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 additional arrays used for dynamic programming.

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 edge cases by ensuring that subarrays are non-empty and by correctly calculating sums using prefix 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.

Use a testing framework like JUnit to automate the testing process.

Thinking and Problem-Solving Tips

When approaching such problems, consider breaking down the problem into smaller subproblems and using dynamic programming to solve them efficiently. Practice similar problems to develop a deeper understanding of the techniques involved.

Conclusion

In this blog post, we discussed how to solve the problem of finding the maximum sum of three non-overlapping subarrays using dynamic programming. We covered the problem definition, approach, algorithm, code implementation, complexity analysis, edge cases, and testing. Understanding and solving such problems is crucial for developing strong problem-solving skills in programming.

Additional Resources

For further reading and practice, consider the following resources: