Max Sum of Two Non-Overlapping Subarrays

The “Max Sum of Two Non-Overlapping Subarrays” is a popular problem in computer science, focusing on how to find two subarrays within a given array such that they do not overlap and their combined sum is maximized. This problem has numerous practical applications, from resource allocation and scheduling to optimizing financial investments over time. In this guide, we’ll explore the concept, break down the approach, and walk through a solution to solve the problem efficiently.

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Problem Statement

Given an array of integers and two integers L and M, the task is to find two non-overlapping subarrays of lengths L and M such that their combined sum is maximized. For example:

Input:
Array = [0, 6, 5, 2, 2, 5, 1, 9, 4]
L = 2
M = 3

Output:
The maximum sum of two non-overlapping subarrays is 31.

Explanation:
The optimal subarrays are [9, 4] (sum = 13, length L=2) and [6, 5, 2] (sum = 18, length M=3).

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Key Observations

1. Non-Overlapping Subarrays: The subarrays must not overlap, meaning their indices must be disjoint. You cannot pick subarrays that share any elements.
2. Order Doesn’t Matter: The order in which you choose the subarrays of lengths L and M is interchangeable, i.e., finding a subarray of length L first and then M is equivalent to finding a subarray of length M first and then L.
3. Maximizing Combined Sum: The main objective is to maximize the sum of both subarrays combined. Hence, you’ll need to efficiently search through the array to find the best possible pairs of subarrays.

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Approach to Solve the Problem

To efficiently solve this problem, we can break down the solution into four main steps:

1. Calculate Prefix Sums

The prefix sum of an array helps in quickly calculating the sum of any subarray. The prefix sum array stores the cumulative sum of elements from the start of the array up to a given index. This way, the sum of any subarray from index i to j can be computed as:

[ \text{sum}(i, j) = \text{prefix}[j] – \text{prefix}[i-1] ]

Example:
Array: [0, 6, 5, 2, 2, 5, 1, 9, 4]
Prefix sum array: [0, 6, 11, 13, 15, 20, 21, 30, 34]

2. Calculate the Maximum Sums for Subarrays of Length L and M

We need to track the maximum sum of subarrays of lengths L and M separately for the left and right sides of the array.

• Left-to-Right Traversal:
  Calculate the maximum sum of subarrays of length L up to each index and store it in a leftMaxL array. Do the same for subarrays of length M and store it in a leftMaxM array.
• Right-to-Left Traversal:
  Calculate the maximum sum of subarrays of length L from each index to the end of the array and store it in a rightMaxL array. Do the same for subarrays of length M and store it in a rightMaxM array.

Example:
If L=2, then leftMaxL[i] stores the maximum sum of any subarray of length 2 up to index i. Similarly, rightMaxL[i] stores the maximum sum of any subarray of length 2 from index i to the end.

3. Calculate the Maximum Combined Sum of Non-Overlapping Subarrays

Now, use the information from leftMaxL, leftMaxM, rightMaxL, and rightMaxM to find the optimal combination of non-overlapping subarrays.

• For L Subarray on the Left and M Subarray on the Right:
  Iterate through each index i and compute the sum as leftMaxL[i] + rightMaxM[i+1].
• For M Subarray on the Left and L Subarray on the Right:
  Iterate through each index i and compute the sum as leftMaxM[i] + rightMaxL[i+1].

Keep track of the maximum sum encountered during both iterations.

4. Return the Maximum Combined Sum

The final result is the maximum combined sum found in the previous step.

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Time Complexity

This approach is efficient with a time complexity of O(n). Calculating prefix sums and maximum sums for subarrays requires O(n) time, and iterating through the array to find the maximum combined sum also takes O(n) time.

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Code Solution

Here’s an example of a Python implementation for the problem:

def maxSumTwoNoOverlap(A, L, M):
    # Prefix sum calculation
    prefix = [0] * (len(A) + 1)
    for i in range(1, len(A) + 1):
        prefix[i] = prefix[i - 1] + A[i - 1]

    def maxSumSubarray(K):
        max_sum = [0] * len(A)
        current_sum = 0

        # Sliding window to calculate max sum of subarray length K
        for i in range(K):
            current_sum += A[i]
        max_sum[K - 1] = current_sum

        for i in range(K, len(A)):
            current_sum += A[i] - A[i - K]
            max_sum[i] = max(max_sum[i - 1], current_sum)

        return max_sum

    # Calculate left and right max sums
    leftMaxL = maxSumSubarray(L)
    rightMaxL = maxSumSubarray(L)[::-1]

    leftMaxM = maxSumSubarray(M)
    rightMaxM = maxSumSubarray(M)[::-1]

    # Max combined sum of two non-overlapping subarrays
    max_sum = 0

    # L subarray on the left, M subarray on the right
    for i in range(L - 1, len(A) - M):
        max_sum = max(max_sum, leftMaxL[i] + rightMaxM[len(A) - (i + 1 + M)])

    # M subarray on the left, L subarray on the right
    for i in range(M - 1, len(A) - L):
        max_sum = max(max_sum, leftMaxM[i] + rightMaxL[len(A) - (i + 1 + L)])

    return max_sum

# Example Usage
A = [0, 6, 5, 2, 2, 5, 1, 9, 4]
L = 2
M = 3
print(maxSumTwoNoOverlap(A, L, M))  # Output: 31

Explanation of the Code

1. Prefix Sum Calculation: We calculate the prefix sum array to facilitate quick subarray sum calculations.
2. Sliding Window to Calculate Max Subarrays: We calculate leftMaxL, rightMaxL, leftMaxM, and rightMaxM to track the maximum sums for subarrays of lengths L and M.
3. Max Sum Calculation: We iterate over the array, finding the maximum possible sum of two non-overlapping subarrays.
4. Output the Result: The function returns the maximum combined sum.

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FAQs

1. Can L and M be the same value?
Yes, L and M can be the same, which means you’re looking for two non-overlapping subarrays of the same length that maximize the combined sum.

2. What if the array has negative numbers?
The algorithm works for any integer values in the array, including negatives. It will find the optimal combination of subarrays to maximize the sum, even if some elements are negative.

3. Is this approach optimal?
Yes, this approach has a time complexity of O(n), which is optimal for this problem as it requires a single pass through the array to calculate the prefix sums and max subarrays.

4. What if the subarrays overlap?
The problem specifically requires non-overlapping subarrays. If the subarrays overlap, this approach would not apply, and a different method would need to be used.

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By understanding the mechanics behind finding the maximum sum of two non-overlapping subarrays, you can apply this technique to solve a variety of similar optimization problems in data and resource allocation!

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