Maximum Sum of Two Non-Overlapping Subarrays

Maximum Sum of Two Non-Overlapping Subarrays

1. What is this problem about?

The Maximum Sum of Two Non-Overlapping Subarrays problem asks you to find two contiguous subarrays, one with length L and the other with length M, such that they do not overlap and the sum of their elements is maximized. The order doesn't matter: the subarray of length L can come before or after the subarray of length M. The goal is to return this maximum possible combined sum.

2. Why is this asked in interviews?

This "Maximum Sum of Two Non-Overlapping Subarrays interview question" is a frequent Amazon interview problem. It tests a candidate's ability to use sliding windows and prefix sums effectively. It specifically evaluates if you can solve a problem in linear time by maintaining a "running maximum" of one subarray as you iterate through the possible positions of the other. It's a great test of efficiency and logical partitioning.

3. Algorithmic pattern used

The "Array, Sliding Window, Dynamic Programming interview pattern" is used to solve this in O(N). You consider two scenarios:

1. Subarray L comes before subarray M.
2. Subarray M comes before subarray L. As you iterate from left to right, you keep track of the maximum sum of a subarray of length L seen so far. For the current index, you calculate the sum of the subarray of length M ending there and add it to the best L sum found previously. You repeat this logic for the second scenario and take the overall maximum.

4. Example explanation

Array: [3, 8, 1, 3, 2, 1, 8, 9, 0], L = 2, M = 3

• Scenario 1 (L before M):
  • Best L so far: [3, 8] = 11.
  • Current M: [3, 2, 1] = 6. Combined: 11 + 6 = 17.
  • Later, current M: [8, 9, 0] = 17. Best L before it was [3, 8] = 11. Combined: 11 + 17 = 28.
• Scenario 2 (M before L):
  • Best M so far: [3, 8, 1] = 12.
  • Current L: [8, 9] = 17. Combined: 12 + 17 = 29. The maximum combined sum is 29.

5. Common mistakes candidates make

In the "Maximum Sum of Two Non-Overlapping Subarrays coding problem," the most common mistake is only considering one order (e.g., L before M) and forgetting the other. Another error is using a nested loop approach (O(N^2)) which is unnecessary. Candidates also frequently make "off-by-one" errors when calculating the sliding window sums or the boundaries for the "best so far" maximums.

6. Interview preparation tip

Practice the "two-pass" or "two-scenario" approach for non-overlapping problems. Whenever you have to find two things that don't overlap, try fixing the position of one and using a pre-computed maximum for the other. This "Max-so-far" technique is a building block for many medium and hard array problems. Also, get comfortable with calculating window sums in O(1) time using either a running sum variable or a prefix sum array.