Maximum Score Of Spliced Array

Maximum Score Of Spliced Array

1. What is this problem about?

The Maximum Score Of Spliced Array interview question is a creative array optimization problem. You are given two arrays, nums1 and nums2. You can choose a single contiguous subarray from nums1 and swap it with the corresponding subarray in nums2 (or vice versa). Your goal is to maximize the sum of either nums1 or nums2 after exactly one such swap.

This problem can be reframed as finding the maximum possible increase in sum by swapping a subarray.

2. Why is this asked in interviews?

Microsoft asks the Maximum Score Of Spliced Array coding problem to test if a candidate can reduce a complex problem to a simpler, well-known one. The key insight is that swapping a subarray to maximize the sum of nums1 is equivalent to finding a subarray in nums2 - nums1 with the maximum sum. This is exactly what Kadane's algorithm does. It evaluates your ability to perform this mathematical reduction and apply a classic algorithm in a new context.

3. Algorithmic pattern used

The Array, Dynamic Programming interview pattern (specifically Kadane's Algorithm) is the core of the solution.

1. Calculate the initial sums: sum1 and sum2.
2. Create two "difference" arrays:
   • diff1[i] = nums2[i] - nums1[i]
   • diff2[i] = nums1[i] - nums2[i]
3. Use Kadane's algorithm to find the maximum subarray sum in diff1. Let this be max_gain1.
4. Use Kadane's algorithm to find the maximum subarray sum in diff2. Let this be max_gain2.
5. The answer is max(sum1 + max_gain1, sum2 + max_gain2).

4. Example explanation

nums1: [10, 20, 30], nums2: [40, 5, 10] sum1 = 60, sum2 = 55.

• diff1 (nums2 - nums1): [30, -15, -20]. Max subarray sum = 30.
• Gain for sum1 = 60 + 30 = 90. (By swapping [10] with [40]).
• diff2 (nums1 - nums2): [-30, 15, 20]. Max subarray sum = 15 + 20 = 35.
• Gain for sum2 = 55 + 35 = 90. (By swapping [5, 10] with [20, 30]). Max score is 90.

5. Common mistakes candidates make

In the Maximum Score Of Spliced Array coding problem, candidates often try to use complex $O(N^2)$ sliding window or DP approaches before realizing the Kadane's reduction. Another mistake is only checking for the gain in nums1 and forgetting that nums2 might yield a higher final sum. Some might also struggle with the logic of "swapping," not realizing it's mathematically equivalent to adding the difference between the elements.

6. Interview preparation tip

Whenever you see a problem involving "maximizing a sum by changing a range," calculate the "difference" the change makes and see if Kadane's algorithm (Maximum Subarray Sum) can be applied. This is a very common trick for transforming range-based optimization problems into simple linear-time tasks.