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Maximum Score Of Spliced Array

Number: 1348

Difficulty: Hard

Paid? No

Companies: Microsoft

Problem Description

Given two arrays of equal length, you are allowed to swap one continuous subarray between nums1 and nums2 at most once. The score is defined as the maximum of the sum of nums1 and the sum of nums2 after the operation. The goal is to maximize this score.

Key Insights

  • The total sums of the arrays can be boosted by swapping a segment that provides a positive net gain.
  • Instead of trying every possible subarray swap directly, compute the gain that would be achieved by swapping a segment.
  • Define two difference arrays:
    • diff1[i] = nums2[i] - nums1[i], which represents the gain in nums1 if the element is swapped.
    • diff2[i] = nums1[i] - nums2[i], which represents the gain in nums2 if the element is swapped.
  • Use Kadane's algorithm on these difference arrays to find the maximum gain achievable by swapping a contiguous segment.
  • If the maximum gain is negative or zero, no swap would increase the score, so simply take the maximum of the original sums.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1) (ignoring input storage; only a few extra variables are used)

Solution

The solution calculates the initial sums of both arrays. It then constructs two difference arrays representing the gain from swapping each element into the other array. Using Kadane’s algorithm on these arrays, we determine the best possible contiguous subarray that maximizes the net gain for each array. Finally, we add this gain to the corresponding original sum and take the maximum between the two cases.

This approach leverages the idea of transforming the problem into a maximum subarray sum problem (Kadane's algorithm) instead of checking all subarray swaps explicitly.

Code Solutions

# Python solution using Kadane's Algorithm

def maximumsSplicedArray(nums1, nums2):
    # Calculate initial sums of both arrays
    total1 = sum(nums1)
    total2 = sum(nums2)
    
    # Helper function: Kadane's algorithm to find maximum subarray sum
    def max_subarray(diff):
        max_gain = curr_gain = 0
        for value in diff:
            curr_gain = max(value, curr_gain + value)
            max_gain = max(max_gain, curr_gain)
        return max_gain
    
    # Create difference arrays for the gain in sum1 and sum2 respectively
    diff1 = [b - a for a, b in zip(nums1, nums2)]
    diff2 = [a - b for a, b in zip(nums1, nums2)]
    
    # Max gain by swapping a subarray to benefit nums1 and nums2
    gain1 = max_subarray(diff1)
    gain2 = max_subarray(diff2)
    
    # The maximum possible score after at most one swap
    return max(total1 + gain1, total2 + gain2)

# Example usage:
# print(maximumsSplicedArray([60,60,60], [10,90,10]))
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