Minimum Sum of Squared Difference

Problem Description

You are given two positive 0-indexed integer arrays nums1 and nums2, both of length n. The sum of squared difference of arrays nums1 and nums2 is defined as the sum of (nums1[i] - nums2[i])² for each 0 <= i < n. You are also given two positive integers k1 and k2. You can modify any of the elements of nums1 by +1 or -1 at most k1 times. Similarly, you can modify any of the elements of nums2 by +1 or -1 at most k2 times. Return the minimum sum of squared difference after modifying array nums1 at most k1 times and modifying array nums2 at most k2 times.

Key Insights

• The objective is to minimize the sum of squared differences between two arrays after allowing a limited number of modifications.
• The modifications can be either increasing or decreasing the values in nums1 and nums2.
• The squared difference is sensitive to larger differences, so it's beneficial to equalize the values in the two arrays as much as possible.
• A greedy approach can be employed to prioritize the largest differences to reduce the overall squared difference effectively.

Space and Time Complexity

Time Complexity: O(n log n) - sorting the differences takes O(n log n) and iterating through them takes O(n). Space Complexity: O(n) - storing the differences requires additional space.

Solution

To find the minimum sum of squared differences, we can follow these steps:

1. Calculate the initial differences between corresponding elements of nums1 and nums2.
2. Store these differences in a list and sort it.
3. Use a greedy approach to reduce the largest differences first.
4. Apply the modifications allowed by k1 and k2 to minimize the squared differences.
5. After applying modifications, recalculate the sum of squared differences.

The algorithm primarily uses sorting and a greedy technique to minimize the squared differences.

Code Solutions