Minimum Cost to Make Arrays Identical

Problem Description

You are given two integer arrays arr and brr of length n, and an integer k. You can perform the following operations on arr any number of times:

• Split arr into any number of contiguous subarrays and rearrange these subarrays in any order. This operation has a fixed cost of k.
• Choose any element in arr and add or subtract a positive integer x to it. The cost of this operation is x.

Return the minimum total cost to make arr equal to brr.

Key Insights

• The primary goal is to make the two arrays identical with the minimum cost.
• The cost of rearranging subarrays is fixed at k, which may be beneficial when significant rearrangements are needed.
• The cost of modifying elements directly is proportional to the absolute difference between corresponding elements in arr and brr.
• Elements need to be compared and possibly adjusted based on their differences to minimize modification costs.
• The problem can be approached by calculating the total cost of making adjustments and comparing it with the cost of rearranging subarrays.

Space and Time Complexity

Time Complexity: O(n log n) - due to sorting operations. Space Complexity: O(1) - no additional space is required beyond input arrays.

Solution

To solve the problem, we can follow these steps:

1. Calculate the absolute differences between corresponding elements in arr and brr.
2. Sum these differences to get the total modification cost.
3. Compare this total modification cost with k to determine if it's cheaper to just rearrange or to perform the modifications.
4. Return the minimum of the modification cost and the cost of rearranging plus any additional modifications needed.

We will use arrays and basic arithmetic operations to achieve this, ensuring efficiency in our approach.

Code Solutions