Minimum Cost to Make Array Equal

Problem Description

You are given two 0-indexed arrays nums and cost consisting each of n positive integers. You can increase or decrease any element of the array nums by 1 any number of times. The cost of doing one operation on the i-th element is cost[i]. Return the minimum total cost such that all the elements of the array nums become equal.

Key Insights

• The problem can be interpreted as finding a target value such that the total cost to make all elements equal to this target is minimized.
• The cost associated with each operation is different, which means some elements have a higher impact on the total cost depending on their values and the corresponding costs.
• A binary search can be employed to efficiently find the optimal target value for which the total cost is minimized.

Space and Time Complexity

Time Complexity: O(n log(max(nums)))
Space Complexity: O(1)

Solution

The problem can be solved using a binary search algorithm on the potential target values that the elements of nums can take. The algorithm works as follows:

1. Determine the range of possible target values, which is from the minimum to the maximum value in nums.
2. For each mid value in this range, calculate the total cost needed to convert all elements of nums to this mid value using the costs provided in the cost array.
3. Adjust the search range based on whether the total cost at the mid value is less than or greater than the cost at the previous mid value.
4. Continue this process until the optimal target value is found.

The key data structures used in this solution are arrays to store the nums and cost, and a simple loop for cost calculation.

Code Solutions