Minimum Number of Operations to Make Arrays Similar

Minimum Number of Operations to Make Arrays Similar

1. What is this problem about?

The Minimum Number of Operations to Make Arrays Similar interview question is a "Hard" difficulty sorting challenge. You are given two arrays, nums and target. In one operation, you can choose two indices i and j in nums, increment nums[i] by 2, and decrement nums[j] by 2. You want to find the minimum number of such operations to make nums a permutation of target.

2. Why is this asked in interviews?

This problem is asked by Amazon and Walmart Labs to evaluate a candidate's ability to handle parity constraints. Because you can only add or subtract 2, an even number will always stay even, and an odd number will always stay odd. This means the even numbers in nums must eventually match the even numbers in target, and similarly for odd numbers.

3. Algorithmic pattern used

This utilizes the Array, Sorting, Greedy interview pattern.

1. Split both nums and target into two groups: Even and Odd.
2. Sort all four resulting sub-arrays.
3. Because we want to minimize operations, we pair the smallest even number in nums with the smallest even in target, and so on.
4. Calculate the total positive difference: sum(abs(nums_even[i] - target_even[i])) / 2 + sum(abs(nums_odd[i] - target_odd[i])) / 2.
5. Since each operation changes two numbers (one up, one down), the final answer is half of the total absolute differences divided by 2.

4. Example explanation

Nums: [8, 12, 5, 9], Target: [4, 16, 7, 7]

1. Nums: Even=[8, 12], Odd=[5, 9]
2. Target: Even=[4, 16], Odd=[7, 7]
3. Sorted: Even: [8, 12] vs [4, 16]. Diff = |8-4| + |12-16| = 4 + 4 = 8. Odd: [5, 9] vs [7, 7]. Diff = |5-7| + |9-7| = 2 + 2 = 4.
4. Total diff = 8 + 4 = 12.
5. Total "steps of 2" = 12 / 2 = 6.
6. Since each op covers 2 steps (one +2, one -2), result is 6 / 2 = 3.

5. Common mistakes candidates make

Failing to separate numbers by parity is the most common mistake. Without this, you might try to match an even number with an odd target, which is impossible using +/- 2. Another mistake is forgetting that each operation accounts for two changes, so you must divide the final sum of differences by 4 (2 for the step size, and 2 for the two indices involved).

6. Interview preparation tip

When you see operations involving +k and -k, always think about the remainder when dividing by k (parity). Elements can only be transformed into others that share the same remainder. This is a powerful way to partition a complex problem into independent sub-problems.