Minimum Moves to Equal Array Elements II

Minimum Moves to Equal Array Elements II

1. What is this problem about?

Unlike the previous version, "Minimum Moves to Equal Array Elements II" allows you to either increment or decrement any single element by 1 in one move. You want to make all elements in the array equal using the minimum total moves.

This is a classic statistical problem: "To what value should you bring all numbers to minimize the sum of absolute differences?"

2. Why is this asked in interviews?

This question tests a candidate's knowledge of Median vs. Mean. Interviewer's look for:

• Mathematical Insight: Does the candidate know that the median minimizes the sum of absolute deviations?
• Sorting: Using sorting to find the median efficiently.
• Quickselect: Knowing that you can find the median in $O(n)$ time on average without fully sorting the array.

3. Algorithmic pattern used

The pattern is Median Optimization.

1. Sort the array.
2. Find the median element (the one at index n/2).
3. The total moves required is the sum of abs(element - median) for every element in the array.

Why the median and not the mean? The mean minimizes the sum of squared differences (useful for least squares regression), but the median minimizes the sum of absolute differences.

4. Example explanation

Array: [1, 2, 10]

• Mean: $(1+2+10)/3 = 4.33$.
  • Moves to 4: $|1-4| + |2-4| + |10-4| = 3 + 2 + 6 = 11$.
• Median: 2.
  • Moves to 2: $|1-2| + |2-2| + |10-2| = 1 + 0 + 8 = 9$. The median is clearly better.

5. Common mistakes candidates make

• Using the Mean: This is the most common mistake. Candidates assume that "averaging" the numbers is the way to go.
• Sorting unnecessarily: While sorting is fine $O(n \log n)$, an expert candidate might mention Quickselect for an $O(n)$ solution.
• Integer Overflow: If the array is very large and the numbers are large, the sum of differences can exceed a 32-bit integer.

6. Interview preparation tip

Remember this rule:

• Minimize $\sum (x_i - k)^2 \rightarrow k = \text{Mean}$
• Minimize $\sum |x_i - k| \rightarrow k = \text{Median}$ This simple fact from statistics solves many "equalization" and "meeting point" problems in coding interviews.