Minimum Operations to Make Array Equal

To solve this problem, quickly identify the target value as the average of all elements. Each operation moves 1 from a larger number to a smaller number, and symmetry in the odd-number sequence allows a direct formula. By summing the differences from the target for half of the array, you can determine the minimum number of operations efficiently.

Problem Statement

You are given an integer n representing the length of an array arr where arr[i] = (2 * i) + 1 for 0 <= i < n. In one operation, select indices x and y, subtract 1 from arr[x], and add 1 to arr[y]. Your goal is to make all elements equal with the fewest operations.

Return the minimum number of operations required to equalize all elements. The array always allows a solution, and n satisfies 1 <= n <= 104. Each operation adjusts elements symmetrically, and recognizing this pattern is key for an efficient solution.

Examples

Example 1

Input: n = 3

Output: 2

arr = [1, 3, 5] First operation choose x = 2 and y = 0, this leads arr to be [2, 3, 4] In the second operation choose x = 2 and y = 0 again, thus arr = [3, 3, 3].

Example 2

Input: n = 6

Output: 9

Example details omitted.

Constraints

• 1 <= n <= 104

Solution Approach

Build the Array and Define Target

Construct arr with arr[i] = (2 * i) + 1. Compute the target value as sum(arr)/n. This step leverages the math-driven pattern of the sequence to simplify the problem.

Calculate Operations Using Symmetry

Observe that the array is symmetric around the target. Only consider the first half of the array. Sum the differences between the target and each element to find the exact number of operations needed.

Return Computed Minimum Operations

Use the sum of differences divided appropriately by the unit adjustment per operation. This guarantees the minimum moves and avoids redundant operations by exploiting the sequential odd-number pattern.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity is O(1) because a direct formula computes the sum of differences for half the array. Space complexity is O(1) since no extra array is stored; calculations are arithmetic based on n.

What Interviewers Usually Probe

• Expect you to notice the array's sequential odd-number pattern.
• Check if you compute target = sum(arr)/n before iterating.
• Look for symmetry to minimize operations, not brute force each pair.

Common Pitfalls or Variants

Common pitfalls

• Trying to simulate each operation instead of using the sum formula.
• Ignoring symmetry and overcounting operations for mirrored elements.
• Miscomputing target as a middle element rather than the true average.

Follow-up variants

• Arrays of even numbers instead of odd numbers but using the same operation rule.
• Allowing operations that adjust elements by more than 1 unit per move.
• Finding minimum operations when only adjacent elements can be adjusted.

FAQ

How is the minimum number of operations determined for this problem?

By summing the differences between each element and the target value for half the array, leveraging symmetry in the sequential odd numbers.

Why do we only need to consider half of the array?

The array is symmetric around the target; each operation on one side mirrors the other, so counting one half suffices.

What if n is very large, up to 104?

The formula approach still works in O(1) time and O(1) space, avoiding loops or simulation regardless of n.

Does the operation always involve subtracting 1 from one element and adding 1 to another?

Yes, each operation is exactly one unit transfer between any two indices to eventually equalize the array.

Is this a Math-driven solution pattern problem?

Exactly, recognizing the math-driven sequential odd-number pattern is key to efficiently computing minimum operations.