Minimum Moves to Equal Array Elements

This problem asks for the minimum number of moves to make all array elements equal, where each move increments n-1 elements. The key insight is transforming the problem into tracking differences relative to the array minimum. Using simple math, you can derive the total moves efficiently without simulating every increment step, ensuring fast execution for large arrays.

Problem Statement

Given an integer array nums of size n, determine the minimum number of moves required to make all elements equal. Each move allows incrementing n-1 elements by 1, so the array gradually balances toward equality.

For example, given nums = [1,2,3], the array can be equalized in three moves: incrementing all except the largest element repeatedly until all values match. Return the total count of moves needed for any input array under these rules.

Examples

Example 1

Input: nums = [1,2,3]

Output: 3

Only three moves are needed (remember each move increments two elements): [1,2,3] => [2,3,3] => [3,4,3] => [4,4,4]

Example 2

Input: nums = [1,1,1]

Output: 0

Example details omitted.

Constraints

• n == nums.length
• 1 <= nums.length <= 105
• -109 <= nums[i] <= 109
• The answer is guaranteed to fit in a 32-bit integer.

Solution Approach

Identify Minimum Element

Find the smallest element in the array, as all other elements must eventually be incremented to match a common target relative to this minimum.

Compute Total Moves Using Sum Difference

Calculate the sum of differences between each element and the minimum. This total directly gives the minimum number of moves required without iterative simulation.

Optimize for Large Arrays

Avoid modifying the array on each move; instead, use the formula total_moves = sum(nums) - n * min(nums). This ensures O(n) time and O(1) extra space complexity.

Complexity Analysis

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

Time complexity is O(n) because it requires a single pass to find the minimum and sum differences. Space complexity is O(1) extra space as no additional structures are needed beyond variables.

What Interviewers Usually Probe

• The candidate quickly identifies the pattern of incrementing n-1 elements as equivalent to decrementing one element.
• They recognize that tracking sum differences relative to the minimum yields an immediate solution.
• They avoid simulating each move, showing awareness of efficient array plus math strategies.

Common Pitfalls or Variants

Common pitfalls

• Simulating every increment individually instead of computing the total mathematically.
• Failing to account for negative numbers in the array when calculating moves.
• Assuming the target is the maximum element rather than aligning all elements relative to the minimum.

Follow-up variants

• Change the move operation to increment k elements instead of n-1 and recalculate minimum moves.
• Allow decrementing n-1 elements in a move and compare with the increment variant for efficiency.
• Apply the pattern to multi-dimensional arrays where each move adjusts n-1 cells along a row or column.

FAQ

What is the minimum number of moves to equal array elements for nums = [1,2,3]?

The total moves are 3, achieved by incrementing n-1 elements per move: [1,2,3] => [2,3,3] => [3,4,3] => [4,4,4].

Why is the array minimum used as the reference for computing moves?

Each move effectively increases all elements except one, so the fastest way to equalize is aligning everything relative to the minimum value.

Can negative numbers in nums affect the calculation?

No, the formula sum(nums) - n * min(nums) correctly accounts for negative numbers without extra handling.

Is it necessary to simulate each move step by step?

No, direct computation using sum differences gives the result in O(n) time, avoiding inefficient simulations.

How does the array plus math pattern apply here?

The problem transforms iterative increments into a sum-of-differences calculation, demonstrating the array plus math pattern explicitly.