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Minimum Operations to Make Array Equal III - Solution & Explanation

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Problem Statement

You are given an integer array nums.

In one operation, you may choose any element nums[i] and perform one of the following:

  • Multiply nums[i] by an integer k, where k >= 2.
  • Divide nums[i] by an integer k, where 2 <= k < nums[i], provided that nums[i] is divisible by k.

Return the minimum number of operations required to make all elements of nums equal.

 

Example 1:

Input: nums = [6,12,8]

Output: 3

Explanation:

We can perform following operates to make all numbers to 6:

  • Divide nums[1] = 12 by 2 to get 6.
  • Divide nums[2] = 8 by 4 to get 2.
  • Multiply nums[2] = 2 by 3 to get 6.

Example 2:

Input: nums = [5,15,20]

Output: 2

Explanation:

We can perform following operates to make all numbers to 5:

  • Divide nums[1] = 15 by 3 to get 5.
  • Divide nums[2] = 20 by 4 to get 5.

Example 3:

Input: nums = [7,7,7]

Output: 0

Explanation:

All elements are already equal, so no operations are needed.

 

Constraints:

  • 1 <= nums.length <= 105
  • 1 <= nums[i] <= 10ā€‹ā€‹ā€‹ā€‹ā€‹ā€‹ā€‹9

Approach Overview

Problem Overview: You need to make two arrays equal using the minimum number of operations where each operation can increment or decrement an element by 1, but only on elements at the same index.

Approach 1: Brute Force (O(n^2))

Check all possible element pairs between arrays. For each mismatch, increment or decrement until equal. This approach iterates through all elements multiple times, leading to quadratic time complexity. Only useful for understanding the problem constraints.

Approach 2: Difference Analysis (O(n))

Calculate the absolute difference between corresponding elements. Sum these differences to get the total operations needed. The key insight is that each mismatch requires exactly one operation per unit difference. This greedy approach works because operations are independent of each other.

Recommended for interviews: The difference analysis approach is expected in interviews. It demonstrates understanding of problem constraints and optimal use of array manipulation. Brute force shows basic comprehension, but the optimal solution highlights efficient problem-solving.

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Detailed Complexity Analysis

ApproachTimeSpaceWhen to Use
Brute ForceO(n^2)O(1)For small arrays or initial problem analysis
Difference AnalysisO(n)O(1)General case, optimal solution

Frequently Asked Questions

Is Minimum Operations to Make Array Equal III easy or hard?
This problem is rated Hard with a 16.6% acceptance rate. It requires understanding of array operations and efficient computation of differences.
Minimum Operations to Make Array Equal III Python/Java solution
Python and Java solutions involve iterating through arrays, calculating absolute differences, and summing them. Both languages achieve O(n) time complexity.
How to solve Minimum Operations to Make Array Equal III in O(n)?
Use a greedy approach by iterating through both arrays once. Sum the absolute differences between corresponding elements to get the minimum operations.
What is the best approach for Minimum Operations to Make Array Equal III?
The difference analysis approach is optimal, running in O(n) time. It calculates absolute differences between array elements and sums them for the total operations needed.
Is Minimum Operations to Make Array Equal III asked at Google/Amazon/Meta?
This problem tests array manipulation skills and is commonly asked at top tech companies like Google and Amazon during coding interviews.
What data structure is used in Minimum Operations to Make Array Equal III?
Arrays are the primary data structure. The solution involves direct element comparison without additional data structures, using O(1) space.
What is the time complexity of Minimum Operations to Make Array Equal III?
The optimal solution has O(n) time complexity. It processes each array element exactly once to compute the total operations required.

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