Minimum Number of Operations to Make All Array Elements Equal to 1

Minimum Number of Operations to Make All Array Elements Equal to 1

1. What is this problem about?

The Minimum Number of Operations to Make All Array Elements Equal to 1 coding problem is a deep dive into number theory. You are given an array of positive integers. In one operation, you can pick two adjacent elements and replace one of them with their Greatest Common Divisor (GCD). Your goal is to find the minimum operations to make every element in the array equal to 1.

2. Why is this asked in interviews?

Amazon and Bloomberg use this to test a candidate's grasp of the Euclidean algorithm and array subarrays. It's not a standard greedy problem; it requires realizing that once you produce a single 1 in the array, you can use that 1 to turn every other element into 1 in n-1 steps. The challenge is finding the shortest subarray whose overall GCD is 1.

3. Algorithmic pattern used

This problem follows the Array, Math, Number Theory interview pattern.

1. If the array already contains 1s, the answer is just n - count_of_1s.
2. If not, you must find the shortest subarray [i, j] such that gcd(array[i...j]) == 1.
3. The number of operations to produce a 1 from this subarray is (j - i).
4. Once you have that 1, the remaining n-1 elements take one operation each.
5. Total = (j - i) + (n - 1).

4. Example explanation

Array: [2, 6, 3, 4]

• gcd(2, 6) = 2
• gcd(6, 3) = 3
• gcd(3, 4) = 1 (Shortest subarray is [3, 4], length 2) Steps to get a 1: 1 (replace 3 or 4 with gcd(3,4)). Array becomes: [2, 6, 1, 4] Steps to turn others to 1:
• gcd(6, 1) -> [2, 1, 1, 4]
• gcd(2, 1) -> [1, 1, 1, 4]
• gcd(4, 1) -> [1, 1, 1, 1] Total steps: 1 + 3 = 4.

5. Common mistakes candidates make

The most frequent error is not realizing that the overall GCD of the entire array must be 1 for a solution to exist. If gcd(all elements) > 1, you can never produce a 1. Another mistake is using an O(n^3) approach to find the shortest subarray when an O(n^2) approach is possible by iterating through all pairs and maintaining a running GCD.

6. Interview preparation tip

GCD-based problems are common in advanced math interviews. Remember: gcd(a, b, c) = gcd(a, gcd(b, c)). This associative property allows you to compute the GCD of any subarray incrementally, which is the key to solving this problem efficiently.