Integer Replacement

Problem

Given a positive integer n and you can do operations as follow:

If n is even, replace n with n/2. If n is odd, you can replace n with either n + 1 or n - 1. What is the minimum number of replacements needed for n to become 1?

Example 1:

Input:
8

Output:
3

Explanation:
8 -> 4 -> 2 -> 1

Example 2:

Input:
7

Output:
4

Explanation:
7 -> 8 -> 4 -> 2 -> 1
or
7 -> 6 -> 3 -> 2 -> 1

Solution

Recursive Solution

public class Solution {
    public int integerReplacement(int n) {
        if (n <= 2) return n - 1;
        if (n <= 4) return 2;
        if (n == Integer.MAX_VALUE) return 32; //max = 2 ^ 31 - 1
        if (n % 2 == 0) return 1 + integerReplacement(n / 2);
        return 1 + Math.min(integerReplacement(n + 1), integerReplacement(n - 1));
    }
}

Iterative Solution

public class Solution {
    public int integerReplacement(int n) {
        if (n <= 2) return n - 1;
        if (n <= 4) return 2;
        if (n == Integer.MAX_VALUE) return 32; //max = 2 ^ 31 - 1

        int count = 0;
        while (n > 1) {
            count++;
            if (n % 2 == 0) n /= 2;
            else if ((n+1) % 4 == 0 && n != 3) n++;
            else n--;
        }
        return count;
    }
}

Analysis

The recursive solution is straightforward
We just need to handle some corner cases then use Math.min() to decide whether move forward or backward

I don't have a proof of iterative solution yet.