397. Integer Replacement

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

1. If n is even, replace n with n/2.
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

Approach #1: Math. [Java]

class Solution {
    public int integerReplacement(int n) {
        int c = 0;
        while (n != 1) {
            if ((n & 1) == 0) n >>>= 1;
            else if (n == 3 || ((n >>> 1) & 1) == 0) n--;
            else n++;
            c++;
        }
        return c;
    }
}

　　

Analysis:

The first step towards solution is to realize that you're allowed to remove the LSB only if it's zero. And to reach the target as far as possible, removing digits is the best way to go. Hence, even numbers are better than odd. This is quite obvious.

What is not so obvious is what to do with odd numbers. One may think that you just need to remove as many 1's as possible to increase the evenness of the number. Wrong! Look at this example:

111011 -> 111010 -> 11101 -> 11100 -> 1110 -> 111 -> 110 -> 11 -> 10 -> 1

And yet, this is not the best way because

111011 -> 111100 -> 11110 -> 1111 -> 10000 -> 1000 -> 100 -> 10 -> 1

Both 111011 -> 111010 and 111011 -> 111100 remove the same number of 1's, but the second way is better.

So, we just need to remove as many 1's as possible, doing +1 in case of a tie? Not quite. The infamous test with n = 3 fails for that stratefy because 11 -> 10 -> 1 is better than 11 -> 100 -> 10 -> 1. Fortunately, that's the only exception (or at least I can't think of any other, and there are none in the tests).

So the logic is:

If n is even, halve it.

If n = 3 or n - 1has less 1's than n+1, decrement n.

Otherwise, increment n.

Reference:

https://leetcode.com/problems/integer-replacement/discuss/87920/A-couple-of-Java-solutions-with-explanations