89. Gray Code

The gray code is a binary numeral system where two successive values differ in only one bit.

Given a non-negative integer n representing the total number of bits in the code, print the sequence of gray code.
A gray code sequence must begin with 0.

For example, given n = 2, return [0,1,3,2]. Its gray code sequence is:

00 - 0
01 - 1
11 - 3
10 - 2
Note:
For a given n, a gray code sequence is not uniquely defined.

For example, [0,2,3,1] is also a valid gray code sequence according to the above definition.

For now, the judge is able to judge based on one instance of gray code sequence. Sorry about that.

Analysis:

Try one more example, n = 3:

000 - 0

001 - 1

011 - 3

010 - 2

110 - 6

111 - 7

101 - 5

100 - 4 

Comparing n = 2: [0,1,3,2] and n=3: [0,1,3,2,6,7,5,4], we found that the first four numbers in case n=3 are the same as the the numbers in case n=4.  Besides, [6,7,5,4] = [2+4,3+4,1+4,0+4].  Which means remaining numbers in case n=3 can also be calculated from the numbers in case n=4 in reversing order.  Therefore, we decided to use recursive approach to form the resulting ArrayList.

divide && conquer

public List<Integer> grayCode(int n) {
        List<Integer> res = new ArrayList<Integer>();
        if(n == 0) {
            res.add(0);
        }
        else {
            List<Integer> pre = grayCode(n-1);
            res.addAll(pre);
            for (int i=pre.size()-1; i>=0; i--) {
                res.add(pre.get(i) + (int)Math.pow(2, n-1));
            }
        }
        return res;
    }

　　Iterative 解法（推荐方法）：每次也不用定义一个 ArrayList<Integer> pre = res, 只用记录当前res 的size就好了

public class Solution {
    public List<Integer> grayCode(int n) {
        List<Integer> res = new ArrayList<Integer>();
        if (n < 0) return res;
        res.add(0);
        if (n == 0) return res;
        for (int i=1; i<=n; i++) {
            int size = res.size();
            for (int j=size-1; j>=0; j--) {
                res.add(res.get(j) + (int)Math.pow(2, i-1));
            }
        }
        return res;
    }
}

算法复杂度是O(2+2^2+...+2^n-1)=O(2^n)，所以是指数量级的，因为是结果数量无法避免。空间复杂度则是结果的大小，也是O(2^n)。　　

 这些代码都要注意一个细节，即input为0时，output为[0]，并不是[]