> For the complete documentation index, see [llms.txt](https://jimmylin1991.gitbook.io/practice-of-algorithm-problems/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://jimmylin1991.gitbook.io/practice-of-algorithm-problems/math/89.-gray-code.md).

# 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.

**Example 1:**

```
Input: 2
Output: [0,1,3,2]
Explanation:
00 - 0
01 - 1
11 - 3
10 - 2

For a given n, a gray code sequence may not be uniquely defined.
For example, [0,2,3,1] is also a valid gray code sequence.

00 - 0
10 - 2
11 - 3
01 - 1
```

**Example 2:**

```
Input: 0
Output: [0]
Explanation: We define the gray code sequence to begin with 0.
             A gray code sequence of n has size = 2n, which for n = 0 the size is 20 = 1.
             Therefore, for n = 0 the gray code sequence is [0].
```

```cpp
// Bit manipulation for gray code
vector<int> grayCode(int n) { // time: O(2^n); space: O(2^n)
    vector<int> res(1 << n);
    for (int i = 0; i < (1 << n); ++i) {
        res[i] = i ^ (i >> 1);
    }
    return res;
}
```

```cpp
// Iteration
// 2 numbers in the middle only differ at their most significant bit
vector<int> grayCode(int n) { // time: O(2^n); space: O(2^n)
    vector<int> res;
    res.reserve(1 << n);
    res.push_back(0);
    for (int i = 0; i < n; ++i) {
        for (int j = res.size() - 1; j >= 0; --j) {
            res.push_back(res[j] | (1 << i));
        }
    }
    return res;
}
```

```cpp
// Backtracking
void helper(vector<int>& res, const int& bits, const int& k) {
    if (k == 0) res.push_back(bits);
    else {
        helper(res, bits, k - 1);
        helper(res, bits ^ ((1 << k) - 1), k - 1);
    }
}
vector<int> grayCode(int n) { // time: O(2^n); space: O(2^n)
    vector<int> res;
    res.reserve(1 << n);
    int bits = 0;
    helper(res, bits, n);
    return res;
}
```