Gray Code

Problem Statement

An n-bit gray code sequence is a specific sequence of 2^n integers, each designated to represent unique patterns where successive entries differ by precisely one bit. Gray code sequences are used in various digital communication and error correction scenarios because they minimize transition errors. The requirements for these sequences are:

• Each integer must fall within the inclusive range [0, 2^n - 1].
• The sequence begins with the integer 0.
• Each integer in the sequence must be unique.
• Adjacent integers in the sequence must differ by exactly one bit in their binary form.
• The first and the last integers in the sequence should also differ by exactly one bit.

Given an integer n, we need to construct a valid n-bit gray code sequence.

Examples

Example 1

Input:

n = 2

Output:

[0,1,3,2]

Explanation:

The binary representation of [0,1,3,2] is [00,01,11,10].
- 00 and 01 differ by one bit
- 01 and 11 differ by one bit
- 11 and 10 differ by one bit
- 10 and 00 differ by one bit
[0,2,3,1] is also a valid gray code sequence, whose binary representation is [00,10,11,01].
- 00 and 10 differ by one bit
- 10 and 11 differ by one bit
- 11 and 01 differ by one bit
- 01 and 00 differ by one bit

Example 2

Input:

n = 1

Output:

[0,1]

Constraints

• 1 <= n <= 16

Approach and Intuition

The problem of generating a Gray code sequence revolves around creating a list of numbers where consecutive numbers differ by exactly one bit, including the loop-back from the last number to the first. Here’s a step-by-step approach based on the examples and constraints:

1. Understanding Bit Differences: In Gray code, only one bit changes at any step between two consecutive numbers. This is crucial as it reduces chances of error in digital transmissions where slight variations can introduce faults.

2. Start with Zero: The sequence starts with 0 by definition, which in binary, irrespective of n, is represented as a series of zeros (000...0 of length n).

3. Recursive Construction: For constructing an n-bit Gray code from an (n-1)-bit code:

   • First, take all the codes from the (n-1)-bit Gray sequence.
   • Then, prefix the mirror image of the (n-1)-bit sequence with a 1 (which means adding 2^(n-1) to these numbers).
   • The resulting sequence will satisfy all conditions regarding bit changes between consecutive entries.

4. Iterative Bit Manipulation: Starting from zero, for each step generating the next number can be achieved by flipping the rightmost bit that will produce a new number not already in the sequence. This can be done using bit manipulation techniques.

5. Handling Large n: Considering the upper limit constraint (n ≤ 16), the number of elements in our sequence can become quite large (65536 for n = 16). Hence, ensuring that computation remains efficient is crucial, and bit manipulation provides a way to handle this effectively.

By adhering to these principles, it's possible to build a valid Gray code sequence for any given n. The pivotal concept is ensuring that the binary representations of consecutive numbers differ by exactly one bit, which is inherent to the nature of Gray code. The recursive approach leverages already generated sequences for smaller bit lengths to construct sequences for higher bit lengths efficiently.

Solutions

class Solution {
public:
    vector<int> generateGrayCode(int bits) {
        vector<int> sequence;
        int total = 1 << bits;
        for (int i = 0; i < total; i++) {
            int gray = i ^ (i >> 1);
            sequence.push_back(gray);
        }
        return sequence;
    }
};