Gray Code

Problem Description

An n-bit gray code sequence is a sequence of 2^n integers where:

• Every integer is in the inclusive range [0, 2^n - 1],
• The first integer is 0,
• An integer appears no more than once in the sequence,
• The binary representation of every pair of adjacent integers differs by exactly one bit,
• The binary representation of the first and last integers differs by exactly one bit.

Given an integer n, return any valid n-bit gray code sequence.

Key Insights

• Gray code sequences are constructed such that each integer differs from its neighbors by one bit.
• The sequence starts with 0, and the total number of integers is 2^n.
• There is a mathematical way to generate gray codes using the formula: gray_code(i) = i ^ (i >> 1) for integers i from 0 to 2^n - 1.
• The properties of gray codes ensure they can wrap around, meaning the first and last elements also differ by one bit.

Space and Time Complexity

Time Complexity: O(2^n) - We need to generate 2^n integers. Space Complexity: O(2^n) - The output array will store 2^n integers.

Solution

To generate an n-bit gray code sequence, we can utilize the property of gray codes that allows us to derive each code from its binary representation. Specifically, for each integer i in the range 0 to 2^n - 1, the gray code can be calculated using the formula gray_code(i) = i ^ (i >> 1). This approach efficiently generates the required sequence.

The algorithm involves:

1. Looping through all integers from 0 to 2^n - 1.
2. Applying the gray code formula to generate the corresponding gray code integer.
3. Storing these integers in a result array which is returned at the end.

Code Solutions