#89

Medium

auto_awesomeBacktracking search with pruning

LeetCode Problem Workspace

Gray Code

Generate an n-bit Gray Code sequence using backtracking with pruning and bit manipulation techniques.

Problem Statement

Given an integer n, generate a valid n-bit Gray Code sequence. The Gray Code is a sequence of 2^n integers where two consecutive numbers differ by exactly one bit in their binary representation. For example, when n = 2, the sequence [0, 1, 3, 2] is a valid Gray Code sequence because each number differs from the previous one by one bit.

The sequence must start with 0, and each step must ensure only a single bit changes. Your task is to return any valid Gray Code sequence for a given n. The problem involves exploring all possible valid sequences and selecting one such that the difference between each pair of adjacent numbers is one bit.

Examples

Example 1

Input: n = 2

Output: [0,1,3,2]

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]

Example details omitted.

Constraints

1 <= n <= 16

Solution Approach

Backtracking Search with Pruning

A backtracking approach is optimal for generating the Gray Code sequence by exploring possible transitions between numbers. The key is pruning the search space by ensuring only valid transitions occur, i.e., consecutive numbers differ by one bit. This avoids unnecessary calculations and ensures efficiency in generating the sequence.

Bit Manipulation

Using bit manipulation, we can efficiently generate the Gray Code sequence by exploiting the relationship between the binary and Gray Code representations. The Gray Code of a number can be obtained by performing a bitwise XOR operation between the number and its right-shifted self.

Iterative Construction of the Sequence

An iterative approach can also be employed to generate the sequence. By iteratively constructing the sequence and applying the Gray Code formula, we ensure that each number in the sequence is valid and differs from the previous one by exactly one bit.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of the Gray Code problem is dependent on the method used. In the backtracking approach, the complexity can be O(2^n) due to the exploration of all potential valid sequences. The space complexity is O(n) for storing the sequence, and additional space may be used for recursion stack in backtracking.

What Interviewers Usually Probe

Ability to implement backtracking with pruning efficiently.
Familiarity with bit manipulation techniques, especially XOR and bit shifting.
Knowledge of iterative methods for constructing sequences.

Common Pitfalls or Variants

Common pitfalls

Failing to ensure the difference between consecutive numbers is exactly one bit.
Not optimizing the backtracking algorithm to prune unnecessary paths.
Incorrect handling of bit manipulation, especially when constructing Gray Code from binary numbers.

Follow-up variants

Generating Gray Code for larger n values (n > 10).
Implementing Gray Code using dynamic programming.
Optimizing for space complexity while maintaining correctness.

FAQ

What is the Gray Code sequence?

Gray Code is a binary numeral system where two successive values differ in only one bit. It is used in various applications, such as error correction and digital communication.

How can Gray Code be efficiently generated?

Gray Code can be efficiently generated using backtracking with pruning, bit manipulation, or iterative construction methods.

What is the backtracking approach in Gray Code generation?

The backtracking approach explores all possible sequences, pruning the search space by ensuring only valid transitions between consecutive numbers, which differ by exactly one bit.

Why is bit manipulation important for solving the Gray Code problem?

Bit manipulation allows for efficient conversion between binary numbers and their Gray Code equivalents, utilizing XOR and bit shifts to achieve the desired sequence.

What is the time complexity of generating Gray Code?

The time complexity is O(2^n) when using backtracking or iterative construction methods, as the number of valid sequences grows exponentially with n.

terminal

Solution

Solution 1: Binary to Gray Code Conversion

Gray code is a type of encoding method that we often encounter in engineering. Its basic feature is that only one bit of binary number is different between any two adjacent codes.

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class Solution:
    def grayCode(self, n: int) -> List[int]:
        return [i ^ (i >> 1) for i in range(1 << n)]

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#996 number of squareful arrays #90 subsets ii #78 subsets