#1238

Medium

auto_awesomeBacktracking search with pruning

LeetCode Problem Workspace

Circular Permutation in Binary Representation

Generate a circular permutation from a range of numbers using backtracking and bit manipulation.

Problem Statement

You are given two integers, n and start. Your task is to return any permutation p of numbers from 0 to 2^n - 1 such that each consecutive pair of numbers in p differ by exactly one bit in their binary representation.

For example, if n = 2 and start = 3, the permutation could be [3, 2, 0, 1], where the binary representations are (11, 10, 00, 01), and each pair differs by one bit. The sequence must be generated efficiently, taking into account the constraints.

Examples

Example 1

Input: n = 2, start = 3

Output: [3,2,0,1]

The binary representation of the permutation is (11,10,00,01).

All the adjacent element differ by one bit. Another valid permutation is [3,1,0,2]

Example 2

Input: n = 3, start = 2

Output: [2,6,7,5,4,0,1,3]

The binary representation of the permutation is (010,110,111,101,100,000,001,011).

Constraints

1 <= n <= 16
0 <= start < 2 ^ n

Solution Approach

Backtracking with Pruning

The problem is most efficiently solved using a backtracking approach, where you explore potential solutions and prune invalid paths early. Starting from the given 'start' number, backtrack through potential permutations while ensuring each next number differs from the previous by one bit.

Gray Code Sequence

Gray code is an optimal method to generate a binary sequence where two consecutive numbers differ by only one bit. By using Gray code, we can generate the desired sequence for any given n, ensuring the problem's condition is met efficiently.

Bit Manipulation for Efficient Search

Using bit manipulation, the problem can be tackled efficiently. Specifically, toggling specific bits can help transition between numbers differing by exactly one bit, which is key to the solution.

Complexity Analysis

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

The time and space complexity depend on the chosen approach. A backtracking approach with pruning has an exponential complexity, but it is reduced by gray code and bit manipulation. In general, the time complexity is O(2^n), while the space complexity is O(n).

What Interviewers Usually Probe

Candidate demonstrates an understanding of Gray code and its application in generating valid sequences.
Candidate efficiently uses bit manipulation to minimize unnecessary computation.
Candidate identifies and explains the pruning strategy used in backtracking for optimization.

Common Pitfalls or Variants

Common pitfalls

Failing to implement pruning in backtracking, leading to inefficient searches.
Incorrectly transitioning between numbers that don't differ by exactly one bit.
Not leveraging Gray code for an optimal solution.

Follow-up variants

Changing the start value, which may require a different traversal order.
Exploring more efficient search algorithms for generating the sequence.
Extending the problem to handle larger values of n (beyond 16).

FAQ

What is the Circular Permutation in Binary Representation problem?

This problem involves generating a permutation of numbers from 0 to 2^n - 1 where each adjacent pair of numbers differs by exactly one bit in their binary representation.

How do you approach solving the Circular Permutation in Binary Representation?

A backtracking approach with pruning works well, leveraging Gray code to generate the sequence. Bit manipulation helps efficiently transition between numbers with one-bit differences.

What is Gray code and how does it help in this problem?

Gray code is a binary numeral system where two consecutive numbers differ by exactly one bit. It helps generate the permutation efficiently while satisfying the problem's constraints.

What are common pitfalls in solving the Circular Permutation in Binary Representation?

Common pitfalls include not implementing pruning in backtracking, incorrectly transitioning between numbers, and failing to use Gray code for efficiency.

How does backtracking work in the Circular Permutation problem?

Backtracking explores all possible solutions while pruning invalid paths early. By ensuring that each transition between numbers differs by exactly one bit, it helps generate the correct permutation.

terminal

Solution

Solution 1: Binary Code to Gray Code

We observe the arrangement in the problem, and find that in its binary representation, only one bit is different between any two (including the first and last) adjacent numbers. This kind of coding method is Gray code, which is a coding method we will encounter in engineering.

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class Solution:
    def circularPermutation(self, n: int, start: int) -> List[int]:
        g = [i ^ (i >> 1) for i in range(1 << n)]
        j = g.index(start)
        return g[j:] + g[:j]

Solution 2: Conversion Optimization

Since $gray(0) = 0$, then $gray(0) \oplus start = start$, and $gray(i)$ is only one binary bit different from $gray(i-1)$, so $gray(i) \oplus start$ is also only one binary bit different from $gray(i-1) \oplus start$.

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class Solution:
    def circularPermutation(self, n: int, start: int) -> List[int]:
        g = [i ^ (i >> 1) for i in range(1 << n)]
        j = g.index(start)
        return g[j:] + g[:j]

Continue Practicing

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