
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:
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.
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 lengthn
).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 adding2^(n-1)
to these numbers). - The resulting sequence will satisfy all conditions regarding bit changes between consecutive entries.
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.
Handling Large n: Considering the upper limit constraint (
n ≤ 16
), the number of elements in our sequence can become quite large (65536
forn = 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
- C++
- Java
- C
- JavaScript
- Python
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;
}
};
The solution provided is a C++ implementation of generating a Gray code sequence. Gray code is a binary numeral system where two successive values differ in only one bit.
In this implementation:
- The
generateGrayCode
function accepts an integerbits
, which represents the number of bits in each Gray code value. - The function initializes an empty vector
sequence
that will store the Gray code sequence. - It calculates the total number of Gray code elements, denoted by
total
, as (2^{bits}). - The function employs a for-loop running from 0 to
total - 1
. For each iteration:- It calculates the Gray code using the formula
i ^ (i >> 1)
, where^
is the XOR operator and>>
is the right shift operator. - The calculated Gray code is then appended to the
sequence
vector.
- It calculates the Gray code using the formula
- Finally, the function returns the vector containing all the Gray codes.
This approach ensures efficient computation of Gray codes for a specified number of bits.
class Solution {
public List<Integer> generateGrayCode(int bits) {
List<Integer> codes = new ArrayList<>();
int total = 1 << bits;
for (int index = 0; index < total; index++) {
int grayCode = index ^ (index >> 1);
codes.add(grayCode);
}
return codes;
}
}
The task is to generate a sequence known as Gray Code for a given number of bits. The Gray Code sequence ensures that two consecutive values differ in only one bit which makes it useful in various digital communications to prevent errors.
Implement the Gray Code generator using Java with the following steps:
- Create a
generateGrayCode
method that accepts an integerbits
, which signifies the number of bits. - Initialize an
ArrayList
calledcodes
to store the sequence of Gray Codes. - Compute the total number of codes possible, which is
1 << bits
(equivalent to 2 raised to the power ofbits
). - Use a for-loop to iterate from
0
tototal - 1
. For each indexi
, calculate the Gray Code using the formulai ^ (i >> 1)
. This formula represents XOR of the number with itself right-shifted by one bit. - Add the computed Gray Code to the
codes
list. - Finally, return the
codes
list containing the complete Gray Code sequence for the specified number of bits.
Ensure the use of bitwise operations like XOR and right shift effectively as they are crucial for calculating the Gray Code values efficiently.
int* generateGrayCode(int bits, int* outputSize) {
int length = 1 << bits;
*outputSize = length;
int* sequence = (int*)malloc(sizeof(int) * length);
for (int index = 0; index < length; index++) {
sequence[index] = index ^ (index >> 1);
}
return sequence;
}
The Gray Code sequence is generated using a C function named generateGrayCode
, which computes sequences where successive values differ in only one bit. Here is how the function operates:
- Define the function
generateGrayCode
that accepts an integerbits
, representing the number of bits in the Gray code, and a pointeroutputSize
used to store the size of the generated sequence. - Calculate the length of the Gray code sequence as (2^{\text{bits}}), and store this value in
outputSize
. - Dynamically allocate an array
sequence
to hold the Gray code usingmalloc
. - Populate the
sequence
array. Conversion of a binary number to its Gray code equivalent is done using the formula ( \text{index} \oplus (\text{index} >> 1) ) in a loop that iterates throughindex
from 0 tolength - 1
. - Return the pointer to the
sequence
array.
This function efficiently outputs a complete sequence of Gray codes for a given number of bits, offering a straightforward method for generating such sequences in C.
var generateGrayCode = function (numBits) {
let sequence = [];
let totalLength = 1 << numBits;
for (let index = 0; index < totalLength; index++) {
let grayValue = index ^ (index >> 1);
sequence.push(grayValue);
}
return sequence;
};
The problem involves generating a sequence known as Gray code for a given number of bits. Gray code is a binary numeral system where two successive values differ in only one bit, making it useful in error correction in digital communications and other applications.
Here’s a breakdown of the JavaScript solution you provided:
- Define the function
generateGrayCode
which takes one parameternumBits
indicating the number of bits in each code. - Instantiate
sequence
as an empty array to store the resulting Gray codes. - Calculate
totalLength
as (2^{numBits}), which represents the total number of Gray codes to generate. - Use a for-loop to iterate through each integer from
0
tototalLength - 1
, computing corresponding Gray codes. - Inside the loop, calculate
grayValue
by XOR-ing the current index with the right-shifted version of the index. The right shift effectively halves the index, ensuring the desired property that consecutive Gray codes differ by only one bit. - Append each
grayValue
to thesequence
array. - Finally, return the
sequence
array containing all Gray codes for the providednumBits
.
This function efficiently generates a sequence of Gray codes for a specified number of bits using bit manipulation which is optimal for such operations. Make sure to call this function with a desired number of bits to obtain the sequence.
class Solution:
def generateGrayCode(self, num_bits: int):
output_sequence = []
total_codes = 1 << num_bits
for idx in range(total_codes):
code = idx ^ idx >> 1
output_sequence.append(code)
return output_sequence
The Gray Code solution provided is implemented in Python 3 and involves generating a sequence where each successive entry differs from the previous one by exactly one bit. This sequence is known as Gray code and is often used in position encoders, digital communications, and error correction. The primary function, generateGrayCode
, takes one parameter num_bits
, which defines the number of bits for the Gray code sequence.
Here's a brief breakdown of the solution approach:
- Initialize an empty list called
output_sequence
to store the Gray codes. - Calculate
total_codes
by left-shifting 1 bynum_bits
. This operation determines the total number of Gray codes that can be formed with the given number of bits. - Use a loop to iterate through the range of
total_codes
. For each indexidx
, compute the Gray code. - The Gray code computation is done by XOR-ing
idx
withidx
shifted right by 1 bit; this is represented asidx ^ (idx >> 1)
. - Append the computed Gray code to
output_sequence
. - Return the list
output_sequence
containing all Gray codes up to the specified number of bits.
This function efficiently generates the list of Gray codes by utilizing bit manipulation techniques which offer a direct and fast method for Gray code conversion.
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