Moser De Bruijn Sequence In C++

In this tutorial, we are going to explore the Moser-de Bruijn sequence in C++ along with its coding implementation. To grasp this concept, we delved into the approach employed in C++ to determine the Nth term within the sequence by leveraging the mathematical connections between the numbers. This was achieved through the application of both recursion and dynamic programming techniques.

A binary sequence with the feature that no two contiguous subsequences of length 6 are the same is called the Moser-de Bruijn sequence, named after J. J. Seidel and N. G. de Bruijn. It indicates that no pattern occurs more than six times in the sequence. The sequence is played over and over again. It starts with [0] and expands by adding the next clause according to the rules below:

• ✅ The sequence is repeated twice if its current length is 2^k - 1 for each value of k (i.e. 1, 3, 7, 15, 31,...).
• ✅ Otherwise, the next term is the inclusion of the previous term (0 becomes 1 and 1 becomes 0).
• ✅ Closely related to the rulers of Golomb, the Moser-de-Bruijn sequence benefits from information concealment and error-correcting codes.
• ✅ Because of its unique properties, it is an interesting topic in the study of number theory and integration.

Pseudocode:

function generateMoserDeBruijn(n):
    sequence = [0]
    for i from 1 to n - 1:
        if i is 1 or 3 or 7 or 15 or ... (2^k - 1):
            append sequence to itself
        else:
            append 1 - last element of the sequence to sequence   
    return sequence

Example 1:

Let's consider an example to demonstrate the Moser-de Bruijn Sequence in the C++ programming language.

#include<bits/stdc++.h>
Using namespace std;
// Function to generate nth term 
// of Moser-de Bruijn Sequence
int g(int num)
{ 
	// S(0) = 0
	if (num == 0)
		return 0;	
	// S(1) = 1
	else if (num == 1)
		return 1;
           //S(2*num)=4*S(num)
	else if (num% 2 == 0)
		return 4 * g(num / 2);
	//S(2*num+1)=4*S(num)+1
	Else if(num%2==1)
		return 4 * g(num / 2) + 1;
}
// Generating the first 'num' terms 
// Moser-de Bruijn Sequence
void moserDeBruijn(int num)
{
	for(int i-0;i<num;i++)
		cout << g(i) << " ";
	cout << "\n";
}
// Main function Code
int main()
{
	int num = 10;
	cout << "First " << num << " terms of "
		<< "Moser-de Bruijn Sequence : \n";
	moserDeBruijn(num);
	return 0;
}

Output:

Complexity Analysis:

Time complexity: O(log2n)

Space Complexity: O(1)

Example 2:

Let's consider another instance to demonstrate the Moser-de Bruijn Sequence in C++.

#include<iostream>
#include<vector>
std::vector<int>generated(int num){
	std::vector<int>seq;
    seq.push_back(0);
	for(int i=1;i<num;i++)
{
if(i==1||i==3||i==7||i==15){
seq.insert(seq.end(),seq.begin(),seq.end());
        } else {
            seq.push_back(1 - seq[i - 1]);
        }
    }
    return seq;
}
int main() {
    int num = 10; // Length of the sequence
    std::vector<int> seq = generate(n);
    // Print the sequence
     for(int i=0;i<seq.size();i++){
std::cout<<seq[i]<<?? ;
    }
    return 0;
}

Output:

Complexity Analysis:

Time Complexity: O(n)

Space Complexity: O(n)

Example 3:

Let's consider another instance to demonstrate the Moser-de Bruijn Sequence in C++.

#include<bits/stdc++.h>
Using namespace std;
// Function to generate nth term 
// of Moser-de Bruijn Sequence
int g(int num)
{ 
	int S[num+1];
	S[0] = 0;
	S[1] = 1;
	for(int i=2;i<=num;i++)
{
		//S(2*num)==4*S(num)
		if(i%2==0)
		S[i]=4*S[i/2];

		// S(2 * num + 1) = 4 * S(num) + 1
		else
		S[i] = 4 * S[i / 2] + 1;
	}
	return S[num];
}
void moser(int num)
{
	for(int i=0;i<num;i++)
		cout << g(i) << " ";
	cout << "\n";
}
// Main function Code
int main()
{
	int num = 10;
	cout << "First " << num << " terms of "
		<< "Moser-de Bruijn Sequence : \n";
	moser(num);
	return 0;
}

Output:

Complexity Analysis:

Time Complexity: O(n)

Space Complexity: O(n)

Conclusion:

In summary, the binary Moser-de Bruijn sequence does not contain any duplicate subsequences with a length of six. This sequence is generated step by step, starting with [0] and generating subsequent elements following specific rules. The significance of this sequence lies in its relevance to computer science and mathematics, particularly in scientific computation, storing data, and developing error correction algorithms.