Given n and k, Construct a palindrome of size n using a binary number of size k repeating itself to wrap into the palindrome. **The palindrome must always begin with 1 and contains maximum number of zeros.**

Examples :

Input : n = 5, k = 3 Output : 11011 Explanation : the 3 sized substring is 110 combined twice and trimming the extra 0 in the end to give 11011. Input : n = 2, k = 8 Output : 11 Explanation : the 8 sized substring is 11...... wrapped to two places to give 11.

The **naive approach** would be to try every palindrome of size k starting with 1 such that a palindrome of size n is formed. This approach has an exponential complexity.

A **better way** to do this is to initialize the k sized binary number with the index and connect the palindrome in the way it should be. Like last character of palindrome should match to first, find which indexes will be present at those locations and link them. Set every character linked with 0th index to 1 and the string is ready. This approach will have a linear complexity.

In this approach, first lay the index of the k sized binary to hold into an array, for example if n = 7, k = 3 arr becomes [0, 1, 2, 0, 1, 2, 0]. Following that in the connectchars graph, connect the indices of the k sized binary which should be same by going through the property of palindrome which is kth and (n – k – 1)th variable should be same, such that 0 is linked to 1(and vice versa), 1 is linked to 2(and vice versa) and so on. After that, check what is linked with 0 in connectchars array and make all of the associated indices one (because the first number should be non-zero) by using dfs approach. In the dfs, pass 0, the final answer string and the graph. Begin by making the parent 1 and checking if its children are zero, if they are make them and their children 1. This makes only the required indices of the k sized string one, others are left zero. Finally, the answer contains the 0 to k – 1 indexes and corresponding to arr the digits are printed.

[sourcecode language=”CPP”]

// CPP code to form binary palindrome

#include <iostream>

#include <vector>

using namespace std;

// function to apply DFS

void dfs(int parent, int ans[], vector<int> connectchars[])

{

// set the parent marked

ans[parent] = 1;

// if the node has not been visited set it and

// its children marked

for (int i = 0; i < connectchars[parent].size(); i++) {

if (!ans[connectchars[parent][i]])

dfs(connectchars[parent][i], ans, connectchars);

}

}

void printBinaryPalindrome(int n, int k)

{

int arr[n], ans[n] = { 0 };

// link which digits must be equal

vector<int> connectchars[k];

for (int i = 0; i < n; i++)

arr[i] = i % k;

// connect the two indices

for (int i = 0; i < n / 2; i++) {

connectchars[arr[i]].push_back(arr[n – i – 1]);

connectchars[arr[n – i – 1]].push_back(arr[i]);

}

// set everything connected to

// first character as 1

dfs(0, ans, connectchars);

for (int i = 0; i < n; i++)

cout << ans[arr[i]];

}

// driver code

int main()

{

int n = 10, k = 4;

printBinaryPalindrome(n, k);

return 0;

}

[/sourcecode]

**Output:**

1100110011

Time Complexity : O(n)

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