Generate Binary String with equal number of 01 and 10 Subsequence

Given an integer N (N > 2), the task is to generate a binary string of size N that consists of equal numbers of "10" & "01" subsequences and also the string should contain at least one '0' and one '1'

Note: If multiple such strings exist, print any.

Examples: 

Input: 4
Output: 0110
Explanation : Here, 0110 string consists equal count of '01' and '10'
subsequence and it consists at least one occurrence  of 0 as well as 1.

Input: 3
Output: 010
Explanation : Here, 010 string consists equal count of '01' and '10' 
subsequence and it consists at least one occurrence  of 0 as well as 1.

Naive approach: 

The idea is to generate every possible binary string(string which consists of only 0's and 1's) and find a substring with equal number of '01' and '10' subsequence.

Time Complexity: O(2^N * 2^N) = O(4^N) because each string also has 2^N subsequences.
Auxiliary Space: O(N)

Efficient approach: To solve the problem follow the below observations:

Case 1 (If N is even): Then add 1s at N/2 and (N/2 - 1) index, and add 0s to the rest of the string. So the count of '10' and '01' will be same which is equal to 2*(N/2 - 1) = (N - 2) and string also consists of at least once occurrence of 0 and 1.

For example:

• If N = 4, that means N is even, then add 1 at index N/2 =2 and (N/2)-1 = 1 . 
• And add 0s at remaining positions from 0 to 3 .
• Hence, string is "0110".

Case 2 (If N is odd): Then add 1s at middle and add 0s to the rest of the string. So the count of '10' and '01' is same which is equal to N/2 and string also consists at least once occurrence of 0 and 1.

For example:

• If N=3, that means N is odd then add 1 at index N/2=1.
• And add 0s at remaining positions from 0 to 2.
• Hence, string is "010".

Follow the below steps to implement the above approach:

• Take an empty string s.
• Now check if N is even or odd:
  • If N is even, iterate from 0 to N - 1 and add '1' only at the N/2 and (N/2 - 1) index and '0' on the other indices of the string.
  • If N is odd, iterate from 0 to N-1 and add '1' only at the N/2 index and '0' in all other indices of the string.
• Now, return the constructed string.

Below is the implementation of the above approach.

// C++ code to implement the approach

#include <bits/stdc++.h>
using namespace std;

string makeString(int n)
{
    string s = "";

    // If n is even then follow this block
    int mid = n / 2;
    if (n % 2 == 0) {
        for (int i = 0; i < n; i++) {

            // If i is at middle of n and
            // previous middle of n
            // then add 1 into the string
            if (i == mid || i == mid - 1) {
                s += "1";
            }

            // Otherwise, add 0 to the string
            else {
                s += "0";
            }
        }
    }

    // If n is odd then follow this block
    else {
        for (int i = 0; i < n; i++) {

            // If i is equal to mid then
            // add 1 to the string
            if (i == mid) {
                s += "1";
            }

            // Otherwise add 0 to the string
            else {
                s += "0";
            }
        }

        // Return the constructed string
        return s;
    }
}

// Driver code
int main()
{
    int N = 4;

    // Function call
    cout << makeString(N);
    return 0;
}

// Java code to implement the approach


import java.util.*;

class GFG{

static String makeString(int n)
{
    String s = "";

    // If n is even then follow this block
    int mid = n / 2;
    if (n % 2 == 0) {
        for (int i = 0; i < n; i++) {

            // If i is at middle of n and
            // previous middle of n
            // then add 1 into the String
            if (i == mid || i == mid - 1) {
                s += "1";
            }

            // Otherwise, add 0 to the String
            else {
                s += "0";
            }
        }
    }

    // If n is odd then follow this block
    else {
        for (int i = 0; i < n; i++) {

            // If i is equal to mid then
            // add 1 to the String
            if (i == mid) {
                s += "1";
            }

            // Otherwise add 0 to the String
            else {
                s += "0";
            }
        }

        // Return the constructed String
        return s;
    }
    return s;
}

// Driver code
public static void main(String[] args)
{
    int N = 4;

    // Function call
    System.out.print(makeString(N));
}
}

// This code contributed by shikhasingrajput 

# Python code to implement the approach
def makeString(n):
    s = ""
    
    # If n is even then follow this block
    mid = n/2
    if(n % 2 == 0):
        for i in range(n):
            # If i is at middle of n and previous 
            # middle of n then add 1 into the string
            if(i == mid or i == mid-1):
                s += "1"
            # Otherwise, add 0 to the string.
            else:
                s += "0"

    # If n is odd then follow this block
    else:
        for i in range(n):
            # If i is equal to mid then add 1 to the string
            if i == mid:
                s += "1"
            # Otherwise add 0 to the string
            else:
                s += "0"
                # Return the constructed String
        return s
    return s

N = 4

# Function call
print(makeString(N))

# This code is contributed by lokeshmvs21.

// C# code to implement the approach

using System;

public class GFG{

static String makeString(int n)
{
    string s = "";

    // If n is even then follow this block
    int mid = n / 2;
    if (n % 2 == 0) {
        for (int i = 0; i < n; i++) {

            // If i is at middle of n and
            // previous middle of n
            // then add 1 into the String
            if (i == mid || i == mid - 1) {
                s += "1";
            }

            // Otherwise, add 0 to the String
            else {
                s += "0";
            }
        }
    }

    // If n is odd then follow this block
    else {
        for (int i = 0; i < n; i++) {

            // If i is equal to mid then
            // add 1 to the String
            if (i == mid) {
                s += "1";
            }

            // Otherwise add 0 to the String
            else {
                s += "0";
            }
        }

        // Return the constructed String
        return s;
    }
    return s;
}

// Driver code
public static void Main(string[] args)
{
    int N = 4;

    // Function call
    Console.WriteLine(makeString(N));
}
}

// This code contributed by AnkThon

<script>
// Javascript program for above approach
function makeString(n)
{
    let s = "";

    // If n is even then follow this block
    let mid = n / 2;
    if (n % 2 == 0) {
        for (let i = 0; i < n; i++) {

            // If i is at middle of n and
            // previous middle of n
            // then add 1 into the String
            if (i == mid || i == mid - 1) {
                s += "1";
            }

            // Otherwise, add 0 to the String
            else {
                s += "0";
            }
        }
    }

    // If n is odd then follow this block
    else {
        for (let i = 0; i < n; i++) {

            // If i is equal to mid then
            // add 1 to the String
            if (i == mid) {
                s += "1";
            }

            // Otherwise add 0 to the String
            else {
                s += "0";
            }
        }

        // Return the constructed String
        return s;
    }
    return s;
}

// Driver Code
    let N = 4;

    // Function call
    document.write(makeString(N));
     
     // This code is contributed by code_hunt.
</script>

0110

Time Complexity: O(N)
Auxiliary Space: O(N)