Minimum removals to segregate all 0s and 1s such that count of 0s and 1s are equal

Given a binary string S of length N, the task is to find the minimum number of removals to segregate all the 0s and all the 1s such that all the 0s are before all the 1s and their count are also same.

Examples:

Input: S = 01001101
Output: 2
Explanation: If you remove the 1 at index 1 and 
0 at index 6, then the string becomes 000111.

Input: S = “01”
Output: 0

Approach: The problem can be solved based on the concept of prefix sum:

Instead of trying to find the minimum number of removals find the maximum length subsequence which satisfies the condition.
To do this,  

• Firstly calculate prefix sum for 0s and suffix sum for 1s and 
• For any index i find the maximum length of the string that can be formed using 0s from the left of i and 1s from the right of i when their count are equal.
• The maximum among these lengths, is the subsequence of maximum size satisfying the criteria.
• The remaining elements must be removed.

Follow the below steps to solve the problem:

• Store prefix count of ‘0‘ in a vector left.
• Store Suffix count of ‘1‘ in a vector right.
• Initialize a variable Max = 0.
• Run a loop from index 1 to N-1.
  • Check if left[i-1] = right[i], then update, Max = max(Max, 2*right[i]).
• Return N-Max.

Below is the implementation of the above approach:

C++

// C++ code to implement the above approach

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

// Function to remove minimum number
// of brackets to make the string beautiful
int findMinRemoval(string s)
{
    int n = s.size();

    // Stores count of 0 from left to right
    vector<int> left(n);

    // Stores count of 1 from right to left
    vector<int> right(n);

    if (s[0] == '0')
        left[0] = 1;
    else
        left[0] = 0;

    // Calculate the prefix count of '0'
    // at every index
    for (int i = 1; i < n; i++) {
        if (s[i] == '0')
            left[i] = left[i - 1] + 1;
        else
            left[i] = left[i - 1];
    }

    if (s[n - 1] == '1')
        right[n - 1] = 1;
    else
        right[n - 1] = 0;

    // Calculate the suffix count of '1'
    // at every index
    for (int i = n - 2; i >= 0; i--) {
        if (s[i] == '1')
            right[i] = right[i + 1] + 1;
        else
            right[i] = right[i + 1];
    }

    int Max = 0;

    // Check for max length beautiful string
    for (int i = 1; i < n; i++) {
        if (left[i - 1] == right[i])
            Max = max(Max, 2 * right[i]);
    }
    return n - Max;
}

// Driver Code
int main()
{
    string S = "01001101";

    // Function call
    cout << findMinRemoval(S);
    return 0;
}

Java

// Java code to implement the above approach
import java.io.*;

class GFG {
    // Function to remove minimum number
    // of brackets to make the string beautiful
    public static int findMinRemoval(String s)
    {
        int n = s.length();

        // Stores count of 0 from left to right
        int left[] = new int[n];

        // Stores count of 1 from right to left
        int right[] = new int[n];

        if (s.charAt(0) == '0')
            left[0] = 1;
        else
            left[0] = 0;

        // Calculate the prefix count of '0'
        // at every index
        for (int i = 1; i < n; i++) {
            if (s.charAt(i) == '0')
                left[i] = left[i - 1] + 1;
            else
                left[i] = left[i - 1];
        }

        if (s.charAt(n - 1) == '1')
            right[n - 1] = 1;
        else
            right[n - 1] = 0;

        // Calculate the suffix count of '1'
        // at every index
        for (int i = n - 2; i >= 0; i--) {
            if (s.charAt(i) == '1')
                right[i] = right[i + 1] + 1;
            else
                right[i] = right[i + 1];
        }

        int Max = 0;

        // Check for max length beautiful string
        for (int i = 1; i < n; i++) {
            if (left[i - 1] == right[i])
                Max = Math.max(Max, 2 * right[i]);
        }
        return n - Max;
    }

    // Driver Code
    public static void main(String[] args)
    {
        String S = "01001101";

        // Function call
        System.out.print(findMinRemoval(S));
    }
}

// This code is contributed by Rohit Pradhan

Python3

# Python3 code to implement the above approach

# Function to remove minimum number
# of brackets to make the string beautiful
def findMinRemoval(s):

    n = len(s)

    # Stores count of 0 from left to right
    left = [0] * n

    # Stores count of 1 from right to left
    right = [0] * n

    if (s[0] == '0'):
        left[0] = 1
    else:
        left[0] = 0

    # Calculate the prefix count of '0'
    # at every index
    for i in range(1, n):
        if (s[i] == '0'):
            left[i] = left[i - 1] + 1
        else:
            left[i] = left[i - 1];

    if (s[n - 1] == '1'):
        right[n - 1] = 1
    else:
        right[n - 1] = 0

    # Calculate the suffix count of '1'
    # at every index
    for i in range(n-2, -1, -1):
        if (s[i] == '1'):
            right[i] = right[i + 1] + 1
        else:
            right[i] = right[i + 1]

    Max = 0

    # Check for max length beautiful string
    for i in range(1,n):
        if (left[i - 1] == right[i]):
            Max = max(Max, 2 * right[i])
    return n - Max;

# Driver Code
if __name__ == "__main__":

    S = "01001101"

    # Function call
    print(findMinRemoval(S))

  # This code is contributed by hrithikgarg03188.

C#

// C# program for above approach:
using System;
class GFG {

  // Function to remove minimum number
  // of brackets to make the string beautiful
  static int findMinRemoval(string s)
  {
    int n = s.Length;

    // Stores count of 0 from left to right
    int[] left = new int[n];

    // Stores count of 1 from right to left
    int[] right = new int[n];

    if (s[0] == '0')
      left[0] = 1;
    else
      left[0] = 0;

    // Calculate the prefix count of '0'
    // at every index
    for (int i = 1; i < n; i++) {
      if (s[i] == '0')
        left[i] = left[i - 1] + 1;
      else
        left[i] = left[i - 1];
    }

    if (s[n - 1] == '1')
      right[n - 1] = 1;
    else
      right[n - 1] = 0;

    // Calculate the suffix count of '1'
    // at every index
    for (int i = n - 2; i >= 0; i--) {
      if (s[i] == '1')
        right[i] = right[i + 1] + 1;
      else
        right[i] = right[i + 1];
    }

    int Maxx = 0;

    // Check for max length beautiful string
    for (int i = 1; i < n; i++) {
      if (left[i - 1] == right[i])
        Maxx = Math.Max(Maxx, 2 * right[i]);
    }
    return n - Maxx;
  }

  // Driver Code
  public static void Main()
  {
    String S = "01001101";

    // Function call
    Console.Write(findMinRemoval(S));
  }
}

// This code is contributed by code_hunt.

Javascript

<script>
    // JavaScript code to implement the above approach


    // Function to remove minimum number
    // of brackets to make the string beautiful
    const findMinRemoval = (s) => {
        let n = s.length;

        // Stores count of 0 from left to right
        let left = new Array(n).fill(0);

        // Stores count of 1 from right to left
        let right = new Array(n).fill(0);

        if (s[0] == '0')
            left[0] = 1;
        else
            left[0] = 0;

        // Calculate the prefix count of '0'
        // at every index
        for (let i = 1; i < n; i++) {
            if (s[i] == '0')
                left[i] = left[i - 1] + 1;
            else
                left[i] = left[i - 1];
        }

        if (s[n - 1] == '1')
            right[n - 1] = 1;
        else
            right[n - 1] = 0;

        // Calculate the suffix count of '1'
        // at every index
        for (let i = n - 2; i >= 0; i--) {
            if (s[i] == '1')
                right[i] = right[i + 1] + 1;
            else
                right[i] = right[i + 1];
        }

        let Max = 0;

        // Check for max length beautiful string
        for (let i = 1; i < n; i++) {
            if (left[i - 1] == right[i])
                Max = Math.max(Max, 2 * right[i]);
        }
        return n - Max;
    }

    // Driver Code

    let S = "01001101";

    // Function call
    document.write(findMinRemoval(S));

// This code is contributed by rakeshsahni

</script>

Output

2

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