Minimum characters required to be removed to sort binary string in ascending order - Set 2

Given binary string str of size N, the task is to remove the minimum number of characters from the given binary string such that the characters in the remaining string are in sorted order.

Examples:

Input: str = “1000101”
Output: 2
Explanation: Removal of the first two occurrences of ‘1’ modifies the string to “00001”, which is a sorted order. The string can also be made "00011" by performing 2 removals. Therefore, the minimum count of characters to be removed is 2.

Input: str = “001111”
Output: 0
Explanation: The string is already sorted. Therefore, the minimum count of character to be removed is 0.

Last Occurrence Approach: The optimized approach in linear time and constant space is discussed in Set 1 of this article. Here, we are discussing the Dynamic Programming Approach.

Dynamic Programming Approach: This problem can be solved using dynamic programming by observing the following facts, if K deletion is required to make the string sorted till i^th index and

• Case1: S[i+1] = 1 then minimum number of deletions required to make string sorted till (i+1)th index will also be K as appending 1 to a sorted string will keep string sorted so no more deletion required.
• Case2: S[i + 1] = 0 then we have two way to make string sorted till (i+1)th index that are
  • either delete all 1's before (i+1)th index, or
  • delete current 0.

Minimum number of deletion to make string valid till (i+1)th index will be minimum of (numbers of 1's before (i+1)th index , K+1).

Follow the steps below to solve the problem:

• Initialize the variables count1 as 0 and N is the length of the string s.
• Initialize the vector dp[n+1] with values 0.
• Iterate over the range [0, n) using the variable i and perform the following tasks:
  • If s[i] equals 0, then set dp[i+1] as the minimum of count1 or 1 + dp[i].
  • Else, set dp[i+1] as dp[i] and increase the value of count1 by 1.
• After performing the above steps, print the value of dp[n] as the answer.

Below is the implementation of the above approach:

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the minimum number
// of deletions to make
// the string sorted
int minDeletions(string s)
{
    int n = s.size();

    // dp[i+1] stores minimum number of
    // deletion to make
    // substring(0, i) valid
    vector<int> dp(n + 1, 0);
    int count1 = 0;

    for (int i = 0; i < n; i++) {
        if (s[i] == '0') {

            // Case 1: remove current 0
            // Case 2: keep current 0
            // then delete all 1 before it
            dp[i + 1] = min(count1, 1 + dp[i]);
        }
        else {

            // Appending 1 is always valid
            // if substring(0, i) is sorted
            dp[i + 1] = dp[i];
            count1++;
        }
    }
    return dp[n];
}

// Driver Code
int main()
{
    string s = "00101101";
    cout << minDeletions(s);
    return 0;
}

// Java program for the above approach
import java.util.*;
public class GFG {

    // Function to find the minimum number
    // of deletions to make
    // the string sorted
    static int minDeletions(String s)
    {
        int n = s.length();

        // dp[i+1] stores minimum number of
        // deletion to make
        // substring(0, i) valid
        int[] dp = new int[n + 1];
        for (int i = 0; i < n + 1; i++) {
            dp[i] = 0;
        }
        int count1 = 0;

        for (int i = 0; i < n; i++) {
            if (s.charAt(i) == '0') {

                // Case 1: remove current 0
                // Case 2: keep current 0
                // then delete all 1 before it
                dp[i + 1] = Math.min(count1, 1 + dp[i]);
            }
            else {

                // Appending 1 is always valid
                // if substring(0, i) is sorted
                dp[i + 1] = dp[i];
                count1++;
            }
        }
        return dp[n];
    }

    // Driver Code
    public static void main(String args[])
    {
        String s = "00101101";
        System.out.println(minDeletions(s));
    }
}

// This code is contributed by Samim Hossain Mondal.

# Python code for the above approach

# Function to find the minimum number
# of deletions to make
# the string sorted


def minDeletions(s):
    n = len(s)

    # dp[i+1] stores minimum number of
    # deletion to make
    # substring(0, i) valid
    dp = [0] * (n + 1)
    count1 = 0

    for i in range(n):
        if (s[i] == '0'):

            # Case 1: remove current 0
            # Case 2: keep current 0
            # then delete all 1 before it
            dp[i + 1] = min(count1, 1 + dp[i])
        else:

            # Appending 1 is always valid
            # if substring(0, i) is sorted
            dp[i + 1] = dp[i]
            count1 += 1
    return dp[n]


# Driver Code
s = "00101101"
print(minDeletions(s))

# This code is contributed by Saurabh Jaiswal

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

    // Function to find the minimum number
    // of deletions to make
    // the string sorted
    static int minDeletions(string s)
    {
        int n = s.Length;

        // dp[i+1] stores minimum number of
        // deletion to make
        // substring(0, i) valid
        int[] dp = new int[n + 1];
        for (int i = 0; i < n + 1; i++) {
            dp[i] = 0;
        }
        int count1 = 0;

        for (int i = 0; i < n; i++) {
            if (s[i] == '0') {

                // Case 1: remove current 0
                // Case 2: keep current 0
                // then delete all 1 before it
                dp[i + 1] = Math.Min(count1, 1 + dp[i]);
            }
            else {

                // Appending 1 is always valid
                // if substring(0, i) is sorted
                dp[i + 1] = dp[i];
                count1++;
            }
        }
        return dp[n];
    }

    // Driver Code
    public static void Main()
    {
        string s = "00101101";
        Console.Write(minDeletions(s));
    }
}

// This code is contributed by Saurabh Jaiswal

<script>
        // JavaScript code for the above approach


        // Function to find the minimum number
        // of deletions to make
        // the string sorted
        function minDeletions(s) {
            let n = s.length;

            // dp[i+1] stores minimum number of
            // deletion to make 
            // substring(0, i) valid
            let dp = new Array(n + 1).fill(0);
            let count1 = 0;

            for (let i = 0; i < n; i++) {
                if (s[i] == '0') {

                    // Case 1: remove current 0
                    // Case 2: keep current 0 
                    // then delete all 1 before it
                    dp[i + 1] = Math.min(count1,
                        1 + dp[i]);
                }
                else {

                    // Appending 1 is always valid
                    // if substring(0, i) is sorted
                    dp[i + 1] = dp[i];
                    count1++;
                }
            }
            return dp[n];
        }

        // Driver Code

        let s = "00101101";
        document.write(minDeletions(s));

  // This code is contributed by Potta Lokesh
    </script>

2

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

Efficient Approach: Space Optimization Approach

In this approach we can improve the space complexity of the algorithm by using two variables to keep track of the minimum and maximum positions of the given element in the array instead of using a 2D vector dp. This will reduce the space complexity from O(N) to O(1).  

Steps:

• Declare function minDeletions that take a string as an argument. 
• Initialize the variable next and curr to 0 that refers to the current and previous element of dp.
• Now the approach is the same as the previous code but to optimize the space we change dp[i] to curr and dp[i+1] to next because dp[i] is only dependent on dp[i+1] and dp[i].

Implementation : 

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the minimum number
// of deletions to make
// the string sorted
int minDeletions(string s)
{
    int n = s.size();

    int next = 0;
    int curr = 0;

    int count1 = 0;

    for (int i = 0; i < n; i++) {
        if (s[i] == '0') {
            // Case 1: remove current 0
            // Case 2: keep current 0
            // then delete all 1 before it

            next = min(count1, 1 + curr);
        }
        else {
            // Appending 1 is always valid
            // if substring(0, i) is sorted
            next = curr;
            count1++;
        }

        // assigning value of next element to current
        // element and traverse further
        curr = next;
    }
    return next;
}

// Driver Code
int main()
{
    string s = "00101101";
    cout << minDeletions(s);
    return 0;
}

// this code is contributed by bhardwajji

# Function to find the minimum number
# of deletions to make the string sorted


def minDeletions(s: str) -> int:
    n = len(s)
    next, curr, count1 = 0, 0, 0

    for i in range(n):
        if s[i] == '0':
            # Case 1: remove current 0
            # Case 2: keep current 0
            # then delete all 1 before it
            next = min(count1, 1 + curr)
        else:
            # Appending 1 is always valid
            # if substring(0, i) is sorted
            next = curr
            count1 += 1

        # assigning value of next element to current
        # element and traverse further
        curr = next
    return next


# Driver code
s = "00101101"
print(minDeletions(s))

using System;

public class Gfg {
    // Function to find the minimum number
    // of deletions to make
    // the string sorted
    static int minDeletions(string s)
    {
        int n = s.Length;
        int next = 0;
        int curr = 0;
        int count1 = 0;

        for (int i = 0; i < n; i++) {
            if (s[i] == '0') {
                // Case 1: remove current 0
                // Case 2: keep current 0
                // then delete all 1 before it

                next = Math.Min(count1, 1 + curr);
            }
            else {
                // Appending 1 is always valid
                // if substring(0, i) is sorted
                next = curr;
                count1++;
            }

            // assigning value of next element to current
            // element and traverse further
            curr = next;
        }
        return next;
    }

    // Driver Code
    public static void Main()
    {
        string s = "00101101";
        Console.WriteLine(minDeletions(s));
    }
}

// Javascript program for the above approach

// Function to find the minimum number of deletions to make the string sorted
function minDeletions(s) {
    let n = s.length;

    let next = 0;
    let curr = 0;
    
    let count1 = 0;

    for (let i = 0; i < n; i++) {
        if (s[i] == '0') {
            // Case 1: remove current 0
            // Case 2: keep current 0 then delete all 1 before it
            next = Math.min(count1,1 + curr);
        } else {
            // Appending 1 is always valid if substring(0, i) is sorted
            next = curr;
            count1++;
        }
        
        // assigning value of next element to current element and traverse further 
        curr = next;
    }
    return next;
}

// Test the code with an example value of s
let s = "00101101";
console.log(minDeletions(s));

import java.util.*;

public class Main {

    // Function to find the minimum number
    // of deletions to make string sorted
    public static int minDeletions(String s)
    {
        int n = s.length();

        int next = 0;
        int curr = 0;

        int count1 = 0;

        for (int i = 0; i < n; i++) {
            if (s.charAt(i) == '0') {

                // Case 1: remove current 0
                // Case 2: keep current 0
                // then delete all 1 before it
                next = Math.min(count1, 1 + curr);
            }
            else {

                // Appending 1 is always valid
                // if substring(0, i) is sorted
                next = curr;
                count1++;
            }

            // assigning value of next element
            // to current element and
            // traverse further
            curr = next;
        }
        return next;
    }

    // Driver Code
    public static void main(String[] args)
    {
        String s = "00101101";
        System.out.println(minDeletions(s));
    }
}

2

Time Complexity:  O(N)
Auxiliary Space: O(1) 

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Article Tags :

• Strings
• Dynamic Programming
• DSA
• binary-string

Practice Tags :

• Dynamic Programming
• Strings