Minimum cost to remove all 1s from given Binary String as per given conditions

Given a binary sequence of 1's and 0's. Our task is to remove all 1s from the sequence in minimum cost by below operations.

• Remove an element from the left end (i.e., remove s[0]) which costs 1 coin.
• Remove an element from the right end (i.e., remove s[s.length - 1]) which costs 1 coin.
• Remove an element from anywhere in the sequence which costs 2 coins.

Return the minimum cost to remove all the 1s from the sequence.

Examples:

Input: s = "1100101"
Output: 5
Explanation:
Remove left most "1100101" in cost of 1 coin, new s = "100101"
Remove left most "100101" in cost of 1 coin, new s = "00101"
Remove right most "00101" in cost of 1 coin, new s = "0010"
Remove middle 1 ("0010") in cost of 2 coin, new s = "000".
So total 5 coins are required

Input: s = "0010"
Output: 2
Explanation: Remove middle 1 ("0010") in cost of 2 coin, new s = "000"

Approach:  The idea is to use DP with sliding window. Use an array dp[N] where dp[i] stores the minimum coin to remove all 1s from index i to n-1, where we can only remove from right. Follow the below steps.

• Loop from the rightmost index. So, for any index i the transition is as follows:
  • If it's a '0' then dp[i]=dp[i+1]
  • If it's a '1' then we have two options - either consider this element as a middle one and add 2 to dp[i+1] or remove all elements. Hence, dp[i]=min(dp[i+1]+2, n-i)
• Now,  also remove elements from left. So,  loop over the vector.
  • When we reach any index i, we will consider that we have removed all elements from 0 to i from the leftside.
  • So, our time for that index will be i+1+dp[i+1].
  • Since, we have removed all elements until i, we need to just consider elements from i+1 and hence we are adding the dp[i+1] to our i+1.
• Finally return minimum answer

Below is the code implementation:

// C++ program for find the minimum cost
// to remove all 1's from the given binary string
#include <bits/stdc++.h>
using namespace std;

// Function to find minimum cost
int minimumCoin(string s)
{
    int n = s.length();
    vector<int> dp(n, 0);
    if (s[n - 1] == '0') {
        dp[n - 1] = 0;
    }
    else {
        dp[n - 1] = 1;
    }
    for (int i = n - 2; i >= 0; i--) {
        if (s[i] == '0')
            dp[i] = dp[i + 1];
        if (s[i] == '1') {

            // consider current index to
            // be a middle one and remove
            dp[i] = 2 + dp[i + 1];

            // or remove all to the right
            dp[i] = min(dp[i], n - i);
        }
    }

    // now go from left to right
    int ans = dp[0];
    for (int i = 0; i < n - 1; i++) {

        // cost of removing all
        // from left and dp[i+1]
        ans = min(ans, i + 1 + dp[i + 1]);
    }

    // taking overall minimum
    ans = min(ans, n);
    return ans;
}

// Driver function
int main()
{

    string str = "1001001";
    int coins = minimumCoin(str);
    cout << coins;
    return 0;
}

//Java program for find the minimum cost
// to remove all 1's from the given binary string
import java.io.*;

class GFG {

  // Function to find minimum cost
  static int minimumCoin(String s)
  {
    int n = s.length();
    int dp[] = new int[n];
    if (s.charAt(n - 1) == '0') {
      dp[n - 1] = 0;
    }
    else {
      dp[n - 1] = 1;
    }
    for (int i = n - 2; i >= 0; i--) {
      if (s.charAt(i) == '0')
        dp[i] = dp[i + 1];
      if (s.charAt(i) == '1') {

        // consider current index to
        // be a middle one and remove
        dp[i] = 2 + dp[i + 1];

        // or remove all to the right
        dp[i] = Math.min(dp[i], n - i);
      }
    }

    // now go from left to right
    int ans = dp[0];
    for (int i = 0; i < n - 1; i++) {

      // cost of removing all
      // from left and dp[i+1]
      ans = Math.min(ans, i + 1 + dp[i + 1]);
    }

    // taking overall minimum
    ans = Math.min(ans, n);
    return ans;
  }

  // Driver function
  public static void main (String[] args) {
    String str = "1001001";
    int coins = minimumCoin(str);
    System.out.println(coins);
  }
}

// This code is contributed by hrithikgarg03188.

# Python code for the above approach

# Function to find minimum cost
def minimumCoin(s):

    n = len(s)
    dp = [0]*n
    if (s[n - 1] == '0'):
        dp[n - 1] = 0

    else:
        dp[n - 1] = 1

    for i in range(n-2, -1, -1):
        if s[i] == '0':
            dp[i] = dp[i + 1]
        if s[i] == '1':

            # consider current index to
            # be a middle one and remove
            dp[i] = 2 + dp[i + 1]

            # or remove all to the right
            dp[i] = min(dp[i], n - i)

    # now go from left to right
    ans = dp[0]
    for i in range(0, n-1):

        # cost of removing all
        # from left and dp[i+1]
        ans = min(ans, i + 1 + dp[i + 1])

    # taking overall minimum
    ans = min(ans, n)
    return ans

# Driver function
str = "1001001"
coins = minimumCoin(str)
print(coins)

# This code is contributed by Potta Lokesh

// C# program for find the minimum cost
// to remove all 1's from the given binary string
using System;

class GFG {

  // Function to find minimum cost
  static int minimumCoin(string s)
  {
    int n = s.Length;
    int[] dp = new int[n];
    if (s[n - 1] == '0') {
      dp[n - 1] = 0;
    }
    else {
      dp[n - 1] = 1;
    }
    for (int i = n - 2; i >= 0; i--) {
      if (s[i] == '0')
        dp[i] = dp[i + 1];
      if (s[i] == '1') {

        // consider current index to
        // be a middle one and remove
        dp[i] = 2 + dp[i + 1];

        // or remove all to the right
        dp[i] = Math.Min(dp[i], n - i);
      }
    }

    // now go from left to right
    int ans = dp[0];
    for (int i = 0; i < n - 1; i++) {

      // cost of removing all
      // from left and dp[i+1]
      ans = Math.Min(ans, i + 1 + dp[i + 1]);
    }

    // taking overall minimum
    ans = Math.Min(ans, n);
    return ans;
  }

  // Driver function
  public static void Main()
  {
    string str = "1001001";
    int coins = minimumCoin(str);
    Console.Write(coins);
  }
}

// This code is contributed by Samim Hossain Mondal.

    <script>
        // JavaScript program for find the minimum cost
        // to remove all 1's from the given binary string

        // Function to find minimum cost
        const minimumCoin = (s) => {
            let n = s.length;
            let dp = new Array(n).fill(0);
            if (s[n - 1] == '0') {
                dp[n - 1] = 0;
            }
            else {
                dp[n - 1] = 1;
            }
            for (let i = n - 2; i >= 0; i--) {
                if (s[i] == '0')
                    dp[i] = dp[i + 1];
                if (s[i] == '1') {

                    // consider current index to
                    // be a middle one and remove
                    dp[i] = 2 + dp[i + 1];

                    // or remove all to the right
                    dp[i] = Math.min(dp[i], n - i);
                }
            }

            // now go from left to right
            let ans = dp[0];
            for (let i = 0; i < n - 1; i++) {

                // cost of removing all
                // from left and dp[i+1]
                ans = Math.min(ans, i + 1 + dp[i + 1]);
            }

            // taking overall minimum
            ans = Math.min(ans, n);
            return ans;
        }

        // Driver function
        let str = "1001001";
        let coins = minimumCoin(str);
        document.write(coins);

    // This code is contributed by rakeshsahni

    </script>

4

Time complexity: O(n)
Space complexity: O(n)