Minimum Cost required to generate a balanced Bracket Sequence

Given a string str of length N, representing a bracket sequence, and two integers A and B, the task is to find the minimum cost required to obtain a regular bracket sequence from str by performing any number of moves(possibly zero) of the following types: 

• Remove a character from the string for a cost A.
• Remove a character from the string and append at the end of the string for a cost B.

A balanced bracket sequence can be of the following types: 

• Empty string
• A string consisting of a closing bracket corresponding to every opening bracket.

Examples:

Input: str = “)()”, A = 1, B = 2 
Output: 1 
Explanation:
Removal of the 0^th character, that is, ‘)’, costs 1, generating a balanced string “()”. Therefore, the minimum cost is 1.

Input: str = “)(“, A = 3, B = 9 
Output: 6 
Explanation:
Removal of the 0^th character and appending at the end of the string generates a balanced string “()”. 
Therefore, cost = 9. 
Removal of both the characters generates an empty string for a cost 6. 
Therefore, the minimum cost to generate a balanced string is 6.

Approach: Follow the steps below to solve the problem: 

• Count the frequencies of opening ‘(‘ and closing ‘)’ brackets in the given string and store the one more frequent of the two.
• Minimum cost will be at least a * (abs(open – count)), as these brackets need to be removed in order to balance the string.
• Count the number of unbalanced open and closing brackets in the string. If the open brackets are excess , then reduce the count of unbalanced open brackets by count of excess open brackets. Similarly, reduce count of unbalanced closing brackets if closing brackets are excess.
• Now, calculate the cost of removing all unbalanced open and unbalanced closed brackets as well as the cost of removing unbalanced closed brackets and adding them to the end. Compare and add the minimum of the two costs to the answer.
• Therefore, the minimum cost required to generate a balanced bracket sequence is given by the following equation:

Minimum Cost to generate a balanced string = a * (abs(open – close)) + min( a*(unbalanced open + unbalanced closed), b*(unbalanced closed parenthesis)) 

Below is the implementation of the above approach:

C++

// C++ Program to implement
// the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to calculate the minimum cost
// required to generate a balanced bracket
// sequence
void minCost(string str, int a, int b)
{
    // Stores the count of
    // unbalanced open brackets
    int openUnbalanced = 0;

    // Stores the count of
    // unbalanced closed brackets
    int closedUnbalanced = 0;

    // Stores the count of
    // open brackets
    int openCount = 0;

    // Stores the count of
    // closed brackets
    int closedCount = 0;

    for (int i = 0; str[i] != '\0'; i++) {

        // If open brace is encountered
        if (str[i] == '(') {
            openUnbalanced++;
            openCount++;
        }

        // Otherwise
        else {

            // If no unbalanced open
            // brackets are present
            if (openUnbalanced == 0)

                // Increase count of
                // unbalanced closed brackets
                closedUnbalanced++;

            // Otherwise
            else

                // Reduce count of
                // unbalanced open brackets
                openUnbalanced--;

            // Increase count of
            // closed brackets
            closedCount++;
        }
    }

    // Calculate lower bound of minimum cost
    int result = a * (abs(openCount
                          - closedCount));

    // Reduce excess open or closed brackets
    // to prevent counting them twice
    if (closedCount > openCount)
        closedUnbalanced
            -= (closedCount - openCount);

    if (openCount > closedCount)
        openUnbalanced
            -= (openCount - closedCount);

    // Update answer by adding minimum of
    // removing both unbalanced open and
    // closed brackets or inserting closed
    // unbalanced brackets to end of string
    result += min(a * (openUnbalanced
                       + closedUnbalanced),
                  b * closedUnbalanced);

    // Print the result
    cout << result << endl;
}

// Driver Code
int main()
{
    string str = "))()(()()(";
    int A = 1, B = 3;
    minCost(str, A, B);

    return 0;
}

Java

// Java program to implement
// the above approach
import java.util.*;

class GFG{

// Function to calculate the minimum cost
// required to generate a balanced bracket
// sequence
static void minCost(String str, int a, int b)
{

    // Stores the count of
    // unbalanced open brackets
    int openUnbalanced = 0;

    // Stores the count of
    // unbalanced closed brackets
    int closedUnbalanced = 0;

    // Stores the count of
    // open brackets
    int openCount = 0;

    // Stores the count of
    // closed brackets
    int closedCount = 0;

    for(int i = 0; i < str.length(); i++)
    {

        // If open brace is encountered
        if (str.charAt(i) == '(')
        {
            openUnbalanced++;
            openCount++;
        }

        // Otherwise
        else
        {

            // If no unbalanced open
            // brackets are present
            if (openUnbalanced == 0)

                // Increase count of
                // unbalanced closed brackets
                closedUnbalanced++;

            // Otherwise
            else

                // Reduce count of
                // unbalanced open brackets
                openUnbalanced--;

            // Increase count of
            // closed brackets
            closedCount++;
        }
    }

    // Calculate lower bound of minimum cost
    int result = a * (Math.abs(openCount -
                             closedCount));

    // Reduce excess open or closed brackets
    // to prevent counting them twice
    if (closedCount > openCount)
        closedUnbalanced -= (closedCount -
                               openCount);

    if (openCount > closedCount)
        openUnbalanced -= (openCount -
                         closedCount);

    // Update answer by adding minimum of
    // removing both unbalanced open and
    // closed brackets or inserting closed
    // unbalanced brackets to end of String
    result += Math.min(a * (openUnbalanced +
                            closedUnbalanced),
                        b * closedUnbalanced);

    // Print the result
    System.out.print(result + "\n");
}

// Driver Code
public static void main(String[] args)
{
    String str = "))()(()()(";
    int A = 1, B = 3;

    minCost(str, A, B);
}
}

// This code is contributed by amal kumar choubey

Python3

# Python3 program to implement
# the above approach

# Function to calculate the minimum cost
# required to generate a balanced bracket
# sequence
def minCost(str, a, b):

    # Stores the count of
    # unbalanced open brackets
    openUnbalanced = 0;

    # Stores the count of
    # unbalanced closed brackets
    closedUnbalanced = 0;

    # Stores the count of
    # open brackets
    openCount = 0;

    # Stores the count of
    # closed brackets
    closedCount = 0;

    for i in range(len(str)):

        # If open brace is encountered
        if (str[i] == '('):
            openUnbalanced += 1;
            openCount += 1;

        # Otherwise
        else:

            # If no unbalanced open
            # brackets are present
            if (openUnbalanced == 0):

                # Increase count of
                # unbalanced closed brackets
                closedUnbalanced += 1;

            # Otherwise
            else:

                # Reduce count of
                # unbalanced open brackets
                openUnbalanced -= 1;

            # Increase count of
            # closed brackets
            closedCount += 1;

    # Calculate lower bound of minimum cost
    result = a * (abs(openCount - closedCount));

    # Reduce excess open or closed brackets
    # to prevent counting them twice
    if (closedCount > openCount):
        closedUnbalanced -= (closedCount - openCount);

    if (openCount > closedCount):
        openUnbalanced -= (openCount - closedCount);

    # Update answer by adding minimum of
    # removing both unbalanced open and
    # closed brackets or inserting closed
    # unbalanced brackets to end of String
    result += min(a * (openUnbalanced +
                       closedUnbalanced),
                   b * closedUnbalanced);

    # Print the result
    print(result);

# Driver Code
if __name__ == '__main__':
    str = "))()(()()(";
    A = 1; B = 3;

    minCost(str, A, B);

# This code is contributed by Rohit_ranjan

C#

// C# program to implement
// the above approach
using System;
class GFG{

// Function to calculate the minimum cost
// required to generate a balanced bracket
// sequence
static void minCost(String str, int a, int b)
{

    // Stores the count of
    // unbalanced open brackets
    int openUnbalanced = 0;

    // Stores the count of
    // unbalanced closed brackets
    int closedUnbalanced = 0;

    // Stores the count of
    // open brackets
    int openCount = 0;

    // Stores the count of
    // closed brackets
    int closedCount = 0;

    for(int i = 0; i < str.Length; i++)
    {

        // If open brace is encountered
        if (str[i] == '(')
        {
            openUnbalanced++;
            openCount++;
        }

        // Otherwise
        else
        {

            // If no unbalanced open
            // brackets are present
            if (openUnbalanced == 0)

                // Increase count of
                // unbalanced closed brackets
                closedUnbalanced++;

            // Otherwise
            else

                // Reduce count of
                // unbalanced open brackets
                openUnbalanced--;

            // Increase count of
            // closed brackets
            closedCount++;
        }
    }

    // Calculate lower bound of minimum cost
    int result = a * (Math.Abs(openCount -
                               closedCount));

    // Reduce excess open or closed brackets
    // to prevent counting them twice
    if (closedCount > openCount)
        closedUnbalanced -= (closedCount -
                             openCount);

    if (openCount > closedCount)
        openUnbalanced -= (openCount -
                           closedCount);

    // Update answer by adding minimum of
    // removing both unbalanced open and
    // closed brackets or inserting closed
    // unbalanced brackets to end of String
    result += Math.Min(a * (openUnbalanced +
                           closedUnbalanced),
                       b * closedUnbalanced);

    // Print the result
    Console.Write(result + "\n");
}

// Driver Code
public static void Main(String[] args)
{
    String str = "))()(()()(";
    int A = 1, B = 3;

    minCost(str, A, B);
}
}

// This code is contributed by gauravrajput1

Javascript

<script>
// JavaScript program for the
// above approach

// Function to calculate the minimum cost
// required to generate a balanced bracket
// sequence
function minCost(str, a, b)
{

    // Stores the count of
    // unbalanced open brackets
    let openUnbalanced = 0;

    // Stores the count of
    // unbalanced closed brackets
    let closedUnbalanced = 0;

    // Stores the count of
    // open brackets
    let openCount = 0;

    // Stores the count of
    // closed brackets
    let closedCount = 0;

    for(let i = 0; i < str.length; i++)
    {

        // If open brace is encountered
        if (str[i] == '(')
        {
            openUnbalanced++;
            openCount++;
        }

        // Otherwise
        else
        {

            // If no unbalanced open
            // brackets are present
            if (openUnbalanced == 0)

                // Increase count of
                // unbalanced closed brackets
                closedUnbalanced++;

            // Otherwise
            else

                // Reduce count of
                // unbalanced open brackets
                openUnbalanced--;

            // Increase count of
            // closed brackets
            closedCount++;
        }
    }

    // Calculate lower bound of minimum cost
    let result = a * (Math.abs(openCount -
                             closedCount));

    // Reduce excess open or closed brackets
    // to prevent counting them twice
    if (closedCount > openCount)
        closedUnbalanced -= (closedCount -
                               openCount);

    if (openCount > closedCount)
        openUnbalanced -= (openCount -
                         closedCount);

    // Update answer by adding minimum of
    // removing both unbalanced open and
    // closed brackets or inserting closed
    // unbalanced brackets to end of String
    result += Math.min(a * (openUnbalanced +
                            closedUnbalanced),
                        b * closedUnbalanced);

    // Print the result
    document.write(result + "\n");
}


// Driver Code

    let str = "))()(()()(";
    let A = 1, B = 3;

    minCost(str, A, B);

</script>

Output:

4

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