Calculate weight of parenthesis based on the given conditions

Given a valid parenthesis string S, the task is to find the weight of the parenthesis based on the following conditions:  

1. Weight of “( )” is 1
2. Weight of “AB” = weight of “A” + weight of “B” (where, A and B are both independent valid parenthesis). e.g. weight of “()()” = weight of “()” + weight of “()”
3. Weight of “(A)” = 2 times the weight of “A” (where A is an independent valid parenthesis). e.g. weight of “(())” is 2 times the weight of “()”

Examples:

Input: S = “()(())” 
Output: 3 
Explanation:
Weight of () = 1 
Weight of (()) = 2 
Hence, the weight of ()(()) = 1 + 2 = 3

Input: S = “(()(()))” 
Output: 6 
Explanation:
Weight of ()(()) = 3 
Weight of (()(())) = 2 * 3 = 6 

Approach:
This problem can be solved using the Divide and Conquer approach. Follow the steps below to solve the problem: 

• It is given that the input parenthesis string is always valid, i.e. balanced. So, any opening bracket ‘(‘ has a corresponding closing bracket ‘)’.
• Consider the opening bracket at the beginning of the input string (The beginning bracket can not be a closing bracket, otherwise it will not be valid). Now, for this opening bracket, the corresponding closing bracket can have any of the following two possible indices. 
  1. At the very end i.e. (n-1)^th index
  2. Somewhere between start and end i.e. [1, n-2]
• If the closing bracket has an index at the end, then according to constraint no. 3, the total weight of the parenthesis will be twice that of the weight of string[1, n-2].
  
• If the closing bracket is somewhere in between start and end, say mid, then according to constraint no. 2, the total weight of parenthesis will be the sum of the weight of string[start, mid] and the sum of the weight of string[mid+1, end]. 
  
• The base case for our recursion will be when we have only two brackets in the string, they will have weight 1 because inherently they will be valid. 
  
• Now, the question is how we can find out the index of the corresponding closing bracket for an opening bracket. The idea is similar to Valid Parenthesis Check. We will use the Stack Data Structure to check and store the index of the closing bracket for the corresponding opening bracket in a HashMap. 
  
• Perform the following steps: 
  • Traverse through the string.
  • If a character is an opening bracket, push its index into the Stack.
  • If it is a closing bracket, pop its index from the Stack and insert the (popped_index, current_index) pairing into the HashMap.

Below is the implementation of the above approach. 

C++

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

// HashMap to store the ending
// index of every opening bracket
unordered_map<int, int> endIndex;

// Function to calculate and store
// the closing index of each opening
// bracket in the parenthesis
void getClosingIndex(string s)
{
    int n = s.length();

    stack<int> st;

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

            // If it's a closing bracket,
            // pop index of it's corresponding
            // opening bracket
            int startIndex = st.top();
            st.pop();

            // Insert the index of opening
            // bracket and closing bracket
            // as key-value pair in the
            // hashmap
            endIndex[startIndex] = i;
        }
        else
        {

            // If it's an opening bracket,
            // push it's index into the stack
            st.push(i);
        }
    }
}

// Function to return the weight of
// parenthesis
int calcWeight(string s, int low, int high)
{

    // Base case
    if (low + 1 == high)
    {
        return 1;
    }

    else
    {

        // Mid refers to ending index of
        // opening bracket at index low
        int mid = endIndex[low];

        if (mid == high)
        {
            return 2 * calcWeight(s, low + 1,
                                    high - 1);
        }
        else
        {
            return calcWeight(s, low, mid) +
                   calcWeight(s, mid + 1,
                              high);
        }
    }
}

// Driver Code
int main()
{
    string input = "(()(()))";
    int n = input.length();

    // Update the closing Index
    getClosingIndex(input);

    cout << (calcWeight(input, 0, n - 1)) << endl;

    return 0;
}

// This code is contributed by divyeshrabadiya07

Java

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

public class GFG {

    // HashMap to store the ending
    // index of every opening bracket
    static HashMap<Integer, Integer> endIndex
        = new HashMap<Integer, Integer>();

    // Function to calculate and store
    // the closing index of each opening
    // bracket in the parenthesis
    public static void getClosingIndex(String s)
    {

        int n = s.length();

        Stack<Integer> st = new Stack<Integer>();

        for (int i = 0; i < n; i++) {

            if (s.charAt(i) == ')') {

                // If it's a closing bracket,
                // pop index of it's corresponding
                // opening bracket
                int startIndex = st.pop();

                // Insert the index of opening
                // bracket and closing bracket
                // as key-value pair in the
                // hashmap
                endIndex.put(startIndex, i);
            }
            else {

                // If it's an opening bracket,
                // push it's index into the stack
                st.push(i);
            }
        }
    }

    // Function to return the weight of
    // parenthesis
    public static int calcWeight(String s,
                                int low,
                                int high)
    {

        // Base case
        if (low + 1 == high) {
            return 1;
        }

        else {

            // Mid refers to ending index of
            // opening bracket at index low
            int mid = endIndex.get(low);
            if (mid == high) {
                return 2 * calcWeight(s, low + 1,
                                    high - 1);
            }
            else {
                return calcWeight(s, low, mid)
                    + calcWeight(s, mid + 1,
                                high);
            }
        }
    }

    public static void main(String[] args)
    {
        String input = "(()(()))";
        int n = input.length();

        // Update the closing Index
        getClosingIndex(input);

        System.out.println(calcWeight(input,
                                    0, n - 1));
    }
}

Python3

# Python3 program to implement the
# above approach

# Function to calculate and store
# the closing index of each opening
# bracket in the parenthesis
def getClosingIndex(string):

    # Dictionary to store
    # the ending index of
    # each opening bracket
    endIndex = dict()


    n = len(string)

    stack = []
    for i in range(n):
        if (string[i]==')'):

            # If it's a closing bracket,
            # pop index of it's
            # corresponding
            # opening bracket
            startIndex = stack.pop()

            # Put the index of opening
            # bracket and closing
            # bracket as key value
            # pair in the Dictionary
            endIndex[startIndex] = i

        else:

            # If it's an opening bracket,
            # push it's index into
            # the stack
            stack.append(i)
    return endIndex

# Function to return the weight
# of parenthesis
def calcWeight(s, low,
                high, endIndex):


    # Base case
    if (low + 1 == high):
        return 1
    else:

        # Mid refers to ending index of
        # opening bracket at index low
        mid = endIndex[low]
        if (mid == high):
            return 2*(calcWeight(s, low + 1,
            high-1, endIndex))

        else:
            return calcWeight(s, low,
            mid, endIndex) + calcWeight(s,
            mid + 1, high, endIndex)


if __name__ == "__main__":
    string = "(()(()))"
    n = len(string)
    endIndex = getClosingIndex(string)
    print(calcWeight(string, 0,
    n-1, endIndex))

C#

// C# program to implement
// the above approach
using System;
using System.Collections.Generic;

class GFG{

// HashMap to store the ending
// index of every opening bracket
static Dictionary<int,
                  int> endIndex = new Dictionary<int,
                                                 int>();

// Function to calculate and store
// the closing index of each opening
// bracket in the parenthesis
public static void getClosingIndex(string s)
{
    int n = s.Length;

    Stack<int> st = new Stack<int>();

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

            // If it's a closing bracket,
            // pop index of it's corresponding
            // opening bracket
            int startIndex = st.Pop();

            // Insert the index of opening
            // bracket and closing bracket
            // as key-value pair in the
            // hashmap
            endIndex.Add(startIndex, i);
        }
        else
        {

            // If it's an opening bracket,
            // push it's index into the stack
            st.Push(i);
        }
    }
}

// Function to return the weight of
// parenthesis
public static int calcWeight(string s, int low,
                                       int high)
{

    // Base case
    if (low + 1 == high)
    {
        return 1;
    }
    else
    {

        // Mid refers to ending index of
        // opening bracket at index low
        int mid = endIndex[low];
        if (mid == high)
        {
            return 2 * calcWeight(s, low + 1,
                                    high - 1);
        }
        else
        {
            return calcWeight(s, low, mid) +
                   calcWeight(s, mid + 1, high);
        }
    }
}

// Driver code
public static void Main(string[] args)
{
    string input = "(()(()))";
    int n = input.Length;

    // Update the closing Index
    getClosingIndex(input);

    Console.Write(calcWeight(input, 0, n - 1));
}
}

// This code is contributed by rutvik_56

Javascript

<script>

// Javascript program to implement
// the above approach

// HashMap to store the ending
// index of every opening bracket
var endIndex = new Map();

// Function to calculate and store
// the closing index of each opening
// bracket in the parenthesis
function getClosingIndex(s)
{
    var n = s.length;

    var st = [];

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

            // If it's a closing bracket,
            // pop index of it's corresponding
            // opening bracket
            var startIndex = st[st.length-1];
            st.pop();

            // Insert the index of opening
            // bracket and closing bracket
            // as key-value pair in the
            // hashmap
            endIndex[startIndex] = i;
        }
        else
        {

            // If it's an opening bracket,
            // push it's index into the stack
            st.push(i);
        }
    }
}

// Function to return the weight of
// parenthesis
function calcWeight(s, low, high)
{

    // Base case
    if (low + 1 == high)
    {
        return 1;
    }

    else
    {

        // Mid refers to ending index of
        // opening bracket at index low
        var mid = endIndex[low];

        if (mid == high)
        {
            return 2 * calcWeight(s, low + 1,
                                    high - 1);
        }
        else
        {
            return calcWeight(s, low, mid) +
                   calcWeight(s, mid + 1,
                              high);
        }
    }
}

// Driver Code
var input = "(()(()))";
var n = input.length;
// Update the closing Index
getClosingIndex(input);
document.write((calcWeight(input, 0, n - 1)));


</script>

Output:

6

Time Complexity: O(N) 
Auxiliary Space: O(N), where N is the length of the string.