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. 

6

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