856. Score of Parentheses

Problem Description

The problem presents us with a string s that contains only balanced parentheses. A balanced parentheses string means every opening parenthesis '(' has a corresponding closing parenthesis ')'. Our goal is to compute the "score" of this string based on certain rules:

1. A pair of parentheses "()" directly next to each other has a score of 1.
2. The score AB for two consecutive balanced parentheses strings A and B is the sum of their individual scores, i.e., score(A) + score(B).
3. The score (A) for a balanced parentheses string A enclosed within another pair of parentheses is double the score of A, i.e., 2 * score(A).

The problem asks us to compute the total score of the given string based on these rules.

Intuition

To solve this problem, we can iterate through the string while keeping track of two things: the depth of nested parentheses d, and the current score ans. The depth increments each time we encounter an opening parenthesis '(' and decrements with a closing parenthesis ')'. The key insight is that a single pair of parentheses "()" contributes score based on the depth of nesting at that point; specifically, it contributes a score of 2^d, where d is the depth just before the pair ends.

Here is the thought process to arrive at the solution:

• Initialize the answer variable ans to zero and depth variable d to zero.
• Loop through each character c in the string s, using its index i to access previous characters if needed.
• If we encounter an opening parenthesis '(', we increase the depth d by 1, indicating we are going deeper in the nested structure.
• If we encounter a closing parenthesis ')', this means we are completing a parenthetical group; so we decrease the depth d by 1.
• Now, if the closing parenthesis is immediately following an opening one, i.e., we have a pair "()", it's time to add the score.
  • The tricky part is that if this pair is nested inside other pairs, its contribution to the score is not merely 1 but is actually 2^d where d is the depth just before this pair.
  • This is because each layer of nesting effectively doubles the score of the pair inside it.
  • We efficiently calculate 2^d as 1 << d because bit shifting to the left is equivalent to multiplying by two.
  • We then add this computed score to ans.
• After processing all characters, ans holds the final score of the string, which we return.

This way, by using a depth variable and a bit of bitwise arithmetic, we compute the score in a single pass over the input string.

Learn more about Stack patterns.

Solution Approach

The provided solution takes advantage of a stack-like behavior but optimizes it by using a depth variable instead to keep track of the levels of nested parentheses. This means that rather than actually pushing and popping from a stack, we can simply increment and decrement a depth counter to simulate the stack's depth tracking behavior.

Here's a step-by-step walkthrough of the implementation using the Reference Solution Approach:

1. Initialization: We initialize two integer variables:

   • ans: To store the cumulative score of the balanced parentheses string.
   • d: To keep track of the depth of the nested parentheses.

2. Traversal: We iterate through each character c of the string s. The loop utilizes enumerate to gain both the index i and the character c.

3. Depth Update:

   • When we encounter an opening parenthesis '(', we are going deeper into a level of nesting, so we increase d by 1.
   • Conversely, for a closing parenthesis ')', we decrease d by 1 as we are coming out of a nested level.

4. Score Calculation:

   • The crucial observation for score computation is made when we encounter a closing parenthesis ')'.
   • We check if the previous character (s[i - 1]) was an opening parenthesis '('. If it was, then we have found a "()" pair.
   • The score of this pair is 1 << d (which is 2 raised to the power of d). This is because each additional depth level doubles the score of a pair of parentheses.
   • We add this score to our answer ans.

5. Result: After the loop completes, ans holds the total score of the string which is then returned.

Algorithm: The algorithm is effectively a linear scan with a depth counting approach. The algorithm runs in O(n) time, with n being length of the string s, as each character is visited once.

Data Structures: While a stack is the intuitive choice for dealing with nested structures, this solution optimizes space complexity by using a single integer for depth tracking instead.

Patterns: This is an example of a single-pass algorithm with a small space optimization. The bit shifting trick (1 << d) is a clever way to compute powers of two that leverages the binary representation of integers.

This approach is efficient both in terms of time and space complexity, performing the calculation in linear time while using constant space.

Example Walkthrough

Let's illustrate the solution approach using a small example string: "(())()".

1. Initialization:

   • ans = 0: To store the cumulative score.
   • d = 0: To keep track of the depth of nested parentheses.

2. Traversal:

   • We start with the first character '('.
     • d becomes 1 because we have entered a new level of nesting.
   • Next character '(':
     • d becomes 2 as we go deeper.
   • We encounter a closing parenthesis ')':
     • d is 2, hence before decreasing d, we check the previous character which is '('.
     • Since it forms a pair "()", we add 1 << (d - 1) to ans which is 1 << 1 = 2.
     • ans = ans + 2 and d decreases to 1.
   • The next character is ')' again:
     • d is 1, and we check the previous character which is not '(', so we just decrease d by 1.
     • ans stays the same since we don't have a new pair, and d decreases to 0.
   • Next character '(':
     • d becomes 1 since we are starting a new nesting level.
   • Lastly, we encounter ')':
     • d is 1, the previous character is '(', so this is a pair "()", and we add 1 << (d - 1) which is 1 << 0 = 1 to ans.
     • ans = ans + 1 and ans becomes 3, then d goes back to 0.

3. Traversal Complete:

   • We have processed all characters, and our final score ans is 3.

This example verifies our approach to scoring the string. Each pair of parentheses that forms "()" directly contributes 1 << (depth - 1) to the score, with the depth being counted just before the closing parenthesis of the pair.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the function scoreOfParentheses is O(n), where n is the length of the string s. This is because there is a single loop that iterates through each character of the string s exactly once.

Space Complexity

The space complexity of the function is O(1). The only extra space used is for the variables ans and d, which store the cumulative score and the current depth respectively. The space used does not scale with the size of the input string, so the space complexity is constant.

Learn more about how to find time and space complexity quickly using problem constraints.