To solve the Valid Parenthesis String problem, apply dynamic programming to manage the parentheses and wildcard '' characters. The solution requires recognizing all valid string combinations where '' can act as '(', ')', or an empty string. This technique ensures that the parentheses are balanced while considering the wildcard's possible contributions.
Problem Statement
Given a string s consisting of three types of characters: '(', ')', and '', determine if the string is valid. A valid string is one where every open parenthesis has a corresponding close parenthesis, and '' can act as a parenthesis or be ignored.
The rules for a valid string are as follows: the '' can represent either a '(', a ')', or be an empty string. The string is valid if, after replacing '' in any possible way, the result is a properly balanced parentheses string.
Examples
Example 1
Input: s = "()"
Output: true
Example details omitted.
Example 2
Input: s = "(*)"
Output: true
Example details omitted.
Example 3
Input: s = "(*))"
Output: true
Example details omitted.
Constraints
- 1 <= s.length <= 100
- s[i] is '(', ')' or '*'.
Solution Approach
State Transition Dynamic Programming
Use dynamic programming to track all valid states during the iteration of the string. At each step, consider how the previous states can transition based on the current character, whether it is '(', ')', or '*'.
Backtracking for Wildcard Handling
Backtrack to explore every possible way the '*' character can be replaced. Consider its three possibilities: '(', ')', or an empty string, and verify if any leads to a valid parentheses string.
Greedy Approach for Parenthesis Matching
While iterating through the string, attempt a greedy approach by matching parentheses as they appear. If the balance of open and close parentheses is maintained, the string can be considered valid.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(n) |
| Space | O(1) |
The time complexity of the solution is O(n) due to the need to iterate through the string once. The space complexity is O(1) as we only need a constant amount of space to store the current state information.
What Interviewers Usually Probe
- Assess the candidate’s understanding of dynamic programming, especially in relation to state transitions.
- Evaluate how well the candidate uses backtracking to explore multiple combinations of wildcard substitutions.
- Look for the candidate's ability to optimize their approach, ensuring they don't exceed O(n) time complexity.
Common Pitfalls or Variants
Common pitfalls
- Failing to account for the wildcard '*' and treating it as a fixed character can lead to incorrect solutions.
- Not properly handling all possible states of the string during backtracking, leading to missed valid configurations.
- Overcomplicating the solution by not leveraging efficient state transitions in dynamic programming.
Follow-up variants
- Modify the problem to disallow the use of the wildcard '*' character.
- Implement the solution in a way that restricts the length of valid parentheses subsequences.
- Consider the problem with additional constraints such as limiting the number of wildcards or parentheses.
How GhostInterview Helps
- GhostInterview helps by providing instant insights into how dynamic programming can be applied to manage multiple states for valid parenthesis problems.
- The solver optimizes the backtracking approach, ensuring all wildcard possibilities are explored efficiently for correct results.
- Our tool emphasizes the importance of staying within O(n) time complexity and offers guidance on handling state transitions properly.
Topic Pages
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FAQ
How do you solve the Valid Parenthesis String problem using dynamic programming?
By keeping track of possible valid states as you iterate through the string, considering how each character affects the balance of parentheses and utilizing dynamic transitions.
What is the role of the wildcard '*' in the Valid Parenthesis String problem?
The wildcard '*' can represent either '(', ')', or be ignored. The solution must explore all three possibilities during the evaluation.
How can backtracking be used in this problem?
Backtracking explores all possible ways the '*' can be substituted to verify if any combination results in a valid parentheses string.
What are some common pitfalls in solving the Valid Parenthesis String problem?
Failing to handle the wildcard properly, missing potential valid configurations during backtracking, or overcomplicating the solution by not using state transitions efficiently.
What is the time complexity of solving the Valid Parenthesis String problem?
The time complexity is O(n), where n is the length of the string, as we only need to iterate through the string once.
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