Dynamic Programming Questions Medium
The time complexity of the Dynamic Programming approach for solving the Longest Valid Parentheses problem is O(n), where n is the length of the input string.
In this approach, we use an array dp of size n to store the length of the longest valid parentheses substring ending at each index i. We initialize all elements of dp to 0.
We iterate through the input string from left to right and update the dp array based on the following conditions:
- If the current character is '(', we set dp[i] to 0 since no valid parentheses substring can end with an opening bracket.
- If the current character is ')' and the previous character (i-1) is '(', we can form a valid parentheses substring ending at index i. In this case, dp[i] is set to dp[i-2] + 2.
- If the current character is ')' and the previous character (i-1) is ')' and the character at index i - dp[i-1] - 1 is '(', we can form a valid parentheses substring ending at index i. In this case, dp[i] is set to dp[i-1] + dp[i - dp[i-1] - 2] + 2.
After iterating through the entire input string, the maximum value in the dp array will represent the length of the longest valid parentheses substring.
Since we only iterate through the input string once and perform constant time operations for each character, the time complexity of this approach is O(n).