Problem Description

The problem is to determine if a given string s is a valid string based on certain rules. The string s consists of only three types of characters: '(', ')' and ''. The characters must form a valid sequence based on the following criteria: every '(' character must have a corresponding ')' character, every ')' must have a corresponding '(', the '(' must occur before its corresponding ')', and the '' character can be interpreted as either a '(', a ')', or an empty string. We need to assess whether we can rearrange the '*' characters in such a way that all the parentheses are correctly matched, and if so, we will return true; otherwise, we return false.

Intuition

Approaching the solution involves considering the flexibility that the '' character affords. The '' can potentially balance out any unmet parentheses if placed correctly, which is crucial to forming a valid string. To solve the problem, we have to deal with the ambiguity of the '*', determining when it should act as a '(', a ')', or be ignored as an empty string ' "" '.

The algorithm employs two passes to ensure that each parenthesis is paired up. In the first pass, we treat every '' as '(', increasing our "balance" or counter whenever we encounter a '(' or '', and decreasing it when we encounter a ')'. If our balance drops to zero, it means we have matched all parentheses thus far. However, if it drops below zero, there are too many ')' characters without a matching '(' or '*', so the string can't be valid.

On the second pass, we reverse the string and now treat every '' as ')'. We then perform a similar count, increasing the balance when we see a ')' or '', and decreasing it for '('. If the balance drops below zero this time, it means that there are too many '(' characters without a matching ')' or '*', and the string is again invalid.

If we complete both passes without the balance going negative, it means that for every '(' there is a corresponding ')', and the string is valid. We have effectively managed the ambiguity of '*' by checking that they can serve as a viable substitute for whichever parenthesis is needed to complete a valid set.

Our method ensures that all '(' have a corresponding ')', and vice versa, by treating '*' as placeholder for the correct parenthesis needed, thus validating the input string.

Learn more about Stack, Greedy and Dynamic Programming patterns.

Solution Approach

The reference solution approach suggests using dynamic programming to solve the problem. Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is typically used when a problem has overlapping subproblems and a recursive structure that allows solutions to these subproblems to be reused.

However, the provided solution code takes a different, more efficient approach by making two single pass scans of the string: one from left to right, and another from right to left.

In the first pass (left to right), we utilize a counter x to track the balance between the '(' and ')' characters. We increment x when we encounter a '(' or a '', and decrement it when we encounter a ')'. If we encounter a ')' but x is already 0, it means there's no '(' or '' preceding it that can be used to make a valid pair, thus the string is invalid and we return false.

x = 0
for c in s:
    if c in '(*':
        x += 1
    elif x:
        x -= 1
    else:
        return False

In the second pass (right to left), we reset the counter x and treat '' as ')'. We increment x for ')' or '', and decrement it for '('. Similar to the first pass, if we encounter a '(' but x is 0, it means there's no ')' or '*' to pair with it, so the string can't be valid.

x = 0
for c in s[::-1]:
    if c in '*)':
        x += 1
    elif x:
        x -= 1
    else:
        return False

After both scans, if our balance has never gone negative, we can conclude that for every '(' there is a matching ')' (after considering that '*' could count as either), so the string is valid.

This solution is more efficient than the dynamic programming approach mentioned in the reference solution approach. While dynamic programming would have a time complexity of O(n^3) and a space complexity of O(n^2), the provided solution has a time complexity of O(n) and a space complexity of O(1), since it simply uses a single integer for tracking and iterates through the string twice, without requiring any additional data structures.

Example Walkthrough

Let's take the string s = "(*))" as an example to illustrate the solution approach.

First Pass (Left to Right):

We will initialize our balance counter x to 0. We iterate through the string one character at a time.

1. s[0] = '(': Since it is a '(', we increment x to 1.
2. s[1] = '*': Since it is a '*', it could be '(', ')', or empty. We treat it as '(', so we increment x to 2.
3. s[2] = ')': We have a ')', so this could potentially pair with the previous '(' or '*'. We decrement x to 1.
4. s[3] = ')': Another ')'. It can pair with the '(' or '*' we assumed in step 2. We decrement x to 0.

At the end of this pass, x is not negative, which means that we have not encountered a ')' that could not be matched with a previous '(' or '*' treated as '('.

Second Pass (Right to Left):

Now, we will reset x to 0 and traverse from right to left, treating '*' as ')'.

1. s[3] = ')': We increment x to 1 since it could pair with an '(', or a '*', treated as '('.
2. s[2] = ')': We increment x to 2—similar reasoning as step 1.
3. s[1] = '*': Treating it as ')', we increment x to 3.
4. s[0] = '(': We have an '(', so we pair it with one of the ')' or '*' we treated as ')'. We decrement x to 2.

After the second pass, x is not negative, confirming that we have never encountered a '(' character that could not be matched with a subsequent ')' or '*' treated as ')'.

Since we finished both passes without the balance x going negative, we can conclude that the string s = "(*))" can be rearranged into a valid sequence. Therefore, our function would return true for this input.

Solution Implementation

Time and Space Complexity

Time Complexity

The provided algorithm consists of two for-loops that scan through the string s. Each loop runs independently from the beginning and from the end of the string, checking conditions and updating the variable x accordingly.

The time complexity of each pass is O(n), where n is the length of string s, since each character is examined exactly once in each pass.

Since there are two passes through the string, the total time complexity of the algorithm is O(n) + O(n) which simplifies to O(n).

Space Complexity

The space complexity of the algorithm depends on the additional space used by the algorithm, excluding the input itself.

In this case, only a single integer x is used to keep track of the balance of parentheses and asterisks, which occupies constant space.

Hence, the space complexity of the algorithm is O(1), as it does not depend on the size of the input string s and only uses a fixed amount of additional space.

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