# Time:  O(n)
# Space: O(1)

# Given a string containing only three types of characters: '(', ')' and '*',
# write a function to check whether this string is valid. We define the validity of a string by these rules:
# 1. Any left parenthesis '(' must have a corresponding right parenthesis ')'.
# 2. Any right parenthesis ')' must have a corresponding left parenthesis '('.
# 3. Left parenthesis '(' must go before the corresponding right parenthesis ')'.
# 4. '*' could be treated as a single right parenthesis ')' or a single left parenthesis '(' or an empty string.
# 5. An empty string is also valid.
#
# Example 1:
# Input: "()"
# Output: True
# Example 2:
# Input: "(*)"
# Output: True
# Example 3:
# Input: "(*))"
# Output: True
# Note:
# The string size will be in the range [1, 100].

class Solution(object):
    def checkValidString(self, s):
        """
        :type s: str
        :rtype: bool
        """
        lower, upper = 0, 0  # keep lower bound and upper bound of '(' counts
        for c in s:
            lower += 1 if c == '(' else -1
            upper -= 1 if c == ')' else -1
            if upper < 0: break
            lower = max(lower, 0)
        return lower == 0  # range of '(' count is valid