Check if a Parentheses String Can Be Valid

2116. Check if a Parentheses String Can Be Valid

Difficulty: Medium

Topics: String, Stack, Greedy

A parentheses string is a non-empty string consisting only of '(' and ')'. It is valid if any of the following conditions is true:

• It is ().
• It can be written as AB (A concatenated with B), where A and B are valid parentheses strings.
• It can be written as (A), where A is a valid parentheses string.

You are given a parentheses string s and a string locked, both of length n. locked is a binary string consisting only of '0's and '1's. For each index i of locked,

• If locked[i] is '1', you cannot change s[i].
• But if locked[i] is '0', you can change s[i] to either '(' or ')'.

Return true if you can make s a valid parentheses string. Otherwise, return false.

Example 1:

• Input: s = "))()))", locked = "010100"
• Output: true
• Explanation: locked[1] == '1' and locked[3] == '1', so we cannot change s[1] or s[3].
  • We change s[0] and s[4] to '(' while leaving s[2] and s[5] unchanged to make s valid.

Example 2:

• Input: s = "()()", locked = "0000"
• Output: true
• Explanation: We do not need to make any changes because s is already valid.

Example 3:

• Input: s = ")", locked = "0"
• Output: false
• Explanation: locked permits us to change s[0].
  • Changing s[0] to either '(' or ')' will not make s valid.

Constraints:

• n == s.length == locked.length
• 1 5
• s[i] is either '(' or ')'.
• locked[i] is either '0' or '1'.

Hint:

1. Can an odd length string ever be valid?
2. From left to right, if a locked ')' is encountered, it must be balanced with either a locked '(' or an unlocked index on its left. If neither exist, what conclusion can be drawn? If both exist, which one is more preferable to use?
3. After the above, we may have locked indices of '(' and additional unlocked indices. How can you balance out the locked '(' now? What if you cannot balance any locked '('?

Solution:

We can approach it step by step, keeping in mind the constraints and the behavior of the locked positions.

Key Points:

1. If the length of the string is odd, we can immediately return false because a valid parentheses string must have an even length (each opening ( needs a closing )).
2. We need to keep track of the number of open parentheses (() and closed parentheses ()) as we iterate through the string. If at any point the number of closing parentheses exceeds the number of opening ones, it's impossible to balance the string and we return false.
3. We must carefully handle the positions that are locked (locked[i] == '1') and unlocked (locked[i] == '0'). Unlocked positions allow us to change the character, but locked positions do not.

Algorithm:

• Step 1: Check if the length of the string s is odd. If so, return false immediately.
• Step 2: Loop through the string from left to right to track the balance of parentheses.
  • Use a counter to track the balance between opening ( and closing ) parentheses.
  • If at any point, the number of closing parentheses exceeds the opening parentheses, check if the locked positions have enough flexibility to balance it.
  • After processing the entire string, check if the parentheses are balanced, i.e., if there are no leftover unmatched opening parentheses.

Let's implement this solution in PHP: 2116. Check if a Parentheses String Can Be Valid

<?php /**
 * @param String $s
 * @param String $locked
 * @return Boolean
 */
function canBeValid($s, $locked) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Example usage:
$s = "))()))";
$locked = "010100";
var_dump(canBeValid($s, $locked));  // Output: bool(true)

$s = "()()";
$locked = "0000";
var_dump(canBeValid($s, $locked));  // Output: bool(true)

$s = ")";
$locked = "0";
var_dump(canBeValid($s, $locked));  // Output: bool(false)
?>

Explanation:

1. First pass (left to right):

   • We iterate through the string and track the balance of open parentheses. Each time we encounter an open parenthesis (, we increment the open counter. For a closed parenthesis ), we decrement the open counter.
   • If the current character is unlocked (locked[i] == '0'), we can assume it to be ( if needed to balance the parentheses.
   • If at any point the open counter goes negative, it means we have more closing parentheses than opening ones, and we return false.

2. Second pass (right to left):

   • We perform a similar operation in reverse to handle the scenario of unmatched opening parentheses that might be at the end of the string.
   • Here we track closing parentheses ()) with the close counter and ensure that no unbalanced parentheses exist.

3. Edge Case: If the string length is odd, we immediately return false because it cannot form a valid parentheses string.

Time Complexity:

• Both passes (left-to-right and right-to-left) take linear time, O(n), where n is the length of the string. Thus, the overall time complexity is O(n), which is efficient for the input size constraints.

This solution correctly handles the problem within the given constraints.

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