LeetCode678. Valid Parenthesis String

Analysis

Solution1

The idea is to use an int count to keep tracking the relationship between left parenthesis and right parenthesis since we don’t care about the exact parenthesis but rather their corresponding counts.

For every character in the input String s, there are 3 possible conditions:

1. Current character is '(', then we increase the count.

2. Current character is ')', then we decrease the count.

3. Current character is '*', there are 3 possible sub-conditions:

   • Treat * as (, increase count by 1.
   • Treat * as ), decrease count by 1.
   • Treat * as empty string, count does not change.

Complexity

The space complexity of this solution is O(1) without considering the recursion stack. The recursion stack space would be linear to the length of the input String.

The time complexity would be O(n) as well. Worst case O(2^n) to O(3^n) when the input string contains only *.

Suppose n is the length of the input string.

Then we have the following solution based on straight-forwardly implement the above thoughts.

class Solution {
    public boolean checkValidString(String s) {
        if (s == null || s.isEmpty() || s.length() == 0) {
            return true;
        }

        char[] charS = s.toCharArray();
        return checkValidStringHelper(charS, 0, 0);
    }

    private boolean checkValidStringHelper(char[] charS, int start, int count) {
        // Base case
        if (start == charS.length) {
            return count == 0 ? true : false;
        }

        // If the count smaller than 0, early stop cause this is invalid.
        if (count < 0) {
            return false;
        }

        if (charS[start] == '(') {
            return checkValidStringHelper(charS, start + 1, count + 1);
        } else if (charS[start] == ')') {
            return checkValidStringHelper(charS, start + 1, count - 1);
        } else { // charS[start] == '*'
            // 1. treat '*' as empty string
            boolean ret1 = checkValidStringHelper(charS, start + 1, count);
            // 2. treat '*' as left parenthesis
            boolean ret2 = checkValidStringHelper(charS, start + 1, count + 1);
            // 3. treat '*' as right parenthesis
            boolean ret3 = checkValidStringHelper(charS, start + 1, count - 1);

            return (ret1 || ret2 || ret3);
        }
    }
}

Solution2 (More optimal)

Another interesting observation is that the resulting count will actually be a continuous sequence of integer.

E.g., for input "(**))"

At step 1, the count would be 1.

At step 2, the count would be 0, 1, or 2

At step 3, the count would be 0+1, 0, 0-1, 1+1, 1, 1-1, 2+1, 2, 2-1, which is -1, 0, 1, 2, 3. And since negative count is always invalid, we early stop for these conditions. ==> 0, 1, 2, 3.

At step 4, the count would be 0, 1, 2 with early pruning.

At step 5, the count would be 0, 1 with early pruning.

Considering that actually the reslut count is a consecutive sequence of integers, we can actually stores only the range of the output rather than iterate through all possible conditions.

Complexity

This algorithm runs in O(1) space and O(n) time where n is the length of the input string.

Resulting to the following solution:

class Solution {
    public boolean checkValidString(String s) {
        if (s == null || s.isEmpty() || s.length() == 0) {
            return true;
        }

        int low = 0;
        int high = 0;
        for (char c : s.toCharArray()) {
            if (c == '(') {
                high++;
                low++;
            } else if (c == ')') {
                if (low > 0) {
                    low--;
                }
                high--;
            } else { //c == '*'
                if (low > 0) {
                    low--;
                }
                high++;
            }

            if (high < 0) {
                return false;
            }
        }

        // Since we have pruned all conditions where low is smaller than 0, the smallest low would be 0
        // if the input string is valid.
        return low == 0;
    }
}