<p>Given a string <code>s</code> containing only three types of characters: <code>&#39;(&#39;</code>, <code>&#39;)&#39;</code> and <code>&#39;*&#39;</code>, return <code>true</code> <em>if</em> <code>s</code> <em>is <strong>valid</strong></em>.</p>

<p>The following rules define a <strong>valid</strong> string:</p>

<ul>
	<li>Any left parenthesis <code>&#39;(&#39;</code> must have a corresponding right parenthesis <code>&#39;)&#39;</code>.</li>
	<li>Any right parenthesis <code>&#39;)&#39;</code> must have a corresponding left parenthesis <code>&#39;(&#39;</code>.</li>
	<li>Left parenthesis <code>&#39;(&#39;</code> must go before the corresponding right parenthesis <code>&#39;)&#39;</code>.</li>
	<li><code>&#39;*&#39;</code> could be treated as a single right parenthesis <code>&#39;)&#39;</code> or a single left parenthesis <code>&#39;(&#39;</code> or an empty string <code>&quot;&quot;</code>.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre><strong>Input:</strong> s = "()"
<strong>Output:</strong> true
</pre><p><strong class="example">Example 2:</strong></p>
<pre><strong>Input:</strong> s = "(*)"
<strong>Output:</strong> true
</pre><p><strong class="example">Example 3:</strong></p>
<pre><strong>Input:</strong> s = "(*))"
<strong>Output:</strong> true
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= s.length &lt;= 100</code></li>
	<li><code>s[i]</code> is <code>&#39;(&#39;</code>, <code>&#39;)&#39;</code> or <code>&#39;*&#39;</code>.</li>
</ul>

class Solution {
public:
    int checkValidString(const string& s, int i, int left_count, std::vector<std::vector<int>>& dp) {
        if (i == s.size()) {
            return left_count == 0;
        }
        if (left_count < 0) {
            return 0;
        }

        if (dp[i][left_count] != -1) {
            return dp[i][left_count];
        }

        int val = 0;
        if (s[i] == '(') {
            val = checkValidString(s, i+1, left_count+1, dp);
        }
        else if (s[i] == ')') {
            val =  checkValidString(s, i+1, left_count-1, dp);
        } else {
            val = checkValidString(s, i+1, left_count, dp) || checkValidString(s, i+1, left_count+1, dp) || checkValidString(s, i+1, left_count-1, dp);
        }
        dp[i][left_count] = val;
        return val;
    }

    bool checkValidString(string s) {
        std::vector<std::vector<int>> dp(s.size(), std::vector<int>(s.size(), -1));

        return checkValidString(s, 0, 0, dp);    
    }
};
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