<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 n ;

    int dp[101][101] ;
    bool f(int ind , int open , string &s)
    {
        if(ind == n)
        {
            return open == 0;
        }

        if(open < 0)
        {
            return false ;
        }
        if(dp[ind][open] != -1)
        {
            return dp[ind][open] ;
        }

        bool flag = false ;

        // trying all 3 possibilities
        if(s[ind] == '*')
        {
            // * -> (
            if(open > 0)
            {
                flag |= f(ind+1 , open - 1 , s) ;
            }

            // * -> )
            flag |= f(ind+1 , open+1 , s) ;
            
            // * -> ""
            flag |= f(ind+1 , open , s) ;
            
        }

        else if(s[ind] == ')')
        {
            flag |= f(ind+1 , open - 1 , s) ;
        }
        else
        {
            flag |= f(ind+1 , open + 1 , s) ;
        }

        return dp[ind][open] = flag ;
    }
    bool checkValidString(string s) {
        n = s.size() ;
        memset(dp,-1,sizeof(dp)) ;
        return f(0 , 0 ,s) ;
    }
};
8180