<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 f(int i,int sum,string s,vector<vector<int>> &dp){
        if(i==0){
            if(sum==1)
            return true;
            else
            return false;
        }
        if(sum==0)
        return false;
        if(dp[i][sum]!=-1)
        return dp[i][sum];
        if(s[i-1]=='(')
            return dp[i][sum]=f(i-1,sum-1,s,dp);    
        else if(s[i-1]==')')
            return dp[i][sum]=f(i-1,sum+1,s,dp);
        else if(s[i-1]=='*'){
            bool a,b,c;
            a=f(i-1,sum+1,s,dp);
            b=f(i-1,sum-1,s,dp);
            c=f(i-1,sum,s,dp);
            return dp[i][sum]=(a||b||c);
        }
        return dp[i][sum]=true;
    }
    bool checkValidString(string s) {
        int n=s.size();
        vector<vector<int>> dp(n+1,vector<int> (n+2,-1));
        return f(s.size(),1,s,dp);
    }
};
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