<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:
    bool f(int ind, int open, string &s, int n, vector<vector<int>> &dp){
        if(open < 0) return false;
        if(ind==n) return open==0;

        if(dp[ind][open] != -1) return dp[ind][open];

        bool ans = false;
        if(s[ind]=='(') return f(ind+1, open+1, s, n, dp);
        else if(s[ind]==')') return f(ind+1, open-1, s, n, dp);
        else{
            ans |= f(ind+1, open, s, n, dp) | f(ind+1, open+1, s, n, dp) | f(ind+1, open-1, s, n, dp) ;
        }
        return dp[ind][open] = ans;
    }
    bool checkValidString(string s) {
        int n = s.size();
        vector<vector<int>> dp(n, vector<int>(101, -1));
        return f(0, 0, s, n, dp);

        // stack<int> star, open;
        // for(int i=0; i<n; i++){
        //     if(s[i]=='('){
        //         open.push(i);
        //     }
        //     else if(s[i]=='*'){
        //         star.push(i);
        //     }
        //     else{
        //         if(star.empty() && open.empty()) return false;
        //         if(!open.empty()) open.pop();
        //         else star.pop();
        //     }
        // }

        // while(!open.empty()){
        //     if(star.empty()) return false;
        //     if(star.top() < open.top()) return false;
        //     star.pop();
        //     open.pop();
        // }
        // return true;

        // int high = 0, low = 0;
        // for(auto x: s){
        //     switch(x){
        //         case '(' :
        //             low++;
        //             high++;
        //             break;
        //         case ')' :
        //             low = max(0, --low);
        //             high--;
        //             break;
        //         case '*' :
        //             low = max(0, --low);
        //             high++;
        //     }
        //     if(high < 0) return false;
        // }
        // return low==0;
    }
};  
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