<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 t[102][102];
    bool f(int ind, string& s, int cnt,int n) {

        if (ind ==n) {
            return cnt == 0;
        }
        if (t[ind][cnt] != -1)
            return t[ind][cnt]==1?true:false;

        bool ans = false;

        if(s[ind]=='*'){
             
            if (cnt>0)
                ans |= f(ind + 1, s, cnt - 1, n);
            ans |= f(ind + 1, s, cnt+1, n);
             ans |= f(ind + 1, s, cnt, n);
        }

       else if (s[ind] == '(') {
            ans |= f(ind + 1, s, cnt + 1, n);

        } else if (s[ind] == ')') {
            if (cnt)
                ans |= f(ind + 1, s, cnt - 1, n);

        } 
        return t[ind][cnt] = ans;
    }

    bool checkValidString(string s) {
        int n = s.size();
        memset(t,-1,sizeof(t));
       // vector<vector<int>> dp(n, vector<int>(n, -1));
        return f(0, s, 0,n);
    }
};

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