<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 checkValidString(string s) {
        if(s.size()==1)return(s[0] == '*');
        if(s[s.size()-1] == '(')return false;
        map<pair<int,int>, bool> dp;
        return(isValid(s,0,0,dp));
    }

    bool isValid(string s, int i, int l_count, map<pair<int,int>, bool>&dp){
        cout<<s[i]<<" "<<i<<" "<<l_count<<endl;
        auto key  = make_pair(i,l_count);
        if(dp.find(key)!= dp.end()) return dp[key];
        if(i == s.size()-1)return dp[key] = (l_count==1 && s[i] !='(') || (l_count == 0 && s[i] == '*');
        if(s[i] == '('){
            return dp[key] = isValid(s,i+1, l_count+1, dp);
        }
        if(s[i]==')'){
            if(!l_count)return false;
            return dp[key] = isValid(s,i+1, l_count-1, dp);
        }
        if(s[i] == '*'){
            return dp[key] = isValid(s,i+1, l_count+1, dp) || (l_count && isValid(s,i+1,l_count-1, dp)) || isValid(s,i+1, l_count,dp);
        }
        return false;
    }
};
28900