<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) {
       return RCheckValidString(0, 0, s);
    }

    map<pair<int, int>, bool> storage;
    bool RCheckValidString(int i, int numOfOpenPar, string& s)
    {
        if (numOfOpenPar < 0) return false;

        for(; i < s.size(); i++)
        {
            if (s[i] == '(')
                numOfOpenPar++;
            else if (s[i] == ')')
                if (numOfOpenPar > 0) 
                    numOfOpenPar--;
                else 
                    return false;
            else
            {
                bool l1, l2, l3;

                if (storage.find({i + 1, numOfOpenPar + 1}) != storage.end())
                {
                    l1 = storage[{i + 1, numOfOpenPar + 1}];
                }
                else
                {
                    l1 = RCheckValidString(i + 1, numOfOpenPar + 1, s);
                    storage[{i + 1, numOfOpenPar + 1}] = l1;
                }

                if (storage.find({i + 1, numOfOpenPar - 1}) != storage.end())
                {
                    l2 = storage[{i + 1, numOfOpenPar - 1}];
                }
                else
                {
                    l2 = RCheckValidString(i + 1, numOfOpenPar - 1, s);
                    storage[{i + 1, numOfOpenPar - 1}] = l2;
                }

                if (storage.find({i + 1, numOfOpenPar}) != storage.end())
                {
                    l3 = storage[{i + 1, numOfOpenPar}];
                }
                else
                {
                    l3 = RCheckValidString(i + 1, numOfOpenPar, s);
                    storage[{i + 1, numOfOpenPar}] = l3;
                }

                return (l1 || l2 || l3); 
            }             
        }

        return numOfOpenPar == 0;
    }
};
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