<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>

unordered_map<int, unordered_map<int, bool>> dp;

class Solution {
public:
    bool checkValidString(const string& s) {
        // Clear the global dp table before each new computation
        dp.clear();
        return f(s, 0, 0);
    }

private:
    bool f(const string& s, int i, int count) {
        if (count < 0) {
            return false;
        }
        if (i == s.size()) {
            return count == 0;
        }

        // Check if the result is already computed
        if (dp[i].find(count) != dp[i].end()) {
            return dp[i][count];
        }

        bool result = false;
        if (s[i] == '(') {
            result = f(s, i + 1, count + 1);
        } else if (s[i] == ')') {
            result = f(s, i + 1, count - 1);
        } else {
            result = f(s, i + 1, count + 1) ||  
                     f(s, i + 1, count - 1) ||  
                     f(s, i + 1, count);        
        }

        // Store the result in the memoization table
        dp[i][count] = result;
        return result;
    }
};
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