<p>Given an input string <code>s</code>&nbsp;and a pattern <code>p</code>, implement regular expression matching with support for <code>&#39;.&#39;</code> and <code>&#39;*&#39;</code> where:</p>

<ul>
	<li><code>&#39;.&#39;</code> Matches any single character.​​​​</li>
	<li><code>&#39;*&#39;</code> Matches zero or more of the preceding element.</li>
</ul>

<p>The matching should cover the <strong>entire</strong> input string (not partial).</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;aa&quot;, p = &quot;a&quot;
<strong>Output:</strong> false
<strong>Explanation:</strong> &quot;a&quot; does not match the entire string &quot;aa&quot;.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;aa&quot;, p = &quot;a*&quot;
<strong>Output:</strong> true
<strong>Explanation:</strong> &#39;*&#39; means zero or more of the preceding element, &#39;a&#39;. Therefore, by repeating &#39;a&#39; once, it becomes &quot;aa&quot;.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;ab&quot;, p = &quot;.*&quot;
<strong>Output:</strong> true
<strong>Explanation:</strong> &quot;.*&quot; means &quot;zero or more (*) of any character (.)&quot;.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= s.length&nbsp;&lt;= 20</code></li>
	<li><code>1 &lt;= p.length&nbsp;&lt;= 20</code></li>
	<li><code>s</code> contains only lowercase English letters.</li>
	<li><code>p</code> contains only lowercase English letters, <code>&#39;.&#39;</code>, and&nbsp;<code>&#39;*&#39;</code>.</li>
	<li>It is guaranteed for each appearance of the character <code>&#39;*&#39;</code>, there will be a previous valid character to match.</li>
</ul>

class Solution {

public:

    bool isMatch(string s, string p) {

        // Create a map to store computed results

        unordered_map<string, bool> memo;

        return isMatchHelper(s, p, memo);

    }

private:

    bool isMatchHelper(const string& s, const string& p, unordered_map<string, bool>& memo) {

        // Check if the result has already been computed

        string key = s + "-" + p;

        if (memo.count(key))

            return memo[key];

        // Base case: pattern is empty

        if (p.empty())

            return s.empty();

        bool match = false;

        // Check if the second character of the pattern is '*'

        if (p.length() > 1 && p[1] == '*') {

            // Case 1: Skip the current pattern character and '*'

            match = isMatchHelper(s, p.substr(2), memo);

            // Case 2: Check if the current string character matches the pattern character (or the pattern is a dot)

            if (!match && !s.empty() && (s[0] == p[0] || p[0] == '.'))

                match = isMatchHelper(s.substr(1), p, memo);

        }

        else {

            // Check if the current string character matches the pattern character (or the pattern is a dot)

            if (!s.empty() && (s[0] == p[0] || p[0] == '.'))

                match = isMatchHelper(s.substr(1), p.substr(1), memo);

        }

        // Store the computed result in the memoization map

        memo[key] = match;

        return match;

    }

};


        
    

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