<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:
    def isMatch(self, s: str, p: str) -> bool:
        i, j = len(s) - 1, len(p) - 1
        return self.backtrack({}, s, p, i, j)

    def backtrack(self, cache, s, p, i, j):
        key = (i, j)
        if key in cache:
            return cache[key]

        if i == -1 and j == -1:
            cache[key] = True
            return True

        if i != -1 and j == -1:
            cache[key] = False
            return cache[key]

        if i == -1 and p[j] == '*':
            k = j
            while k != -1 and p[k] == '*':
                k -= 2
            
            if k == -1:
                cache[key] = True
                return cache[key]
            
            cache[key] = False
            return cache[key]
        
        if i == -1 and p[j] != '*':
            cache[key] = False
            return cache[key]

        if p[j] == '*':
            if self.backtrack(cache, s, p, i, j - 2):
                cache[key] = True
                return cache[key]
            
            if p[j - 1] == s[i] or p[j - 1] == '.':
                if self.backtrack(cache, s, p, i - 1, j):
                    cache[key] = True
                    return cache[key]
        
        if p[j] == '.' or s[i] == p[j]:
            if self.backtrack(cache, s, p, i - 1, j - 1):
                cache[key] = True
                return cache[key]

        cache[key] = False
        return cache[key]
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