<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) {
        int n = s.size();
        int m = p.size();
        // dp[i][j] will be true if s[i:] matches p[j:]
        vector<vector<int>> dp(n + 1, vector<int>(m + 1, -1)); // -1 indicates not computed yet

        function<bool(int, int)> dfs = [&](int i, int j) {
            if (dp[i][j] != -1)
                return dp[i][j];

            if (j == m)
                return dp[i][j] = (i == n);

            bool first_match = (i < n && (p[j] == s[i] || p[j] == '.'));

            if (j + 1 < m && p[j + 1] == '*') {
                // Try to match zero occurrence of p[j]
                bool match_zero = dfs(i, j + 2);
                // Try to match one or more occurrence of p[j]
                bool match_one_or_more = first_match && dfs(i + 1, j);
                return dp[i][j] = match_zero || match_one_or_more;
            } else {
                // Move to the next character if there's a match
                return dp[i][j] = first_match && dfs(i + 1, j + 1);
            }
        };

        return dfs(0, 0);
    }
};

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