<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>

template <>
struct std::hash<std::pair<size_t, size_t>>{
    size_t operator()(const std::pair<size_t, size_t>& p) const noexcept {
        const auto hash1 = std::hash<size_t>{}(p.first);
        const auto hash2 = std::hash<size_t>{}(p.second);
        return hash1 ^ hash2 << 1;
    }
};

class Solution {
public:
    bool isMatch(const std::string& s, const std::string& p) {
        std::unordered_map<std::pair<size_t, size_t>, bool> cache;
        std::function<bool(size_t i, size_t j)> dfs = [&](size_t i, size_t j) {
            if (cache.contains({i, j})) {
                return cache[{i, j}];
            }
            if (i >= s.size() && j >= p.size()) {
                return true;
            }
            if (j >= p.size()) {
                return false;
            }
            const bool are_matched = i < s.size()
                && (s[i] == p[j] || p[j] == '.');
            if (j + 1 < p.size() && p[j + 1] == '*') {
                cache[{i, j}] = dfs(i, j + 2) || are_matched && dfs(i + 1, j);
                return cache[{i, j}];
            }
            if (are_matched) {
                cache[{i, j}] = dfs(i + 1, j + 1);
                return cache[{i, j}];
            }
            return cache[{i, j}] = false;
            return cache[{i, j}];
        };
        return dfs(0, 0);
    }
};
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