<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>

struct hasher {
    size_t operator()(pair<int, int> p) const noexcept {
        size_t h = 5381;
        h = (h * 33) + p.first;
        h = (h * 33) + p.second;
        return h;
    }
};

class Solution {
public:
    unordered_map<pair<int, int>, bool, hasher> memo;

    bool isMatch(string s, string p) { return this->aux(s, p, 0, 0); }

    bool aux(const string& s, const string& p, int i, int j) {
        pair<int, int> q = make_pair(i, j);
        auto mitr = memo.find(q);
        if (mitr != memo.end()) {
            return mitr->second;
        }
        if (j == p.size()) {
            // note - if pattern not ended
            // but there is string left, it's still possible to succeed
            // becaues of star - for example "" matches re "a*"
            // but if pattern ended and there is string left then we fail
            return (i == s.size());
        }

        // does first character match?
        bool first_match = (i < s.size()) && (p[j] == '.' || p[j] == s[i]);

        // is there a star?
        if (j < p.size() - 1 && p[j + 1] == '*') {
            // option 1 - skip
            bool match1 = this->aux(s, p, i, j + 2);
            if (match1) {
                // cout << " zero * char match\n";
                return true;
            }
            // option 2 - match first token (already done) and recurse
            bool out = first_match && this->aux(s, p, i + 1, j);
            memo[make_pair(i, j)] = out;
            return out;
        } else {
            // cout << "one char match\n";
            // if no star, keep going with simple match
            bool out = first_match && this->aux(s, p, i + 1, j + 1);
            memo[make_pair(i, j)] = out;
            return out;
        }
        return true;
    }
};
16118