<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) {
        
        if (s == p) {
            return true;
        }
        
        vector<vector<int>> cache(s.size(), vector<int>(p.size(), -1));
        return DFS(s, p, s.size() - 1, p.size() - 1);
    }

    bool DFS(string& s, string& p, int index_s, int index_p) {
        if (index_s == -1 && index_p == -1) {
            return true;
        }

        if (index_p == -1) {
            return false;
        }

        if (index_s == -1) {
            while (index_p > 0 && p[index_p] == '*') {
                index_p -= 2;
            }

            if (index_p == -1) {
                return true;
            }
            
            return false;
        }

        if (s[index_s] == p[index_p] || p[index_p] == '.') {
            return DFS(s, p, index_s - 1, index_p - 1);
        }

        if (p[index_p] == '*') {
            
            if (s[index_s] == p[index_p - 1] || p[index_p - 1] == '.') {
                return DFS(s, p, index_s - 1, index_p) || DFS(s, p, index_s, index_p - 2);
            }

            else {
                return DFS(s, p, index_s, index_p - 2);
            }
        }

        if (p[index_p] != '.' && p[index_p] != '*' && s[index_s] != p[index_p]) {
            return false;
        }

        return true;
    }
};
8993