<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 isAllStars(string p, int ind)
    {
        if(ind & 2 == 0)
            return false;
        while(ind >= 0)
        {
            if(p[ind] != '*')
                return false;
            ind -= 2;
        }
        return true;
    }

    bool isPatternMatching(string s, string p, int ind1, int ind2, vector<vector<int>> &dp)
    {
        if(ind1 < 0)
            return isAllStars(p, ind2);
        if(ind2 < 0)
            return false;
        if(dp[ind1][ind2] != -1)
            return dp[ind1][ind2] == 1;
        bool result = false;
        if(s[ind1] == p[ind2] || p[ind2] == '.')
            result =  isPatternMatching(s, p, ind1-1, ind2-1, dp);
        else if(p[ind2] == '*') {
            result = isPatternMatching(s, p, ind1, ind2-2, dp); //Ignore the stars
            if(s[ind1] == p[ind2-1] || p[ind2-1] == '.')
                result = result || isPatternMatching(s, p, ind1-1, ind2, dp);
        }
        dp[ind1][ind2] = result ? 1 : 0;
        return result;
    }

    bool isMatch(string s, string p) 
    {
        int s_length = s.length(), p_length = p.length();
        vector<vector<int>> dp(s_length, vector<int>(p_length, -1));
        return isPatternMatching(s, p, s_length-1, p_length-1, dp);
    }
};
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