<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:
    // Top Down Dynamic Programming Approach

    // if j >= len(p) --> no match
    // if i >= len(s) --> Still Possible match: i.e  s= "a", p = "a*b*"
    // p[j] == '.' or s[i] == p[j] --> (i+1, j+1)
    // match if i < len(s) and (s[i] == p[j] or p[j] == '.')
    //
    // if (j+1 < len(p) and p[j+1] == '*')

    bool dfs(string s, string p, int i, int j, vector<pair<pair<int, int>, bool>>&dp)
    {
        // if (i,j) in dp
        pair<int,int> a = make_pair(i,j);
        pair<pair<int,int>, bool> b = make_pair(a, false);
        pair<pair<int,int>, bool> c = make_pair(a, true);
        if (std::find(dp.begin(), dp.end(), b) != dp.end())
            return false;
        if (std::find(dp.begin(), dp.end(), c) != dp.end())
            return true;
        // both reached end of string
        if (i >= s.length() &&  j >= p.length())
        {
            dp.push_back(c);
            return true;
        }
        if (j >= p.length())
        {
            dp.push_back(b);
            return false;
        }
        bool match = (i < s.length() && (s[i] == p[j] || p[j] == '.'));

        if (j+1 < p.length() && p[j+1] == '*')
        {
            // Case DON'T USE star or Case USE Star
            //     (i,j+2)                 (i+1, j)
            dp.push_back(make_pair(a, dfs(s, p, i, j+2, dp) || match && dfs(s, p, i+1, j, dp)));
            if (std::find(dp.begin(), dp.end(), b) != dp.end())
                return false;
            if (std::find(dp.begin(), dp.end(), c) != dp.end())
                return true;
        }

        // Case . or matching character
        if (match)
        {
            dp.push_back(make_pair(a, dfs(s, p, i+1, j +1, dp)));

            if (std::find(dp.begin(), dp.end(), b) != dp.end())
                return false;
            if (std::find(dp.begin(), dp.end(), c) != dp.end())
                return true;
        }
        
        dp.push_back(b);
        return false;
    
    }


    bool isMatch(string s, string p) {
        vector<pair<pair<int, int>, bool>> dp;
        return dfs(s, p, 0, 0, dp);
    }
};
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