<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:
    void printNFA(const vector<vector<int>>& next, const vector<vector<char>>& cNeeded) {
        for (int i = 0; i < next.size(); ++i) {
            cout << "State " << i << " goes to (state, character):";
            for (int j = 0; j < next[i].size(); ++j)
                cout << " (" << next[i][j] << ',' << cNeeded[i][j] << ')';
            cout << endl;
        }
    }

    bool isMatch(string s, string p) {
        //build NFA with 
        vector<vector<int>> next; 
        vector<vector<char>> cNeeded;
        buildNFA(p, next, cNeeded);
        printNFA(next, cNeeded);

        function<void(int, unordered_set<int>&)> addEmpty = [&next, &cNeeded, &addEmpty] (int state, unordered_set<int>& q) {
            q.insert(state);
            for (int i = 0; i < next[state].size(); ++i) {
                if (cNeeded[state][i] == ' '){
                    addEmpty(next[state][i], q);
                }
            }
        };

        function<void(int, char, unordered_set<int>&)> addChild = [&next, &cNeeded, &addChild, &addEmpty] (int state, char c, unordered_set<int>& q) {
            for (int i = 0; i < next[state].size(); ++i) {
                // if (cNeeded[state][i] == ' '){
                //     //q.push_back(next[state][i]);
                //     addChild(next[state][i], c, q);
                // }
                if (cNeeded[state][i] == '.' || cNeeded[state][i] == c){
                    addEmpty(next[state][i], q);
                }
            }
        };

        unordered_set<int> old, states;
        addEmpty(0, old);
        for (int i = 0; i < s.size(); ++i) {
            cout << "iteration ";
            for (int s: old)
                cout << ' ' << s;
            cout << endl;
            int n = states.size();
            for (int o : old) {
                addChild(o, s[i], states);
            }
            old = move(states);
            states.clear();
        }
        cout << "iteration ";
        for (int s: old)
            cout << ' ' << s;
        cout << endl;
        for (int s : old)
            if (s == next.size() - 1)
                return true;
        return false;
    }

    //assumes next and cNeeded is empty
    void buildNFA(const string& p, vector<vector<int>>& next, vector<vector<char>>& cNeeded) {
        //0 is the begin state
        int currState = 0, prevState;
        //' ' is the empty string (\lambda)
        for (int i = 0; i < p.size(); ++i) {
            next.emplace_back();
            cNeeded.emplace_back();
            
            prevState = currState++;
            next[prevState].push_back(currState);
            cNeeded[prevState].push_back(p[i]);
            if (i < p.size() - 1 && p[i+1] == '*'){
                next[prevState].push_back(currState);
                cNeeded[prevState].push_back(' ');
                next[prevState].push_back(prevState);
                cNeeded[prevState].push_back(p[i]);
                ++i;
            }
        }
        next.emplace_back();
        cNeeded.emplace_back();
    }

};
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