<p>Given a string <code>s</code> containing only three types of characters: <code>&#39;(&#39;</code>, <code>&#39;)&#39;</code> and <code>&#39;*&#39;</code>, return <code>true</code> <em>if</em> <code>s</code> <em>is <strong>valid</strong></em>.</p>

<p>The following rules define a <strong>valid</strong> string:</p>

<ul>
	<li>Any left parenthesis <code>&#39;(&#39;</code> must have a corresponding right parenthesis <code>&#39;)&#39;</code>.</li>
	<li>Any right parenthesis <code>&#39;)&#39;</code> must have a corresponding left parenthesis <code>&#39;(&#39;</code>.</li>
	<li>Left parenthesis <code>&#39;(&#39;</code> must go before the corresponding right parenthesis <code>&#39;)&#39;</code>.</li>
	<li><code>&#39;*&#39;</code> could be treated as a single right parenthesis <code>&#39;)&#39;</code> or a single left parenthesis <code>&#39;(&#39;</code> or an empty string <code>&quot;&quot;</code>.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre><strong>Input:</strong> s = "()"
<strong>Output:</strong> true
</pre><p><strong class="example">Example 2:</strong></p>
<pre><strong>Input:</strong> s = "(*)"
<strong>Output:</strong> true
</pre><p><strong class="example">Example 3:</strong></p>
<pre><strong>Input:</strong> s = "(*))"
<strong>Output:</strong> true
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= s.length &lt;= 100</code></li>
	<li><code>s[i]</code> is <code>&#39;(&#39;</code>, <code>&#39;)&#39;</code> or <code>&#39;*&#39;</code>.</li>
</ul>

class Solution {
public:
    bool checkValidString(string s) {
        int n = s.size();
        vector<vector<bool>> dp(s.size()+1, vector<bool>(s.size()+1, false));
        dp[n][0] = true;
        for(int i = s.size()-1; i >= 0;i--) {
            for(int count = 0; count<s.size();count++) {
                bool isValid = false;

                if(s[i] == '*') {
                    isValid |= dp[i+1][count+1];

                    if(count > 0) {
                        isValid |= dp[i+1][count-1];
                    }
                    isValid |= dp[i+1][count];
                } else {
                    if(s[i] == '(') {
                        isValid |= dp[i+1][count+1];
                    } else if (s[i] == ')') {
                        if(count > 0) isValid |= dp[i+1][count-1];
                    }
                }
                dp[i][count] = isValid;
            }
        }
        return dp[0][0];
        int count = 0;
        //return backtrack(0, s, count, dp);
    }
    bool backtrack(int i , string s, int count, vector<vector<int>> &dp) {
        if(i == s.size()) {
            if(count == 0) return true;
            return false;
        }
        if(dp[i][count] != -1) return dp[i][count];
        bool ans = false;
        if(s[i] == '(') ans = backtrack(i+1, s, count+1, dp);
        else if(s[i] == ')') {
              if(count > 0) ans = backtrack(i+1, s, count-1, dp);
        }
        else {
            //cout<<i<<endl;
            ans |= backtrack(i+1, s, count, dp);
            if(count > 0) ans |= backtrack(i+1, s, count-1, dp);
            ans |= backtrack(i+1, s, count+1, dp);

        }
        dp[i][count] = ans;
        //cout<<i<<" "<<ans<<endl;
        return dp[i][count];
        

    }

};

/*
()
(*
((*)
(*))
*/
9020