<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 {
    bool topDown (const string &s, int start, int balance, vector<vector<int>> &dp) {
        if (start == s.size()) {
            return balance == 0;
        }
        if (balance < 0)
            return false;
        if (dp[start][balance]!=-1){
            return dp[start][balance];
        }
        if (s[start] == '(') {
            
            return dp[start][balance] = topDown (s, start+1, balance+1,dp);
        } else if (s[start] == ')'){
            return dp[start][balance] = topDown (s, start+1, balance-1,dp);
        } else {
            return dp[start][balance] = (topDown (s, start+1, balance+1,dp) || topDown (s, start+1, balance-1,dp) || topDown(s, start+1, balance,dp));
        }
    }
    bool bottomUp(const string &s) {
        vector<vector<bool>> dp(s.size()+1, vector<bool>(s.size()+1));
        dp[s.size()][0] = true;
        for (int i = s.size()-1; i >= 0; i--) {
            for (int openBracket = 0; openBracket <= s.size(); openBracket++) {
                if (s[i]=='(') {
                    dp[i][openBracket] = dp[i+1][openBracket+1];                    
                } else if (s[i]==')') {
                    dp[i][openBracket] = openBracket > 0 ? dp[i+1][openBracket-1] : false;
                } else {
                    dp[i][openBracket] = dp[i+1][openBracket] || dp[i+1][openBracket+1] || (openBracket > 0 ? dp[i+1][openBracket-1] : false);
                }
            }    
        }
        return dp[0][0];
    }
public:
    bool checkValidString(string s) {
        vector<vector<int>> dp(s.size(), vector<int>(s.size(),-1));
        int method = 1;
        switch (method) {
            case 0:
                return topDown(s, 0, 0, dp);
            case 1:
                return bottomUp(s);
            default:
                break;
        }
        return 0;
    }
};
10980