<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 {
private:
int dp[102][102];
bool solve(string s , int index , int left){
    // if(left<0){
    //     return false;
    // }
    // if(left==0 && index==s.length()){
    //     return true;
    // }
    // if(left!=0 && index==s.length()){
    //     return false;
    // }
    // if(dp[index][left] != -1){
    //     return dp[index][left];
    // }
    // vector<vector<int>> dp(s.length()+1 , vector<int>(s.length()+1,0));
    vector<int> curr(s.length()+1,0);
    vector<int> next(s.length()+1,0);
    next[0]=1;
    // dp[s.length()][0]=1;
    for(int index=s.length()-1 ; index>=0 ; index--){
        for(int left=s.length()-1 ; left>=0 ; left--){
            bool ans = 0;
            if(s[index]=='('){
            ans = ans | next[left+1];
            }
            if(s[index]==')'){
                if(left-1>=0)
                    ans = ans | next[left-1];
            }
            if(s[index]=='*'){
                
                ans = ans | next[left+1] ;
                if(left-1>=0)
                    ans = ans | next[left-1] ;
                ans = ans | next[left];
            }
             curr[left]=ans;
        }
        next = curr;
    }
    return curr[0];



    
}
public:
    bool checkValidString(string s) {
        memset(dp,-1,sizeof(dp));
        return solve(s,0,0);
        
    }
};
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