[22]. 括号生成

• • 题目
  • 算法设计：回溯
  • 算法设计：空间换时间

 

---

题目

传送门：https://leetcode.cn/problems/generate-parentheses/

 

---

算法设计：回溯

括号问题可以分成俩类：

• 括号的合法性判断，主要是用栈
• 括号的合法生成，主要是回溯

搞懂回溯算法，最重要的是理解回溯框架。

回溯框架：

ans = []
void recall( 路径，选择列表 )
	if 满足结束条件：
		ans.add( 路径 )
		return
	for 选择 in 选择列表：
		做选择
		recall( 路径，选择列表 )
		撤销选择

回溯框架，本质是遍历一颗决策树。

• 路径：已经做出的选择
• 选择列表：当前可以做的选择
• 结束条件：到了决策树底层，无法再做选择

核心在于 for 循环里面的递归，在递归之前做选择，在递归之后撤销选择。

• for 循环，如果可视化就是在遍历一颗 N 叉树

问题是，选择和撤销选择是在这颗树上做什么呢？

• 选择：是在这棵树上做前序遍历
• 撤销选择：是在这颗树上做后序遍历

所以，选择的可视化是在进入树的某一节点前执行，撤销选择的可视化是在离开树的某一节点后执行。

做选择：在进入节点前，从选择列表拿出一个选择，将它放入路径。

撤销选择：在离开节点后，从路径中拿出一个选择，将它恢复到选择列表中。

 

---

套用回溯框架，打印所有括号：

• 结束条件：当前位置 == 2 * n
• 选择列表：每个位置要么左括号，要么右括号 [‘(’, ‘)’]

int n;
void recall( int x, string& ans ) {
		if( x == 2 * n )	
			print(ans);
			return;
			
		for c in ['(', ')']:
			ans.push(c);
			recall(x + 1, ans);
			ans.pop(c);	
}

不过，这题不是生成所有括号，而是生成所有合法括号。

那么合法的定义是什么？

• 左括号数量 = 右括号数量
• 必然是左括号在前，右括号在后
• 在添加左括号时，之前的左括号一定大于或等于右括号
• 在添加右括号时，之前的左括号数量必然大于右括号
• 组合长度等于 2n（n对括号，一对括号是2，等于2n）

对于 2n 个括号，必然有 n 个左括号、n 个右括号

这里把穷举位置 x，改成用 l 记录还可以使用多少个左括号，用 r 记录还可以使用多少个右括号。

class Solution {
	vector<string> res;               // 记录答案 
public:
    vector<string> generateParenthesis(int n) {
        if( n == 0 ) return res;
        recall(n , n, "");            // 当前可用左括号和右括号数量为 n   
        return res;
    }
    void recall(int l, int r, string str) {
        if( l < 0 || r < 0 ) return;  // 剪枝：数量小于0，不合法
        if( r < l ) return;           // 剪枝：左括号剩下的多，不合法
        if( l == 0 && r == 0 ) {      // 所有括号恰好用完，得到一个合法括号
        	res.push_back(str);  
        	return;                                   
        }                    
        str.push_back('(');           // 选择左括号
        recall(l-1, r, str);          // 进入下一层
        str.pop_back();               // 撤销选择

        str.push_back(')');
        recall(l, r-1, str);
        str.pop_back();
    }
};

当 n = 2 时，下图是代码的可视化：

 

---

算法设计：空间换时间

回溯算法的时间复杂度太高了。

因为这题的输入数据 < 8，所以可以采用空间换时间策略。

class Solution {
public:
    vector<string> generateParenthesis(int n) {
        switch(n){
            case 1:
                return vector<string>{"()"};
            case 2:
                return vector<string>{"()()","(())"};
            case 3:
                return vector<string>{"((()))","(()())","(())()","()(())","()()()"};
            case 4:
                return vector<string>{"(((())))","((()()))","((())())","((()))()","(()(()))","(()()())","(()())()","(())(())","(())()()","()((()))","()(()())","()(())()","()()(())","()()()()"};
            case 5:
                return vector<string>{"((((()))))","(((()())))","(((())()))","(((()))())","(((())))()","((()(())))","((()()()))","((()())())","((()()))()","((())(()))","((())()())","((())())()","((()))(())","((()))()()","(()((())))","(()(()()))","(()(())())","(()(()))()","(()()(()))","(()()()())","(()()())()","(()())(())","(()())()()","(())((()))","(())(()())","(())(())()","(())()(())","(())()()()","()(((())))","()((()()))","()((())())","()((()))()","()(()(()))","()(()()())","()(()())()","()(())(())","()(())()()","()()((()))","()()(()())","()()(())()","()()()(())","()()()()()"};
            case 6:
                return vector<string>{"(((((())))))","((((()()))))","((((())())))","((((()))()))","((((())))())","((((()))))()","(((()(()))))","(((()()())))","(((()())()))","(((()()))())","(((()())))()","(((())(())))","(((())()()))","(((())())())","(((())()))()","(((()))(()))","(((()))()())","(((()))())()","(((())))(())","(((())))()()","((()((()))))","((()(()())))","((()(())()))","((()(()))())","((()(())))()","((()()(())))","((()()()()))","((()()())())","((()()()))()","((()())(()))","((()())()())","((()())())()","((()()))(())","((()()))()()","((())((())))","((())(()()))","((())(())())","((())(()))()","((())()(()))","((())()()())","((())()())()","((())())(())","((())())()()","((()))((()))","((()))(()())","((()))(())()","((()))()(())","((()))()()()","(()(((()))))","(()((()())))","(()((())()))","(()((()))())","(()((())))()","(()(()(())))","(()(()()()))","(()(()())())","(()(()()))()","(()(())(()))","(()(())()())","(()(())())()","(()(()))(())","(()(()))()()","(()()((())))","(()()(()()))","(()()(())())","(()()(()))()","(()()()(()))","(()()()()())","(()()()())()","(()()())(())","(()()())()()","(()())((()))","(()())(()())","(()())(())()","(()())()(())","(()())()()()","(())(((())))","(())((()()))","(())((())())","(())((()))()","(())(()(()))","(())(()()())","(())(()())()","(())(())(())","(())(())()()","(())()((()))","(())()(()())","(())()(())()","(())()()(())","(())()()()()","()((((()))))","()(((()())))","()(((())()))","()(((()))())","()(((())))()","()((()(())))","()((()()()))","()((()())())","()((()()))()","()((())(()))","()((())()())","()((())())()","()((()))(())","()((()))()()","()(()((())))","()(()(()()))","()(()(())())","()(()(()))()","()(()()(()))","()(()()()())","()(()()())()","()(()())(())","()(()())()()","()(())((()))","()(())(()())","()(())(())()","()(())()(())","()(())()()()","()()(((())))","()()((()()))","()()((())())","()()((()))()","()()(()(()))","()()(()()())","()()(()())()","()()(())(())","()()(())()()","()()()((()))","()()()(()())","()()()(())()","()()()()(())","()()()()()()"};
            case 7:
                return vector<string>{"((((((()))))))","(((((()())))))","(((((())()))))","(((((()))())))","(((((())))()))","(((((()))))())","(((((())))))()","((((()(())))))","((((()()()))))","((((()())())))","((((()()))()))","((((()())))())","((((()()))))()","((((())(()))))","((((())()())))","((((())())()))","((((())()))())","((((())())))()","((((()))(())))","((((()))()()))","((((()))())())","((((()))()))()","((((())))(()))","((((())))()())","((((())))())()","((((()))))(())","((((()))))()()","(((()((())))))","(((()(()()))))","(((()(())())))","(((()(()))()))","(((()(())))())","(((()(()))))()","(((()()(()))))","(((()()()())))","(((()()())()))","(((()()()))())","(((()()())))()","(((()())(())))","(((()())()()))","(((()())())())","(((()())()))()","(((()()))(()))","(((()()))()())","(((()()))())()","(((()())))(())","(((()())))()()","(((())((()))))","(((())(()())))","(((())(())()))","(((())(()))())","(((())(())))()","(((())()(())))","(((())()()()))","(((())()())())","(((())()()))()","(((())())(()))","(((())())()())","(((())())())()","(((())()))(())","(((())()))()()","(((()))((())))","(((()))(()()))","(((()))(())())","(((()))(()))()","(((()))()(()))","(((()))()()())","(((()))()())()","(((()))())(())","(((()))())()()","(((())))((()))","(((())))(()())","(((())))(())()","(((())))()(())","(((())))()()()","((()(((())))))","((()((()()))))","((()((())())))","((()((()))()))","((()((())))())","((()((()))))()","((()(()(()))))","((()(()()())))","((()(()())()))","((()(()()))())","((()(()())))()","((()(())(())))","((()(())()()))","((()(())())())","((()(())()))()","((()(()))(()))","((()(()))()())","((()(()))())()","((()(())))(())","((()(())))()()","((()()((()))))","((()()(()())))","((()()(())()))","((()()(()))())","((()()(())))()","((()()()(())))","((()()()()()))","((()()()())())","((()()()()))()","((()()())(()))","((()()())()())","((()()())())()","((()()()))(())","((()()()))()()","((()())((())))","((()())(()()))","((()())(())())","((()())(()))()","((()())()(()))","((()())()()())","((()())()())()","((()())())(())","((()())())()()","((()()))((()))","((()()))(()())","((()()))(())()","((()()))()(())","((()()))()()()","((())(((()))))","((())((()())))","((())((())()))","((())((()))())","((())((())))()","((())(()(())))","((())(()()()))","((())(()())())","((())(()()))()","((())(())(()))","((())(())()())","((())(())())()","((())(()))(())","((())(()))()()","((())()((())))","((())()(()()))","((())()(())())","((())()(()))()","((())()()(()))","((())()()()())","((())()()())()","((())()())(())","((())()())()()","((())())((()))","((())())(()())","((())())(())()","((())())()(())","((())())()()()","((()))(((())))","((()))((()()))","((()))((())())","((()))((()))()","((()))(()(()))","((()))(()()())","((()))(()())()","((()))(())(())","((()))(())()()","((()))()((()))","((()))()(()())","((()))()(())()","((()))()()(())","((()))()()()()","(()((((())))))","(()(((()()))))","(()(((())())))","(()(((()))()))","(()(((())))())","(()(((()))))()","(()((()(()))))","(()((()()())))","(()((()())()))","(()((()()))())","(()((()())))()","(()((())(())))","(()((())()()))","(()((())())())","(()((())()))()","(()((()))(()))","(()((()))()())","(()((()))())()","(()((())))(())","(()((())))()()","(()(()((()))))","(()(()(()())))","(()(()(())()))","(()(()(()))())","(()(()(())))()","(()(()()(())))","(()(()()()()))","(()(()()())())","(()(()()()))()","(()(()())(()))","(()(()())()())","(()(()())())()","(()(()()))(())","(()(()()))()()","(()(())((())))","(()(())(()()))","(()(())(())())","(()(())(()))()","(()(())()(()))","(()(())()()())","(()(())()())()","(()(())())(())","(()(())())()()","(()(()))((()))","(()(()))(()())","(()(()))(())()","(()(()))()(())","(()(()))()()()","(()()(((()))))","(()()((()())))","(()()((())()))","(()()((()))())","(()()((())))()","(()()(()(())))","(()()(()()()))","(()()(()())())","(()()(()()))()","(()()(())(()))","(()()(())()())","(()()(())())()","(()()(()))(())","(()()(()))()()","(()()()((())))","(()()()(()()))","(()()()(())())","(()()()(()))()","(()()()()(()))","(()()()()()())","(()()()()())()","(()()()())(())","(()()()())()()","(()()())((()))","(()()())(()())","(()()())(())()","(()()())()(())","(()()())()()()","(()())(((())))","(()())((()()))","(()())((())())","(()())((()))()","(()())(()(()))","(()())(()()())","(()())(()())()","(()())(())(())","(()())(())()()","(()())()((()))","(()())()(()())","(()())()(())()","(()())()()(())","(()())()()()()","(())((((()))))","(())(((()())))","(())(((())()))","(())(((()))())","(())(((())))()","(())((()(())))","(())((()()()))","(())((()())())","(())((()()))()","(())((())(()))","(())((())()())","(())((())())()","(())((()))(())","(())((()))()()","(())(()((())))","(())(()(()()))","(())(()(())())","(())(()(()))()","(())(()()(()))","(())(()()()())","(())(()()())()","(())(()())(())","(())(()())()()","(())(())((()))","(())(())(()())","(())(())(())()","(())(())()(())","(())(())()()()","(())()(((())))","(())()((()()))","(())()((())())","(())()((()))()","(())()(()(()))","(())()(()()())","(())()(()())()","(())()(())(())","(())()(())()()","(())()()((()))","(())()()(()())","(())()()(())()","(())()()()(())","(())()()()()()","()(((((())))))","()((((()()))))","()((((())())))","()((((()))()))","()((((())))())","()((((()))))()","()(((()(()))))","()(((()()())))","()(((()())()))","()(((()()))())","()(((()())))()","()(((())(())))","()(((())()()))","()(((())())())","()(((())()))()","()(((()))(()))","()(((()))()())","()(((()))())()","()(((())))(())","()(((())))()()","()((()((()))))","()((()(()())))","()((()(())()))","()((()(()))())","()((()(())))()","()((()()(())))","()((()()()()))","()((()()())())","()((()()()))()","()((()())(()))","()((()())()())","()((()())())()","()((()()))(())","()((()()))()()","()((())((())))","()((())(()()))","()((())(())())","()((())(()))()","()((())()(()))","()((())()()())","()((())()())()","()((())())(())","()((())())()()","()((()))((()))","()((()))(()())","()((()))(())()","()((()))()(())","()((()))()()()","()(()(((()))))","()(()((()())))","()(()((())()))","()(()((()))())","()(()((())))()","()(()(()(())))","()(()(()()()))","()(()(()())())","()(()(()()))()","()(()(())(()))","()(()(())()())","()(()(())())()","()(()(()))(())","()(()(()))()()","()(()()((())))","()(()()(()()))","()(()()(())())","()(()()(()))()","()(()()()(()))","()(()()()()())","()(()()()())()","()(()()())(())","()(()()())()()","()(()())((()))","()(()())(()())","()(()())(())()","()(()())()(())","()(()())()()()","()(())(((())))","()(())((()()))","()(())((())())","()(())((()))()","()(())(()(()))","()(())(()()())","()(())(()())()","()(())(())(())","()(())(())()()","()(())()((()))","()(())()(()())","()(())()(())()","()(())()()(())","()(())()()()()","()()((((()))))","()()(((()())))","()()(((())()))","()()(((()))())","()()(((())))()","()()((()(())))","()()((()()()))","()()((()())())","()()((()()))()","()()((())(()))","()()((())()())","()()((())())()","()()((()))(())","()()((()))()()","()()(()((())))","()()(()(()()))","()()(()(())())","()()(()(()))()","()()(()()(()))","()()(()()()())","()()(()()())()","()()(()())(())","()()(()())()()","()()(())((()))","()()(())(()())","()()(())(())()","()()(())()(())","()()(())()()()","()()()(((())))","()()()((()()))","()()()((())())","()()()((()))()","()()()(()(()))","()()()(()()())","()()()(()())()","()()()(())(())","()()()(())()()","()()()()((()))","()()()()(()())","()()()()(())()","()()()()()(())","()()()()()()()"};
            case 8:
                return vector<string>{"(((((((())))))))","((((((()()))))))","((((((())())))))","((((((()))()))))","((((((())))())))","((((((()))))()))","((((((())))))())","((((((()))))))()","(((((()(()))))))","(((((()()())))))","(((((()())()))))","(((((()()))())))","(((((()())))()))","(((((()()))))())","(((((()())))))()","(((((())(())))))","(((((())()()))))","(((((())())())))","(((((())()))()))","(((((())())))())","(((((())()))))()","(((((()))(()))))","(((((()))()())))","(((((()))())()))","(((((()))()))())","(((((()))())))()","(((((())))(())))","(((((())))()()))","(((((())))())())","(((((())))()))()","(((((()))))(()))","(((((()))))()())","(((((()))))())()","(((((())))))(())","(((((())))))()()","((((()((()))))))","((((()(()())))))","((((()(())()))))","((((()(()))())))","((((()(())))()))","((((()(()))))())","((((()(())))))()","((((()()(())))))","((((()()()()))))","((((()()())())))","((((()()()))()))","((((()()())))())","((((()()()))))()","((((()())(()))))","((((()())()())))","((((()())())()))","((((()())()))())","((((()())())))()","((((()()))(())))","((((()()))()()))","((((()()))())())","((((()()))()))()","((((()())))(()))","((((()())))()())","((((()())))())()","((((()()))))(())","((((()()))))()()","((((())((())))))","((((())(()()))))","((((())(())())))","((((())(()))()))","((((())(())))())","((((())(()))))()","((((())()(()))))","((((())()()())))","((((())()())()))","((((())()()))())","((((())()())))()","((((())())(())))","((((())())()()))","((((())())())())","((((())())()))()","((((())()))(()))","((((())()))()())","((((())()))())()","((((())())))(())","((((())())))()()","((((()))((()))))","((((()))(()())))","((((()))(())()))","((((()))(()))())","((((()))(())))()","((((()))()(())))","((((()))()()()))","((((()))()())())","((((()))()()))()","((((()))())(()))","((((()))())()())","((((()))())())()","((((()))()))(())","((((()))()))()()","((((())))((())))","((((())))(()()))","((((())))(())())","((((())))(()))()","((((())))()(()))","((((())))()()())","((((())))()())()","((((())))())(())","((((())))())()()","((((()))))((()))","((((()))))(()())","((((()))))(())()","((((()))))()(())","((((()))))()()()","(((()(((()))))))","(((()((()())))))","(((()((())()))))","(((()((()))())))","(((()((())))()))","(((()((()))))())","(((()((())))))()","(((()(()(())))))","(((()(()()()))))","(((()(()())())))","(((()(()()))()))","(((()(()())))())","(((()(()()))))()","(((()(())(()))))","(((()(())()())))","(((()(())())()))","(((()(())()))())","(((()(())())))()","(((()(()))(())))","(((()(()))()()))","(((()(()))())())","(((()(()))()))()","(((()(())))(()))","(((()(())))()())","(((()(())))())()","(((()(()))))(())","(((()(()))))()()","(((()()((())))))","(((()()(()()))))","(((()()(())())))","(((()()(()))()))","(((()()(())))())","(((()()(()))))()","(((()()()(()))))","(((()()()()())))","(((()()()())()))","(((()()()()))())","(((()()()())))()","(((()()())(())))","(((()()())()()))","(((()()())())())","(((()()())()))()","(((()()()))(()))","(((()()()))()())","(((()()()))())()","(((()()())))(())","(((()()())))()()","(((()())((()))))","(((()())(()())))","(((()())(())()))","(((()())(()))())","(((()())(())))()","(((()())()(())))","(((()())()()()))","(((()())()())())","(((()())()()))()","(((()())())(()))","(((()())())()())","(((()())())())()","(((()())()))(())","(((()())()))()()","(((()()))((())))","(((()()))(()()))","(((()()))(())())","(((()()))(()))()","(((()()))()(()))","(((()()))()()())","(((()()))()())()","(((()()))())(())","(((()()))())()()","(((()())))((()))","(((()())))(()())","(((()())))(())()","(((()())))()(())","(((()())))()()()","(((())(((())))))","(((())((()()))))","(((())((())())))","(((())((()))()))","(((())((())))())","(((())((()))))()","(((())(()(()))))","(((())(()()())))","(((())(()())()))","(((())(()()))())","(((())(()())))()","(((())(())(())))","(((())(())()()))","(((())(())())())","(((())(())()))()","(((())(()))(()))","(((())(()))()())","(((())(()))())()","(((())(())))(())","(((())(())))()()","(((())()((()))))","(((())()(()())))","(((())()(())()))","(((())()(()))())","(((())()(())))()","(((())()()(())))","(((())()()()()))","(((())()()())())","(((())()()()))()","(((())()())(()))","(((())()())()())","(((())()())())()","(((())()()))(())","(((())()()))()()","(((())())((())))","(((())())(()()))","(((())())(())())","(((())())(()))()","(((())())()(()))","(((())())()()())","(((())())()())()","(((())())())(())","(((())())())()()","(((())()))((()))","(((())()))(()())","(((())()))(())()","(((())()))()(())","(((())()))()()()","(((()))(((()))))","(((()))((()())))","(((()))((())()))","(((()))((()))())","(((()))((())))()","(((()))(()(())))","(((()))(()()()))","(((()))(()())())","(((()))(()()))()","(((()))(())(()))","(((()))(())()())","(((()))(())())()","(((()))(()))(())","(((()))(()))()()","(((()))()((())))","(((()))()(()()))","(((()))()(())())","(((()))()(()))()","(((()))()()(()))","(((()))()()()())","(((()))()()())()","(((()))()())(())","(((()))()())()()","(((()))())((()))","(((()))())(()())","(((()))())(())()","(((()))())()(())","(((()))())()()()","(((())))(((())))","(((())))((()()))","(((())))((())())","(((())))((()))()","(((())))(()(()))","(((())))(()()())","(((())))(()())()","(((())))(())(())","(((())))(())()()","(((())))()((()))","(((())))()(()())","(((())))()(())()","(((())))()()(())","(((())))()()()()","((()((((()))))))","((()(((()())))))","((()(((())()))))","((()(((()))())))","((()(((())))()))","((()(((()))))())","((()(((())))))()","((()((()(())))))","((()((()()()))))","((()((()())())))","((()((()()))()))","((()((()())))())","((()((()()))))()","((()((())(()))))","((()((())()())))","((()((())())()))","((()((())()))())","((()((())())))()","((()((()))(())))","((()((()))()()))","((()((()))())())","((()((()))()))()","((()((())))(()))","((()((())))()())","((()((())))())()","((()((()))))(())","((()((()))))()()","((()(()((())))))","((()(()(()()))))","((()(()(())())))","((()(()(()))()))","((()(()(())))())","((()(()(()))))()","((()(()()(()))))","((()(()()()())))","((()(()()())()))","((()(()()()))())","((()(()()())))()","((()(()())(())))","((()(()())()()))","((()(()())())())","((()(()())()))()","((()(()()))(()))","((()(()()))()())","((()(()()))())()","((()(()())))(())","((()(()())))()()","((()(())((()))))","((()(())(()())))","((()(())(())()))","((()(())(()))())","((()(())(())))()","((()(())()(())))","((()(())()()()))","((()(())()())())","((()(())()()))()","((()(())())(()))","((()(())())()())","((()(())())())()","((()(())()))(())","((()(())()))()()","((()(()))((())))","((()(()))(()()))","((()(()))(())())","((()(()))(()))()","((()(()))()(()))","((()(()))()()())","((()(()))()())()","((()(()))())(())","((()(()))())()()","((()(())))((()))","((()(())))(()())","((()(())))(())()","((()(())))()(())","((()(())))()()()","((()()(((())))))","((()()((()()))))","((()()((())())))","((()()((()))()))","((()()((())))())","((()()((()))))()","((()()(()(()))))","((()()(()()())))","((()()(()())()))","((()()(()()))())","((()()(()())))()","((()()(())(())))","((()()(())()()))","((()()(())())())","((()()(())()))()","((()()(()))(()))","((()()(()))()())","((()()(()))())()","((()()(())))(())","((()()(())))()()","((()()()((()))))","((()()()(()())))","((()()()(())()))","((()()()(()))())","((()()()(())))()","((()()()()(())))","((()()()()()()))","((()()()()())())","((()()()()()))()","((()()()())(()))","((()()()())()())","((()()()())())()","((()()()()))(())","((()()()()))()()","((()()())((())))","((()()())(()()))","((()()())(())())","((()()())(()))()","((()()())()(()))","((()()())()()())","((()()())()())()","((()()())())(())","((()()())())()()","((()()()))((()))","((()()()))(()())","((()()()))(())()","((()()()))()(())","((()()()))()()()","((()())(((()))))","((()())((()())))","((()())((())()))","((()())((()))())","((()())((())))()","((()())(()(())))","((()())(()()()))","((()())(()())())","((()())(()()))()","((()())(())(()))","((()())(())()())","((()())(())())()","((()())(()))(())","((()())(()))()()","((()())()((())))","((()())()(()()))","((()())()(())())","((()())()(()))()","((()())()()(()))","((()())()()()())","((()())()()())()","((()())()())(())","((()())()())()()","((()())())((()))","((()())())(()())","((()())())(())()","((()())())()(())","((()())())()()()","((()()))(((())))","((()()))((()()))","((()()))((())())","((()()))((()))()","((()()))(()(()))","((()()))(()()())","((()()))(()())()","((()()))(())(())","((()()))(())()()","((()()))()((()))","((()()))()(()())","((()()))()(())()","((()()))()()(())","((()()))()()()()","((())((((())))))","((())(((()()))))","((())(((())())))","((())(((()))()))","((())(((())))())","((())(((()))))()","((())((()(()))))","((())((()()())))","((())((()())()))","((())((()()))())","((())((()())))()","((())((())(())))","((())((())()()))","((())((())())())","((())((())()))()","((())((()))(()))","((())((()))()())","((())((()))())()","((())((())))(())","((())((())))()()","((())(()((()))))","((())(()(()())))","((())(()(())()))","((())(()(()))())","((())(()(())))()","((())(()()(())))","((())(()()()()))","((())(()()())())","((())(()()()))()","((())(()())(()))","((())(()())()())","((())(()())())()","((())(()()))(())","((())(()()))()()","((())(())((())))","((())(())(()()))","((())(())(())())","((())(())(()))()","((())(())()(()))","((())(())()()())","((())(())()())()","((())(())())(())","((())(())())()()","((())(()))((()))","((())(()))(()())","((())(()))(())()","((())(()))()(())","((())(()))()()()","((())()(((()))))","((())()((()())))","((())()((())()))","((())()((()))())","((())()((())))()","((())()(()(())))","((())()(()()()))","((())()(()())())","((())()(()()))()","((())()(())(()))","((())()(())()())","((())()(())())()","((())()(()))(())","((())()(()))()()","((())()()((())))","((())()()(()()))","((())()()(())())","((())()()(()))()","((())()()()(()))","((())()()()()())","((())()()()())()","((())()()())(())","((())()()())()()","((())()())((()))","((())()())(()())","((())()())(())()","((())()())()(())","((())()())()()()","((())())(((())))","((())())((()()))","((())())((())())","((())())((()))()","((())())(()(()))","((())())(()()())","((())())(()())()","((())())(())(())","((())())(())()()","((())())()((()))","((())())()(()())","((())())()(())()","((())())()()(())","((())())()()()()","((()))((((()))))","((()))(((()())))","((()))(((())()))","((()))(((()))())","((()))(((())))()","((()))((()(())))","((()))((()()()))","((()))((()())())","((()))((()()))()","((()))((())(()))","((()))((())()())","((()))((())())()","((()))((()))(())","((()))((()))()()","((()))(()((())))","((()))(()(()()))","((()))(()(())())","((()))(()(()))()","((()))(()()(()))","((()))(()()()())","((()))(()()())()","((()))(()())(())","((()))(()())()()","((()))(())((()))","((()))(())(()())","((()))(())(())()","((()))(())()(())","((()))(())()()()","((()))()(((())))","((()))()((()()))","((()))()((())())","((()))()((()))()","((()))()(()(()))","((()))()(()()())","((()))()(()())()","((()))()(())(())","((()))()(())()()","((()))()()((()))","((()))()()(()())","((()))()()(())()","((()))()()()(())","((()))()()()()()","(()(((((()))))))","(()((((()())))))","(()((((())()))))","(()((((()))())))","(()((((())))()))","(()((((()))))())","(()((((())))))()","(()(((()(())))))","(()(((()()()))))","(()(((()())())))","(()(((()()))()))","(()(((()())))())","(()(((()()))))()","(()(((())(()))))","(()(((())()())))","(()(((())())()))","(()(((())()))())","(()(((())())))()","(()(((()))(())))","(()(((()))()()))","(()(((()))())())","(()(((()))()))()","(()(((())))(()))","(()(((())))()())","(()(((())))())()","(()(((()))))(())","(()(((()))))()()","(()((()((())))))","(()((()(()()))))","(()((()(())())))","(()((()(()))()))","(()((()(())))())","(()((()(()))))()","(()((()()(()))))","(()((()()()())))","(()((()()())()))","(()((()()()))())","(()((()()())))()","(()((()())(())))","(()((()())()()))","(()((()())())())","(()((()())()))()","(()((()()))(()))","(()((()()))()())","(()((()()))())()","(()((()())))(())","(()((()())))()()","(()((())((()))))","(()((())(()())))","(()((())(())()))","(()((())(()))())","(()((())(())))()","(()((())()(())))","(()((())()()()))","(()((())()())())","(()((())()()))()","(()((())())(()))","(()((())())()())","(()((())())())()","(()((())()))(())","(()((())()))()()","(()((()))((())))","(()((()))(()()))","(()((()))(())())","(()((()))(()))()","(()((()))()(()))","(()((()))()()())","(()((()))()())()","(()((()))())(())","(()((()))())()()","(()((())))((()))","(()((())))(()())","(()((())))(())()","(()((())))()(())","(()((())))()()()","(()(()(((())))))","(()(()((()()))))","(()(()((())())))","(()(()((()))()))","(()(()((())))())","(()(()((()))))()","(()(()(()(()))))","(()(()(()()())))","(()(()(()())()))","(()(()(()()))())","(()(()(()())))()","(()(()(())(())))","(()(()(())()()))","(()(()(())())())","(()(()(())()))()","(()(()(()))(()))","(()(()(()))()())","(()(()(()))())()","(()(()(())))(())","(()(()(())))()()","(()(()()((()))))","(()(()()(()())))","(()(()()(())()))","(()(()()(()))())","(()(()()(())))()","(()(()()()(())))","(()(()()()()()))","(()(()()()())())","(()(()()()()))()","(()(()()())(()))","(()(()()())()())","(()(()()())())()","(()(()()()))(())","(()(()()()))()()","(()(()())((())))","(()(()())(()()))","(()(()())(())())","(()(()())(()))()","(()(()())()(()))","(()(()())()()())","(()(()())()())()","(()(()())())(())","(()(()())())()()","(()(()()))((()))","(()(()()))(()())","(()(()()))(())()","(()(()()))()(())","(()(()()))()()()","(()(())(((()))))","(()(())((()())))","(()(())((())()))","(()(())((()))())","(()(())((())))()","(()(())(()(())))","(()(())(()()()))","(()(())(()())())","(()(())(()()))()","(()(())(())(()))","(()(())(())()())","(()(())(())())()","(()(())(()))(())","(()(())(()))()()","(()(())()((())))","(()(())()(()()))","(()(())()(())())","(()(())()(()))()","(()(())()()(()))","(()(())()()()())","(()(())()()())()","(()(())()())(())","(()(())()())()()","(()(())())((()))","(()(())())(()())","(()(())())(())()","(()(())())()(())","(()(())())()()()","(()(()))(((())))","(()(()))((()()))","(()(()))((())())","(()(()))((()))()","(()(()))(()(()))","(()(()))(()()())","(()(()))(()())()","(()(()))(())(())","(()(()))(())()()","(()(()))()((()))","(()(()))()(()())","(()(()))()(())()","(()(()))()()(())","(()(()))()()()()","(()()((((())))))","(()()(((()()))))","(()()(((())())))","(()()(((()))()))","(()()(((())))())","(()()(((()))))()","(()()((()(()))))","(()()((()()())))","(()()((()())()))","(()()((()()))())","(()()((()())))()","(()()((())(())))","(()()((())()()))","(()()((())())())","(()()((())()))()","(()()((()))(()))","(()()((()))()())","(()()((()))())()","(()()((())))(())","(()()((())))()()","(()()(()((()))))","(()()(()(()())))","(()()(()(())()))","(()()(()(()))())","(()()(()(())))()","(()()(()()(())))","(()()(()()()()))","(()()(()()())())","(()()(()()()))()","(()()(()())(()))","(()()(()())()())","(()()(()())())()","(()()(()()))(())","(()()(()()))()()","(()()(())((())))","(()()(())(()()))","(()()(())(())())","(()()(())(()))()","(()()(())()(()))","(()()(())()()())","(()()(())()())()","(()()(())())(())","(()()(())())()()","(()()(()))((()))","(()()(()))(()())","(()()(()))(())()","(()()(()))()(())","(()()(()))()()()","(()()()(((()))))","(()()()((()())))","(()()()((())()))","(()()()((()))())","(()()()((())))()","(()()()(()(())))","(()()()(()()()))","(()()()(()())())","(()()()(()()))()","(()()()(())(()))","(()()()(())()())","(()()()(())())()","(()()()(()))(())","(()()()(()))()()","(()()()()((())))","(()()()()(()()))","(()()()()(())())","(()()()()(()))()","(()()()()()(()))","(()()()()()()())","(()()()()()())()","(()()()()())(())","(()()()()())()()","(()()()())((()))","(()()()())(()())","(()()()())(())()","(()()()())()(())","(()()()())()()()","(()()())(((())))","(()()())((()()))","(()()())((())())","(()()())((()))()","(()()())(()(()))","(()()())(()()())","(()()())(()())()","(()()())(())(())","(()()())(())()()","(()()())()((()))","(()()())()(()())","(()()())()(())()","(()()())()()(())","(()()())()()()()","(()())((((()))))","(()())(((()())))","(()())(((())()))","(()())(((()))())","(()())(((())))()","(()())((()(())))","(()())((()()()))","(()())((()())())","(()())((()()))()","(()())((())(()))","(()())((())()())","(()())((())())()","(()())((()))(())","(()())((()))()()","(()())(()((())))","(()())(()(()()))","(()())(()(())())","(()())(()(()))()","(()())(()()(()))","(()())(()()()())","(()())(()()())()","(()())(()())(())","(()())(()())()()","(()())(())((()))","(()())(())(()())","(()())(())(())()","(()())(())()(())","(()())(())()()()","(()())()(((())))","(()())()((()()))","(()())()((())())","(()())()((()))()","(()())()(()(()))","(()())()(()()())","(()())()(()())()","(()())()(())(())","(()())()(())()()","(()())()()((()))","(()())()()(()())","(()())()()(())()","(()())()()()(())","(()())()()()()()","(())(((((())))))","(())((((()()))))","(())((((())())))","(())((((()))()))","(())((((())))())","(())((((()))))()","(())(((()(()))))","(())(((()()())))","(())(((()())()))","(())(((()()))())","(())(((()())))()","(())(((())(())))","(())(((())()()))","(())(((())())())","(())(((())()))()","(())(((()))(()))","(())(((()))()())","(())(((()))())()","(())(((())))(())","(())(((())))()()","(())((()((()))))","(())((()(()())))","(())((()(())()))","(())((()(()))())","(())((()(())))()","(())((()()(())))","(())((()()()()))","(())((()()())())","(())((()()()))()","(())((()())(()))","(())((()())()())","(())((()())())()","(())((()()))(())","(())((()()))()()","(())((())((())))","(())((())(()()))","(())((())(())())","(())((())(()))()","(())((())()(()))","(())((())()()())","(())((())()())()","(())((())())(())","(())((())())()()","(())((()))((()))","(())((()))(()())","(())((()))(())()","(())((()))()(())","(())((()))()()()","(())(()(((()))))","(())(()((()())))","(())(()((())()))","(())(()((()))())","(())(()((())))()","(())(()(()(())))","(())(()(()()()))","(())(()(()())())","(())(()(()()))()","(())(()(())(()))","(())(()(())()())","(())(()(())())()","(())(()(()))(())","(())(()(()))()()","(())(()()((())))","(())(()()(()()))","(())(()()(())())","(())(()()(()))()","(())(()()()(()))","(())(()()()()())","(())(()()()())()","(())(()()())(())","(())(()()())()()","(())(()())((()))","(())(()())(()())","(())(()())(())()","(())(()())()(())","(())(()())()()()","(())(())(((())))","(())(())((()()))","(())(())((())())","(())(())((()))()","(())(())(()(()))","(())(())(()()())","(())(())(()())()","(())(())(())(())","(())(())(())()()","(())(())()((()))","(())(())()(()())","(())(())()(())()","(())(())()()(())","(())(())()()()()","(())()((((()))))","(())()(((()())))","(())()(((())()))","(())()(((()))())","(())()(((())))()","(())()((()(())))","(())()((()()()))","(())()((()())())","(())()((()()))()","(())()((())(()))","(())()((())()())","(())()((())())()","(())()((()))(())","(())()((()))()()","(())()(()((())))","(())()(()(()()))","(())()(()(())())","(())()(()(()))()","(())()(()()(()))","(())()(()()()())","(())()(()()())()","(())()(()())(())","(())()(()())()()","(())()(())((()))","(())()(())(()())","(())()(())(())()","(())()(())()(())","(())()(())()()()","(())()()(((())))","(())()()((()()))","(())()()((())())","(())()()((()))()","(())()()(()(()))","(())()()(()()())","(())()()(()())()","(())()()(())(())","(())()()(())()()","(())()()()((()))","(())()()()(()())","(())()()()(())()","(())()()()()(())","(())()()()()()()","()((((((()))))))","()(((((()())))))","()(((((())()))))","()(((((()))())))","()(((((())))()))","()(((((()))))())","()(((((())))))()","()((((()(())))))","()((((()()()))))","()((((()())())))","()((((()()))()))","()((((()())))())","()((((()()))))()","()((((())(()))))","()((((())()())))","()((((())())()))","()((((())()))())","()((((())())))()","()((((()))(())))","()((((()))()()))","()((((()))())())","()((((()))()))()","()((((())))(()))","()((((())))()())","()((((())))())()","()((((()))))(())","()((((()))))()()","()(((()((())))))","()(((()(()()))))","()(((()(())())))","()(((()(()))()))","()(((()(())))())","()(((()(()))))()","()(((()()(()))))","()(((()()()())))","()(((()()())()))","()(((()()()))())","()(((()()())))()","()(((()())(())))","()(((()())()()))","()(((()())())())","()(((()())()))()","()(((()()))(()))","()(((()()))()())","()(((()()))())()","()(((()())))(())","()(((()())))()()","()(((())((()))))","()(((())(()())))","()(((())(())()))","()(((())(()))())","()(((())(())))()","()(((())()(())))","()(((())()()()))","()(((())()())())","()(((())()()))()","()(((())())(()))","()(((())())()())","()(((())())())()","()(((())()))(())","()(((())()))()()","()(((()))((())))","()(((()))(()()))","()(((()))(())())","()(((()))(()))()","()(((()))()(()))","()(((()))()()())","()(((()))()())()","()(((()))())(())","()(((()))())()()","()(((())))((()))","()(((())))(()())","()(((())))(())()","()(((())))()(())","()(((())))()()()","()((()(((())))))","()((()((()()))))","()((()((())())))","()((()((()))()))","()((()((())))())","()((()((()))))()","()((()(()(()))))","()((()(()()())))","()((()(()())()))","()((()(()()))())","()((()(()())))()","()((()(())(())))","()((()(())()()))","()((()(())())())","()((()(())()))()","()((()(()))(()))","()((()(()))()())","()((()(()))())()","()((()(())))(())","()((()(())))()()","()((()()((()))))","()((()()(()())))","()((()()(())()))","()((()()(()))())","()((()()(())))()","()((()()()(())))","()((()()()()()))","()((()()()())())","()((()()()()))()","()((()()())(()))","()((()()())()())","()((()()())())()","()((()()()))(())","()((()()()))()()","()((()())((())))","()((()())(()()))","()((()())(())())","()((()())(()))()","()((()())()(()))","()((()())()()())","()((()())()())()","()((()())())(())","()((()())())()()","()((()()))((()))","()((()()))(()())","()((()()))(())()","()((()()))()(())","()((()()))()()()","()((())(((()))))","()((())((()())))","()((())((())()))","()((())((()))())","()((())((())))()","()((())(()(())))","()((())(()()()))","()((())(()())())","()((())(()()))()","()((())(())(()))","()((())(())()())","()((())(())())()","()((())(()))(())","()((())(()))()()","()((())()((())))","()((())()(()()))","()((())()(())())","()((())()(()))()","()((())()()(()))","()((())()()()())","()((())()()())()","()((())()())(())","()((())()())()()","()((())())((()))","()((())())(()())","()((())())(())()","()((())())()(())","()((())())()()()","()((()))(((())))","()((()))((()()))","()((()))((())())","()((()))((()))()","()((()))(()(()))","()((()))(()()())","()((()))(()())()","()((()))(())(())","()((()))(())()()","()((()))()((()))","()((()))()(()())","()((()))()(())()","()((()))()()(())","()((()))()()()()","()(()((((())))))","()(()(((()()))))","()(()(((())())))","()(()(((()))()))","()(()(((())))())","()(()(((()))))()","()(()((()(()))))","()(()((()()())))","()(()((()())()))","()(()((()()))())","()(()((()())))()","()(()((())(())))","()(()((())()()))","()(()((())())())","()(()((())()))()","()(()((()))(()))","()(()((()))()())","()(()((()))())()","()(()((())))(())","()(()((())))()()","()(()(()((()))))","()(()(()(()())))","()(()(()(())()))","()(()(()(()))())","()(()(()(())))()","()(()(()()(())))","()(()(()()()()))","()(()(()()())())","()(()(()()()))()","()(()(()())(()))","()(()(()())()())","()(()(()())())()","()(()(()()))(())","()(()(()()))()()","()(()(())((())))","()(()(())(()()))","()(()(())(())())","()(()(())(()))()","()(()(())()(()))","()(()(())()()())","()(()(())()())()","()(()(())())(())","()(()(())())()()","()(()(()))((()))","()(()(()))(()())","()(()(()))(())()","()(()(()))()(())","()(()(()))()()()","()(()()(((()))))","()(()()((()())))","()(()()((())()))","()(()()((()))())","()(()()((())))()","()(()()(()(())))","()(()()(()()()))","()(()()(()())())","()(()()(()()))()","()(()()(())(()))","()(()()(())()())","()(()()(())())()","()(()()(()))(())","()(()()(()))()()","()(()()()((())))","()(()()()(()()))","()(()()()(())())","()(()()()(()))()","()(()()()()(()))","()(()()()()()())","()(()()()()())()","()(()()()())(())","()(()()()())()()","()(()()())((()))","()(()()())(()())","()(()()())(())()","()(()()())()(())","()(()()())()()()","()(()())(((())))","()(()())((()()))","()(()())((())())","()(()())((()))()","()(()())(()(()))","()(()())(()()())","()(()())(()())()","()(()())(())(())","()(()())(())()()","()(()())()((()))","()(()())()(()())","()(()())()(())()","()(()())()()(())","()(()())()()()()","()(())((((()))))","()(())(((()())))","()(())(((())()))","()(())(((()))())","()(())(((())))()","()(())((()(())))","()(())((()()()))","()(())((()())())","()(())((()()))()","()(())((())(()))","()(())((())()())","()(())((())())()","()(())((()))(())","()(())((()))()()","()(())(()((())))","()(())(()(()()))","()(())(()(())())","()(())(()(()))()","()(())(()()(()))","()(())(()()()())","()(())(()()())()","()(())(()())(())","()(())(()())()()","()(())(())((()))","()(())(())(()())","()(())(())(())()","()(())(())()(())","()(())(())()()()","()(())()(((())))","()(())()((()()))","()(())()((())())","()(())()((()))()","()(())()(()(()))","()(())()(()()())","()(())()(()())()","()(())()(())(())","()(())()(())()()","()(())()()((()))","()(())()()(()())","()(())()()(())()","()(())()()()(())","()(())()()()()()","()()(((((())))))","()()((((()()))))","()()((((())())))","()()((((()))()))","()()((((())))())","()()((((()))))()","()()(((()(()))))","()()(((()()())))","()()(((()())()))","()()(((()()))())","()()(((()())))()","()()(((())(())))","()()(((())()()))","()()(((())())())","()()(((())()))()","()()(((()))(()))","()()(((()))()())","()()(((()))())()","()()(((())))(())","()()(((())))()()","()()((()((()))))","()()((()(()())))","()()((()(())()))","()()((()(()))())","()()((()(())))()","()()((()()(())))","()()((()()()()))","()()((()()())())","()()((()()()))()","()()((()())(()))","()()((()())()())","()()((()())())()","()()((()()))(())","()()((()()))()()","()()((())((())))","()()((())(()()))","()()((())(())())","()()((())(()))()","()()((())()(()))","()()((())()()())","()()((())()())()","()()((())())(())","()()((())())()()","()()((()))((()))","()()((()))(()())","()()((()))(())()","()()((()))()(())","()()((()))()()()","()()(()(((()))))","()()(()((()())))","()()(()((())()))","()()(()((()))())","()()(()((())))()","()()(()(()(())))","()()(()(()()()))","()()(()(()())())","()()(()(()()))()","()()(()(())(()))","()()(()(())()())","()()(()(())())()","()()(()(()))(())","()()(()(()))()()","()()(()()((())))","()()(()()(()()))","()()(()()(())())","()()(()()(()))()","()()(()()()(()))","()()(()()()()())","()()(()()()())()","()()(()()())(())","()()(()()())()()","()()(()())((()))","()()(()())(()())","()()(()())(())()","()()(()())()(())","()()(()())()()()","()()(())(((())))","()()(())((()()))","()()(())((())())","()()(())((()))()","()()(())(()(()))","()()(())(()()())","()()(())(()())()","()()(())(())(())","()()(())(())()()","()()(())()((()))","()()(())()(()())","()()(())()(())()","()()(())()()(())","()()(())()()()()","()()()((((()))))","()()()(((()())))","()()()(((())()))","()()()(((()))())","()()()(((())))()","()()()((()(())))","()()()((()()()))","()()()((()())())","()()()((()()))()","()()()((())(()))","()()()((())()())","()()()((())())()","()()()((()))(())","()()()((()))()()","()()()(()((())))","()()()(()(()()))","()()()(()(())())","()()()(()(()))()","()()()(()()(()))","()()()(()()()())","()()()(()()())()","()()()(()())(())","()()()(()())()()","()()()(())((()))","()()()(())(()())","()()()(())(())()","()()()(())()(())","()()()(())()()()","()()()()(((())))","()()()()((()()))","()()()()((())())","()()()()((()))()","()()()()(()(()))","()()()()(()()())","()()()()(()())()","()()()()(())(())","()()()()(())()()","()()()()()((()))","()()()()()(()())","()()()()()(())()","()()()()()()(())","()()()()()()()()"};
        }
        return {};
    }
};