[22]. 括号生成
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[22]. 括号生成
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>"((((((()))))))","(((((()())))))","(((((())()))))","(((((()))())))","(((((())))()))","(((((()))))())","(((((())))))()","((((()(())))))","((((()()()))))","((((()())())))","((((()()))()))","((((()())))())","((((()()))))()","((((())(()))))","((((())()())))","((((())())()))","((((())()))())","((((())())))()","((((()))(())))","((((()))()()))","((((()))())())","((((()))()))()","((((())))(()))","((((())))()())","((((())))())()","((((()))))(())","((((()))))()()","(((()((())))))","(((()(()()))))","(((()(())())))","(((()(()))()))","(((()(())))())","(((()(()))))()","(((()()(()))))","(((()()()())))","(((()()())()))","(((()()()))())","(((()()())))()","(((()())(())))","(((()())()()))","(((()())())())","(((()())()))()","(((()()))(()))","(((()()))()())","(((()()))())()","(((()())))(())","(((()())))()()","(((())((()))))","(((())(()())))","(((())(())()))","(((())(()))())","(((())(())))()","(((())()(())))","(((())()()()))","(((())()())())","(((())()()))()","(((())())(()))","(((())())()())","(((())())())()","(((())()))(())","(((())()))()()","(((()))((())))","(((()))(()()))","(((()))(())())","(((()))(()))()","(((()))()(()))","(((()))()()())","(((()))()())()","(((()))())(())","(((()))())()()","(((())))((()))","(((())))(()())","(((())))(())()","(((())))()(())","(((())))()()()","((()(((())))))","((()((()()))))","((()((())())))","((()((()))()))","((()((())))())","((()((()))))()","((()(()(()))))","((()(()()())))","((()(()())()))","((()(()()))())","((()(()())))()","((()(())(())))","((()(())()()))","((()(())())())","((()(())()))()","((()(()))(()))","((()(()))()())","((()(()))())()","((()(())))(())","((()(())))()()","((()()((()))))","((()()(()())))","((()()(())()))","((()()(()))())","((()()(())))()","((()()()(())))","((()()()()()))","((()()()())())","((()()()()))()","((()()())(()))","((()()())()())","((()()())())()","((()()()))(())","((()()()))()()","((()())((())))","((()())(()()))","((()())(())())","((()())(()))()","((()())()(()))","((()())()()())","((()())()())()","((()())())(())","((()())())()()","((()()))((()))","((()()))(()())","((()()))(())()","((()()))()(())","((()()))()()()","((())(((()))))","((())((()())))","((())((())()))","((())((()))())","((())((())))()","((())(()(())))","((())(()()()))","((())(()())())","((())(()()))()","((())(())(()))","((())(())()())","((())(())())()","((())(()))(())","((())(()))()()","((())()((())))","((())()(()()))","((())()(())())","((())()(()))()","((())()()(()))","((())()()()())","((())()()())()","((())()())(())","((())()())()()","((())())((()))","((())())(()())","((())())(())()","((())())()(())","((())())()()()","((()))(((())))","((()))((()()))","((()))((())())","((()))((()))()","((()))(()(()))","((()))(()()())","((()))(()())()","((()))(())(())","((()))(())()()","((()))()((()))","((()))()(()())","((()))()(())()","((()))()()(())","((()))()()()()","(()((((())))))","(()(((()()))))","(()(((())())))","(()(((()))()))","(()(((())))())","(()(((()))))()","(()((()(()))))","(()((()()())))","(()((()())()))","(()((()()))())","(()((()())))()","(()((())(())))","(()((())()()))","(()((())())())","(()((())()))()","(()((()))(()))","(()((()))()())","(()((()))())()","(()((())))(())","(()((())))()()","(()(()((()))))","(()(()(()())))","(()(()(())()))","(()(()(()))())","(()(()(())))()","(()(()()(())))","(()(()()()()))","(()(()()())())","(()(()()()))()","(()(()())(()))","(()(()())()())","(()(()())())()","(()(()()))(())","(()(()()))()()","(()(())((())))","(()(())(()()))","(()(())(())())","(()(())(()))()","(()(())()(()))","(()(())()()())","(()(())()())()","(()(())())(())","(()(())())()()","(()(()))((()))","(()(()))(()())","(()(()))(())()","(()(()))()(())","(()(()))()()()","(()()(((()))))","(()()((()())))","(()()((())()))","(()()((()))())","(()()((())))()","(()()(()(())))","(()()(()()()))","(()()(()())())","(()()(()()))()","(()()(())(()))","(()()(())()())","(()()(())())()","(()()(()))(())","(()()(()))()()","(()()()((())))","(()()()(()()))","(()()()(())())","(()()()(()))()","(()()()()(()))","(()()()()()())","(()()()()())()","(()()()())(())","(()()()())()()","(()()())((()))","(()()())(()())","(()()())(())()","(()()())()(())","(()()())()()()","(()())(((())))","(()())((()()))","(()())((())())","(()())((()))()","(()())(()(()))","(()())(()()())","(()())(()())()","(()())(())(())","(()())(())()()","(()())()((()))","(()())()(()())","(()())()(())()","(()())()()(())","(()())()()()()","(())((((()))))","(())(((()())))","(())(((())()))","(())(((()))())","(())(((())))()","(())((()(())))","(())((()()()))","(())((()())())","(())((()()))()","(())((())(()))","(())((())()())","(())((())())()","(())((()))(())","(())((()))()()","(())(()((())))","(())(()(()()))","(())(()(())())","(())(()(()))()","(())(()()(()))","(())(()()()())","(())(()()())()","(())(()())(())","(())(()())()()","(())(())((()))","(())(())(()())","(())(())(())()","(())(())()(())","(())(())()()()","(())()(((())))","(())()((()()))","(())()((())())","(())()((()))()","(())()(()(()))","(())()(()()())","(())()(()())()","(())()(())(())","(())()(())()()","(())()()((()))","(())()()(()())","(())()()(())()","(())()()()(())","(())()()()()()","()(((((())))))","()((((()()))))","()((((())())))","()((((()))()))","()((((())))())","()((((()))))()","()(((()(()))))","()(((()()())))","()(((()())()))","()(((()()))())","()(((()())))()","()(((())(())))","()(((())()()))","()(((())())())","()(((())()))()","()(((()))(()))","()(((()))()())","()(((()))())()","()(((())))(())","()(((())))()()","()((()((()))))","()((()(()())))","()((()(())()))"LeetCode 22. 括号生成（Generate Parentheses）