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22. Generate Parentheses

Description

Given n pairs of parentheses, write a function to generate all combinations of well-formed parentheses.

 

Example 1:

Input: n = 3
Output: ["((()))","(()())","(())()","()(())","()()()"]

Example 2:

Input: n = 1
Output: ["()"]

 

Constraints:

  • 1 <= n <= 8

Solutions

Solution 1: DFS + Pruning

The range of $n$ in the problem is $[1, 8]$, so we can directly solve this problem through "brute force search + pruning".

We design a function $dfs(l, r, t)$, where $l$ and $r$ represent the number of left and right brackets respectively, and $t$ represents the current bracket sequence. Then we can get the following recursive structure:

  • If $l \gt n$ or $r \gt n$ or $l \lt r$, then the current bracket combination $t$ is invalid, return directly;
  • If $l = n$ and $r = n$, then the current bracket combination $t$ is valid, add it to the answer array ans, and return directly;
  • We can choose to add a left bracket, and recursively execute dfs(l + 1, r, t + "(");
  • We can also choose to add a right bracket, and recursively execute dfs(l, r + 1, t + ")").

The time complexity is $O(2^{n\times 2} \times n)$, and the space complexity is $O(n)$.

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class Solution:
    def generateParenthesis(self, n: int) -> List[str]:
        def dfs(l, r, t):
            if l > n or r > n or l < r:
                return
            if l == n and r == n:
                ans.append(t)
                return
            dfs(l + 1, r, t + '(')
            dfs(l, r + 1, t + ')')

        ans = []
        dfs(0, 0, '')
        return ans

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class Solution {
    private List<String> ans = new ArrayList<>();
    private int n;

    public List<String> generateParenthesis(int n) {
        this.n = n;
        dfs(0, 0, "");
        return ans;
    }

    private void dfs(int l, int r, String t) {
        if (l > n || r > n || l < r) {
            return;
        }
        if (l == n && r == n) {
            ans.add(t);
            return;
        }
        dfs(l + 1, r, t + "(");
        dfs(l, r + 1, t + ")");
    }
}

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class Solution {
public:
    vector<string> generateParenthesis(int n) {
        vector<string> ans;
        function<void(int, int, string)> dfs = [&](int l, int r, string t) {
            if (l > n || r > n || l < r) return;
            if (l == n && r == n) {
                ans.push_back(t);
                return;
            }
            dfs(l + 1, r, t + "(");
            dfs(l, r + 1, t + ")");
        };
        dfs(0, 0, "");
        return ans;
    }
};

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func generateParenthesis(n int) (ans []string) {
    var dfs func(int, int, string)
    dfs = func(l, r int, t string) {
        if l > n || r > n || l < r {
            return
        }
        if l == n && r == n {
            ans = append(ans, t)
            return
        }
        dfs(l+1, r, t+"(")
        dfs(l, r+1, t+")")
    }
    dfs(0, 0, "")
    return ans
}

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function generateParenthesis(n: number): string[] {
    function dfs(l, r, t) {
        if (l > n || r > n || l < r) {
            return;
        }
        if (l == n && r == n) {
            ans.push(t);
            return;
        }
        dfs(l + 1, r, t + '(');
        dfs(