77. Combinations

Description

Given two integers n and k, return all possible combinations of k numbers chosen from the range [1, n].

You may return the answer in any order.

Example 1:

Input: n = 4, k = 2
Output: [[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]
Explanation: There are 4 choose 2 = 6 total combinations.
Note that combinations are unordered, i.e., [1,2] and [2,1] are considered to be the same combination.

Example 2:

Input: n = 1, k = 1
Output: [[1]]
Explanation: There is 1 choose 1 = 1 total combination.

Constraints:

1 <= n <= 20
1 <= k <= n

Solutions

Solution 1: Backtracking (Two Ways)

We design a function $dfs(i)$, which represents starting the search from number $i$, with the current search path as $t$, and the answer as $ans$.

The execution logic of the function $dfs(i)$ is as follows:

If the length of the current search path $t$ equals $k$, then add the current search path to the answer and return.
If $i \gt n$, it means the search has ended, return.
Otherwise, we can choose to add the number $i$ to the search path $t$, and then continue the search, i.e., execute $dfs(i + 1)$, and then remove the number $i$ from the search path $t$; or we do not add the number $i$ to the search path $t$, and directly execute $dfs(i + 1)$.

The above method is actually enumerating whether to select the current number or not, and then recursively searching the next number. We can also enumerate the next number $j$ to be selected, where $i \leq j \leq n$. If the next number to be selected is $j$, then we add the number $j$ to the search path $t$, and then continue the search, i.e., execute $dfs(j + 1)$, and then remove the number $j$ from the search path $t$.

In the main function, we start the search from number $1$, i.e., execute $dfs(1)$.

The time complexity is $(C_n^k \times k)$, and the space complexity is $O(k)$. Here, $C_n^k$ represents the combination number.

Similar problems:

Python Code

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class Solution:
    def combine(self, n: int, k: int) -> List[List[int]]:
        def dfs(i: int):
            if len(t) == k:
                ans.append(t[:])
                return
            if i > n:
                return
            t.append(i)
            dfs(i + 1)
            t.pop()
            dfs(i + 1)

        ans = []
        t = []
        dfs(1)
        return ans

Java Code

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

    public List<List<Integer>> combine(int n, int k) {
        this.n = n;
        this.k = k;
        dfs(1);
        return ans;
    }

    private void dfs(int i) {
        if (t.size() == k) {
            ans.add(new ArrayList<>(t));
            return;
        }
        if (i > n) {
            return;
        }
        t.add(i);
        dfs(i + 1);
        t.remove(t.size() - 1);
        dfs(i + 1);
    }
}

C++ Code

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class Solution {
public:
    vector<vector<int>> combine(int n, int k) {
        vector<vector<int>> ans;
        vector<int> t;
        function<void(int)> dfs = [&](int i) {
            if (t.size() == k) {
                ans.emplace_back(t);
                return;
            }
            if (i > n) {
                return;
            }
            t.emplace_back(i);
            dfs(i + 1);
            t.pop_back();
            dfs(i + 1);
        };
        dfs(1);
        return ans;
    }
};

Go Code

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func combine(n int, k int) (ans [][]int) {
	t := []int{}
	var dfs func(int)
	dfs = func(i int) {
		if len(t) == k {
			ans = append(ans, slices.Clone(t))
			return
		}
		if i > n {
			return
		}
		t = append(t, i)
		dfs(i + 1)
		t = t[:len(t)-1]
		dfs(i + 1)
	}
	dfs(1)
	return
}

TypeScript Code

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function combine(n: number, k: number): number[][] {
    const ans: number[][] = [];
    const t: number[] = [];
    const dfs = (i: number) => {
        if (t.length === k) {
            ans.push(t.slice());
            return;
        }
        if (i > n) {
            return;
        }
        t.push(i);
        dfs(i + 1);
        t.pop();
        dfs(i + 1);
    };
    dfs(1);
    return ans;
}

Rust Code

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impl Solution {
    fn dfs(i: i32, n: i32, k: i32, t: &mut Vec<i32>, ans: &mut Vec<Vec<i32>>) {
        if t.len() == (k as usize) {
            ans.push(t.clone());
            return;
        }
        if i > n {
            return;
        }
        t.push(i);
        Self::dfs(i + 1, n, k, t, ans);
        t.pop();
        Self::dfs(i + 1, n, k, t, ans);
    }

    pub fn combine(n: i32, k: i32) -> Vec<Vec<i32>> {
        let mut ans = vec![];
        Self::dfs(1, n, k, &mut vec![], &mut ans);
        ans
    }
}

C# Code

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public class Solution {
    private List<IList<int>> ans = new List<IList<int>>();
    private List<int> t = new List<int>();
    private int n;
    private int k;

    public IList<IList<int>> Combine(int n, int k) {
        this.n = n;
        this.k = k;
        dfs(1);
        return ans;
    }

    private void dfs(int i) {
        if (t.Count == k) {
            ans.Add(new List<int>(t));
            return;
        }
        if (i > n) {
            return;
        }
        t.Add(i);
        dfs(i + 1);
        t.RemoveAt(t.Count - 1);
        dfs(i + 1);
    }
}

Solution 2