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216. Combination Sum III

Description

Find all valid combinations of k numbers that sum up to n such that the following conditions are true:

  • Only numbers 1 through 9 are used.
  • Each number is used at most once.

Return a list of all possible valid combinations. The list must not contain the same combination twice, and the combinations may be returned in any order.

 

Example 1:

Input: k = 3, n = 7
Output: [[1,2,4]]
Explanation:
1 + 2 + 4 = 7
There are no other valid combinations.

Example 2:

Input: k = 3, n = 9
Output: [[1,2,6],[1,3,5],[2,3,4]]
Explanation:
1 + 2 + 6 = 9
1 + 3 + 5 = 9
2 + 3 + 4 = 9
There are no other valid combinations.

Example 3:

Input: k = 4, n = 1
Output: []
Explanation: There are no valid combinations.
Using 4 different numbers in the range [1,9], the smallest sum we can get is 1+2+3+4 = 10 and since 10 > 1, there are no valid combination.

 

Constraints:

  • 2 <= k <= 9
  • 1 <= n <= 60

Solutions

Solution 1: Pruning + Backtracking (Two Approaches)

We design a function $dfs(i, s)$, which represents that we are currently enumerating the number $i$, and there are still numbers with a sum of $s$ to be enumerated. The current search path is $t$, and the answer is $ans$.

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

Approach One:

  • If $s = 0$ and the length of the current search path $t$ is $k$, it means that a group of answers has been found. Add $t$ to $ans$ and then return.
  • If $i \gt 9$ or $i \gt s$ or the length of the current search path $t$ is greater than $k$, it means that the current search path cannot be the answer, so return directly.
  • Otherwise, we can choose to add the number $i$ to the search path $t$, and then continue to search, i.e., execute $dfs(i + 1, s - i)$. After the search is completed, remove $i$ from the search path $t$; we can also choose not to add the number $i$ to the search path $t$, and directly execute $dfs(i + 1, s)$.
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class Solution:
    def combinationSum3(self, k: int, n: int) -> List[List[int]]:
        def dfs(i: int, s: int):
            if s == 0:
                if len(t) == k:
                    ans.append(t[:])
                return
            if i > 9 or i > s or len(t) >= k:
                return
            t.append(i)
            dfs(i + 1, s - i)
            t.pop()
            dfs(i + 1, s)

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

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

    public List<List<Integer>> combinationSum3(int k, int n) {
        this.k = k;
        dfs(1, n);
        return ans;
    }

    private void dfs(int i, int s) {
        if (s == 0) {
            if (t.size() == k) {
                ans.add(new ArrayList<>(t));
            }
            return;
        }
        if (i > 9 || i > s || t.size() >= k) {
            return;
        }
        t.add(i);
        dfs(i + 1, s - i);
        t.remove(t.size() - 1);
        dfs(i + 1, s);
    }
}

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class Solution {
public:
    vector<vector<int>> combinationSum3(int k, int n) {
        vector<vector<int>> ans;
        vector<int> t;
        function<void(int, int)> dfs = [&](int i, int s) {
            if (s == 0) {
                if (t.size() == k) {
                    ans.emplace_back(t);
                }
                return;
            }
            if (i > 9 || i > s || t.size() >= k) {
                return;
            }
            t.emplace_back(i);
            dfs(i + 1, s - i);
            t.pop_back();
            dfs(i + 1, s);
        };
        dfs(1, n);
        return ans;
    }
};

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func combinationSum3(k int, n int) (ans [][]int) {
    t := []int{}
    var dfs func(i, s int)
    dfs = func(i, s int) {
        if s == 0 {
            if len(t) == k {
                ans = append(ans, slices.Clone(t))
            }
            return
        }
        if i > 9 || i > s || len(t) >= k {
            return
        }
        t = append(t, i)
        dfs(i+1, s-i)
        t = t[:len(t)-1]
        dfs(i+1, s)
    }
    dfs(1, n)
    return
}

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function combinationSum3(k: number, n: number): number[][] {
    const ans: number[][] = [];
    const t: number[] = [];
    const dfs = (i: number, s: number) => {
        if (s === 0) {
            if (t.length === k) {
                ans.push(t.slice());
            }
            return;
        }
        if (i > 9 || i > s || t.length >= k) {
            return;
        }
        t.push(i);
        dfs(i + 1, s - i);
        t.pop();
        dfs(i + 1, s);
    };
    dfs(1, n);
    return ans;
}

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impl Solution {
    #[allow(dead_code)]
    pub fn combination_sum3(k: i32, n: i32) -> Vec<Vec<i32>> {
        let mut ret = Vec::new();
        let mut candidates = (1..=9).collect();
        let mut cur_vec = Vec::new();
        Self::dfs(n, k, 0, 0, &mut cur_vec, &mut candidates, &mut ret);
        ret
    }

    #[allow(dead_code)]
    fn dfs(
        target: i32,
        length: i32,
        cur_index: usize,
        cur_sum: i32,
        cur_vec: &mut Vec<i32>,
        candidates: &Vec<i32>,
        ans: &mut Vec<Vec<i32>>
    ) {
        if cur_sum > target || cur_vec.len() > (length as usize) {
            // No answer for this
            return;
        }
        if cur_sum == target && cur_vec.len() == (length as usize) {
            // We get an answer
            ans.push(cur_vec.clone());
            return;
        }
        for i in cur_index..candidates.len() {
            cur_vec.push(candidates[i]);
            Self::dfs(target, length, i + 1, cur_sum + candidates[i], cur_vec, candidates, ans);
            cur_vec.pop().unwrap();
        }
    }
}

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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 k;

    public IList<IList<int>> CombinationSum3(int k, int n) {
        this.k = k;
        dfs(1, n);
        return ans;
    }

    private void dfs(int i, int s) {
        if (s == 0) {
            if (t.Count == k) {
                ans.Add(new List<int>(t));
            }
            return;
        }
        if (i > 9 || i > s || t.Count >= k) {
            return;
        }
        t.Add(i);
        dfs(i + 1, s - i);
        t.RemoveAt(t.Count - 1);
        dfs(i + 1, s);
    }
}

Another approach:

  • If $s = 0$ and the length of the current search path $t$ is $k$, it means that a group of answers has been found. Add $t$ to $ans$ and then return.
  • If $i \gt 9$ or $i \gt s$ or the length of the current search path $t$ is greater than $k$, it means that the current search path cannot be the answer, so return directly.
  • Otherwise, we enumerate the next number $j$, i.e., $j \in [i, 9]$, add the number $j$ to the search path $t$, and then continue to search, i.e., execute $dfs(j + 1, s - j)$. After the search is completed, remove $j$ from the search path $t$.

In the main function, we call $dfs(1, n)$, i.e., start enumerating from the number $1$, and the remaining numbers with a sum of $n$ need to be enumerated. After the search is completed, we can get all the answers.

The time complexity is $(C_{9}^k \times k)$, and the space complexity is $O(k)$.

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class Solution:
    def combinationSum3(self, k: int, n: int) -> List[List[int]]:
        def dfs(i: int, s: int):
            if s == 0:
                if len(t) == k:
                    ans.append(t[:])
                return
            if i > 9 or i > s or len(t) >= k:
                return
            for j in range(i, 10):
                t.append(j)
                dfs(j + 1, s - j)
                t.pop()

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

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

    public List<List<Integer>> combinationSum3(int k, int n) {
        this.k = k;
        dfs(1, n);
        return ans;
    }

    private void dfs(int i, int s) {
        if (s == 0) {
            if (t.size() == k) {
                ans.add(new ArrayList<>(t));
            }
            return;
        }
        if (i > 9 || i > s || t.size() >= k) {
            return;
        }
        for (int j = i; j <= 9; ++j) {
            t.add(j);
            dfs(j + 1, s - j);
            t.remove(t.size() - 1);
        }
    }
}

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class Solution {
public:
    vector<vector<int>> combinationSum3(int k, int n) {
        vector<vector<int>> ans;
        vector<int> t;
        function<void(int, int)> dfs = [&](int i, int s) {
            if (s == 0) {
                if (t.size() == k) {
                    ans.emplace_back(t);
                }
                return;
            }
            if (i > 9 || i > s || t.size() >= k) {
                return;
            }
            for (int j = i; j <= 9; ++j) {
                t.emplace_back(j);
                dfs(j + 1, s - j);
                t.pop_back();
            }
        };
        dfs(1, n);
        return ans;
    }
};

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func combinationSum3(k int, n int) (ans [][]int) {
    t := []int{}
    var dfs func(i, s int)
    dfs = func(i, s int) {
        if s == 0 {
            if len(t) == k {
                ans = append(ans, slices.Clone(t))
            }
            return
        }
        if i > 9 || i > s || len(t) >= k {
            return
        }
        for j := i; j <= 9; j++ {
            t = append(t, j)
            dfs(j+1, s-j)
            t = t[:len(t)-1]
        }
    }
    dfs(1, n)
    return
}

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function combinationSum3(k: number, n: number): number[][] {
    const ans: number[][] = [];
    const t: number[] = [];
    const dfs = (i: number, s: number) => {
        if (s === 0) {
            if (t.length === k) {
                ans.push(t.slice());
            }
            return;
        }
        if (i > 9 || i > s || t.length >= k) {
            return;
        }
        for (let j = i; j <= 9; ++j) {
            t.push(j);
            dfs(j + 1, s - j);
            t.pop();
        }
    };
    dfs(1, n);
    return ans;
}

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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 k;

    public IList<IList<int>> CombinationSum3(int k, int n) {
        this.k = k;
        dfs(1, n);
        return ans;
    }

    private void dfs(int i, int s) {
        if (s == 0) {
            if (t.Count == k) {
                ans.Add(new List<int>(t));
            }
            return;
        }
        if (i > 9 || i > s || t.Count >= k) {
            return;
        }
        for (int j = i; j <= 9; ++j) {
            t.Add(j);
            dfs(j + 1, s - j);
            t.RemoveAt(t.Count - 1);
        }
    }
}

Solution 2: Binary Enumeration

We can use a binary integer of length $9$ to represent the selection of numbers $1$ to $9$, where the $i$-th bit of the binary integer represents whether the number $i + 1$ is selected. If the $i$-th bit is $1$, it means that the number $i + 1$ is selected, otherwise, it means that the number $i + 1$ is not selected.

We enumerate binary integers in the range of $[0, 2^9)$. For the currently enumerated binary integer $mask$, if the number of $1$s in the binary representation of $mask$ is $k$, and the sum of the numbers corresponding to $1$ in the binary representation of $mask$ is $n$, it means that the number selection scheme corresponding to $mask$ is a group of answers. We can add the number selection scheme corresponding to $mask$ to the answer.

The time complexity is $O(2^9 \times 9)$, and the space complexity is $O(k)$.

Similar problems:

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class Solution:
    def combinationSum3(self, k: int, n: int) -> List[List[int]]:
        ans = []
        for mask in range(1 << 9):
            if mask.bit_count() == k:
                t = [i + 1 for i in range(9) if mask >> i & 1]
                if sum(t) == n:
                    ans.append(t)
        return ans

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class Solution {
    public List<List<Integer>> combinationSum3(int k, int n) {
        List<List<Integer>> ans = new ArrayList<>();
        for (int mask = 0; mask < 1 << 9; ++mask) {
            if (Integer.bitCount(mask) == k) {
                List<Integer> t = new ArrayList<>();
                int s = 0;
                for (int i = 0; i < 9; ++i) {
                    if ((mask >> i & 1) == 1) {
                        s += (i + 1);
                        t.add(i + 1);
                    }
                }
                if (s == n) {
                    ans.add(t);
                }
            }
        }
        return ans;
    }
}

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class Solution {
public:
    vector<vector<int>> combinationSum3(int k, int n) {
        vector<vector<int>> ans;
        for (int mask = 0; mask < 1 << 9; ++mask) {
            if (__builtin_popcount(mask) == k) {
                int s = 0;
                vector<int> t;
                for (int i = 0; i < 9; ++i) {
                    if (mask >> i & 1) {
                        t.push_back(i + 1);
                        s += i + 1;
                    }
                }
                if (s == n) {
                    ans.emplace_back(t);
                }
            }
        }
        return ans;
    }
};

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func combinationSum3(k int, n int) (ans [][]int) {
    for mask := 0; mask < 1<<9; mask++ {
        if bits.OnesCount(uint(mask)) == k {
            t := []int{}
            s := 0
            for i := 0; i < 9; i++ {
                if mask>>i&1 == 1 {
                    s += i + 1
                    t = append(t, i+1)
                }
            }
            if s == n {
                ans = append(ans, t)
            }
        }
    }
    return
}

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function combinationSum3(k: number, n: number): number[][] {
    const ans: number[][] = [];
    for (let mask = 0; mask < 1 << 9; ++mask) {
        if (bitCount(mask) === k) {
            const t: number[] = [];
            let s = 0;
            for (let i = 0; i < 9; ++i) {
                if (mask & (1 << i)) {
                    t.push(i + 1);
                    s += i + 1;
                }
            }
            if (s === n) {
                ans.push(t);
            }
        }
    }
    return ans;
}

function bitCount(i: number): number {
    i = i - ((i >>> 1) & 0x55555555);
    i = (i & 0x33333333) + ((i >>> 2) & 0x33333333);
    i = (i + (i >>> 4)) & 0x0f0f0f0f;
    i = i + (i >>> 8);
    i = i + (i >>> 16);
    return i & 0x3f;
}

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public class Solution {
    public IList<IList<int>> CombinationSum3(int k, int n) {
        List<IList<int>> ans = new List<IList<int>>();
        for (int mask = 0; mask < 1 << 9; ++mask) {
            if (bitCount(mask) == k) {
                List<int> t = new List<int>();
                int s = 0;
                for (int i = 0; i < 9; ++i) {
                    if ((mask >> i & 1) == 1) {
                        s += i + 1;
                        t.Add(i + 1);
                    }
                }
                if (s == n) {
                    ans.Add(t);
                }
            }
        }
        return ans;
    }

    private int bitCount(int x) {
        int cnt = 0;
        while (x > 0) {
            x -= x & -x;
            ++cnt;
        }
        return cnt;
    }
}

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