Array     Backtracking     C++     Depth-First Search     Medium     Recursion    

Problem Statement:

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

Solution:

The approach is to run DFS starting with value 1. We can either,

  • include current value and run DFS with reduced target and next value (since repetition is not allowed), OR
  • skip current value and run DFS with with same target from next value.
 
class Solution {
public:
    void dfs(int curr, int target, int count,  vector<vector<int>> &res, vector<int> &v)
    {
        if (count==0 && target==0){res.push_back(v);return;}
        if (count<0 || target<0 || curr>=10) return;
        v.push_back(curr);
        dfs(curr+1, target-curr, count-1, res, v);
        v.pop_back();
        dfs(curr+1, target, count, res, v);
    }
    vector<vector<int>> combinationSum3(int k, int n) {
        vector<vector<int>> res;
        vector<int>v;
        dfs(1,n,k,res,v);
        return res;
    }
};

$TC: O(n^k), SC:O(n^k)$