377. Combination Sum IV

Given an integer array with all positive numbers and no duplicates, find the number of possible combinations that add up to a positive integer target.

Example:

nums = [1, 2, 3]
target = 4

The possible combination ways are:
(1, 1, 1, 1)
(1, 1, 2)
(1, 2, 1)
(1, 3)
(2, 1, 1)
(2, 2)
(3, 1)

Note that different sequences are counted as different combinations.

Therefore the output is 7.

Follow up:
What if negative numbers are allowed in the given array?
How does it change the problem?
What limitation we need to add to the question to allow negative numbers?

Credits:
Special thanks to @pbrother for adding this problem and creating all test cases.

Approach #1: DFS. [C++] (TLE)

class Solution {
public:
    int combinationSum4(vector<int>& nums, int target) {
        int ans = 0;
        dfs(nums, ans, target, 0);
        return ans;
    }
    
private:
    void dfs(const vector<int>& nums, int& ans, const int target, int curval) {
        if (curval > target) return ;
        if (curval == target) ans++;
        for (int i = 0; i < nums.size(); ++i) {
            dfs(nums, ans, target, curval+nums[i]);
        }
    }
};

　　

Approach #2: Recursive With Memoization. [C++]

class Solution {
public:
    int combinationSum4(vector<int>& nums, int target) {
        // memo.resize(target+1, 0);
        memo = vector<int>(target + 1, -1);
        memo[0] = 1;
        return dp(nums, target);
    }
    
private:
    vector<int> memo;
    int dp(const vector<int>& nums, int target) {
        if (target < 0) return 0;
        if (memo[target] != -1) return memo[target];
        int ans = 0;
        for (int num : nums) 
            ans += dp(nums, target-num);
        return memo[target] = ans;
    }
};

　　

Approach #3: DP. [C++]

    int combinationSum4(vector<int>& nums, int target) {
        vector<int> dp(target+1, 0);
        dp[0] = 1;
        for (int i = 1; i <= target; ++i) {
            for (const int num : nums) {
                if (i-num >= 0)
                    dp[i] += dp[i-num];
            }
        }
        return dp[target];
    }

　　

Reference:

http://zxi.mytechroad.com/blog/dynamic-programming/leetcode-377-combination-sum-iv/