Problem Description

In this problem, you are given an array of unique integers nums and a target integer target. Your task is to find out how many different combinations of elements from nums can be added together to sum up to the target value. Importantly, combinations may use the same element from nums multiple times if needed.

For example, if nums = [1, 2, 3] and target = 4, there are seven combinations that you could use to get a sum of 4:

• 1+1+1+1
• 1+1+2
• 1+2+1
• 2+1+1
• 2+2
• 1+3
• 3+1

The results must fit within a 32-bit integer range, meaning they should be between -(2^31) and (2^31)-1.

Intuition

The intuition for solving this problem comes from recognizing that it is similar to computing the number of ways to make change for a certain amount of money given different coin denominations. Here, each integer in nums can be thought of as a coin denomination.

To arrive at the solution, we apply dynamic programming because the problem asks for counting combinations, which suggests overlapping subproblems and the need for memorization of previous results to optimize the computation. Each subproblem here is finding the number of ways to reach a smaller target, which we can use to build up the solution to a larger target.

The dp array is created where each index i represents the target sum, and the value at that index represents the number of ways to reach that sum using elements from nums. The base case is dp[0] = 1, indicating there's one way to reach a sum of 0, which is using no elements.

We iterate through each sub-target from 1 to the desired target, and for each, we look at all elements in nums. If the current element is less than or equal to the sub-target, we add to dp[i] the value of dp[i - x], where x is the value of the current element from nums. This signifies that all the ways to reach the sum i-x can contribute to ways to reach the sum i by adding x.

By the end of the iterations, dp[target] contains the desired number of combinations.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution approach utilizes dynamic programming, a method for solving complex problems by breaking them down into simpler sub-problems. It is a helpful strategy when a problem has overlapping subproblems and optimal substructure, meaning the problem can be broken down into smaller, simpler subproblems which can be solved independently.

In our case, we use an array dp to store the number of combinations that sum up to each value up to target. We initialize dp[0] = 1 because there's exactly one combination to achieve a sum of 0: using no numbers from nums.

The outer loop iterates through all sub-targets from 1 to target, and the inner loop goes through all the available numbers in nums. Whenever the current number x is less than or equal to the sub-target i, we update dp[i] by adding the number of ways to form the sum i-x. This is based on the assumption that if we have a way to arrive at a sum of i-x, then by adding x to it, we can get to i.

In simpler terms, to solve for dp[i] which represents the number of combinations to reach a sum of i, we look at all numbers x in nums that could contribute to i and sum up all the combinations that make up the sum i-x (all of which are valid ways to reach i when adding x).

This solution's algorithm can be summarized in the following steps:

1. Define an array dp of length target + 1 and initialize it with zeros. dp[0] is set to 1 since there's one combination that results in a sum of zero, which is not using any numbers.

2. Loop through each sub-target i from 1 to target.

   • Inside this loop, iterate through each number x in nums.
     • If x is less than or equal to i, increment dp[i] by dp[i - x].

3. After the loops finish, dp[target] will hold the number of ways to combine the numbers in nums to sum up to the target.

The Python code implementing the solution is as follows:

class Solution:
    def combinationSum4(self, nums: List[int], target: int) -> int:
        dp = [1] + [0] * target
        for i in range(1, target + 1):
            for x in nums:
                if i >= x:
                    dp[i] += dp[i - x]
        return dp[target]

In this implementation, dp is initialized with size target+1 to accommodate sums from 0 to target inclusive. The two nested loops are used to populate the dp array according to the dynamic programming strategy outlined above.

Example Walkthrough

Let's illustrate the solution approach with a smaller example.

Suppose nums = [1, 3, 4] and target = 5. We want to find out how many different combinations of elements from nums can be added together to sum up to 5.

Here's how we would complete the task step by step:

1. Initialize a dp array with target + 1 zeros and set dp[0] to 1, because there is exactly one way to achieve the sum of 0 (using no elements).

   After initialization, our dp array looks like this: dp = [1, 0, 0, 0, 0, 0]

2. Loop through each sub-target i from 1 to target.

3. For each sub-target i, consider each number x from nums.

Let's fill the dp array:

• For i = 1: The possible number from nums to use is 1.

  • We update dp[1] to be dp[1] + dp[1 - 1] = dp[1] + dp[0] = 0 + 1 = 1.

• For i = 2: The possible number from nums to use is 1.

  • We update dp[2] to be dp[2] + dp[2 - 1] = dp[2] + dp[1] = 0 + 1 = 1.

• For i = 3: We can use 1 and 3 from nums.

  • We can update dp[3] to be dp[3] + dp[3 - 1] (using 1) = dp[3] + dp[2] = 0 + 1 = 1.
  • We can also update dp[3] using 3 to be dp[3] + dp[3 - 3] = dp[3] + dp[0] = 1 + 1 = 2.

• For i = 4: We can use 1, 3, and 4 from nums.

  • Update dp[4] with 1 to be dp[4] + dp[4 - 1] = dp[4] + dp[3] = 0 + 2 = 2.
  • Update dp[4] with 3: No change, since dp[4 - 3] is zero.
  • Update dp[4] with 4: dp[4] + dp[4 - 4] = dp[4] + dp[0] = 2 + 1 = 3.

• For i = 5: We can use 1, 3, and 4.

  • Update dp[5] with 1: dp[5] + dp[5 - 1] = dp[5] + dp[4] = 0 + 3 = 3.
  • Update dp[5] with 3: dp[5] + dp[5 - 3] = dp[5] + dp[2] = 3 + 1 = 4.
  • Update dp[5] with 4: dp[5] + dp[5 - 4] = dp[5] + dp[1] = 4 + 1 = 5.

The final dp array is [1, 1, 1, 2, 3, 5], and dp[5] is 5, meaning there are five different combinations to reach the target sum of 5 using elements from nums [1, 3, 4] with repetition allowed.

Those combinations are:

• 1 + 1 + 1 + 1 + 1
• 1 + 1 + 3
• 1 + 3 + 1
• 3 + 1 + 1
• 1 + 4

No other combinations are possible without exceeding the target, so the final answer is 5.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code can be analyzed based on the nested loops.

We have an outer loop:

for i in range(1, target + 1):

This loop runs once for every integer from 1 to target, inclusive, so it runs target times.

Inside the outer loop, we have an inner loop:

for x in nums:
    if i >= x:
        f[i] += f[i - x]

This loop iterates over all elements in nums. If n is the number of elements in nums, the inner loop runs n times for each iteration of the outer loop.

However, not all iterations of the inner loop will execute the update f[i] += f[i - x]. The condition i >= x needs to be met. In the worst case, though, where i is always greater than or equal to every element x in nums, the body of the inner loop will execute.

Therefore, the worst-case time complexity is the product of the number of iterations of both loops, which is O(target * n).

Space Complexity

Analyzing space complexity involves looking at how much additional memory the algorithm uses as a function of the input size.

The space complexity is driven by the list f that has target + 1 elements:

f = [1] + [0] * target

This means we need a space proportional to (and linear with) target. Hence, the space complexity of the algorithm is O(target).

No other data structures that scale with the input size is used, so the fixed-size variables and inputs do not affect the overall space complexity significantly compared to the list f which dominates the space usage.

Learn more about how to find time and space complexity quickly using problem constraints.