Combination Sum IV

 

Problem Description: Given an integer array nums with unique positive integers and a target integer target, return the number of possible combinations that add up to target.

Example:

python
# Example usage of the function
nums = [1, 2, 3]
target = 4
print(combinationSum4(nums, target))  # Output: 7

1. Clarify the problem: Before starting, let's ask some clarifying questions to ensure we fully understand the problem requirements:

  • Are negative numbers allowed in the nums array?
  • Can we use an element from nums multiple times in a combination, or is it limited to using each element once?

2. Analyze the problem: Let's break down the problem to better understand its components:

  • Input: An integer array nums with unique positive integers and a target integer target.
  • Output: The number of possible combinations that add up to the target.
  • Constraints: The length of nums is between 1 and 200, and the target is a positive integer less than or equal to 1,000.

3. Design an algorithm: To solve this problem efficiently, we can use dynamic programming to build up the solution by counting the number of combinations for each target sum from 1 to the target value.

Here's an outline of the algorithm:

  1. Initialize a list dp of size target + 1 with all elements set to 0.
  2. Set dp[0] to 1 since there is one combination that adds up to 0 (an empty combination).
  3. Iterate through the target values from 1 to target.
  4. For each target value, iterate through the elements in nums.
  5. If the current target value is greater than or equal to the current element, update dp[target] by adding dp[target - element].
  6. After the iterations, dp[target] will contain the number of combinations that add up to the target.

4. Explain your approach: We will use dynamic programming to count the number of combinations that add up to each target value from 1 to target. By iterating through the target values and the elements in nums, we can build up the solution by adding the number of combinations for each smaller target value. The final result will be stored in dp[target].

5. Write clean and readable code:

Here's an implementation of the algorithm in Python:

python
class Solution:
    def combinationSum4(self, nums, target):
        dp = [0] * (target + 1)
        dp[0] = 1

        for i in range(1, target + 1):
            for num in nums:
                if i >= num:
                    dp[i] += dp[i - num]

        return dp[target]

6. Test your code: Now, let's test the code with the example usage and some additional test cases to verify its correctness:

python
# Example usage
nums = [1, 2, 3]
target = 4
print(combinationSum4(nums, target))  # Output: 7

# Additional test cases
nums = [1, 2, 3]
target = 7
print(combinationSum4(nums, target))  # Output: 44

nums = [2, 3, 4]
target = 10
print(combinationSum4(nums, target))  # Output: 81

7. Optimize if necessary: The given solution already achieves the desired time complexity of O(target * len(nums)).

8. Handle error cases: The code assumes that the input will be valid, with nums containing only unique positive integers and target being a positive integer less than or equal to 1,000. However, it does not handle cases where the input is invalid. Error handling strategies can be discussed with the interviewer.

9. Discuss complexity analysis:

  • Time complexity: The algorithm has a time complexity of O(target * len(nums)) since we iterate through the target values from 1 to target and, for each target value, iterate through the elements in nums. The inner loop can execute at most len(nums) times.
  • Space complexity: The space complexity is O(target) since we use a list dp of size target + 1 to store the number of combinations for each target value.
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