Combination Sum

Clarify the problem:

• The problem requires us to find all unique combinations of numbers in a given list that add up to a target value.
• We need to implement a function that takes in a list of distinct integers and a target value and returns a list of all possible combinations.
• The same number can be chosen from the input list multiple times to form a combination.
• The combinations should not contain duplicate combinations.

Analyze the problem:

• Input: List of distinct integers, target value.
• Output: List of all possible combinations.
• Constraints:
  • The length of the input list is between 1 and 30.
  • The integers in the input list are distinct and positive.
  • The target value is a positive integer.

Design an algorithm:

• We can solve this problem using backtracking, which is a depth-first search (DFS) algorithm.
• We start with an empty combination list.
• We iterate through each number in the input list.
• For each number, we add it to the combination list and recursively generate combinations for the remaining target value reduced by the current number.
• After generating the combinations, we backtrack by removing the current number from the combination list.
• We repeat this process for all numbers in the input list.
• The base case of the recursion is when the target value becomes zero, indicating that we have found a valid combination that sums up to the target.
• We store the valid combinations in a result list and return it as the output.

Explain your approach:

• We will implement a backtracking algorithm to generate all possible combinations.
• First, we define a recursive helper function called generate_combinations that takes in the input list, the current combination list, the current target value, the current index, and a result list.
• Inside the helper function, we check the base case: if the target value is zero, we have found a valid combination, so we add it to the result list.
• Otherwise, we iterate through each number in the input list starting from the current index.
• We subtract the current number from the target value and recursively call the generate_combinations function with the updated combination list, target value, and index.
• After the recursive call, we remove the current number from the combination list to backtrack.
• Finally, we define the main function combinationSum that initializes the combination list, the result list, and calls the generate_combinations function.
• The combinationSum function returns the result list of combinations.

Write clean and readable code:

python

class Solution:
    def combinationSum(self, candidates, target):
        def generate_combinations(candidates, combination, target, index, result):
            if target == 0:
                result.append(combination[:])
                return

            for i in range(index, len(candidates)):
                if candidates[i] > target:
                    continue
                combination.append(candidates[i])
                generate_combinations(candidates, combination, target - candidates[i], i, result)
                combination.pop()

        result = []
        generate_combinations(candidates, [], target, 0, result)
        return result

Test your code:

python

solution = Solution()

candidates = [2, 3, 6, 7]
target = 7
combinations = solution.combinationSum(candidates, target)
print(combinations)  # Output: [[2, 2, 3], [7]]

candidates = [2, 3, 5]
target = 8
combinations = solution.combinationSum(candidates, target)
print(combinations)  # Output: [[2, 2, 2, 2], [2, 3, 3], [3, 5]]

candidates = [2]
target = 1
combinations = solution.combinationSum(candidates, target)
print(combinations)  # Output: []

candidates = [1]
target = 1
combinations = solution.combinationSum(candidates, target)
print(combinations)  # Output: [[1]]

Optimize if necessary:

• The backtracking algorithm already explores all possible combinations, so it is unlikely to be optimized further.
• However, we can optimize the algorithm by sorting the input list beforehand.
• Sorting the list allows us to skip certain branches during the backtracking process, avoiding duplicate combinations.
• Additionally, we can add an early termination condition to exit the loop if the current number is greater than the remaining target value.

Handle error cases:

• The code assumes that the input for the combinationSum function is a valid list of distinct positive integers and a positive target value.
• If the input list is empty or the target value is zero, the function will return an empty list.

Discuss complexity analysis:

• Let N be the length of the input list (candidates) and T be the target value.
• The time complexity of the solution is O(N^T) in the worst case since we have N choices for each target value from T to 0.
• The space complexity is O(T) for the recursion stack, as the maximum depth of recursion is T.
• The solution is efficient considering the combinatorial nature of the problem, but the time complexity can grow exponentially with larger input sizes.