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This approach uses backtracking to explore all possible combinations. Start with an empty combination, then add numbers incrementally and explore subsequent possibilities. If the sum of combination exceeds n or if it reaches the required length k, backtracking occurs to explore other potential combinations.
Time Complexity: O(2^n) because each number has a choice of being included or not.
Space Complexity: O(k) for the recursion stack.
1def combinationSum3(k, n):
2 def backtrack(k, n, start, path, results):
3 if k == 0 and n == 0:
4 results.append(list(path))
5 return
6 for i in range(start, 10):
7 if n < i:
8 break
9 path.append(i)
10 backtrack(k - 1, n - i, i + 1, path, results)
11 path.pop()
12
13 results = []
14 backtrack(k, n, 1, [], results)
15 return results
16
17# Example Usage:
18# print(combinationSum3(3, 7))This Python example similarly utilizes a recursive backtracking approach. The helper function backtrack attempts all combinations from start to 9 using recursion. Solutions are added to results when valid combinations are identified.
This approach leverages bitmasking to generate all subsets of numbers {1,2,...,9} and filters conceivable combinations by size and sum criteria. It's potentially less intuitive, yet eliminates recursion.
Time Complexity: O(2^n), iterating through bitmask combinations for validity checks.
Space Complexity: O(k) due to maintained size of data array.
1
In Python, bitmasking iterates across all 512 combinations of using numbers 1 through 9. Post-filtering by sum and size k, successful combinations residing in result meet the objectives.