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This approach utilizes dynamic programming along with sorting to find the largest subset where each pair of elements is divisible by the other. The core idea is to sort the array, then use a dynamic programming array to keep track of the size of the largest divisible subset that ends at each element. We also maintain a path array to help reconstruct the subset.
Time Complexity: O(n^2), where n is the number of elements in the input set due to the nested loop.
Space Complexity: O(n), for arrays used to store intermediate results.
1def largestDivisibleSubset(nums):
2 if not nums:
3 return []
4 nums.sort()
5 dp = [[num] for num in nums]
6 for i in range(len(nums)):
7 for j in range(i):
8 if nums[i] % nums[j] == 0 and len(dp[j]) + 1 > len(dp[i]):
9 dp[i] = dp[j] + [nums[i]]
10 return max(dp, key=len)This Python function begins by handling the edge case of an empty nums. It sorts the array and initializes a dynamic list dp where each entry starts with the single number subset. For each number, previous elements are checked, and if they divide the current number, the subset is expanded by appending the current number. Finally, the longest subset from dp is returned.
This approach leverages backtracking with pruning to explore subsets and constrain exploration using the divisibility constraint. It uses a sorted array to systematically explore subsets and prune paths early when constraints are no longer satisfied, allowing potentially faster exploration compared to the dynamic programming approach, especially in tightly constrained subsets.
Time and space complexities are challenging to define precisely for a backtracking solution as they depend on many factors including input distribution.
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