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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.
1import java.util.*;
2
3public class Solution {
4 public List<Integer> largestDivisibleSubset(int[] nums) {
5 if (nums.length == 0) return new ArrayList<>();
6 Arrays.sort(nums);
7 List<Integer>[] dp = new ArrayList[nums.length];
8 List<Integer> maxSubset = new ArrayList<>();
9 for (int i = 0; i < nums.length; i++) {
10 List<Integer> subset = new ArrayList<>();
11 for (int k = 0; k < i; k++) {
12 if (nums[i] % nums[k] == 0 && dp[k].size() > subset.size()) {
13 subset = new ArrayList<>(dp[k]);
14 }
15 }
16 subset.add(nums[i]);
17 dp[i] = subset;
18 if (dp[i].size() > maxSubset.size()) {
19 maxSubset = dp[i];
20 }
21 }
22 return maxSubset;
23 }
24}In this Java implementation, we make use of an array of lists dp, where each index i stores the largest divisible subset ending with nums[i]. The Arrays.sort() helps in sequential processing of elements for building dp. Subsets are expanded by iterating over previous numbers and checking divisibility, updating the current list when necessary. The largest subset is then picked and returned. The list storing the maximum subset is returned at the end.
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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