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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.
1var largestDivisibleSubset = function(nums) {
2 if (nums.length === 0) return [];
3 nums.sort((a, b) => a - b);
4 let dp = Array.from({ length: nums.length }, () => []);
5 for (let i = 0; i < nums.length; i++) {
6 dp[i] = [nums[i]];
7 for (let j = 0; j < i; j++) {
8 if (nums[i] % nums[j] === 0 && dp[j].length + 1 > dp[i].length) {
9 dp[i] = dp[j].concat(nums[i]);
10 }
11 }
12 }
13 return dp.reduce((max, curr) => curr.length > max.length ? curr : max, []);
14};This JavaScript implementation sorts the array and uses a dynamic array dp to devise possible subsets. Starting from each element nums[i], it checks all previous elements nums[j] to extend the subset if divisible. Through concatenation, it constructs potential subsets and identifies the largest one by comparing lengths with reduce.
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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