Largest divisible pairs subset

Given an array of n distinct elements, find length of the largest subset such that every pair in the subset is such that the larger element of the pair is divisible by smaller element. 

Examples: 

Input : arr[] = {10, 5, 3, 15, 20} 
Output : 3 
Explanation: The largest subset is 10, 5, 20.
10 is divisible by 5, and 20 is divisible by 10.
Input : arr[] = {18, 1, 3, 6, 13, 17} 
Output : 4
Explanation: The largest subset is 18, 1, 3, 6,
In the subsequence, 3 is divisible by 1, 
6 by 3 and 18 by 6.

Brute Force Approach: The brute force approach generates all possible subsets of the input array using bit manipulation and checks if each subset contains only pairs of elements that are divisible. It keeps track of the largest subset that satisfies this condition and returns it as the result. 

• Initialize maxSubset = 1
• For each subset S of the input array a:
  •  Initialize subsetSize = 0 and validSubset = true
  • For each pair of elements (a[i], a[j]) in S:
    • If a[i] % a[j] != 0 and a[j] % a[i] != 0, set validSubset = false and break out of the loop.
    •  Otherwise, continue to the next pair. 
  • If validSubset is still true, increment subsetSize by 1
  • If subsetSize is greater than maxSubset, set maxSubset = subsetSize
• Return maxSubset as the result

3

Time Complexity: O(2ⁿ*n²)
Space Complexity:  O(1)

Dynamic programming Approach: This can be solved using Dynamic Programming. We traverse the sorted array from the end. For every element a[i], we compute dp[i] where dp[i] indicates size of largest divisible subset where a[i] is the smallest element. We can compute dp[i] in array using values from dp[i+1] to dp[n-1]. Finally, we return the maximum value from dp[].

Below is the implementation of the above approach:  

4

Time Complexity: O(n²)
Space Complexity: O(n)

Largest divisible pairs subset in c:

Approach

1. Create an array dp of size n to store the size of the largest divisible subset ending with each element of the array arr.

2. Initialize all elements of the dp array to 1, as every element is a divisible subset of itself.

3. For every element arr[i] in the array, check all previous elements arr[j] (where j < i) to see if arr[i] is divisible by arr[j] or arr[j] is divisible by arr[i].

4. If arr[i] is divisible by arr[j] or arr[j] is divisible by arr[i], then we can extend the divisible subset ending with arr[j] by including the arr[i] element as well. We will choose the arr[j] element which gives the largest divisible subset ending with arr[j].

5. Update the dp array with the size of the largest divisible subset ending with arr[i].

6. Keep track of the maximum element in the dp array, which gives the largest divisible subset among all elements.

7. Return the maximum element in the dp array.

8. In the main() function, create an array arr of integers and call the largest_divisible_pairs_subset() function with the array and its size.

9. Print the result returned by the largest_divisible_pairs_subset() function, which gives the size of the largest divisible subset.

3

Time Complexity:
The time complexity of the above algorithm is O(n^2), as we need to check all previous elements for each element.

Auxiliary Space:
The space complexity of the above algorithm is O(n), as we are only using an array of size n to store the largest divisible subset ending with each element.