# How To Find The Largest Divisible Subset of a Given Integer Set

## Problem statement

You have a collection of positive integers called nums, where each integer is distinct. Your task is to find the largest subset answer from this collection, such that for every pair of elements (answer[i], answer[j]) within this subset:

• Either answer[i] is a multiple of answer[j] (i.e., answer[i] % answer[j] == 0), or
• answer[j] is a multiple of answer[i] (i.e., answer[j] % answer[i] == 0).

If there are multiple possible solutions, you can return any of them.

### Example 1

Input: nums = [1,2,3]
Output: [1,2]
Explanation: [1,3] is also accepted.


### Example 2

Input: nums = [1,2,4,8]
Output: [1,2,4,8]


### Constraints

• 1 <= nums.length <= 1000.
• 1 <= nums[i] <= 2 * 10^9.
• All the integers in nums are unique.

## Solution 1: Bruteforce with Dynamic programming

Note that the condition a % b == 0 is called a is divisible by b. In mathematics, it can also be called b divides a and be written as b | a.

The symmetry of the divisibility criteria means it does not count the ordering of the answer. You could sort the vector nums before trying to find the longest subset answer = [answer, answer, ..., answer[m]] where answer[i] | answer[j] with all 0 <= i <= j <= m.

Now assuming the nums were sorted. For each i, you need to find the largest subset maxSubset[i] starting from nums[i]. And the final answer is the largest one among them.

### Example 3

Input: nums = [2, 4, 3, 9, 8].
Sorted nums = [2, 3, 4, 8, 9].
maxSubset = [2, 4, 8].
maxSubset = [3, 9].
maxSubset = [4, 8].
maxSubset = .
maxSubset = .
Output: [2, 4, 8].


Note that for a sorted nums, if nums[i] | nums[j] for some i < j, then maxSubset[j] is a subset of maxSubset[i].

For example, maxSubset is a subset of maxSubset in Example 3 because nums = 2 | 4 = nums.

That might lead to some unnecessary recomputing. To avoid it, you could use dynamic programming to store the maxSubset[j] you have already computed.

### Code

#include <iostream>
#include <vector>
#include <unordered_map>
#include <algorithm>
using namespace std;

//! @brief compute the maxSubset[i] starting from nums[i]
//!         and store it to _map[i]
//! @note nums is sorted
vector<int> largestDivisibleSubsetOf(vector<int>& nums,
int i, unordered_map<int, vector<int> >& _map)
{
if (_map.find(i) != _map.end())
{
return _map[i];
}
vector<int> maxSubset{nums[i]};
if (i == nums.size() - 1)
{
_map.insert({i, maxSubset});
return maxSubset;
}
for (int j = i + 1; j < nums.size(); j++)
{
if (nums[j] % nums[i] == 0)
{
auto subset = largestDivisibleSubsetOf(nums, j, _map);
subset.push_back(nums[i]);
if (maxSubset.size() < subset.size())
{
maxSubset = subset;
}
}
}
_map.insert({i, maxSubset});
return maxSubset;
}
vector<int> largestDivisibleSubset(vector<int>& nums)
{
if (nums.size() <= 1)
{
return nums;
}
unordered_map<int, vector<int> > _map;
sort(nums.begin(), nums.end());
for (int i = 0; i < nums.size(); i++)
{
auto maxSubset = largestDivisibleSubsetOf(nums, i, _map);
{
}
}
}
void printSolution(vector<int>& result)
{
cout << "[";
for (auto& v : result)
{
cout << v << ",";
}
cout << "]" << endl;
}
int main()
{
vector<int> nums{2,1,3};
nums = {1,2,4,8};
}

Output:
[2,1,]
[8,4,2,1,]


### Code explanation

This solution utilizes dynamic programming and recursive functions to efficiently identify and store the largest divisible subsets.

Here's how the code works:

1. It starts by sorting the input vector nums in ascending order, as it simplifies the process of finding divisibility relationships.

2. The largestDivisibleSubsetOf function is a recursive helper function. It takes three parameters: the input vector nums, the current index i, and a map _map that stores previously computed subsets for each index. This function aims to find the largest divisible subset starting from index i.

3. If the function has already computed the largest divisible subset for index i and stored it in _map, it returns that precomputed subset.

4. Otherwise, it initializes maxSubset with the current element at index i. If i is the last index in nums, it inserts this element into _map and returns it as the maximum subset.

5. If i is not the last index, the function iterates through the elements with indices greater than i and checks if each element is divisible by the element at index i. If it is divisible, it calls largestDivisibleSubsetOf recursively for the next index, appends the current element to the subset obtained from the recursive call, and updates maxSubset if the new subset is larger.

6. After processing all elements, the function inserts maxSubset into _map for index i and returns it.

7. The largestDivisibleSubset function is the main function. It iterates through the sorted nums vector, calling largestDivisibleSubsetOf for each index and keeping track of the largest subset encountered so far.

8. Finally, it returns the largest divisible subset found.

### Complexity

The time complexity of this solution is O(n^2), where n is the number of elements in the nums vector. The recursive function may be called multiple times for each element in nums, leading to a quadratic time complexity. However, it optimizes the process by storing and reusing previously computed subsets using the _map data structure.

• Runtime: O(n^2).
• Extra space: O(n^2).

## Solution 2: Store only the representative of the maxSubset

In the brute-force solution above, you used a big map to log all maxSubset[i] though you need only the largest one at the end.

One way to save memory (and eventually improve performance) is just storing the representative of the chain relationship between the values nums[i] of the maxSubset through their indices mapping.

That means if maxSubset[i] = [nums[i0] | nums[i1] | ... | nums[iN1]] | nums[iN]], you could log pre[iN] = iN1, ..., prev[i1] = i0.

Then all you need to find is only the last index iN of the largest maxSubset.

### Example 3

Input: nums = [2, 4, 3, 9, 8].
sorted nums = [2, 3, 4, 8, 9].
pre = -1 (there is no nums[i] | nums).
pre = -1 (there is no nums[i] | nums).
pre = 0 (nums is the only divisor of nums).
pre = 2 (for the largest subset though nums and nums are both divisors of nums).
pre = 1 (nums is the only divisor of nums).
iN = 3 ([2 | 4 | 8] is the largest maxSubset).
Output: [8, 4, 2].


### Code

#include <iostream>
#include <vector>
#include <algorithm>
using namespace std;
vector<int> largestDivisibleSubset(vector<int>& nums) {
if (nums.size() <= 1) {
return nums;
}
sort(nums.begin(), nums.end());
int maxSize = 0;    // the size of the resulting subset
int maxindex = 0;   // nums[maxindex] is the largest value of the resulting subset
vector<int> subsetSize(nums.size(), 1); // subsetSize[i] stores the size
// of the largest subset having
// biggest number nums[i]
vector<int> pre(nums.size(), -1);       // pre[i] stores the previous
// index of i in their largest subset
for (int i = 0; i < nums.size(); i++) {
// find the previous nums[j] that make subsetSize[i] largest
for (int j = i - 1; j >= 0; j--) {

if (nums[i] % nums[j] == 0 && subsetSize[j] + 1 > subsetSize[i]) {
subsetSize[i] = subsetSize[j] + 1;
pre[i] = j;
}
}
// update the largest subset
if (maxSize < subsetSize[i]) {
maxSize = subsetSize[i];
maxindex = i;
}
}
vector<int> result;
while (maxindex != -1) {
result.push_back(nums[maxindex]);
maxindex = pre[maxindex];
}
return result;
}
void printSolution(vector<int>& result) {
cout << "[";
for (auto& v : result) {
cout << v << ",";
}
cout << "]" << endl;
}
int main() {
vector<int> nums{2,1,3};
auto result = largestDivisibleSubset(nums);
printSolution(result);
nums = {1,2,4,8};
result = largestDivisibleSubset(nums);
printSolution(result);
}

Output:
[2,1,]
[8,4,2,1,]


### Code explanation

1. The code starts by sorting the input vector nums in ascending order. Sorting simplifies the process of finding divisibility relationships and allows us to build the largest divisible subset incrementally.

2. The algorithm initializes some variables:

• maxSize represents the size of the largest divisible subset encountered.
• maxindex stores the index of the largest element in the largest subset.
• subsetSize is a vector where subsetSize[i] stores the size of the largest subset ending with nums[i].
• pre is a vector where pre[i] stores the index of the previous element in the largest subset ending with nums[i].
3. It iterates through the sorted nums vector. For each element at index i, it searches for a previous element nums[j] (where j < i) such that nums[i] is divisible by nums[j] and subsetSize[j] + 1 is greater than subsetSize[i]. If such a condition is met, it updates subsetSize[i] and pre[i] accordingly.

4. While iterating, it also tracks the largest subset encountered by updating maxSize and maxindex whenever a larger subset is found.

5. After processing all elements, the code constructs the largest divisible subset by starting from maxindex (the largest element) and backtracking through pre until it reaches -1. It appends each element to the result vector.

6. Finally, it returns the result vector containing the largest divisible subset.

### Complexity

The time complexity of this solution is O(n^2), where n is the number of elements in the nums vector. The nested loop searches for previous elements with divisibility relationships, which may lead to quadratic time complexity in the worst case. However, it efficiently maintains information about subset sizes and elements, reducing redundant calculations and improving performance.

• Runtime: O(n^2).
• Extra space: O(n).

## Key takeaway

In this interesting problem, we use index mapping to simplify everything. That improves the performance in both runtime and memory.

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