<p>You are given an integer array of unique positive integers <code>nums</code>. Consider the following graph:</p>

<ul>
	<li>There are <code>nums.length</code> nodes, labeled <code>nums[0]</code> to <code>nums[nums.length - 1]</code>,</li>
	<li>There is an undirected edge between <code>nums[i]</code> and <code>nums[j]</code> if <code>nums[i]</code> and <code>nums[j]</code> share a common factor greater than <code>1</code>.</li>
</ul>

<p>Return <em>the size of the largest connected component in the graph</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/01/ex1.png" style="width: 500px; height: 97px;" />
<pre>
<strong>Input:</strong> nums = [4,6,15,35]
<strong>Output:</strong> 4
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/01/ex2.png" style="width: 500px; height: 85px;" />
<pre>
<strong>Input:</strong> nums = [20,50,9,63]
<strong>Output:</strong> 2
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/01/ex3.png" style="width: 500px; height: 260px;" />
<pre>
<strong>Input:</strong> nums = [2,3,6,7,4,12,21,39]
<strong>Output:</strong> 8
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= nums.length &lt;= 2 * 10<sup>4</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>5</sup></code></li>
	<li>All the values of <code>nums</code> are <strong>unique</strong>.</li>
</ul>

class Solution:
    def largestComponentSize(self, A: List[int]) -> int:

        dsu = DisjointSetUnion(max(A))

        # attribute each element in A
        #   to all the groups that lead by its factors.
        for a in A:
            for factor in range(2, int(sqrt(a))+1):
                if a % factor == 0:
                    dsu.union(a, factor)
                    dsu.union(a, a // factor)

        # count the size of group one by one
        max_size = 0
        group_count = defaultdict(int)
        for a in A:
            group_id = dsu.find(a)
            group_count[group_id] += 1
            max_size = max(max_size, group_count[group_id])

        return max_size


class DisjointSetUnion(object):

    def __init__(self, size):
        # initially, each node is an independent component
        self.parent = [i for i in range(size+1)]
        # keep the size of each component
        self.size = [1] * (size+1)
    
    def find(self, x):
        """ return the component id that the element x belongs to. """
        if self.parent[x] != x:
            self.parent[x] = self.find(self.parent[x])
        return self.parent[x]
    
    def union(self, x, y):
        """ merge the two components that x, y belongs to respectively,
              and return the merged component id as the result.
        """
        px, py = self.find(x), self.find(y)
        
        # the two nodes share the same set
        if px == py:
            return px
        
        # otherwise, connect the two sets (components)
        if self.size[px] > self.size[py]:
            # add the node to the union with less members.
            # keeping px as the index of the smaller component
            px, py = py, px
        # add the smaller component to the larger one
        self.parent[px] = py
        self.size[py] += self.size[px]
        # return the final (merged) group
        return py
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