<p>You are given a string <code>s</code>, and an array of pairs of indices in the string&nbsp;<code>pairs</code>&nbsp;where&nbsp;<code>pairs[i] =&nbsp;[a, b]</code>&nbsp;indicates 2 indices(0-indexed) of the string.</p>

<p>You can&nbsp;swap the characters at any pair of indices in the given&nbsp;<code>pairs</code>&nbsp;<strong>any number of times</strong>.</p>

<p>Return the&nbsp;lexicographically smallest string that <code>s</code>&nbsp;can be changed to after using the swaps.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;dcab&quot;, pairs = [[0,3],[1,2]]
<strong>Output:</strong> &quot;bacd&quot;
<strong>Explaination:</strong> 
Swap s[0] and s[3], s = &quot;bcad&quot;
Swap s[1] and s[2], s = &quot;bacd&quot;
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;dcab&quot;, pairs = [[0,3],[1,2],[0,2]]
<strong>Output:</strong> &quot;abcd&quot;
<strong>Explaination: </strong>
Swap s[0] and s[3], s = &quot;bcad&quot;
Swap s[0] and s[2], s = &quot;acbd&quot;
Swap s[1] and s[2], s = &quot;abcd&quot;</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;cba&quot;, pairs = [[0,1],[1,2]]
<strong>Output:</strong> &quot;abc&quot;
<strong>Explaination: </strong>
Swap s[0] and s[1], s = &quot;bca&quot;
Swap s[1] and s[2], s = &quot;bac&quot;
Swap s[0] and s[1], s = &quot;abc&quot;
</pre>

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

<ul>
	<li><code>1 &lt;= s.length &lt;= 10^5</code></li>
	<li><code>0 &lt;= pairs.length &lt;= 10^5</code></li>
	<li><code>0 &lt;= pairs[i][0], pairs[i][1] &lt;&nbsp;s.length</code></li>
	<li><code>s</code>&nbsp;only contains lower case English letters.</li>
</ul>

class UnionFind:
    def __init__(self, n):
        self.father = [-1] * n

    def union(self, u, v, s):
        root_u, root_v = self.find(u), self.find(v)
        if root_u == root_v:
            return
        
        if s[root_u] > s[root_v]:
            root_u, root_v = root_v, root_u
        
        self.father[root_v] = root_u

    def find(self, x):
        root = x
        while self.father[root] != -1:
            root = self.father[root]
        # optimization
        while root != x:
            original_father = self.father[x]
            self.father[x] = root
            x = original_father
        return root

class Solution:
    def smallestStringWithSwaps(self, s: str, pairs: List[List[int]]) -> str:
        # Use Union Find. Need to record the location.
        # Find all connected components and compare the value of the root
        n = len(s)
        uf = UnionFind(n)
             
        # connect the nodes and use union find
        for pair in pairs:
            u, v = pair[0], pair[1]
            uf.union(u, v, s)
        
        # map the index into the corresponding root
        mapping = defaultdict(list)
        for i in range(n):
            root = uf.find(i)
            if root == -1:
                mapping[i].append((i, s[i]))
            mapping[root].append((i, s[i]))
        
        # for each list of a root, sort
        for root, components in mapping.items():
            components.sort(key = lambda x: x[1]) # sort increasingly based on value
        
        res = [''] * n
        for root in mapping:
            indices = sorted( [component[0] for component in mapping[root]])
            values = sorted( [component[1] for component in mapping[root]])
            for i, v in zip(indices, values):
                res[i] = v
        
        return ''.join(res)

510