<p>You are given an array of strings <code>equations</code> that represent relationships between variables where each string <code>equations[i]</code> is of length <code>4</code> and takes one of two different forms: <code>&quot;x<sub>i</sub>==y<sub>i</sub>&quot;</code> or <code>&quot;x<sub>i</sub>!=y<sub>i</sub>&quot;</code>.Here, <code>x<sub>i</sub></code> and <code>y<sub>i</sub></code> are lowercase letters (not necessarily different) that represent one-letter variable names.</p>

<p>Return <code>true</code><em> if it is possible to assign integers to variable names so as to satisfy all the given equations, or </em><code>false</code><em> otherwise</em>.</p>

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

<pre>
<strong>Input:</strong> equations = [&quot;a==b&quot;,&quot;b!=a&quot;]
<strong>Output:</strong> false
<strong>Explanation:</strong> If we assign say, a = 1 and b = 1, then the first equation is satisfied, but not the second.
There is no way to assign the variables to satisfy both equations.
</pre>

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

<pre>
<strong>Input:</strong> equations = [&quot;b==a&quot;,&quot;a==b&quot;]
<strong>Output:</strong> true
<strong>Explanation:</strong> We could assign a = 1 and b = 1 to satisfy both equations.
</pre>

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

<ul>
	<li><code>1 &lt;= equations.length &lt;= 500</code></li>
	<li><code>equations[i].length == 4</code></li>
	<li><code>equations[i][0]</code> is a lowercase letter.</li>
	<li><code>equations[i][1]</code> is either <code>&#39;=&#39;</code> or <code>&#39;!&#39;</code>.</li>
	<li><code>equations[i][2]</code> is <code>&#39;=&#39;</code>.</li>
	<li><code>equations[i][3]</code> is a lowercase letter.</li>
</ul>

class Solution {
public:
    class DSU{
        private:
            vector<int>parent;
            vector<int>size;
        public:
            DSU(){
                parent.resize(26,0);
                size.resize(26,1);

                for(int i=0; i<26; i++){
                    parent[i] = i;
                }
            }
            int find(int x){
                if(x == parent[x])  return x;
                
                return parent[x] = find(parent[x]);
            }
            void unionKro(int x,int y){
                int parent_x = find(x);
                int parent_y = find(y);
                
                if(parent_x == parent_y)    return;
                else if(size[parent_x] > size[parent_y]){
                    parent[parent_y] = parent_x;
                    size[parent_x] += size[parent_y];
                }
                else{
                    parent[parent_x] = parent_y;
                    size[parent_y] += size[parent_x];
                }
            }
    };
    bool equationsPossible(vector<string>& equations) {
        DSU dsu;
        for(auto equation : equations){
            int x = equation[0]-'a';
            int y = equation[3]-'a';
            
            if(equation[1]=='='){
                dsu.unionKro(x,y);
            }
        }

        for(auto equation : equations){
            int x = equation[0]-'a';
            int y = equation[3]-'a';
            
            if(equation[1]=='!'){
                int parentX = dsu.find(x);
                int parentY = dsu.find(y);
                if(parentX == parentY)  return false;
            }
        }
        return true;
    }
};
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