<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:
    int find(int i, vector<int>&parent){
        if(i==parent[i])return i;

        return parent[i]=find(parent[i],parent);
    }

    void Union(int x, int y, vector<int>&rank, vector<int>&parent){
        int x_parent=find(x,parent);
        int y_parent=find(y,parent);

        if(x_parent==y_parent)return;

        if(rank[x_parent]>rank[y_parent]){
            parent[y_parent]=x_parent;
        }
        else if(rank[x_parent]<rank[y_parent]){
            parent[x_parent]=y_parent;
        }
        else {
            parent[x_parent]=y_parent;
            rank[y_parent]++;
        }

        return;
    }
    bool equationsPossible(vector<string>& equations) {
        vector<int>parent(26);
        vector<int>rank(26,1);

        vector<string> equalities;
        vector<string> nonEqualities;

        for(auto it:equations){
            if(it[1]=='='){
                equalities.push_back(it);
            }
            else nonEqualities.push_back(it);
        }
        int i=0;
        for(auto it:equalities)equations[i++]=it;
        for(auto it:nonEqualities)equations[i++]=it;

        for(int i=0;i<26;i++){
            parent[i]=i;
        }

        for(auto it:equations){
            char ch1=it[0];
            char ch2=it[3];
            char equality=it[1];

            if(equality=='='){
               Union(ch1-'a',ch2-'a',rank,parent);
            }
            else{
                int ch1_parent=find(ch1-'a',parent);
                int ch2_parent=find(ch2-'a',parent);

                if(ch1_parent==ch2_parent)return false;
            }
        }

        return true;
    }
};
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