<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 DisjointSet{
    
public:
    vector<int> parent, size;

    DisjointSet(int n){
        parent.resize(n+1);
        size.resize(n+1);

        for(int i=0; i<=n; i++){
            parent[i] = i;
            size[i] = 1;
        }
    }

    int ultParent(int node){
        if(node == parent[node]) return node;
        return parent[node] = ultParent(parent[node]);
    }

    void unionBySize(int u, int v){
        int uPar_u = ultParent(u);
        int uPar_v = ultParent(v);
        if(uPar_u == uPar_v) return;
        if(size[uPar_u] < size[uPar_v]){
            parent[uPar_u] = uPar_v;
            size[uPar_v] += size[uPar_u];
        } else{
            parent[uPar_v] = uPar_u;
            size[uPar_u] += size[uPar_v];
        }
    }
};

class Solution {
public:
    bool equationsPossible(vector<string>& equations) {
        // solve by DSU
        int n = equations.size();
        DisjointSet ds(1000);
        for(int i=0; i<n; i++){
            if(equations[i][1] == '='){
                ds.unionBySize(equations[i][0]-'a', equations[i][3]-'a');
            }
        }

        for(int i=0; i<n; i++){
            if(equations[i][1] == '!'){
                if(ds.ultParent(equations[i][0]-'a') == ds.ultParent(equations[i][3]-'a')) return false;
            }
        }

        return true;

    }
};
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