<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 {
    vector<int> parent;
    vector<int> rank;
    vector<int> size;
    public:
    DisjointSet(int n){
        // 0 based indexing
        parent.resize(n, 0);
        rank.resize(n, 0);
        size.resize(n,0);
        for(int i=0;i<n;i++){
            parent[i] = i;
            size[i] = 1;
        }
    }
    
    int findUpar(int node){
        if(parent[node] == node) return node;
        // path compression
        return parent[node] = findUpar(parent[node]);
    }
    
    void unionByRank(int u, int v){
        int p_u = findUpar(u);
        int p_v = findUpar(v);
        if(p_u == p_v) return;
        if(rank[p_u]>rank[p_v]){
            parent[p_v] = p_u;
        }else if(rank[p_u]<rank[p_v]){
            parent[p_u] = p_v;
        }else {
            parent[p_v] = p_u;
            rank[p_u]++;
        }
    }
    
    void unionBySize(int u, int v){
        int p_u = findUpar(u);
        int p_v = findUpar(v);
        if(p_u == p_v) return;
        if(size[p_u]>size[p_v]){
            parent[p_v] = p_u;
            size[p_u] += size[p_v];
        }else {
            parent[p_u] = p_v;
            size[p_v] += size[p_u];
        }
    }
};

class Solution {
public:
    bool equationsPossible(vector<string>& equations) {
        DisjointSet ds(27);

        for(auto eq : equations){
            int u  = eq[0] - 'a';
            int v  = eq[3] - 'a';
            if(eq[1] == '='){
                ds.unionByRank(u,v);
            }
        }

        for(auto eq : equations){
            int u  = eq[0] - 'a';
            int v  = eq[3] - 'a';
            if(eq[1] == '!'){
               if(ds.findUpar(u) == ds.findUpar(v)) return false;
            }
        }
        return true;
    }
};
15500