<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:
    template<int N>
    struct DSU {
        array<int, N> arr;
        array<int, N> sizes;

        DSU() {
            for (int i = 0; i < N; i++) {
                arr[i] = i;
                sizes[i] = 1;
            }
        }

        int find(int u) {
            if (arr[u] == u) {
                return u;
            }
            int parent = find(arr[u]);
            arr[u] = parent;
            return parent;
        }

        void join(int u, int v) {
            int u_rep = find(u);
            int v_rep = find(v);
            if (sizes[u_rep] < sizes[v_rep]) {
                arr[v_rep] = u_rep;
                sizes[v_rep] += sizes[u_rep];
                sizes[u_rep] = 0;
            } else {
                arr[u_rep] = v_rep;
                sizes[u_rep] += sizes[v_rep];
                sizes[v_rep] = 0;
            }
        }
    };

    bool equationsPossible(vector<string>& equations) {
        DSU<256> dsu;
        for (const string& s : equations) {
            if (s[1] == '=') {
                dsu.join(s[0], s[3]);
            }
        }
        for (const string& s : equations) {
            if (s[1] == '!') {
                if (dsu.find(s[0]) == dsu.find(s[3])) {
                    return false;
                }
            }
        }
        return true;
    }
};
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