Satisfiability of Equality Equations
<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>
Union Find; Graph; Array; String
class UnionFind {
    private parent: number[];

    constructor() {
        this.parent = Array.from({ length: 26 }).map((_, i) => i);
    }

    find(index: number) {
        if (this.parent[index] === index) {
            return index;
        }
        this.parent[index] = this.find(this.parent[index]);
        return this.parent[index];
    }

    union(index1: number, index2: number) {
        this.parent[this.find(index1)] = this.find(index2);
    }
}

function equationsPossible(equations: string[]): boolean {
    const uf = new UnionFind();
    for (const [a, s, _, b] of equations) {
        if (s === '=') {
            const index1 = a.charCodeAt(0) - 'a'.charCodeAt(0);
            const index2 = b.charCodeAt(0) - 'a'.charCodeAt(0);
            uf.union(index1, index2);
        }
    }
    for (const [a, s, _, b] of equations) {
        if (s === '!') {
            const index1 = a.charCodeAt(0) - 'a'.charCodeAt(0);
            const index2 = b.charCodeAt(0) - 'a'.charCodeAt(0);
            if (uf.find(index1) === uf.find(index2)) {
                return false;
            }
        }
    }
    return true;
}
