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Analysis

The worst case scenario is:

Make-Set(x₁)
...
Make-Set(x_n) {{x₁}, {x₂}, {x₃}, $\ldots$ , {x_n}}
Union(x₁, x₂) {{ $x_1 \rightarrow x_2$ }, {x₃}, $\ldots$ , {x_n}}
Union(x₂, x₃) {{ $x_1 \rightarrow x_2 \rightarrow x_3$ }, $\ldots$ , {x_n}}
...
Union(x_q-1, x_q) {{ $x_1 \rightarrow x_2 \rightarrow x_3 \rightarrow \ldots \rightarrow x_n$ }

n = #Make-Set operations

m = #Make-Set, Union, and Find-Set operations
m = n + (q - 1) operations

T(m) = $\Theta(n) \;+\; \sum_{i=1}^{q-1} i$

= $\Theta(n \;+\; q^2)$

n = $\Theta(m)$ and q = $\Theta(m)$

Therefore, T(m) = $\Theta(m^2)$ and the amortized cost is $\Theta(m)$ per operation.

Can we do better?

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