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Analysis:

Union by Rank Only:

$\Theta(m lg n)$

m = #operations

n ( $\leq$ m) = #MakeSet operations in m

Path Compression Only:

$\Theta(f \log_{(1 + f/n)} n)$ if f $\geq$ n

$\Theta(n \;+\; f \lg n)$ if f < n

n = #MakeSet operations

There are $\leq$ n-1 Unions

f = #FindSet operations

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