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Changing Variables

Example: T(n) = 2T( $\sqrt{n}$ ) + lg n

Let $m = lg n, \;\; 2^m = n, \;\; n^{1/2} \;=\; (2^m)^{1/2} \;=\; 2^{m/2}$

$T(2^m) \;\; = \;\; 2T(2^{m/2}) \; + \; m$

Let $S(m) \;\;=\;\; T(2^m) \;\;=\;\; T(n)$

$S(m) \;\;=\;\; 2S(m/2) + m$

$S(m) \;\;=\;\; O(mlgm)$

$T(n) \;\;=\;\; O(mlgm) \;\;=\;\; O(lgn \;lglgn)$

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