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Analysis

From this result we can analyze the run time of BuildHeap.

$\begin{displaymath}\sum_{h=0}^{\lfloor lg n \rfloor} \lceil \frac{n}{2^{h+1}} \r... ... O(n*1/2 \; \sum_{h=0}^{\lfloor lg n \rfloor} \; \frac{h}{2^h})\end{displaymath}$

Note that $\sum_{k=0}^{\infty} x^k \;=\; \frac{1}{1 - x}$ , for $\mid x \mid \;<\; 1$ . The derivative of both sides,
$\frac{d}{dx}(\sum_{k=0}^{\infty} x^k) \;=\; \frac{d}{dx}(\frac{1}{1 - x})$ , is equal to $\sum_{k=0}^{\infty} k x^{k-1} \;=\; \frac{1}{(1 - x)^2}$ .

Multiplying both sides of the equivalence by x we get
$\sum_{k=0}^{\infty} k x^k \;=\; \frac{x}{(1 - x)^2}$ .
In our case k = h and x = 1/2.
Thus $\sum_{h=0}^{\infty} \frac{h}{2^h} \;=\; \frac{1/2}{(1 - 1/2)^2}$ = 2, x = 1/2.

Thus the run time is O(n * 1/2 * 2) = O(n).

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