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Bounding Maximum Degree D(n)

Lemma 20.2

\(F_{k+2} \;=\; 1 \;+\; \sum_{i=0}^k F_i\), $\;\;\;\;\;$where Fk is a Fibonacci number.


\begin{displaymath}F_k = \left\{ \begin{array}{ll}
k & {\rm if} \; k < 2 \\
F_{k-1} + F_{k-2} & {\rm if} \; k \geq 2
\end{array} \right. \end{displaymath}



Lemma 20.3

For a node x in a Fibonacci heap, where k = degree(x),

size(x) $\geq F_{k+2} \geq \phi^k$, where \(\phi \;=\; \frac{1 + \sqrt{5}}{2}\)

size(x) = #nodes in subtree rooted at x



Corollary 20.4

D(n) = O(lgn)

By Lemma 20.3, n $\geq$ size(x) $\geq \phi^k$, where n = nodes in Fibonacci Heap and k = degree of any node x.

Then \(\log_\phi n \;\geq\; k\), and k = \(O(\log_\phi n)\) = O(lg n).


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