Introduction to Algorithms
Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 19.4, Problem 1E
Program Plan Intro

Show that for any positive integer n, a sequence of Fibonacci-heap operations that creates a Fibonacci heap consisting of one tree is a linear chain of n nodes.

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Think about the challenge of finding the smallest element in a maximum heap. A max heap's smallest components must be one of the n/2 leaves. (If not, the tree cannot be a max heap since there must be a nonleaf smaller than one of its descendants.) Therefore, a thorough leaf search is sufficient. Show, at the very least, that leaf-by-leaf searching is required.
Consider the problem of determining the smallest element in a maxheap. The smallest elements of a max heap must be one of the n/2 leaves. (Otherwise,there must be a nonleaf that is smaller than one of its descendants, which means thetree is not a max heap.) Thus, it is sufficient to search all leaves. Prove a lower boundthat searching all the leaves is necessary
Let's assume that a binary heap is represented using a binary tree such that each node may have a left child node and a right child node. For this type of representation, we can still label the nodes of the tree in the same way as we label the nodes for an array representation. That is, the root node has a label 1. In general, for a node with label i, its left child node will have a label 2i and the right child  has a label 2i+1.  For any i with 1 <= I <= n , Terry says that the following easy algorithm will walk you from the root node to the node with label i: First find the binary representation P of i.   Start with the rightmost bit (least significant bit) of P, walk down from the root as follow: For a 0 bit, walk to the left child, for a 1 bit walk to the right child. At the end, you’ll reach the node with label i.  Which of the following is the most appropriate? A. Terry’s algorithm is wrong and not fixable. B. Terry’s algorithm is right. C. Terry's algorithm can be…
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