chap6-solutions - Selected Solutions for Chapter 6 Heapsort...

Selected Solutions for Chapter 6:
Heapsort
Solution to Exercise 6.1-1
Since a heap is an almost-complete binary tree (complete at all levels except pos-
sibly the lowest), it has at most 2 h C 1 NUL 1 elements (if it is complete) and at least
2 h NUL 1 C 1 D 2 h elements (if the lowest level has just 1 element and the other levels
are complete).
Solution to Exercise 6.1-2
Given an n -element heap of height h , we know from Exercise 6.1-1 that
2
h
± n ± 2
h C 1
NUL 1 < 2
h C 1
:
Thus, h ± lg n < h C 1 . Since h is an integer, h D b lg n c (by definition of b c ).
Solution to Exercise 6.2-6
If you put a value at the root that is less than every value in the left and right
subtrees, then M
AX
-H
EAPIFY
will be called recursively until a leaf is reached. To
make the recursive calls traverse the longest path to a leaf, choose values that make
M
AX
-H
EAPIFY
always recurse on the left child. It follows the left branch when
the left child is greater than or equal to the right child, so putting 0 at the root
and 1 at all the other nodes, for example, will accomplish that. With such values,