Description
1. A number of different sized pancakes are stacked. A sorted pancake stack is defined
as having the smallest pancake on top, the second smallest pancake under the smallest
pancake, etc. The only tool provided is a spatula that will flip any top partition of the
stack. There only ever exists one stack of pancakes, not multiple smaller, independent
stacks. If every flip took one unit of time to complete, exactly how many flips or units
of time are required in the worst-case (i.e., for the worst-case arrangement of
pancakes) to sort the stack? Aim for a good algorithm. Express your answer, T(n), as
a function of the number of pancakes. Then, give a Big-Θ estimate. What is the
minimum number of flips needed to sort a worst-case arrangement of 4 pancakes?
(Note that your algorithm may not necessarily give the true minimum number of
flips.)
2. Let T be a full binary tree, and assume that the nodes of T are ordered by a preorder
traversal. This traversal assigns the label 1 to all internal nodes, and the label 0 to
each leaf (a node is internal if it is not a leaf). The sequence of 0’s and 1’s that results
from the preorder traversal of T is called the tree’s characteristic sequence.
a) Determine the full binary trees for the characteristic sequences:
i. 10110010100
ii. 1011100100100
3. In class we employed a version of the Heapify algorithm in which an unordered
sequence of elements was converted to a heap by the repeated insertion of the
elements into a heap (beginning with the empty heap). After each insertion, the
ReheapUp algorithm was applied to the new element to reheapify the tree structure.
Here’s another way to Heapify a sequence of elements. First, just put all the elements
into a complete binary tree, ignoring any notion of “heapness”. Apply the
ReheapDown algorithm to the roots of the smallest subtrees, then to the roots of the
subtrees whose left and right subtrees have been heapified, and so on up the tree.
Use this new version of Heapify to convert the following trees to heaps, and then sort
them using Heapsort (either ascending or descending order for the results is fine).
Clearly show each step.
a) Create a minimum heap (then sort):
10
/ \
15 8
/ \ / \
7 2 9 1
b) Create a maximum heap (then sort):
1
/ \
2 31
/ \ / \
4 5 16 7
4. The degree of a node in a tree refers to the number of edges incident on it (that is, the
number of edges that connect to it). In the case of a binary tree, the maximum degree
of any node is 3. The total degree refers to the sum total of every node’s degree.
For the following questions, either draw a tree with the given specification or explain
why no such tree exists:
a) Tree, 12 vertices, 15 edges
b) Tree, 5 vertices, total degree 10
c) Full binary tree, 5 internal (non-leaf) vertices
d) Full binary tree, 8 internal (non-leaf) vertices, 7 leaves
e) Complete binary tree, 4 leaves
f) Nearly complete binary tree, 7 leaves
5. What is the height of the tallest possible nearly complete binary tree with 240 leaves?
(Do not draw the tree– instead show your calculation.)
