Download Computer Science Exam Questions - Cork Institute of Technology and more Exams Computer Science in PDF only on Docsity!
Cork Institute of Technology
Bachelor of Science (Honours) in Computer Applications – Stage 3
(Bachelor of Science in Computer Applications – Stage 3)
(NFQ – Level 8)
Autumn 2005
COMPUTER SCIENCE
(Time : 3 Hours)
Answer any four questions. Examiners : Dr. D. Chambers Mr. P. O' Connor Mr. T. Parslow Mr. V. Ryan
Question 1 a) List all elementary subdivisions of the following graph:
[4 Marks]
b) Define the term homeomorphic. [3 Marks]
Give an example of a graph which is homeomorphic to the graph shown in part (a) above, but is not equal to this graph. [2 Marks]
c) Prove the following result: In any graph or multigraph, the number of vertices of odd degree is even. [6 Marks]
d) Show that the HAMILTON CYCLE problem is reducible in polynomial-time to the TRAVELLING SALESMAN problem (TSP). [10 Marks]
d
a • b
c
Question 2 a) Determine if the following graphs G 1 and G 2 are isomorphic or not. If they are not isomorphic, show this to be the case. If they are isomorphic, list any one of the isomorphisms? Prove that the one you list is an isomorphism. Detail the steps that you used in arriving at your answer. Most marks are awarded for your approach to answering the question. A correct yes/no type of answer without the steps used to arrive at the answer fully explained and detailed will attract zero marks. [10 Marks]
b) How many edges has the hypercube Qn? Justify your answer. [5 Marks]
c) What is the complexity of the following code segment? for (i=0; i<n ; i++)
for(j=n; j>i ; j--)
for (k=1; k<y; k++)
S where S is a statement whose complexity is O(1) and y is another variable?
(Assume that the input size is n)
Show detailed workings carefully. [10 Marks]
A
G
B
C
F E D
T
Z
Y
U
X W
V
G 1 :
G 2 :
Question 4
a) Define what is meant by a rooted tree. [2 Marks]
Define what is meant by a rooted m-ary tree being complete. [2 Marks]
Define what is meant by a rooted tree being balanced. [2 Marks]
What is the relationship between the height and the number of leaves in a complete m-ary tree? [2 Marks]
What is the relationship between the height and the number of leaves in a complete m-ary tree which is also balanced? [2 Marks]
b) The management structure in organization X is a tree structure, in which every manager manages 4 people and may have another manager managing him/herself. If the organization has 865 employees in total, how many employees are not managers? [5 Marks]
c) Solve the following recurrence relation h (^) n = 4h (^) n-1 - 3h (^) n-2 , n ≥ 2 h 0 = 2 h 1 = 4 [6 Marks]
Check your solution by computing h 3. [4 Marks]
Question 5
a) Find the Chromatic number of the following graphs
[3 Marks]
[3 Marks]
A
H
L
F
I
M
K
B
J
D
G C
E
A
F B
E D C
Question 6
a) Using Dijkstra’s algorithm, find the shortest path from vertex N to vertex R in the following graph. Demonstrate clearly that you have used the algorithm to arrive at the solution.
In particular, list the order in which the labels on the nodes are made permanent. [7 Marks]
N
f
b i e
m d
j g c
k
h R
b) Using Kruskal’s Algorithm, find a minimum spanning tree for the following graph Show all workings clearly. Demonstrate clearly that you have used the algorithm to arrive at the solution.
[8 Marks]
c) What is the prefix problem? Give an example to illustrate your answer. [2 Marks]
Give a sequential algorithm to address the parallel prefix problem. [3 Marks]
Work out its complexity? [3 Marks]
Describe a parallel algorithm to address the parallel prefix problem. [2 Marks]
A
B
C
D
E
G
F
