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Cork Institute of Technology
Bachelor of Science (Honours) in Software Development - Stage 3
Autumn 2007
Computer Science
(Time : 3 Hours)
Answer any four questions. Examiners : Mr. V. Ryan Dr. M. O Cinneide Mr. M. Donnelly
Question 1 a) Which of the following graphs are Hamiltonian (have a Hamilton Cycle) and explain why? i. the complete graphs K (^) n ii. the complete biparite graphs K (^) i,j iii. trees [7 Marks]
b) Using the four rules discussed in class for determining whether or not a graph has a Hamilton Cycle, determine whether the following graph a Hamilton Cycle or not. If it has a Hamilton Cycle, find one such cycle.
[8 Marks]
c) Explain how we deal with repeated roots in the solving of linear recurrence relations. [3 Marks]
d) Solve the following linear recurrence relation: h 0 = 0 h 1 = 2
h (^) n = 5h (^) n-1 + 9h (^) n-2 , n>1. Verify your answer by checking the value of h (^4) .. [7 Marks]
Question 2 a) Explain the following terms as they relate to graph theory: i. loop ii. distance iii. diameter of a graph iv. walk v. trail vi. path vii. cycle [7 Marks]
b) Prove the following result: In any graph or multigraph, the number of vertices of odd degree is even. [8 Marks]
c) Describe the SUBSET-SUM Problem. [2 Marks]
d) Show that the SUBSET-SUM Problem is in the class NP. [3 Marks]
e) Explain what is meant by the problem SUBSET^ ā^ SUM. [2 Marks]
f) Explain why SUBSET^ ā^ SUM is probably not in the class NP. [3 Marks]
Question 4 a) Give an example of a planar graph, which has a drawing which is non-planar. [2 Marks] b) What is meant by two graphs being homeomorphic? [2 Marks]
c) Consider the following graph. Test and decide if it is planar or not. If it is planar, verify Eulerās Formula for the graph. If it is non-planar, verify Kuratowskiās theorem for this graph. [6 Marks]
d) State Euler's Formula, as it applies to a planar graph. [2 Marks]
e) What can you say about the degrees of the vertices of a graph which is loop-free, connected, and is planar. [3 Marks]
f) What is the definition of big-Oh? [3 Marks]
g) What is the complexity of the following algorithmic structures with respect to the problem size n. Assume that S is some operation in the order of O(1) and a is a constant greater than 1. Show all workings clearly.
(i) for ( i=1 ; i<=n ; i++ ) for ( j=1 ; j<=a ; j++ ) S; [2 Marks]
(ii) for ( i=1 ; i<=a ; i++ ) for ( j=1 ; j<=i ; j++ ) S; [3 Marks]
(iii) for ( i=1 ; i<=n ; i+=a ) S; [2 Marks]
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Question 5 a) Virus X spreads by email in the following manner: The author of the virus launched it from his/her computer to 7 victims. It searches the victimās address book for others that the person corresponds with, and automatically sends itself on to 7 of those people. We refer to the process of being sent to another victim and installing itself on the victimās computer as an infection. These 7 infections take place in parallel. Assume that everybody has at least 7 entries in their address book. Assume that no person will be infected twice by this process. If it takes an average of 5 minutes for an infection, how long before 40,005 PCs are infected? Use a tree to model the problem, and use the results and/or algorithms developed in class relating to trees in arriving at a solution to the problems. State clearly any results that you use. [8 Marks]
b) Explain what is meant by a prefix code. [2 Marks]
c) Give an example of a prefix code as represented by a Huffman Tree. [2 Marks]
d) Why will a Huffman Tree with codes at the leaves as we used in class always yield a prefix code? [3 Marks]
e) What is the definition of NP-Complete? [3 Marks]
f) Consider the following statement āIf A <=^ p^ B where B in NP , and A is NP-complete , then B is NP-complete also.ā Explain in detail why this is the case. [4 Marks]
g) List three well-known NP-Complete problems. [3 Marks]
c) Using an example, explain why some algorithms may be parallelised and others canāt. [3 Marks ]
d) Outline the parallel Enumeration Sort algorithm. Give a time complexity analysis of the algorithm. What would its complexity be if it were run on a uniprocessor? Explain your answer. [7 Marks ]
APPENDIX
Refers to Question 6
Please detach and submit with your answer book.
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