: COMP108
DEPARTMENT Computer Science Tel. No. 795-4257
MAY 2007 EXAMINATIONS
Bachelor of Arts : Year 1 Bachelor of Science : Year 1 Bachelor of Science : Year 2 Master of Engineering : Year 1 No qualication aimed for: Year 1
ALGORITHMIC FOUNDATIONS
TIME ALLOWED : TWO hours
INSTRUCTIONS TO CANDIDATES
Answer FOUR questions.
If you attempt to answer more questions than the required number of questions (in any section), the marks awarded for the excess questions answered will be discarded (starting with your lowest mark). All logarithms are to the base 2.
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Question 1
1A. State (without proof) the order of magnitude, in the form of O(f (n)), of the following functions. Use the simplest f (n) possible in your answers. n2 + n log3 n + 5n + 10 II. 6 n3 + n5 + 7n2 + n III. 5 log n + n + log3 n + 1 I. IV. 3n + n4 + 2n + n2 [4 marks] 1B. Consider the following graph G. The label of an edge is the cost of the edge. a 1 e 2 4 3 f 5 b 6 10 c 12 g 7 14 8 d 8 h
Using Kruskals algorithm, draw a minimum spanning tree (MST) of the graph G. Write down the order in which the edges are selected. Is the MST drawn unique? (i.e., is it the one and only MST for the graph?) [8 marks] 1C. Referring to the same graph above, nd the shortest paths from the vertex a to all other vertices in the graph G using Dijkstras algorithm. Show the changes of the labels of the vertices step by step and give the order in which edges are selected. [13 marks]
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Question 2
2A. Consider the following graph G. The label of an edge is the edge number. Give the incidence matrix of the graph G. a 3 1 4 e 7 b 2 5 c 6
d
f
[4 marks] 2B. The time complexity of the merge sort algorithm to sort n numbers, denoted by T (n), can be described by the following recurrence. T (n) = 1 if n = 1, 2T (n/2) + n if n > 1.
Show that T (n) = O(n log n) by the substitution method. (Hint: show that T (n) 2n log n for n 2 by Mathematical Induction.) [8 marks] 2C. A length-n sequence S of characters S[0], S[1], , S[n1] is called a palindrome if S is the same as its reverse. For example, CIVIC, DAD, NOON, RADAR are all palindromes. Design and write a pseudo code algorithm to determine if a sequence S of characters is a palindrome or not. What is the worst case time complexity of your algorithm (in big-O notation)? Explain briey. [13 marks]
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Question 3
3A. I. Briey describe the idea of the greedy method. II. Briey describe the idea of the divide-and-conquer method. [4 marks] 3B. Show that the function n3 log n + 2n3 + 3n2 log n + 2n2 + 5n + 2 is O(n3 log n). [8 marks] 3C. Suppose there are two assembly lines each with 4 stations, Si,j . The assembly time is given in the circle representing the station and the transfer time is given next to the arrow from one station to another.
S1,1 0 10 10 45 5 S1,2 0 20 10 S1,3 0 10 S1,4
start
5 30 S2,1 0 20 S2,2 5 0 30 S2,3 5 0 10 S2,4
nish
I.
Using dynamic programming, ll in the table of the minimum time fi [j] needed to get through station Si,j . Show all the intermediate steps in computing these values. j f1 [j] [j] f2 1 2 3 4
II. What is the minimum time f needed to get through the assembly line? III. Which stations should be chosen? [13 marks]
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Question 4
4A. Which of the following problems is/are NP-complete problem(s)? I. Finding minimum spanning tree (MST) in a weighted undirected graph.
II. Vertex Cover Problem. III. 0/1 Knapsack problem. IV. Finding the n-th Fibonacci number. 4B. I. [4 marks]
Describe what a decision problem is and what an optimisation problem is.
II. An optimisation problem can be turned into a decision problem if we add a parameter k; and then ask whether the optimal value in the optimisation problem is at most or at least k. State the decision version of the following optimisation problems: a. Given an undirected graph G, nd the minimum number of colours that is needed to colour the edges in G such that no two edges with the same colour share a common endpoint. b. Given an undirected graph G, nd the maximum number of vertices such that for any two such vertices, there is no edge connecting them. [8 marks] 4C. Consider the following recursive algorithm to compute Fibonacci numbers. Algorithm F (n) if 0 n 1 then result = 1 else if n > 1 then result = F (n 1) + F (n 2) return result Suppose f (n) denote the time complexity of the above algorithm. f (n) satises the following recurrence f (n) = I. 1 if n 1, f (n 1) + f (n 2) + 1 if n > 1.
II. Using the iterative method, show that f (n) is exponential in n. III. The above algorithm is not efcient. Design and write the pseudo code of a faster (non-recursive) algorithm using the concept of dynamic programming. What is the time complexity of the faster algorithm (in big-O notation)? [13 marks]
Show that f (n) > 2f (n 2).
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Question 5
5A. What are the time complexities (in big-O notation) of the followin...