Enrolment number (student number):

The University of Melbourne

Practice Exam Paper

School of Computing and Information Systems

COMP90038 Algorithms and Complexity

Reading Time: 15 minutes

Exam Duration: 3 hours

This paper has 11 pages, including this front page.

Authorised Materials:

None. This is a closed book exam. Electronic devices, including calculators and laptop

computers are not permitted.

Instructions to Invigilators:

Students will provide answers in the exam paper itself. The exam paper must remain in

the exam venue and must be returned to the examiner.

Instructions to Students:

This is not an actual exam paper. It is a practice paper which has been put together

to show you the format that you can expect in the exam. Many aspects of this paper’s

contents do not necessarily reflect the contents of the actual exam paper: The selection

of topics, the number of questions or sub-questions, the perceived difficulty of individual

questions, and the distribution of weights are all aspects that may be different. When

preparing for the exam, you should cover the entire syllabus and not focus only on topics

or question types used in this practice paper. If anything, the exam paper can be expected

to be harder than this practice paper.

There are 12 questions. As in the exam, you should attempt them all. Of course

your answers must be readable. Any unreadable parts will be considered wrong. You will

find some questions easier than others; in the actual exam you should allocate your time

accordingly. Marks are indicated for each question, adding to a total of 70.

The actual exam paper will be printed single-sided, so you will have plenty of space

for rough work on the flip sides. Only what you write inside the allocated boxes will be

marked. Page 11 is overflow space, in case you need more writing space for some question.

Examiners’ use:

1 2 3 4 5 6 7 8 9 10 11 12

Page 2 of 11

Question 1 (4 marks)

A. Give the names of two stable sorting algorithms, together with their worst-case time

complexities. Write the names and complexities in the box:

B. Give the names of two unstable sorting algorithms, together with their worst-case time

complexities. Write the names and complexities in the box:

Question 2 (4 marks)

We are given an array A holding n integers, for some large n. The array is sorted, and the

values in A range from -2147483648 to 2147483647, evenly distributed. Give Θ expressions

for the following tasks:

A. Running the insertion sort algorithm on the array A:

B. Running the selection sort algorithm on the array A:

C. Performing binary search for integer k which is not in A:

D. Performing interpolation search for integer k not in A:

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Page 3 of 11

Question 3 (4 marks)

For the directed graph below, list the order in which the nine nodes are visited during a

depth-first (DFS) traversal, as well as the order in which they are visited during a breadth-

first (BFS) traversal. As always, assume that any ties are resolved by taking nodes in

alphabetical order. Write the answers in the boxes given.

a

b c d

e

fgh

x

DFS sequence:

BFS sequence:

Question 4 (4 marks)

Given the pattern A T G A and the text

T C A T C A T C C A T G C A C A A T G A C T T T

how many character comparisons will Horspool’s algorithm make before locating the pattern

in the text? Write the number in the box:

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Page 4 of 11

Question 5 (4 marks)

Assume the array A holds the keys 77, 64, 15, 43, 28, 91, 80, 32, 56 in index positions 1 to 9.

Show the heap that results after application of the linear-time bottom-up heap construction

algorithm. You may show the heap as a tree or as an array.

Question 6 (4 marks)

The functions A–D are defined recursively as follows (all divisions round down, to the closest

integer):

A(n) = 2 A(n/3) + 2, with A(1) = 1

B(n) = B(n/2) + n/2, with B(1) = 1

C(n) = 512 C(n/8) + 4n2, with C(1) = 4

D(n) = 4 D(n/2) + n2, with D(1) = 2

In the following table, for each of the four functions, tick the most precise correct statement

about the function’s rate of growth:

O(n) Θ(n) O(n logn) Θ(n2) O(n2 logn) Θ(n2

√

n) O(n3)

A

B

C

D

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Page 5 of 11

Question 7 (4 marks)

For each of A–D below, answer yes or no, and, in each case, briefly explain your reasoning

(just a justification of your answer, rather than detailed calculations). A yes/no answer that

is not justified will not attract marks, even if correct.

Question Answer/explanation

A.

Is

√

n ∈ Ω(n)?

B.

Is n2 logn ∈ O(nlogn)?

C.

Is Θ(log(n2 logn)) =

Θ(log(nlogn))?

D.

Is Θ(log(2n)) =

Θ(log(3n))?

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Page 6 of 11

Question 8 (6 marks)

A. The box below contains a weighted undirected graph with eight nodes. Give a minimum

spanning tree for the graph. You may do that either by outlining a minimum spanning tree

on the graph itself, or by drawing the tree in the empty space next to the graph.

a b

c

d

ef

g

h

5

74

46

9

73

5

8

4

8

6

B. Given a weighted graph G = 〈V,E〉, a subgraph 〈V,E ′〉 (that is, E ′ ⊆ E) which is a tree

with minimal weight is a maximum spanning tree for G.

We want a transformation of the graph G so that we can run Prim’s algorithm on the

transformed graph G′, and the algorithm will find a maximum spanning tree for G.

Describe such a (systematic) transformation from G to G′.

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Page 7 of 11

Question 9 (6 marks)

Consider the function F below. The function takes as input an integer array A, and the size

n of A. The array indices run from 1 to n. The division used is integer division, that it, it

rounds down to the closest smaller (or equal) integer value.

In the box, give a Θ expression for the function’s time complexity.

function F(A[·], n)

s← 0

m← n

while m > 0 do

for i← 1 to m do

s← s+ A[i]

m← m/2

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Page 8 of 11

Question 10 (10 marks)

Using pseudo-code, give an algorithm for deleting the smallest element of a binary search

tree (a BST). Assume a non-empty binary tree T has attributes T.left , T.right , and T.root

which denote T ’s left sub-tree, right sub-tree, and the key of T ’s root node, respectively.

You can use these tests if they seem useful: IsLeaf(T ) tests whether the binary tree T is

a leaf, and IsEmpty(T ) tests whether it is empty.

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Page 9 of 11

Question 11 (10 marks)

Consider an array A of n distinct integers (that is, all elements are different). It is known

that A was originally sorted in ascending order, but A was then right-rotated r places, where

0 < r < n. In other words, the last r elements were moved from the end of the array to the

beginning, with all other elements being pushed r positions to the right. For example, for

n = 7 and r = 3, the result may look like this:

[43, 46, 58, 12, 20, 29, 34]

For r = 5, the result, based on the same original array, would be

[29, 34, 43, 46, 58, 12, 20]

You know that the given A[0..n−1] has this particular form, that is, for some r, the sequence

A[r], . . . , A[n − 1], A[0], . . . A[r − 1] is in ascending order, but you do not know what r is.

Design an algorithm to find the largest integer in A. Full marks are given for an algorithm

that works in time O(logn); half marks are given for a solution that is correct, but less

efficient.

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Page 10 of 11

Question 12 (10 marks)

Two programmers face the following problem. Given an array containing n random integers

in random order, find the largest integer. The integers are placed in cells A[1] . . .A[n].

ProgrammerX has come up with the code shown below, on the left. (In the programming

language used, arrays are indexed from 0, but X ’s method does not use A[0].)

function X(A[·], n)

max ← A[1]

i← 2

while i ≤ n do

if A[i] > max then

max ← A[i]

i← i+ 1

return max

function Y(A[·], n)

i← n

while i > 0 do

A[0]← A[i]

i← i− 1

while A[0] > A[i] do

i← i− 1

return A[0]

Programmer Y has solved the same problem differently, as shown above on the right.

Compare the two solutions using three criteria: Correctness, time complexity class, and the

number of comparisons performed. Write your analysis in the box:

[COMP90038] [end of exam]

Page 11 of 11

Overflow space

Use this page if you ran out of writing space in some question. Make sure to leave a pointer

to this page from the relevant question.

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