## Description

2 Your Tasks

The questions on this assignment are about timing analysis and ADT specification. There is no programming on this assignment.

Question 1 ():

For each of the following functions, give the tightest upper bound chosen from among the usual

simple functions listed in Section 3.5 of the course readings. Answers should be expressed in big-O

notation.

(a) (1 point) f1(n) = n log2

n + n

4

log280 n + 2

n

42

(b) (1 point) f3(n) = 0.4n

4 + n

2

log n

2 + log2

(2

n

)

(c) (1 point) f2(n) = 4n

0.7 + 29n log2

n + 280

Question 2 ():

Suppose the exact time required for an algorithm A in both the best and worst cases is given by the

function

TA(n) = 1

280n

2 + 42 log n + 12n

3 + 280√

n

(a) (2 points) For each of the following statements, indicate whether the statement istrue or false.

1. Algorithm A is O(log n)

2. Algoirthm A is O(n

2

)

3. Algoirthm A is O(n

3

)

4. Algoirthm A is O(2

n

)

(b) (1 point) What is the time complexity of algorithm A in big-Θ notation.

Question 3 ():

If possible, simplify the following expressions. Hint: See slide 11 of topic 4 of the lecture slides!

(a) (1 point) O(n

2

) + O(log n) + O(n log n)

(b) (1 point) O(2

n

) · O(n

2

)

(c) (1 point) 42O(n log n) + 18O(n

3

)

(d) (1 point) O(n

2

log2

n

2

) + O(m) (yes, that’s an ‘m’, not a typo; note that m is independent of n)

Question 4 ():

Consider the following Java code fragment:

// Print out all ordered pairs of numbers between 1 and n

for ( i = 1; i <= n; i ++) {

for ( j = 1; j <= n; j ++) {

System . out . println ( i + ” ,␣ ” + j ) ;

}

}

(a) (3 points) Use the statement counting approach to determine the exact number of statements that

are executed when we run this code fragment as a function of n. Show all of your calculations.

(b) (1 point) Express the answer you obtained in part (a) in big-Θ notation.

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Question 5 ():

Consider the following pseudocode:

Algorithm roundRobinTournament ( a)

This algorithm generates the list of matches that must be

played in a round – robin pirate – dueling tournament ( a tournament where

each pirate duels each other pirate exactly once ).

a is an array of strings containing names of pirates in the tournament

n = a . length

for i = 0 to n -1

for j = i +1 to n -1

print a [ i ] + ” ␣ duels ␣ ” + a [ j ] + ” ,␣ Yarrr !”

(a) (6 points) Use the statement counting approach to determine the exact number of statements that

are executed by this pseudocode as a function of n. Show all of your calculations..

(b) (1 point) Express the answer you obtained in part a) in big-Θ notation.

Question 6 (3 points):

Using the active operation approach, determine the time complexity of the pseudocode in question 5.

Show all your work and express your final answer in Big-Θ notation.

Question 7 (6 points):

Consider the following pseudocode.

Algorithm multiSearch ( data , target ):

data : an list of arrays of integers ; in each array the

integers are sorted in ascending order ; the list

’ data ’ has a cursor .

target : an integer

// Iterate over the arrays in the list ’ data ’ using

// its cursor :

data . goFirst ()

found = false

while ( ! data . after () and ! found ) {

// search for integer ’ target ’ in A

found = binarySearch ( data . currentItem () , target )

data . goForth ()

Using the active operation approach to timing analysis determine the time complexity of this pseudocode in the worst case. Assume that the list of arrays contains n arrays and that each array has

exactly m items in it. Show all your work and express your final answer in Big-O notation.

Question 8 (17 points):

A priority queue is a queue where a numeric priority is associated with each element. Access to

elements that have been inserted into the queue is limited to inspection and removal of the elements

with smallest and largest priority only. A priority queue may have multiple items that are of equal

priority.

Give the ADT specification for a bounded priority queue using the specification method described

in Topic 7 of the lecture notes. By “bounded”, it is meant that the priority queue has a maximum

capacity specified when it is created, and it can never contain more than that number of items.

Your specification must specify the following operations:

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newPriorityQueue: make a new queue

insert: inserts an element with a certain priority

isEmpty: test if the queue is empty

isFull: test if the queue is full

maxItem: obtain the item in the queue with the highest priority

minItem: obtain the item in the queue with the lowest priority

deleteMax: remove from the queue the item with the highest priority

deleteAllMax: remove from the queue all items that are tied for the highest priority

deleteMin: remove from the queue the item with the lowest priority

frequency: obtain the number of times a certain item occurs in the queue (with any priority)

3 Files Provided

None.

4 What to Hand In

You must submit the following files:

assignment2.doc/docx/rtf/pdf/txt – your answers to questions 1 to 8. Digital images of handwritten pages

are also acceptable, provided that they are clearly legible.

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