THE UNIVERSITY OF WESTERN ONTARIO

DEPARTMENT OF COMPUTER SCIENCE

LONDON CANADA

Analysis of Algorithms

(Computer Science 3340b)

ASSIGNMENT 1

Due date: Tuesday, Feburary 2, 2021, 11:55 PM

1. A complete binary tree is defined inductively as follows. A complete binary tree of

height 0 consists of 1 node which is the root. A complete binary tree of height h+ 1 consists

of two complete binary trees of height h whose roots are connected to a new root. Let T be

a complete binary tree of height h. Prove that the number of leaves of the tree is 2h and the

size of the tree (number of nodes in T ) is 2h+1 − 1.

2. Question 2-1 (pp. 39) a., b., and c. in the text book.

3. Question 2-4 (pp. 41) in the text book.

4. Read the text book for the definition of o and ω and answer question 3-2 (pp. 61) in the

text book.

5. Question 4-2 (pp. 107) in the text book.

6. Suppose that the running time of a recursive program is represented by the following

recurrence relation:

T (2) ≤ c

T (n) ≤ 2T (n/2) + cn log2(n)

Determine the time complexity of the program using recurrence tree method and then prove

your answer.

1

7. Consider the Fibonacci numbers:

F (n) = F (n− 1) + F (n− 2), n > 1; F (1) = 1; F (0) = 0.

a). Write a recursive function to compute F (n) using the above definition directly. Imple-

ment your solution and print F (i ∗ 5), where 0 ≤ i ≤ 10, as output.

b). Write a recursive function/procedure to compute F (n) with time complexity O(n) (more

precisely, the time complexity should be O(nA(n)) when n is large, where A(n) is the com-

plexity of adding F (n−1) and F (n−2)). Implement your solution and print F (i∗20), where

0 ≤ i ≤ 25, as output. This program must be able to compute F (n) precisely for n ≤ 500.

Hint 1:

Let G(n) =

(

F (n)

F (n− 1)

)

: G(n) =

(

1 1

1 0

)

G(n−1), n > 1; G(1) =

(

1

0

)

; G(0) =

(

0

0

)

Design a recursive algorithm for G(n), that means the algorithm will return both F (n) and

F (n− 1) with input parameter n.

Hint 2: Can you use a primitive type to store F (500)?

c) (optional for bonus credit)

Write a recursive function/procedure to compute F (n) with time complexity O(log(n)) (more

precisely, the time complexity should be O(log(n)A(n)) when n is large, where A(n) is the

complexity of adding or multiplying F (i) and F (j)).

Hint : G(n) =

(

1 1

1 0

)

G(n− 1) =

(

1 1

1 0

)n−1

G(1), n > 1

For programs in a), b), and c) of this question, you are NOT allowed to use Python. For

C++ and Java, you can only use primitive types such as char, int, long and long long. You

are not allowed to use large integer, such as BigInteger in Java, from the language library.

You have to write your own large integer data type or object, if needed.

d) Use the Unix time facility (bash time command) to track the time needed for each algo-

rithm. Compare the results and state your conclusion in two or three sentences.

e) Can you use your program in a) to compute F (50) if int type of 4 bytes is used? Briefly

explain your answer. Explain why your program in b) computes F (500) precisely?

Algorithms and answers for 6 b) to 6 e) should be in asn1 solutions.pdf

For this question, for C++, when compiling option O2 should be considered and a makefile

should be written such that the command ”make clean” will remove all the ”*.o” files, the

command ”make asn1 a” will generate an executable file ”asn1 a” for 6 a) that can be run

by typing ”asn1 a”; the command ”make asn1 b” will generate an executable file ”asn1 b”

for 6 b) that can be run by typing ”asn1 b”; and the command ”make asn1 c” will generate

an executable file ”asn1 c” for 6 c) that can be run by typing ”asn1 c”. If you are using Java,

you may not need the makefile. In that case, you should have shell script files, ”asn1 a”,

”asn1 b”, and ”asn1 c” such that by typing, ”asn1 a”, ”asn1 b”, and ”asn1 c”, your java

programs will run.

2

You should use unix command ”script” to capture the screen of the execution of your pro-

gram.

Your programs have to be able to run on compute.gaul.csd.uwo.ca as our TAs will be

marking your programs there.

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