Analysis of Algorithms ASSIGNMENT 1

Analysis of Algorithms
(Computer Science 3340b)
ASSIGNMENT 1

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.
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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. Implement 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 complexity 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 algorithm. 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.
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You should use unix command ”script” to capture the screen of the execution of your program.
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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