Objectives Experiment #1 Introduction to Algorithms

Cairo University
Faculty of Engineering
3 rd Year Computer Engineering Department
CMPN302 Winter 2012
Objectives
Experiment #1
Introduction to Algorithms
After this lab, the student should be able to
Understand that any problem may have different algorithms to solve it
Understand the Pseudo Code of an algorithm
Comment about different algorithms' performance
Problem Definitions:
Greatest Common Divisor:
The Greatest Common Divisor (gcd(m,n)) of any 2 non negative, not both zeros
integers is defined as the largest integer that divides both m and n evenly with a
reminder of zero.
Examples of gcd problem:
1. gcd(60,24) = 12
2. gcd(60,0) = 0
There are different algorithms used to solve gcd problem. Two of them are:
Requirements
1- Euclid's algorithm
o Is based on repeated application of the equality
gcd(m,n) = gcd(n, m mod n) until the second number become 0
2- Consecutive integer checking algorithm
o Is based on the fact that the greatest possible common divisor is the
smallest of m and n
1- Implement the two algorithms described above.
2- Run both algorithms on the following data
a. 1050,2100
b. 30,45
c. 13,11
d. 99,60
e. 300,200
f. 1000001,99999
Winter 2012 Experiment #1 1/5

3- For the previous data fill the following table.
input Euclid Brute Force
# of iterations # of iterations
1050,2100
30,45
13,11
99,60
300,200
1000001,99999
Winter 2012 Experiment #1 2/5

Prime Numbers:
A prime number (or a prime) is a natural number greater than 1 that has no positive divisors
other than 1 and itself. One of the most common problems is finding the Prime numbers in a
certain range, this can be done using:
1. Brute Force algorithm:
By testing all numbers one by one.
2. Sieve of Eratosthenes
Algorithm:
To find all the prime numbers less than or equal to a given integer n by Sieve method:
Requirements
1. Create a list of consecutive integers from 2 to n: (2, 3, 4, ..., n).
2. Initially, let p equal 2, the first prime number.
3. Starting from p, count up in increments of p and mark each of these numbers greater
than p itself in the list. These numbers will be 2p, 3p, 4p, etc.; note that some of them
may have already been marked.
4. Find the first number greater than p in the list that is not marked.If there was no such
number, stop. Otherwise, let p now equal this number (which is the next prime), and
repeat from step 3.
When the algorithm terminates, all the numbers in the list that are not marked are
prime.
1- Implement the two algorithms described above.
2- Run both algorithms on the following data
a. 100
b. 1000
c. 10000
d. 100000
e. 200000
3- For the previous data fill the following table.
input Brute Force Sieve
Time in MS Time in MS
100
1000
10000
100000
200000
Winter 2012 Experiment #1 3/5

AnotherFibonacci
AnotherFibonacci sequence begins by defining the first three elements A[0], A[1]
and A[2]. The remaining elements are calculated using the following recurrence:
A[i] = A[i-1] + A[i-2] + A[i-3]
You are given a int[] A which contains exactly one element that is equal to -1, you must
replace this element with a positive number in a way that the sequence becomes a
AnotherFibonacci sequence. Return this number.
If no such positive number exists, return -1.
It’s required that you implement and test this following class by the example data, and write the
complexity of your algorithm if you can (in terms of time, iterations, or number of basic operations)
Definition
Class: AnotherFibonacci
Method:
complete
Parameters: array of integers
Returns:
int
Method signature (in C#): int complete(int[] A)
Notes
- The constraints for the elements of the input int[] A do not necessarily apply for the
replacement to the missing element.
Constraints
- A will contain between 4 and 20 elements, inclusive.
- Each element of A will be -1 or between 1 and 1000000, inclusive.
- Exactly one element of A will be -1.
Examples
0)
input = {1,2,3,-1}
Returns: 6
Winter 2012 Experiment #1 4/5

1)
input = {10, 20, 30, 60, -1 , 200}
Returns: 110
2)
input = {1, 2, 3, 5, -1}
Returns: -1
No replacement can make this sequence AnotherFibonacci as 5 is not equal to 1+2+3.
3)
{1, 1, -1, 2, 3}
Returns: -1
The missing element must be 0 for this sequence to be AnotherFibonacci. Since this is not a positive
integer, return -1.
4)
{-1, 7, 8, 1000000}
Returns: 999985
Winter 2012 Experiment #1 5/5