1. Introduction to the Design and Analysis of Algorithms
ElSayed Badr
Benha University
September 20, 2016

2. Lecture 1: Introduction
Official course information ( Mark distribution: )
– Oral (assignments 10%)
Laboratory ( 15% )
– Midterm 10% each (in class, 60 minutes)
– Final 65% (3 hours)

3. • References: 1- Cormen, Leiserson, Rivest and Stein,
Lecture 1: Introduction
• References:
1- Cormen, Leiserson, Rivest and Stein,
Introduction to Algorithms, 2nd Edition, 2005.
2- C. H. Papadimitriou, and U. V. Vazirani,
Algorithms, 1th Edition,2006.
3- Anany Levitin,
Introduction to: the Design and analysis of Algorithms,
2nd Edition, 2007

4. • Algorithm I, Topics covered: – Introduction to algorithms
Lecture 1: Introduction
Course overview:
• Algorithm I, Topics covered:
– Introduction to algorithms
– Algorithms: sorting, matrices, graphs, sets
– Design: divide-and-conquer, dynamic programming, greedy
– Analysis: model assumptions, worst/average/best case,
asymptotic, reduction, complexity classes P and NP,
hard problems

5. Lecture 1: Introduction
Basic concepts
• Problem
• Instance
• Algorithm
• Issues for a given algorithm
– Correctness — often via Loop Invariants, proved by induction
– Analysis — measuring resource requirement
* Running time
* Space
– Optimality
• Algorithm design concepts
– Divide and conquer
– Data structure
– Dynamic programming
– Exhaustive enumeration
– Greedy
– Reduction (useful for showing problems are hard)
– Branch and bound
– Others (probably won’t cover)

6. Lecture 1: Introduction
Summations 1- 2- 3- 4- 5- 6-

7. Lecture 1: Introduction
Products
If n = 0 then
We can convert a formula with a product to a formula with a summation by using the identity :

9. 2- Prove by induction that

10. Basic Definitions What is an algorithm?
A- An algorithm is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time.
B- An algorithm is any well-defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values, as output. An algorithm is thus a sequence of computational steps that transform the input into the output.

11. What kinds of problems are solved by algorithms?
1- Human Genome Project (DNA Sequence)
2- Electronic Commerce ( cryptograph )
3- Networks
4- Optimization problems
5- Image Processing
6- Bioinformatics
7- Medical expert system .

12. Algorithms as Technology !
106 . 1
Algorithm A
Insertion sort ( c1 n2 )
Supercomputer (108 )ins/sec
Good programmer
Algorithm B
Merge sort ( c2 n log n )
PC (106 )ins/sec
Bad programmer
Computer B runs 20 times faster than computer A .
What happens, if the size of array = 107 numbers.
2.3 days mints

13. What is an algorithm?
An algorithm is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time.
Can be represented various forms
Unambiguity/clearness
Effectiveness
Finiteness/termination
Correctness

14. Historical Perspective
Euclid’s algorithm for finding the greatest common divisor
Muhammad ibn Musa al-Khwarizmi – 9th century mathematician
Euclid’s algorithm is good for introducing the notion of an algorithm because it
makes a clear separation from a program that implements the algorithm.
It is also one that is familiar to most students.
Al Khowarizmi (many spellings possible...) – “algorism” (originally) and then
later “algorithm” come from his name.

15. Euclid’s Algorithm
Problem: Find gcd(m,n), the greatest common divisor of two
nonnegative, not both zero integers m and n
Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ?
Euclid’s algorithm is based on repeated application of equality
gcd(m,n) = gcd(n, m mod n)
until the second number becomes 0, which makes the problem
trivial.
Example: gcd(60,24) = gcd(24,12) = gcd(12,0) = 12
Euclid’s algorithm is good for introducing the notion of an algorithm because it
makes a clear separation from a program that implements the algorithm.
It is also one that is familiar to most students.
Al Khowarizmi (many spellings possible...) – “algorism” (originally) and then
later “algorithm” come from his name.

16. Two descriptions of Euclid’s algorithm
Step 1 If n = 0, return m and stop; otherwise go to Step 2
Step 2 Divide m by n and assign the value of the remainder to r
Step 3 Assign the value of n to m and the value of r to n. Go to Step 1.
while n ≠ 0 do
r ← m mod n
m← n
n ← r
return m
How do we know that Euclid’s algorithm coms to stop ?

17. Other methods for computing gcd(m,n)
Consecutive integer checking algorithm
Step 1 Assign the value of min{m,n} to t
Step 2 Divide m by t. If the remainder is 0, go to Step 3; otherwise, go to Step 4
Step 3 Divide n by t. If the remainder is 0, return t and stop; otherwise, go to Step 4
Step 4 Decrease t by 1 and go to Step 2
Is this slower than Euclid’s algorithm? How much slower?
O(n), if n <= m , vs O(log n)

18. Other methods for gcd(m,n) [cont.]
Middle-school procedure
Step 1 Find the prime factorization of m
Step 2 Find the prime factorization of n
Step 3 Find all the common prime factors
Step 4 Compute the product of all the common prime
factors and return it as gcd(m,n)
Is this an algorithm?
How efficient is it?
Time complexity: O(sqrt(n))

19. Sieve of Eratosthenes Input: Integer n ≥ 2
Output: List of primes less than or equal to n
for p ← 2 to n do A[p] ← p
for p ← 2 to n do
if A[p]  0 //p hasn’t been previously eliminated from the list j ← p* p
while j ≤ n do
A[j] ← 0 //mark element as eliminated
j ← j + p
Example :
Time complexity: O(n)

20. Functions ordered by Growth Rate
Sieve of Eratosthenes
Functions ordered by Growth Rate
log n
log2n
sqrt(n) n
n log n n2 n3 2n