This document introduces algorithms and asymptotic notation used to analyze algorithm performance. It defines common asymptotic notations like O, Ω, and Θ notation, and gives examples of standard functions like logarithms and factorials. It notes that accurately measuring algorithm performance requires specifying the programming language, working program, computer used, and compiler options.

1. INTRODUCTION TO ALGORITHM
Prepared by
R. Gomthijayam
Asst. Prof. of CA

2. Algorithms
• The word algorithm comes from the name of a Persian
mathematician Abu Ja’far Mohammed ibn-i Musa al
Khowarizmi.
• In computer science, this word refers to a special method
useable by a computer for solution of a problem. The
statement of the problem specifies in general terms the
desired input/output relationship.
• For example, sorting a given sequence of numbers into
nondecreasing order provides fertile ground for introducing
many standard design techniques and analysis tools.

3. The problem of sorting

4. Asymptotic Notation
The notation we use to describe the asymptotic running time
of an algorithm are defined in terms of functions whose
domains are the set of natural numbers
 ...,2,1,0N

5. O-notation
• For a given function , we denote by the set of
functions
• We use O-notation to give an asymptotic upper bound of a
function, to within a constant factor.
• means that there existes some constant c s.t.
is always for large enough n.
)(ng ))(( ngO








0
0
allfor)()(0
s.t.andconstantspositiveexistthere:)(
))((
nnncgnf
ncnf
ngO
))(()( ngOnf 
)(ncg)(nf

6. Ω-Omega notation
• For a given function , we denote by the set of
functions
• We use Ω-notation to give an asymptotic lower bound on a
function, to within a constant factor.
• means that there exists some constant c s.t.
is always for large enough n.
)(ng ))(( ng








0
0
allfor)()(0
s.t.andconstantspositiveexistthere:)(
))((
nnnfncg
ncnf
ng
))(()( ngnf 
)(nf )(ncg

7. -Theta notation
• For a given function , we denote by the set of
functions
• A function belongs to the set if there exist
positive constants and such that it can be “sand- wiched”
between and or sufficienly large n.
• means that there exists some constant c1 and c2
s.t. for large enough n.
)(ng ))(( ng








021
021
allfor)()()(c0
s.t.and,,constantspositiveexistthere:)(
))((
nnngcnfng
nccnf
ng
)(nf ))(( ng
1c 2c
)(1 ngc )(2 ngc
Θ
))(()( ngnf 
)()()( 21 ngcnfngc 

8. Standard notations and common functions
• Logarithms:
)lg(lglglg
)(loglog
logln
loglg 2
nn
nn
nn
nn
kk
e





9. Standard notations and common functions
• Factorials
For the Stirling approximation:



















ne
n
nn
n
1
12!
0n
)lg()!lg(
)2(!
)(!
nnn
n
non
n
n




10. Performance Measurement Needs
 programming language
 working program
 computer
 compiler and options to use
javac -o