Algorithm analysis insertion sort and asymptotic notations

Algorithms
Dr. AMIT KUMAR @JUET

Introduction
• The methods of algorithm design form one of the core
practical technologies of computer science.
• The main aim of this lecture is to familiarize the student
with the framework we shall use through the course
about the design and analysis of algorithms.
• We start with a discussion of the algorithms needed to
solve computational problems. The problem of sorting is
used as a running example.
• We introduce a pseudocode to show how we shall
specify the algorithms.

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.

The problem of sorting

Insertion Sort

Example of Insertion Sort

Analysis of algorithms
The theoretical study of computer-program
performance and resource usage.
What’s more important than performance?
• modularity
• correctness
• maintainability
• functionality
• robustness
• user-friendliness
• programmer time
• simplicity
• extensibility
• reliabilityDr. AMIT KUMAR @JUET

Analysis of algorithms
Why study algorithms and performance?
• Algorithms help us to understand scalability.
• Performance often draws the line between what is feasible
and what is impossible.
• Algorithmic mathematics provides a language for talking
about program behavior.
• The lessons of program performance generalize to other
computing resources.
• Speed is fun!

Running Time
• The running time depends on the input: an already
sorted sequence is easier to sort.
• Parameterize the running time by the size of the
input, since short sequences are easier to sort than
long ones.
• Generally, we seek upper bounds on the running
time, because everybody likes a guarantee.

Kinds of analyses
Worst-case: (usually)
• T(n) = maximum time of algorithm on any input of
size n.
Average-case: (sometimes)
• T(n) = expected time of algorithm over all inputs of
size n.
• Need assumption of statistical distribution of inputs.
Best-case:
• Cheat with a slow algorithm that works fast on some
input.

Machine-Independent time
The RAM Model
Machine independent algorithm design depends on a
hypothetical computer called Random Acces Machine (RAM).
Assumptions:
• Each simple operation such as +, -, if ...etc takes exactly
one time step.
• Loops and subroutines are not considered simple
operations.
• Each memory acces takes exactly one time step.

Machine-independent time
What is insertion sort’s worst-case time?
• It depends on the speed of our computer,
• relative speed (on the same machine),
• absolute speed (on different machines).
BIG IDEA:
• Ignore machine-dependent constants.
• Look at growth of
“Asymptotic Analysis”
nnT as)(

Machine-independent time: An example
A pseudocode for insertion sort ( INSERTION SORT ).
INSERTION-SORT(A)
1 for j  2 to length [A]
2 do key  A[ j]
3  Insert A[j] into the sortted sequence A[1,..., j-1].
4 i  j – 1
5 while i > 0 and A[i] > key
6 do A[i+1]  A[i]
7 i  i – 1
8 A[i +1]  key

Analysis of INSERTION-SORT(contd.)
1]1[8
)1(17
)1(][]1[6
][05
114
10]11[sequence
sortedtheinto][Insert3
1][2
][21
timescostSORT(A)-INSERTION
8
27
26
25
4
2
1















nckeyiA
tcii
tciAiA
tckeyiAandi
ncji
njA
jA
ncjAkey
ncAlengthj
n
j j
n
j j
n
j j
do
while
do
tofor

)1()1()1()(
2
6
2
5421  

n
j
j
n
j
j tctcncnccnT
).1()1( 8
2
7  

nctc
n
j
j
The total running time is

The best case: The array is already sorted.
(tj =1 for j=2,3, ...,n)
)1()1()1()1()( 85421  ncncncncncnT
).()( 854285421 ccccnccccc 

•The worst case: The array is reverse sorted
(tj =j for j=2,3, ...,n).
)12/)1(()1()( 521  nncncncnT
)1()2/)1(()2/)1(( 876  ncnncnnc
ncccccccnccc )2/2/2/()2/2/2/( 8765421
2
765 
2
)1(
1



nn
j
n
j
cbnannT  2
)(

Growth of Functions
Although we can sometimes determine the exact
running time of an algorithm, the extra precision is not
usually worth the effort of computing it.
For large inputs, the multiplicative constants and
lower order terms of an exact running time are
dominated by the effects of the input size itself.

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

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

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

-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 

Asymptotic notation
Graphic examples of and . ,, O

2
2
22
1 3
2
1
ncnnnc 
21
3
2
1
c
n
c 
Example 1.
Show that
We must find c1 and c2 such that
Dividing bothsides by n2 yields
For
)(3
2
1
)( 22
nnnnf 
)(3
2
1
,7 22
0 nnnn 

Theorem
• For any two functions and , we have
if and only if
)(ng
))(()( ngnf 
)(nf
)).(()(and))(()( ngnfngOnf 

Because :
)2(5223 nnn 
Example 2.
)2(5223)( nnnnf 
)2(5223 nOnn 

Example 3.
610033,3forsince)(61003 2222
 nnncnOnn

Example 3.
3when61003,1forsince)(61003
610033,3forsince)(61003
2332
2222


nnnncnOnn
nnncnOnn

Example 3.
cnncncnOnn
nnnncnOnn
nnncnOnn



when3,anyforsince)(61003
610033,3forsince)(61003
22
2332
2222

Example 3.
610033,3forsince)(61003
2222
22
2332
2222




nnnncnnn
cnncncnOnn
nnnncnOnn
nnncnOnn

Example 3.
610033,3forsince)(61003
3232
2222
22
2332
2222





nnnncnnn
nnnncnnn
cnncncnOnn
nnnncnOnn
nnncnOnn

Example 3.
100when61003,anyforsince)(61003
610033,3forsince)(61003
22
3232
2222
22
2332
2222






nnncncnnn
nnnncnnn
nnnncnnn
cnncncnOnn
nnnncnOnn
nnncnOnn

Example 3.
apply.andbothsince)(61003
610033,3forsince)(61003
22
22
3232
2222
22
2332
2222







Onnn
nnncncnnn
nnnncnnn
nnnncnnn
cnncncnOnn
nnnncnOnn
nnncnOnn

Example 3.
applies.onlysince)(61003
610033,3forsince)(61003
32
22
22
3232
2222
22
2332
2222
Onnn
Onnn
nnncncnnn
nnnncnnn
nnnncnnn
cnncncnOnn
nnnncnOnn
nnncnOnn









Example 3.
610033,3forsince)(61003
2
32
22
22
3232
2222
22
2332
2222









nnn
Onnn
Onnn
nnncncnnn
nnnncnnn
nnnncnnn
cnncncnOnn
nnnncnOnn
nnncnOnn

Standard notations and common functions
• Floors and ceilings
    11  xxxxx

• Logarithms:
)lg(lglglg
)(loglog
logln
loglg 2
nn
nn
nn
nn
kk
e





• Logarithms:
For all real a>0, b>0, c>0, and n
b
a
a
ana
baab
ba
c
c
b
b
n
b
ccc
ab
log
log
log
loglog
loglog)(log
log





• Logarithms:
b
a
ca
aa
a
b
ac
bb
bb
log
1
log
log)/1(log
loglog




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

Algorithm analysis insertion sort and asymptotic notations

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