Introduction to Algorithms Lecture 1. Introduction The methods of algorithm design form one of the...

Introduction to AlgorithmsIntroduction to Algorithms

Lecture 1

IntroductionIntroduction

• 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.

AlgorithmsAlgorithms

• 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 sortingThe problem of sorting

Insertion SortInsertion Sort

Example of Insertion SortExample of Insertion Sort

Analysis of algorithmsAnalysis of algorithms

The theoretical study of computer-programperformance and resource usage.

What’s more important than performance?• modularity

• correctness• maintainability• functionality• robustness

• user-friendliness• programmer time

• simplicity• extensibility

• reliability

Analysis of algorithmsAnalysis 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 TimeRunning 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 analysesKinds 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-Machine-IIndependent timendependent 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 timeMachine-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 exampleMachine-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 – 15 while i > 0 and A[i] > key6 do A[i+1] A[i]7 i i – 18 A[i +1] key

Analysis of INSERTION-SORT(contd.)Analysis of INSERTION-SORT(contd.)

1]1[8

)1(17

)1(][]1[6

][05

114

10]11[ sequence

sorted theinto][Insert 3

1][2

][21

timescost SORT(A)-INSERTION

8

27

26

25

4

2

1

nckeyiA

tcii

tciAiA

tckeyiAandi

ncji

njA

jA

ncjAkey

ncAlengthj

nj j

nj j

nj j

do

while

do

tofor


)1()1()1()(2

62

5421

n

jj

n

jj tctcncnccnT

).1()1( 82

7

nctcn

jj

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/( 87654212

765

2

)1(1

nnj

n

j

cbnannT 2)(

Growth of FunctionsGrowth 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 NotationAsymptotic 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

...,2,1,0N

O-notationO-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.and constants positiveexist there:)())((

nnncgnf

ncnfngO

))(()( ngOnf

)(ncg)(nf

ΩΩ--Omega Omega notationnotation


• 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.and constants positiveexist there:)())((

nnnfncg

ncnfng

))(()( ngnf

)(nf )(ncg

--Theta Theta notationnotation


• 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,, constants positiveexist there:)())((

nnngcnfng

nccnfng

)(nf ))(( ng1c 2c

)(1 ngc )(2 ngc

Θ

))(()( ngnf )()()( 21 ngcnfngc

Asymptotic notationAsymptotic notation

Graphic examples of and . ,, O

22

221 3

2

1ncnnnc

213

2

1c

nc

Example 1. Example 1.

Show that

We must find c1 and c2 such that

Dividing bothsides by n2 yields

For

)(32

1)( 22 nnnnf

)(32

1,7 22

0 nnnn

Theorem Theorem

• For any two functions and , we have

if and only if

)(ng

))(()( ngnf

)(nf

)).(()( and ))(()( ngnfngOnf

Because :

)2(5223 nnn

Example 2.Example 2.

)2(5223)( nnnnf

)2(5223 nOnn


610033,3forsince)(61003 2222 nnncnOnn


3when61003,1forsince)(61003

610033,3forsince)(610032332

2222

nnnncnOnn

nnncnOnn


cnncncnOnn

nnnncnOnn

nnncnOnn

when3,any forsince)(61003


610033,3forsince)(61003

22

2332

2222





610033,3forsince)(61003

2222

22

2332

2222

nnnncnnn

cnncncnOnn

nnnncnOnn

nnncnOnn






610033,3forsince)(61003

3232

2222

22

2332

2222

nnnncnnn

nnnncnnn

cnncncnOnn

nnnncnOnn

nnncnOnn


100when61003,any forsince)(61003





610033,3forsince)(61003

22

3232

2222

22

2332

2222

nnncncnnn

nnnncnnn

nnnncnnn

cnncncnOnn

nnnncnOnn

nnncnOnn


apply. and both since)(61003






610033,3forsince)(61003

22

22

3232

2222

22

2332

2222

Onnn

nnncncnnn

nnnncnnn

nnnncnnn

cnncncnOnn

nnnncnOnn

nnncnOnn


applies. only since)(61003







610033,3forsince)(61003

32

22

22

3232

2222

22

2332

2222

Onnn

Onnn

nnncncnnn

nnnncnnn

nnnncnnn

cnncncnOnn

nnnncnOnn

nnncnOnn










610033,3forsince)(61003

2

32

22

22

3232

2222

22

2332

2222

nnn

Onnn

Onnn

nnncncnnn

nnnncnnn

nnnncnnn

cnncncnOnn

nnnncnOnn

nnncnOnn

oo-notation-notation

• We use (small o) o-notation to denote an upper bound that is not asymptotically tight.

• We formally define as the set))(( ngo

0

0

allfor )()(0

s.t.0 constants aexist there

0constant positiveany for :)(

))((

nnncgnf

n

cnf

ngo

0)(

)(lim

ng

nf

n


)(2)(

But

)(2)(

isThat

ally tightasymptoticnot Is:)(2)(

ally tightAsymptotic:)(2)(

22

2

2

22

nonnf

nonnf

nOnnf

nOnnf

ωω-notation-notation

• We use ω-notation to denote an upper bound that is not asymptotically tight.

• We formally define as the set))(( ng

0

0

allfor )()(0

s.t.0 constants aexist there

0constant positiveany for :)(

))((

nnnfncg

n

cnf

ng

∞=

=relation The

∞→ g(n)

f(n)lim

tmplies thaω(g(n)) i f(n)

n

Example Example

)(2

)(

But

)(2

)(

22

2

nn

nf

nn

nf

Standard notations and common functionsStandard notations and common functions

• Floors and ceilings

11 xxxxx


• Modular arithmetic

For any integer a and positive integer n

nnaana /mod


• Polynomials:

Given a nonnegative integer d, a polynomial in n of degree d is

d

i

i

i nanp0

)(


• Exponentials:

!3!2

132 xx

xe x


• 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

aa

ana

baab

ba

c

cb

b

n

b

ccc

ab

log

loglog

loglog

loglog)(log

log


• Logarithms:

ba

ca

aa

a

b

ac

bb

bb

log

1log

log)/1(logloglog


• Series expansion:

For

For

5432

)1ln(5432 xxxx

xx

1x

1x

xxx

x

)1ln(

)1(


• Factorials

For the Stirling approximation:

ne

nnn

n1

12!

0n

)lg()!lg(

)2(!

)(!

nnn

n

nonn

n

Designing algorithms Designing algorithms

There are many ways to design algorithms:

• Insertion sort uses an incremental approach• Merge sort uses divide-and-conquer approach

Insertion sort analysisInsertion sort analysis

Merge SortMerge Sort

Merge SortMerge Sort

MERGE_SORT(A,p,r)

1if p<r2 then q← (p+r)/2 3 MERGE_SORT(A,p,q)4 MERGE_SORT(A,q+1, r)5 MERGE(A,p,q,r)

Merging two sorted arraysMerging two sorted arrays

Analyzing merge sortAnalyzing merge sort

Recurrence for merge sortRecurrence for merge sort

Recursion treeRecursion tree

Introduction to Algorithms Lecture 1. Introduction The methods of algorithm design form one of the...

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