Algorithms

Lecture 1

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

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Analysis 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 algorithms

Why study algorithms andperformance?

• Algorithms help us to understand scalability.• Performance often draws the line between whatis feasible and what is impossible.• Algorithmic mathematics provides a languagefor talking about program behavior.• The lessons of program performance generalizeto other computing resources.• Speed is fun!

Running Time

• The running time depends on the input: analready sorted sequence is easier to sort.

• Parameterize the running time by the size ofthe input, since short sequences are easier to

sort than long ones.

• Generally, we seek upper bounds on therunning 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

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 – 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.)

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

Analysis of INSERTION-SORT(contd.)

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

62

5421

n

jj

n

jj tctcncnccnT

).1()1( 82

7

nctcn

jj

•The total running time is

Analysis of INSERTION-SORT(contd.)

The best case: The array is already sorted. (tj =1 for j=2,3, ...,n)

)1()1()1()1()( 85421 ncncncncncnT

).()( 854285421 ccccnccccc

Analysis of INSERTION-SORT(contd.)

•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 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

...,2,1,0N

-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 “sandwiched” between and for sufficienly large n.

)(ng ))(( ng

021

021

allfor )()()(c0

s.t.and,, constants positiveexist there:)())((

nnngcnfng

nccnfng

)(nf ))(( ng1c 2c

)(1 ngc)(2 ngc

Θ

O-notation

• For a given function , we denote by the set of functions

• We use O-notation to give an asymptotic upper bound on a function, to within a constant factor.

)(ng ))(( ngO

0

0

allfor )()(0

s.t.and constants positiveexist there:)())((

nnncgnf

ncnfngO

Ω-notation

• For a given function , we denote by the set of functions

• We use O-notation to give an asymptotic lower bound on a function, to within a constant factor.

)(ng ))(( ng

0

0

allfor )()(0

s.t.and constants positiveexist there:)())((

nnnfncg

ncnfng

Asymptotic notation

Graphic examples of and . ,, O

22

221 3

2

1ncnnnc

213

2

1c

nc

Example

Show that

We must find c1 and c2 such that

Dividing bothsides by n2 yields

For

)(32

1)( 22 nnnnf

)(32

17 22

0 nnnn

Theorem

• For any two functions and , we have

if and only if

)(ng

))(()( ngnf

)(nf

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

Olması

olduğunu gösterir

)2(5223 nnn

Example

)2(5223)( nnnnf

)2(5223 nOnn

o-notation

• We use 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

ω-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

)(

)(lim ng

nf

n

Insertion sort analysis

Merge Sort

Merging two sorted arrays

Merging two sorted arrays

Merging two sorted arrays

Merging two sorted arrays

Merging two sorted arrays

Merging two sorted arrays

Merging two sorted arrays

Merging two sorted arrays

Merging two sorted arrays

Merging two sorted arrays

Merging two sorted arrays

Merging two sorted arrays

Merging two sorted arrays

Analyzing merge sort

Recurrence for merge sort

Recursion tree

Recursion tree

Recursion tree

Recursion tree

Recursion tree

Recursion tree

Recursion tree

Recursion tree

Recursion tree

Recursion tree