Algorithms and Pseudo code

8/9/2019 Algorithms and Pseudo code

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An Assignment on

Algorithms and Pseudo code

Submitted to Submitted by

Dr. Budhayash Gautam Anand

Kumar Maurya

13MTBI!""#


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$ontents

1.Introduction2.Algorithm design techniques3.Pseudo codes%. Pseudo code conventions with an

example&. References


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Algorithms and Pseudo code

Introduction

The word Algorithm! comes from the Persian author A"dullah

#afar $uhammad i"n $usa Al%&howari'mi in ninth centur() who

has given the de*nition of algorithm as follows+ An Algorithm is

a set of rules for carr(ing out calculation either "( hand or on a

machine. An Algorithm is a well de*ned computational

procedure that ta&es input and produces output. An Algorithm

is a *nite sequence of instructions or steps ,i.e. inputs- to

achieve some particular output.

An( Algorithm must satisf( the following criteria ,or Properties-

• Input+ It generall( requires *nite no. of inputs

• utput+ It must produce at least one output

• /niqueness+ 0ach instruction should "e clear and

unam"iguous

• initeness+ It must terminate oer a *nite no. of steps.

Algorithm design techni'ues

There are "asicall( fundamental techniques which are used to

design an algorithm e4cientl(+

1. 5ivide%and%6onquer2. 7reed( method3. 5(namic Programming8. 9ac&trac&ing. 9ranch%and%9ound

1. Di(ide ) con'uer techni'ue is a top%down approach to

solve a pro"lem. The algorithm which follows divide and

conquer technique involves 3 steps+

• 5ivide the original pro"lem into a set of su" pro"lems.


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• 6onquer ,or :olve- ever( su"%pro"lem individuall()

recursive.

• 6om"ine the solutions of these su" pro"lems to get the

solution of original pro"lem.

*. Greedy techni'ue is used to solve an optimi'ation

pro"lem. An ptimi'ation pro"lem is one in which) we are given

a set of input values) which are required to "e either maximi'ed

or minimi'ed ,&nown as o";ective function- w. r. t. some

constraints or conditions. 7reed( algorithm alwa(s ma&es the

choice ,greed( criteria- that loo&s "est at the moment) to

optimi'e a given o";ective function. That is) it ma&es a locall(

optimal choice in the hope that this choice will lead to a overall

glo"all( optimal solution. The greed( algorithm does not alwa(s

guaranteed the optimal solution "ut it generall( produces

solutions that are ver( close in value to the optimal.

3. Dynamic +rogramming technique is similar to divide and

conquer approach. 9oth solve a pro"lem "( "rea&ing it down

into a several su" pro"lems that can "e solved recursivel(. Thedierence "etween the two is that in d(namic programming

approach) the results o"tained from solving smaller su"

pro"lems are reused ,"( maintaining a ta"le of results- in the

calculation of larger su" pro"lems. Thus d(namic programming

is a 9ottom%up approach that "egins "( solving the smaller su"%

pro"lems) saving these partial results) and then reusing them

to solve larger su"%pro"lems until the solution to the original

pro"lem is o"tained. Reusing the results of su"%pro"lems ,"(maintaining a ta"le of results- is the ma;or advantage of

d(namic programming "ecause it avoids the recomputations

,computing results twice or more- of the same pro"lem.

Thus 5(namic programming approach ta&es much less time

than na


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%. Bac,trac,ing % The term "ac&trac&! was coined "(

American mathematician 5.=. >ehmer in the 1?@s.

9ac&trac&ing can "e applied onl( for pro"lems which admit theconcept of a partial candidate solution! and relativel( quic&

test of whether it can possi"l( "e completed to a valid solution.

9ac&trac& algorithms tr( each possi"ilit( until the( *nd the

right one. It is a depth%*rst%search of the set of possi"le

solutions. 5uring the search) if an alternative doesnt wor&) the

search "ac&trac&s to the choice point) the place which

presented dierent alternatives) and tries the next alternative.

Bhen the alternatives are exhausted) the search returns to theprevious choice point and tr( the next alternative there. If there

are no more choice points) the search fails.

&. Branch-and-Bound B)B/ is a rather general optimi'ation

technique that applies where the greed( method and d(namic

programming fail. 9C9 design strateg( is ver( similar to

"ac&trac&ing in that a state%spacetree is used to solve apro"lem. 9ranch and "ound is a s(stematic method for solving

optimi'ation pro"lems. =owever) it is much slower. Indeed) it

often leads to exponential time complexities in the worst case.

n the other hand) if applied carefull() it can lead to algorithms

that run reasona"l( fast on average. The general idea of 9C9 is

a 9:%li&e search for the optimal solution) "ut not all nodes get

expanded ,i.e.) their children generated-. Rather) a carefull(

selected criterion determines which node to expand and when)and another criterion tells the algorithm when an optimal

solution has "een found. 9ranch and 9ound ,9C9- is the most

widel( used tool for solving large scale DP%hard com"inatorial

optimi'ation pro"lems.

The following ta"le summari'es these techniques with some

common pro"lems that follows these techniques with their


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running time. 0ach technique has dierent running time ,Etime

complexit(-.


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Design strategy Problem that 0ollos5ivide C 6onqer • 9inar( :earch

•$ultiplication of two n%"its num"ers

• Fuic& :ort

• =eap :ort

• $erge :ort

7reed( $ethod • Gnapsac& ,fractional-

Pro"lem $inim cost :panning

tree Grus&alHs algorithm

• PrimHs algorithm

• :ingle source shortest

path pro"lem 5i;i&straHs algorithm

5(namic Programming • All pair shortest path

lo(ed algorithm

•

6hain matrixmultiplication

• >ongest common

su"sequence ,>6:-

• @J1 Gnapsac& Pro"lem

• Travelling salesman

pro"lem ,T:P-9ac&trac&ing • D% queenHs pro"lem

• :um% of su"set

9ranch C 9ound • Assignment pro"lem

• Travelling salesman

pro"lem ,T:P-

Pseudo codes


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Pseudo%code ,derived from pseudo%code- is a compact and

informal high level description of a computer programming

algorithm that uses the structural conventions of some

programming language. /nli&e actual computer language such

as 6)6KK or #ALA) Pseudo%code t(picall( omits details that are

not essential for understanding the algorithm) such as functions

,or su"routines-) varia"le declaration) semicolons) special

words and so on. An( version of pseudo%code is accepta"le as

long as its instructions are unam"iguous and is resem"les in

form. Pseudo%code is independent of an( programming

language. Pseudo %code cannot "e compiled nor executed) and

not following an( s(ntax rules.

low charts can "e thought of as a graphical alternative to

pseudo%code. A Mowchart is a schematic representation of an

algorithm) or the step%"(%step solution of a pro"lem) using

some geometric *gures ,called Mowchart s(m"ols- connected

"( Mow%lines for the purpose of designing or documenting a

program.

The purpose of using pseudo%code is that it ma( "e easier to

read than conventional programming languages that ena"les

,or helps- the programmers to concentrate on the algorithms

without worr(ing a"out all the s(ntactic details of a particular

programming language. In fact) one can write a pseudocode for

an( given pro"lem without even &nowing what programming

language one will use for the *nal implementation.

Pseudo code $on(entions

1. 6omments "egin with JJ and continue until the end of the

line.2. 9loc&s are indicated with matching "races Nand O.3. An identi*er "egins with a letter.8. Assignment of values to varia"les is done using the

assignment statement varia"leQ expressionQS

. There are 2 "oolean values true and false and relationaloperators ) )Q)Q etc are used.


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. 0lements of multidimensional arra(s are accessed using

U and V.W. The looping statements availa"le are for) while and repeat

until.

X. A conditional statement has the following forms.If conditionQ then statementQif conditionQ then statement 1Q else statement 2Q

?. Input and output are done using read and write

statements.1@. Algorithm is the one t(pe of procedure) which

consists of a heading and a "od(.Algorithm Dame ,parameter listQ-

2am+le4 To devise an algorithm that sorts a collection of

nQ1 elements of ar"itrar( t(pe.

/sing the natural language li&e 0nglish rom those elements

that are currentl( unsorted) *nd the smallest and place it next

in the sorted list.

Pseudo code representation %

Algorithm :election:ort,a)n-

JJ:ort the arra( aU1 + nV into non%decreasing order.

N

for i 1 to n do

N

; iS

for & i K1 to n do

if ,aU&V aU;V- then ; &S


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t aUiVS aUiV aU;VS aU;V tS

O

O


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5e0erences

1. An Introduction to 9ioinformatics Algorithms % Deil 6.

#ones) Pavel A. Pev'ner2. Introduction to Algorithms) :econd 0dition % Thomas =.

6ormen) 6harles 0. >eiserson) Ronald >. Rivest) 6liord

:tein

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Algorithms and Pseudo code