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PDM MCA 2

What is Data Structure

In computer science, a data structure is a particular

way of storing and organizing data in a computer so

that it can e used efficiently!

Data may e organized in many different ways, the

logical or mathematical model of aparticular

organization of data in memory or ondis" is called

Data Structure!Algorithms are used for manipulationof data!

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Data Structure #perations

$he data appearing in our data structure is processed y means of certainoperations! $he following four operations play a ma%or role&

$rans'ersing

Accessing each record e(actly once so that certain items in the record may

eprocessed!)$his accessing or processing is sometimes called *'isiting+ the

records! Searching

-inding the location of the record with a gi'en "ey 'alue, or finding the

locations of all records, which satisfy one or more conditions!

Inserting.

Adding new records to the structure!

Deleting

/emo'ing a record from the structure!

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$ypes of Data structure

$here are two types of data structure

0! 1inear Data Structure

A data structure is said to e linear if its elements form a se2uence, or in other

words a linear list!

I! Array

II! Stac"

III! 3ueue

I4! 1in"ed 1ist

5! 6on7 1inear Data Structure

A non7linear structure is mainly used to represent data containing a hierarchicalrelationship etween elements!

I! $ree

II! 8raph

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A18#/I$9M

Algorithm is a step7y7step procedure for calculations or -inite step

y step to sol'e the finite prolem with finite amount of time! Some

other definition of algorithm are

An algorithm is a set of rules for carrying out calculation either y

hand or on a machine!

An algorithm is a finite step7y7step procedure to achie'e a re2uired

result!

An algorithm is a se2uence of computational steps that transform

the input into the output! An algorithm is a se2uence of operations performed on data that

ha'e to e organized in data structures!

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Algorithm Analysis

$he comple(ity of an algorithm is a functiong)n that gi'es the

upper ound of the numer of operation )or running time

performed y an algorithm when the input size is n!

$here are two interpretations of upper ound!

Worst7case Comple(ity

$he running time for any gi'en size input will e lower than the

upper ound e(cept possily for some 'alues of the input where

the ma(imum is reached!

A'erage7case Comple(ity$he running time for any gi'en size input will e the a'erage

numer of operations o'er all prolem instances for a gi'en size!

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Algorithm Analysis

:ecause, it is 2uite difficult to estimate the statistical eha'ior

of the input, most of the time we content oursel'es to a worst

case eha'ior! Most of the time, the comple(ity of g)n is

appro(imated y its family o)f)n where f)n is one of the

following functions! n )linear comple(ity, log n )logarithmic

comple(ity, na where a ; 5 )polynomial comple(ity, an

)e(ponential comple(ity!

#ptimality

#nce the comple(ity of an algorithm has een estimated, the

2uestion arises whether this algorithm is optimal! An algorithm

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Algorithm Analysis

for a gi'en prolem is optimal if its comple(ity reaches the

lower ound o'er all the algorithms sol'ing this prolem! -or

e(ample, any algorithm sol'ing )n5 comple(ity! If one finds an

#)n5 algorithm that sol'e this prolem, it will e optimal and of

comple(ity ?)n5!

/eductionAnother techni2ue for estimating the comple(ity of a prolem is

the transformation of prolems, also called prolem reduction!

As an e(ample, suppose we "now a lower ound for

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Algorithm Analysis

PDM MCA 9

a prolem A, and that we would li"e to estimate a lower ound

for a prolem :! If we can transform A into : y a

transformation step whose cost is less than that for sol'ing A,

then : has the same ound as A!

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Comple(ity

measurement

space comple(ity is also important& $his is essentially the

numer of memory cells which an algorithm needs

$ime comple(ity of an algorithm 2uantifies the amount of time

ta"en y an algorithm to run

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:ig # 6otation

# notation appro(imates the cost function of an algorithm

@ $he appro(imation is usually good enough, especially whenconsidering the efficiency of algorithm as ngets 'ery large

@ Allows us to estimate rate of function growth

Instead of computing the entire cost function we only need to countthe numer of times that an algorithm e(ecutes its barometerinstruction(s)

@ $he instruction that is e(ecuted the most numer of times in analgorithm )the highest order term

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:ig # 6otation

$he cost function of an algorithm A, tA(n), can e appro(imated yanother, simpler, functiong(n)which is also a function with only 0'ariale, the data size n.

$he functiong(n)is selected such that it represents an upper bound

on the efficiency of the algorithm A )i!e! an upper ound on the'alue of tA(n)!

$his is e(pressed using the ig7# notation& #(g(n))!

-or e(ample, if we consider the time efficiency of algorithm A then

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$op down approach

A top-down approach is essentially the rea"ing down of a

system to gain insight into its compositional su7systems! In a

top7down approach an o'er'iew of the system is formulated,

specifying ut not detailing any first7le'el susystems! ach

susystem is then refined in yet greater detail, sometimes inmany additional susystem le'els, until the entire specification is

reduced to ase elements! A top7down model is often specified

with the assistance of +lac" o(es+, these ma"e it easier to

manipulate! 9owe'er, lac" o(es may fail to elucidate

elementary mechanisms or e detailed enough to realistically

'alidate the model! $op down approach starts with the ig

picture! It rea"s down from there into smaller segments!

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:ottom up approach

A bottom-up approach is the piecing together of systems to

gi'e rise to grander systems, thus ma"ing the original systems

su7systems of the emergent system! :ottom7up processing is

a type of information processing ased on incoming data from

the en'ironment to form a perception! Information enters theeyes in one direction )input, and is then turned into an image

y the rain that can e interpreted and recognized as a

perception )output! In a ottom7up approach the indi'idual

ase elements of the system are first specified in great detail

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String processing

String & A finite se2uence S of zero or more

characters is called string!

String with zero characters is called empty

string!

String will e denoted y enclosing their

characters in single 2uotation mar"s!

!g! B$# :

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Storing Strings

Strings are stored in types of structures

0! -i(ed 1ength Structure

5! 4ariale 1ength Structure with fi(edma(imum

0! 1in"ed Structures!

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String #perations

1. Substring &Accessing a sustring from a gi'en string@ SE:S$/I68)string,initial,length

2.Indexing& Also called pattern matching i!e! position where a string

pattern P first appears in agi'en string $!

@I6DF)te(t,pattern

3. Concatenation & string consists of characters of first string followed y

characters of 5ndstring!

S0GGS5

4. Length & the numer of characters in a string!

168$9)string

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Word Processing

-ollowing operations are there&

Repacement& replacing one string in the te(t y

another!

@ /P1AC)te(t,pattern0,pattern5

Insertion & inserting a string in the middle of the

te(t!

@I6S/$)te(t,position,stringDeetion& deleting a string from the te(t

@D1$)$e(t,position,length

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Arrays

Array 7 a collection of a fi(ed numer of componentswherein all of the components ha'e the same data type

#ne7dimensional array 7 an array in which the componentsare arranged in a list form

$he general form of declaring a one7dimensional array is&

dataType arrayName[intExp];

where intExpis any e(pression that e'aluates to apositi'e integer

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Declaring an array

$he statement

intnum[5];

declares an array numof 5components of thetype int

$he components are num[0], num[1],

num[2], num[3], and num[4]

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Accessing Array

Components

$he general form )synta( of accessing an array component is&

arrayName[indexExp]

where indexExp, called index, is any e(pression whose'alue is a nonnegati'e integer

Inde( 'alue specifies the position of the component in the array

$he []operator is called the arra! subscriptingoperator

$he array inde( always starts at 0

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1in"ed 1ist

Definition of 1in"ed 1ists

(amples of 1in"ed 1ists

#perations on 1in"ed 1ists

1in"ed 1ist as a Class

1in"ed 1ists as Implementations of Stac"s, Sets, etc!

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De"inition o" Lin#ed

Lists

A lin"ed list is a se2uence of items )o%ects where e'ery item is

lin"ed to the ne(t!

8raphically&

data data data data

head_ptr tail_ptr

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De"inition Detais

ach item has a data part )one or more data memers, and alin" that points to the ne(t item

#ne natural way to implement the lin" is as a pointerH that is, thelin" is the address of the ne(t item in the list

It ma"es good sense to 'iew each item as an o%ect, that is, as aninstance of a class!

We call that class& 6ode

$he last item does not point to anything! We set its lin" memerto $%LL! $his is denoted graphically y a self7loop

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&xampe o" Lin#ed List

'( )aiting Line*

A waiting line of customers& ohn, Mary, Dan, Sue )from the head to

the tail of the line

A lin"ed list of strings can represent this line&

John Mary Dan Sue

head_ptrtail_ptr

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pointer to a

next node

pointer to

an eleent

node

!llu"tration o# a lin$ed li"t in eory%

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pointer to a

next node

pointer to

an eleent

node

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pointer to a

next node

pointer to

an eleent

node

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pointer to a

next node

pointer to

an eleent

node

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Singly 1in"ed 1ists

and Arrays

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+perations on Lin#ed

Lists Inserta new item

@ At the head of the list, or

@ At the tail of the list, or

@ Inside the list, in some designated position

Searchfor an item in the list

@ $he item can e specified y position, or y some 'alue Deetean item from the list

@ Search for and locate the item, then remo'e the item, andfinally ad%ust the surrounding pointers

si,e) H is&mpt!)

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Insert (t the ead

Insert a new data A! Call new/ newPtr 1ist efore insertion&

After insertion to head&

data data data data

head_ptr tail_ptr

&

data data data data

head_ptr tail_ptr

&

'(he lin$ )alue in the ne* ite + old head_ptr

'(he ne* )alue o# head_ptr + ne*,tr

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Insert at the 0ai

Insert a new data A! Call new/ newPtr1ist efore insertion

After insertion to tail&

data data data data

head_ptr tail_ptr

&

data data data data

head_ptr tail_ptr

&

'(he lin$ )alue in the ne* ite + -.

'(he lin$ )alue o# the old la"t ite + ne*,tr

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Insert inside the List

Insert a new data A! Call new/ newPtr1ist efore insertion&

After insertion in rdposition&

data data data data

head_ptr tail_ptr

data

data & data data

head_ptr tail_ptr

data

'(he lin$)alue in the ne* ite + lin$)alue o# 2ndite

'(he ne* lin$)alue o# 2ndite + ne*,tr

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Deete the ead Item

1ist efore deletion&

1ist after deletion of the head item&

data data data data

head_ptr tail_ptr

data data data data

head_ptr tail_ptr

data

'(he ne* )alue o# head_ptr + lin$)alue o# the old head ite

'(he old head ite i" deleted and it" eory returned

data

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Deete the 0ai Item

1ist efore deletion&

1ist after deletion of the tail item&

data data data data

head_ptr tail_ptr

data data data

head_ptr tail_ptr

'-e* )alue o# tail_ptr + lin$)alue o# the 3rd#ro la"t ite

'-e* lin$)alue o# ne* la"t ite + NULL

data

datadata

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Deete an inside Item

1ist efore deletion&

1ist after deletion of the 5nditem&

data data data data

head_ptr tail_ptr

data data

head_ptr tail_ptr

'-e* lin$)alue o# the ite loated e#ore the deleted one +

the lin$)alue o# the deleted ite

data

data datadata

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Impementation o"

Lin#ed List

A lin"ed list is a collection of 6ode o%ects, and must support a

numer of operations

$herefore, it is sensile to implement a lin"ed list as a class

$he class name for it is List

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Circular 1in"ed 1ist

In linear lin"ed lists if a list is tra'ersed )all the elements 'isited ane(ternal pointer to the list must e preser'ed in order to e ale toreference the list again!

Circular lin"ed lists can e used to help the tra'erse the same listagain and again if needed! A circular list is 'ery similar to the linearlist where in the circular list the pointer of the last node points not

6E11 ut the first node!

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Contd!

A Linear Linked List

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Contd!

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Contd!

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Circular 1in"ed 1ist

In a circular lin"ed list there are two methods to "nowif a node is the first node or not!

@ither a e(ternal pointer, list, points the first node

or@A header node is placed as the first node of thecircular list!

$he header node can e separated from the others y

either hea'ing a sentinel value as the info part orha'ing a dedicatedflag 'ariale to specify if the node

is a header node or not!

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Douly 1in"ed 1ist

A doule ended is 2ueue where we can do the

insertion and deletion at oth the front and rear

ends

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Contd!

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Implementation

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Stac"s

A stac" is a list of elements in which anelement may e inserted or deleted only at one

end, called the top of the stac"!

$he elements are remo'ed from a stac" in there'erse order of that in which they were

inserted into the stac"!

Stac" is also "nown as a 1I-# )1ast in -astout list or Push down list!

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:asic Stac" #perations

%S& It is the term used to insert an elementinto a stac"!

,.S operation" on "ta$

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:asic Stac" #perations

+& It is the term used to delete an element

from a stac"!

,, operation #ro a "ta$

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Standard rror Messages

in Stac"

$wo standard error messages of stac" are

@Stac# +er"ow& If we attempt to add new element

eyond the ma(imum size, we will encounter a

stack overflowcondition!@Stac# %nder"ow& If we attempt to remo'e

elements eyond the ase of the stac", we will

encounter astack underflowcondition!

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Stac" #perations

PES9 )S$ACJ, $#P, MAFS$/, I$M& $his procedure pushes

an I$M onto a stac"

0! If $#P K MAFSIL, then Print& #4/-1#W, and /eturn!

5! Set $#P &K $#P 0 NIncreases $#P y 0O

! Set S$ACJ N$#PO &K I$M! NInsert I$M in $#P positionO

! /eturn

P#P )S$ACJ, $#P, I$M& $his procedure deletes the top

element of S$ACJ and assign it to the 'ariale I$M

0! If $#P K Q, then Print& E6D/-1#W, and /eturn!

5! Set I$M &K S$ACJN$#PO

! Set $#P &K $#P 7 0 NDecreases $#P y 0O

! /eturn

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Applications of Stac"

Con'erting algeraic e(pressions from one

form to another! !g! Infi( to Postfi(, Infi( to

Prefi(, Prefi( to Infi(, Prefi( to Postfi(,

Postfi( to Infi( and Postfi( to prefi(!

'aluation of Postfi( e(pression!

Parenthesis :alancing in Compilers!

Depth -irst Search $ra'ersal of 8raph!

/ecursi'e Applications!

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(gebraic &xpressions

In"ix/It is the form of an arithmetic e(pression in which

we fi( )place the arithmetic operator in etween the twooperands! !g!& )A : R )C 7 D

re"ix/ It is the form of an arithmetic notation in which

we fi( )place the arithmetic operator efore )pre itstwo operands! $he prefi( notation is called as polish

notation! !g!& R A : @ C D

ost"ix/ It is the form of an arithmetic e(pression in

which we fi( )place the arithmetic operator after )postits two operands! $he postfi( notation is called as suffix

notationand is also referred to reverse polish notation!

!g& A : C D 7 R

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Conersion "rom In"ix to ost"ix

on)ert the #ollo*in in#ix expre""ion & : ; D / < : into it" e=ui)alent

po"t#ix expre""ion

& ti " t"i

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&auation o" ost"ix

&xpression

Postfi( e(pression& T 5 U R R

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ueue

A 2ueue is a data structure where items areinserted at one end called the rear and deleted

at the other end called the front!

Another name for a 2ueue is a

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Representation o"

ueue!nitially the =ueue i" epty

-o*> in"ert 11 to the =ueue (hen =ueue "tatu" *ill e%

-ext> in"ert 22 to the =ueue (hen the =ueue "tatu" i"%

Representation o"

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Representation o"

ueue-o*> delete an eleent 11

-ext in"ert another eleent> "ay 66 to the =ueue ?e annot in"ert 66 to the

=ueue a" it "inal" =ueue i" #ull (he =ueue "tatu" i" a" #ollo*"%

ueue +perations using

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ueue +perations using

(rra!

4arious operations of 3ueue are&@insert'*/ inserts an element at the end of 2ueue 3!

@deete'*/ deletes the first element of 3!

@dispa!'*/ displays the elements in the 2ueue! $here are two prolems associated with linear

2ueue! $hey are&

@0ime consuming& linear time to e spent in shiftingthe elements to the eginning of the 2ueue!

@Signaing ueue "u& e'en if the 2ueue is ha'ing

'acant position!

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Circuar ueue

A circular 2ueue is one in which the insertionof new element is done at the 'ery first

location of the 2ueue if the last location of the

2ueue is full!

Suppose if we ha'e a 3ueue of n elements

then after adding the element at the last inde(

i!e! )n70th , as 2ueue is starting with Q inde(,

the ne(t element will e inserted at the 'ery

first location of the 2ueue which was not

possile in the simple linear 2ueue!

Circular 3ueue

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Circular 3ueue

operations $he :asic #perations of a circular 2ueue are

@InsertionC& Inserting an element into a circular

2ueue results in /ear K )/ear 0 V MAF, where

MAF is the ma(imum size of the array!

@DeetionC& Deleting an element from a circular2ueue results in -ront K )-ront 0 V MAF,

where MAF is the ma(imum size of the array!

@0raersC& Displaying the elements of a circular3ueue!

Circular 3ueue mpty& -rontK/earKQ!

Circular 3ueue

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/epresentation using

Arrayset u" on"ider a irular =ueue> *hih an hold axiu @M&AB o# "ixeleent" !nitially the =ueue i" epty

Insertion and Deletion operations on a

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Insertion and Deletion operations on a

Circular 3ueue!n"ert ne* eleent" 11> 22> 33> 44 and 55 into the irular =ueue (he irular

=ueue "tatu" i"%

-o*> delete t*o eleent" 11> 22 #ro the irular =ueue (he irular =ueue

"tatu" i" a" #ollo*"%

Insertion and Deletion operations on a

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Insertion and Deletion operations on a

Circular 3ueue&ain> in"ert another eleent 66 to the irular =ueue (he "tatu" o# the irular

=ueue i"%

&ain> in"ert 77 and 88 to the irular =ueue (he "tatu" o# the irular =ueue

i"%

Doule nded 3ueue

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Doule nded 3ueue

)D3E

It is a special 2ueue li"e data structure thatsupports insertion and deletion at oth the front

and the rear of the 2ueue!

Such an e(tension of a 2ueue is called adoube-ended ueue5 or deue5 which is

usually pronounced 6dec#6 to a'oid confusion

with the de2ueue method of the regular 2ueue,

which is pronounced li"e the are'iation

6D..6

It is also often called a head-tai in#ed ist!

D3E /epresentation

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D3E /epresentation

using arrays

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$ypes of D3E

$here are two 'ariations of de2ue! $hey are&@Input restricted de2ue )I/D

@#utput restricted de2ue )#/D

An Input restricted de2ue is a de2ue, whichallows insertions at one end ut allows

deletions at oth ends of the list!

An output restricted de2ue is a de2ue, whichallows deletions at one end ut allows

insertions at oth ends of the list!

i i

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riorit! ueue

A priority 2ueue is a collection of elementsthat each element has een assigned a priority

and such that order in which elements are

deleted and processed comes from the

following riles&

@An element of higher priority is processed efore

any element of lower priority!

@$wo element with the same priority are processedaccording to the order in which they were added to

the 2ueue!

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Priority 3ueue #perations and Esage

Inserting new elements! /emo'ing the largest or smallest element!

Priority 3ueue Esages are&

@Simuations/ 'ents are ordered y the time atwhich they should e e(ecuted!

@7ob scheduing in computer systems& 9igher

priority %os should e e(ecuted first!@Constraint s!stems/ 9igher priority constraints

should e satisfied efore lower priority constraints!

$ree terminology and

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$ree terminology and

defination

Composed of nodesand edges

Node@ Any distinguishale o%ect

Edge@ An ordered pair

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Contd!

0! A single node, with no edges, is a tree! $heroot of the tree is its uni2ue node!

5! 1et "#, , "k)k XK0 e trees with no nodes

in common, and let r0, ,r"e the roots ofthose trees, respecti'ely! 1et re a new node!$hen there is a tree $ consisting of the nodesand edges of "#, , "k, the new node r andthe edges

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Contd!

ris called theparentof r#, $, rk

r#, $, rkare the childrenof r

r#

, $, rk

are the silingsof one another 6ode ' is a descendantof u if

@u % v or

@vis a descendant of a child of u@A path e(ists from uto v!

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$ree $erminology

'ery node is a descendant of itself!

If vis a descendantof u, then uis an ancestorof v!

Properdescendants&@$he descendants of a node, other than the node

itself!

Proper ancestors@$he ancestors of a node, other than the node itself!

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properties

1eaf

@a node with no children

6umYnodes K 6umYedges 0

@'ery tree has e(actly one more node than edge

@'ery edge, e(cept the root, is the head of e(actly

one of the edges

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$ree $erminology

9eight of a node in a tree

@1ength of the longest path from that node to a leaf

Depthof a node in a tree

@1ength of the path from the root of the tree to the

node

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$ree terminology

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$ree terminology

w has depth 5

uhas height

tree has height u

w

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$rees

A tree is a collection of nodes

@$he collection can e empty

@)recursi'e definition If not empty, a tree consists

of a distinguished node r )the root, and zero ormore nonemptysubtrees$0, $5, !!!!, $", each of

whose roots are connected y a directed edgefrom

r

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Some $erminologies

&hildandparent

@ 'ery node e(cept the root has one parent

@ A node can ha'e an aritrary numer of children

'eaves

@ 6odes with no children

ibling

@ nodes with same parent

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Some $erminologies

ath

'ength*numer of edges on the path

+epthof a node

@ length of the uni2ue path from the root to that node

@ $he depth of a treeis e2ual to the depth of the deepest leaf

eightof a node

@ length of the longest path from that node to a leaf

@ all lea'es are at height Q

@ $he height of a treeis e2ual to the height of the root

Ancestorand descendant

@ roper ancestorandproper descendant

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(ample& E6IF

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(ample& E6IF

Directory

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i

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:inary $rees

A tree in which no node can ha'e more than two children

$he depth of an

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Contd!

An ordered tree with at most two children for

each node

If node has two child, the 0stis called the left

child, the 5ndis called the right child

If only one child, it is either the right child or

the left child

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(ample& (pression

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(ample& (pression

$rees

1ea'es are operands )constants or 'ariales

$he other nodes )internal nodes contain operators

Will not e a inary tree if some operators are not inary

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$ree tra'ersal

Esed to print out the data in a tree in a certain

order

Pre7order tra'ersal

@Print the data at the root

@/ecursi'ely print out all data in the left sutree

@/ecursi'ely print out all data in the right sutree

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Preorder, Postorder

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eo de , osto de

and Inorder

Preorder tra'ersal@node, left, right

@prefi( e(pression

aRcRRdefg

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Preorder, Postorder

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,

and Inorder

Postorder tra'ersal@left, right, node

@postfi( e(pression

acRdeRfgR

Inorder tra'ersal@ left, node, right!

@ infi( e(pression aRcdRefRg

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:i $

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:inary $rees

Possile operations on the :inary $ree AD$@ parent

@ leftYchild, rightYchild

@ siling

@ root, etc Implementation

@ :ecause a inary tree has at most two children, we can "eep direct

pointers to them

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8 h

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8raph

A graph -is defined as follows&

-%(,E)

(-)/a finite, nonempty set of 'ertices

E(-)/a set of edges )pairs of 'ertices

Directed 's! undirected

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graphs

When the edges in a graph ha'e no

direction, the graph is called undirected

Directed 's! undirected

8 h t i l

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When the edges in a graph ha'e a direction,the graph is called directed)or digraph

graphs )cont!

3B @3>1B @5>9B @9>11B @5>7B

0arning& if the graph is

directed, the order of the

'ertices in each edge isimportant ZZ

8raph terminology

$ h

8 h t i l

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$rees are special cases of graphsZZ

$rees 's graphs8raph terminology

8 h t i l

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8raph terminology

Ad%acent nodes& two nodes are ad%acent if they areconnected y an edge

Path& a se2uence of 'ertices that connect two

nodes in a graph

Complete graph& a graph in which e'ery 'erte( is

directly connected to e'ery other 'erte(

5 i" adEaentto7

7 i" adEaent #ro5

8raph Properties 77 Paths

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07[\

and Connecti'ity

Paths@ A path from 'erte( u to ' of a graph 8 is defined as a se2uence of

ad%acent )connected y an edge 'ertices that starts with u and ends with'!

@ Simple paths& All edges of a path are distinct!

@ Path lengths& the numer of edges, or the numer of 'ertices @ 0! Connected graphs

@ A graph is said to e connected if for e'ery pair of its 'ertices u and 'there is a path from u to '!

Connected component

@ $he ma(imum connected sugraph of a gi'en graph!

8 h t i l ) t

8raph terminology

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What is the numer of edges in a completedirected graph with 6 'ertices]

N 1 (N*#)

8raph terminology )cont!

5) ,2 N

8raph terminology

8 h t i l ) t

8raph terminology

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What is the numer of edges in a completeundirected graph with 6 'ertices]

N 1 (N*#) 3 4

8raph terminology )cont!

5

) ,2 N

8raph terminology

8raph terminology

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Weighted graph& a graph in which each edgecarries a 'alue

8raph terminology )cont!8raph terminology

8 h i l i

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8raph implementation

Array7ased implementation@A 0D array is used to represent the 'ertices

@A 5D array )ad%acency matri( is used to

represent the edges

Array7ased

i l i

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implementation

8raph implementation

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p p

)cont!

1in"ed7list implementation@A 0D array is used to represent the 'ertices

@A list is used for each 'erte( vwhich contains the

'ertices which are ad%acent from v )ad%acency list

1in"ed7listimplementation

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implementation

Ad%acency matri( 's!ad%acency list

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ad%acency listrepresentation

(d8acenc! matrix@8ood for dense graphs 77E_2)^5

@Memory re2uirements& 2)^4^ E5 K #)^5

@Connecti'ity etween two 'ertices can e tested2uic"ly

(d8acenc! ist

@8ood for sparse graphs 77 E_2)^@Memory re2uirements& 2(55 6 5E5)%2(55)

@ 4ertices ad%acent to another 'erte( can e found