Chapter 3 data structure introductionn

8/18/2019 Chapter 3 data structure introductionn

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McGraw-Hill © The McGraw-Hill Companies, Inc., 2000Dr. Fares Saab

Data Structures - CSC313C H A P T E R 3 – A B S T R A C T D A T A T Y P E S ( A D T )


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Define the concept of an abstract data type (ADT).

Define a stack, the basic operations on stacks, their applications and how

they can be implemented.

Define a queue, the basic operations on queues, their applications and how

they can be implemented.

Define a eneral linear list, the basic operations on lists, their applications

and how they can be implemented.

Define a eneral tree and its application.

Define a binary tree!a special kind of tree!and its applications.

Define a binary search tree ("ST) and its applications.

Define a raph and its applications.

#b$ecti%esAfter studyin this chapter, the student should be able to&

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BACKGROUND

Problem solving with a computer means processing

data. To process data, we need to define the data type

and the operation to be performed on the data. The

definition of the data type and the definition of theoperation to be applied to the data is part of the idea

behind an abstract data type (ADT) —to hide how the

operation is performed on the data. In other words, the

user of an ADT needs only to know that a set ofoperations are available for the data type, but does not

need to know how they are applied.

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Simple ADTs

any programming languages already define some simple

ADTs as integral parts of the language. !or e"ample, the #

language defines a simple ADT as an integer. The type of

this ADT is an integer with predefined ranges. # also defines

several operations that can be applied to this data type

$addition, subtraction, multiplication, division and so on%. #e"plicitly defines these operations on integers and what we

e"pect as the results. A programmer who writes a # program

to add two integers should know about the integer ADT and

the operations that can be applied to it.

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'omple ADTs

Although several simple ADTs, such as integer, real,

character, pointer and so on, have been implemented and are

available for use in most languages, many useful comple"

ADTs are not. As we will see in this chapter, we need a list

ADT, a stack ADT, a &ueue ADT and so on. To be efficient,

these ADTs should be created and stored in the library of thecomputer to be used.

The concept of abstraction means&

. *e know what a data type can do.

+. ow it is done is hidden.

i

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Definition

'et us now define an ADT. An abstract data type is a data

type packaged with the operations that are meaningful for the

data type. (e then encapsulate the data and the operations on

the data and hide them from the user.

Abstract data type&

. Definition of data.

+. Definition of operations.-. ncapsulation of data and operation.

i

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/odel for an abstract data type

The ADT model is shown in !igure ).*. Inside the ADT are

two different parts of the model+ data structure and

operations $public and private%.

Fiure -. The model for an ADTDr. Fares Saab


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0mplementation

#omputer languages do not provide comple" ADT packages.

To create a comple" ADT, it is first implemented and kept in

a library. The main purpose of this chapter is to introduce

some common userdefined ADTs and their applications.

-owever, we also give a brief discussion of each ADT

implementation for the interested reader. (e offer the pseudocode algorithms of the implementations as

challenging e"ercises.

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-2 STACKS

A stack is a restricted linear list in which all additionsand deletions are made at one end, the top. If we insert a

series of data items into a stack and then remove them,

the order of the data is reversed. This reversing attribute

is why stacks are known as last in, first out $'I!% data

structures.

Fiure -.+ Three representations of stacksDr. Fares Saab


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#perations on stacks

There are four basic operations, stack , push, pop and empty,

that we define in this chapter.

The stack operation

The stack operation creates an empty stack. The following

shows the format.

Fiure -.- Stack operationDr. Fares Saab


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The push operation

The push operation inserts an item at the top of the stack.

The following shows the format.

Fiure -.1 2ush operationDr. Fares Saab


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The pop operation

The pop operation deletes the item at the top of the stack.

The following shows the format.

Fiure -.3 2op operationDr. Fares Saab


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The empty operation

The empty operation checks the status of the stack. The

following shows the format.

This operation returns true if the stack is empty and false ifthe stack is not empty.

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Stack ADT

(e define a stack as an ADT as shown below+

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Stack applications

0tack applications can be classified into four broad

categories+ reversing data, pairing data, postponing data

usage and backtracking steps. (e discuss the first two inthe sections that follow.

5e%ersin data items

1eversing data items re&uires that a given set of data items

be reordered so that the first and last items are e"changed,

with all of the positions between the first and last also being

relatively e"changed. !or e"ample, the list $2, 3, 4, *, /, 5%

becomes $5, /, *, 4, 3, 2%.

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ample -.+

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ample -.+ $#ontinued%

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2airin data items

(e often need to pair some characters in an e"pression. !or

e"ample, when we write a mathematical e"pression in a

computer language, we often need to use parentheses to

change the precedence of operators. The following two

e"pressions are evaluated differently because of the

parentheses in the second e"pression+

(hen we type an e"pression with a lot of parentheses, we

often forget to pair the parentheses. ne of the duties of a

compiler is to do the checking for us. The compiler uses a

stack to check that all opening parentheses are paired with a

closing parentheses.Dr. Fares Saab


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ample -.-

The below algorithm shows how we can check if all opening

parentheses are paired with a closing parenthesis.

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ample -.- $#ontinued%

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Stack implementation

At the ADT level, we use the stack and its four operations6 at

the implementation level, we need to choose a data structure

to implement it. 0tack ADTs can be implemented usingeither an array or a linked list. !igure ).4 shows an e"ample

of a stack ADT with five items. The figure also shows how

we can implement the stack.

In our array implementation, we have a record that has two

fields. The first field can be used to store information about

the array. The linked list implementation is similar+ we have

an e"tra node that has the name of the stack. This node also

has two fields+ a counter and a pointer that points to the top

element.

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Fiure -.6 Stack implementations

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-3 QUEUES

A queue is a linear list in which data can only beinserted at one end, called the rear , and deleted from the

other end, called the front . These restrictions ensure that

the data is processed through the &ueue in the order in

which it is received. In other words, a &ueue is a first

in, first out (F0F#) structure.

Fiure -.7 Two representation of queuesDr. Fares Saab

# i


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#perations on queues

Although we can define many operations for a &ueue, four

are basic+ &ueue, en&ueue, de&ueue and empty, as defined

below.The queue operation

The &ueue operation creates an empty &ueue. The following

shows the format.

Fiure -.8 The queue operationDr. Fares Saab

Th ti


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The enqueue operation

The en&ueue operation inserts an item at the rear of the

&ueue. The following shows the format.

Fiure -.9 The enqueue operationDr. Fares Saab

Th d ti


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The dequeue operation

The de&ueue operation deletes the item at the front of the

&ueue. The following shows the format.

Fiure -. The dequeue operationDr. Fares Saab

Th t ti


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The empty operation

The empty operation checks the status of the &ueue. The

following shows the format.

This operation returns true if the &ueue is empty and false ifthe &ueue is not empty.

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: ADT


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:ueue ADT

(e define a &ueue as an ADT as shown below+

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ample -.1

!igure ).*2 shows a segment of an algorithm that applies the

previously defined operations on a &ueue 7.

Fiure -.+ ample -.1Dr. Fares Saab

: li ti


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:ueue applications

7ueues are one of the most common of all data processing

structures. They are found in virtually every operating

system and network and in countless other areas. !ore"ample, &ueues are used in online business applications

such as processing customer re&uests, 8obs and orders. In a

computer system, a &ueue is needed to process 8obs and for

system services such as print spools.

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ample -.3

7ueues can be used to organi9e databases by some characteristic

of the data. !or e"ample, imagine we have a list of sorted data

stored in the computer belonging to two categories+ less than

*:::, and greater than *:::. (e can use two &ueues to separate

the categories and at the same time maintain the order of data in

their own category. Algorithm ).) shows the pseudocode for this

operation.

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ample -.3 $#ontinued%

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Alorithm -.-


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Alorithm -.- #ontinued

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ample -.4

Another common application of a &ueue is to ad8ust and create a

balance between a fast producer of data and a slow consumer of

data. !or e"ample, assume that a #P; is connected to a printer.

The speed of a printer is not comparable with the speed of a #P;.

If the #P; waits for the printer to print some data created by the

#P;, the #P; would be idle for a long time. The solution is a

&ueue. The #P; creates as many chunks of data as the &ueue canhold and sends them to the &ueue. The #P; is now free to do

other 8obs. The chunks are de&ueued slowly and printed by the

printer. The &ueue used for this purpose is normally referred to as

a spool &ueue.

Dr. Fares Saab

:ueue implementation


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:ueue implementation

At the ADT level, we use the &ueue and its four operations at

the implementation level. (e need to choose a data structure

to implement it. A &ueue ADT can be implemented usingeither an array or a linked list. !igure ).*) shows an e"ample

of a &ueue ADT with five items. The figure also shows how

we can implement it. In the array implementation we have a

record with three fields. The first field can be used to storeinformation about the &ueue.

The linked list implementation is similar+ we have an

e"tra node that has the name of the &ueue. This node also has

three fields+ a count, a pointer that points to the front element

and a pointer that points to the rear element.

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Fiure -.- :ueue implementationDr. Fares Saab


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GENERAL LINEAR LISTS

0tacks and &ueues defined in the two previous sectionsare restricted linear lists. A eneral linear list is a list

in which operations, such as insertion and deletion, can

be done anywhere in the list—at the beginning, in the

middle or at the end. !igure ).*3 shows a general linear

list.

Fiure -.1 ;eneral linear listDr. Fares Saab

#perations on eneral linear lists


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#perations on eneral linear lists

Although we can define many operations on a general linear

list, we discuss only si" common operations in this chapter+

list , insert , delete, retrieve, traverse and empty.

The list operation

The list operation creates an empty list. The following shows

the format+

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The insert operation


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The insert operation

0ince we assume that data in a general linear list is sorted,

insertion must be done in such a way that the ordering of the

elements is maintained. To determine where the element is to be placed, searching is needed. -owever, searching is done

at the implementation level, not at the ADT level.

Fiure -.3 The insert operationDr. Fares Saab

The delete operation


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The delete operation

Deletion from a general list $!igure ).*/% also re&uires that

the list be searched to locate the data to be deleted. After the

location of the data is found, deletion can be done. Thefollowing shows the format+

Fiure -.4 The dequeue operationDr. Fares Saab

The retrieve operation


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The retrieve operation


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The traverse operation

=ach of the previous operations involves a single element in

the list, randomly accessing the list. 'ist traversal, on the

other hand, involves se&uential access. It is an operation inwhich all elements in the list are processed one by one. The

following shows the format+

The empty operation

The empty operation checks the status of the list. Thefollowing shows the format+

Dr. Fares Saab

The empty operation


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The empty operation

The empty operation checks the status of the list. The

following shows the format+

This operation returns true if the list is empty, or false if the

list is not empty.

Dr. Fares Saab

;eneral linear list ADT


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;eneral linear list ADT

(e define a general linear list as an ADT as shown below+

Dr. Fares Saab

ample - 6


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ample -.6

!igure ).*5 shows a segment of an algorithm that applies the

previously defined operations on a list '. >ote that the third and

fifth operation inserts the new data at the correct position, because the insert operation calls the search algorithm at the

implementation level to find where the new data should be

inserted. The fourth operation does not delete the item with value

) because it is not in the list.

Fiure -. 7 ample -.6Dr. Fares Saab

;eneral linear list applications


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;eneral linear list applications

?eneral linear lists are used in situations in which the

elements are accessed randomly or se&uentially. !or

e"ample, in a college a linear list can be used to storeinformation about students who are enrolled in each

semester.

Dr. Fares Saab

ample - 7


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ample -.7

Assume that a college has a general linear list that holds

information about the students and that each data element is a

record with three fields+ ID, >ame and ?rade. Algorithm ).3

shows an algorithm that helps a professor to change the grade for

a student. The delete operation removes an element from the list,

but makes it available to the program to allow the grade to be

changed. The insert operation inserts the changed element backinto the list. The element holds the whole record for the student,

and the target is the ID used to search the list.

Dr. Fares Saab

ample - 7 $#ontinued%


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ample - 8


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ample -.8

#ontinuing with ="ample ).5, assume that the tutor wants to print

the record of all students at the end of the semester. Algorithm

*2.@ can do this 8ob. (e assume that there is an algorithm calledPrint that prints the contents of the record. !or each node, the list

traverse calls the Print algorithm and passes the data to be printed

to it.

Dr. Fares Saab

ample - 8 $#ontinued%


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Dr. Fares Saab

;eneral linear list implementation


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;eneral linear list implementation

At the ADT level, we use the list and its si" operations but at

the implementation level we need to choose a data structure

to implement it. A general list ADT can be implementedusing either an array or a linked list. !igure ).* shows an

e"ample of a list ADT with five items. The figure also shows

how we can implement it.

The linked list implementation is similar+ we have an e"tra

node that has the name of the list. This node also has two

fields, a counter and a pointer that points to the first element.

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Fiure +.8 ;eneral linear list implementationDr. Fares Saab


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12-5 TREES

A tree consists of a finite set of elements, called nodes

$or %ertices% and a finite set of directed lines, called

arcs, that connect pairs of the nodes.

Fiure +.+9 Tree representationDr. Fares Saab

(e can divided the vertices in a tree into three categories+


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(e can divided the vertices in a tree into three categories+

the root , leaves and the internal nodes. Table *2.* shows the

number of outgoing and incoming arcs allowed for each

type of node.

Dr. Fares Saab

=ach node in a tree may have a subtree The subtree of each


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=ach node in a tree may have a subtree. The subtree of each

node includes one of its children and all descendents of that

child. !igure *2.2* shows all subtrees for the tree in !igure

*2.2:.

Fiure +.+ SubtreesDr. Fares Saab


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12-6 BINARY TREES

A binary tree is a tree in which no node can have morethan two subtrees. In other words, a node can have 9ero,

one or two subtrees.

Fiure +.++ A binary treeDr. Fares Saab

5ecursi%e definition of binary trees


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y

In #hapter 5 we introduced the recursive definition of an

algorithm. (e can also define a structure or an ADT

recursively. The following gives the recursive definition of a binary tree. >ote that, based on this definition, a binary tree

can have a root, but each subtree can also have a root.

Dr. Fares Saab

!igure *2 2) shows eight trees the first of which is an empty


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!igure *2.2) shows eight trees, the first of which is an empty

binary tree $sometimes called a null binary tree%.

Fiure +.+- amples of binary treesDr. Fares Saab

#perations on binary trees


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p y

The si" most common operations defined for a binary tree

are tree $creates an empty tree%, insert , delete, retrieve,

empty and traversal . The first five are comple" and beyondthe scope of this book. (e discuss binary tree traversal in

this section.

Dr. Fares Saab

"inary tree tra%ersals


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A binary tree traversal re&uires that each node of the tree be

processed once and only once in a predetermined se&uence.

The two general approaches to the traversal se&uence aredepth


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p

!igure *2.2@ shows how we visit each node in a tree using

preorder traversal. The figure also shows the walking order. In

preorder traversal we visit a node when we pass from its left side.The nodes are visited in this order+ A,


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p

!igure *2.2/ shows how we visit each node in a tree using

breadthfirst traversal. The figure also shows the walking order.

The traversal order is A,


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y pp


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pression trees

An arithmetic e"pression can be represented in three

different formats+ infi, postfi and prefi. In an infi"

notation, the operator comes between the two operands. In

postfi" notation, the operator comes after its two operands,

and in prefi" notation it comes before the two operands.

These formats are shown below for addition of two operandsA and


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Fiure +.+6 pression treeDr. Fares Saab


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12-7 BINARY SEARCH TREES

A binary search tree $


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p

!igure *2.2 shows some binary trees that are ote that a tree is a


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A very interesting property of a


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use a version of the binary search we used in #hapter 5 for a

binary search tree. !igure *2.): shows the ;' for a


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The ADT for a binary search tree is similar to the one we

defined for a general linear list with the same operation. As a

matter of fact, we see more


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12-8 GRAPHS

A graph is an ADT made of a set of nodes, called%ertices, and set of lines connecting the vertices, called

edes or arcs. (hereas a tree defines a hierarchical

structure in which a node can have only one single

parent, each node in a graph can have one or more

parents. ?raphs may be either directed or undirected. In

a directed graph, or diraph, each edge, which connects

two vertices, has a direction from one verte" to theother. In an undirected graph, there is no direction.

!igure *2.)2 shows an e"ample of both a directed graph

$a% and an undirected graph $b%.

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Fiure +.-+ ;raphs

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ample +.-


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A map of cities and the roads connecting the cities can be

represented in a computer using an undirected graph. The cities

are vertices and the undirected edges are the roads that connectthem. If we want to show the distances between the cities, we can

use weighted graphs, in which each edge has a weight that

represents the distance between two cities connected by that

edge.

ample +.1

Another application of graphs is in computer networks $#hapter

/%. The vertices can represent the nodes or hubs, the edges can

represent the route. =ach edge can have a weight that defines thecost of reaching from one hub to an ad8acent hub. A router can

use graph algorithms to find the shortest path between itself and

the final destination of a packet.

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Chapter 3 data structure introductionn