Chapter 5 Stack and Queue.pdf

5. Stacks and Queues
5.1. Basic Stack Operations
5.2. Basic Queue Operations
5.3. Implementation of Stacks and queues

Stack
• Linked list and arrays allow one to insert and delete elements
at any place in the list
– At the beginning
– At the end or
– In the middle
• There are frequent situations where one wants to restrict
insertions and deletions so they can take place only at the
beginning or the end of the list, not in the middle
• Two of the data structures that are useful in such situations
are
– stacks and
– queues
3

Stack
• A stack is a linear data structure (a list of elements) that has
restricted data access
• Items (or elements) may be added (or inserted) and removed
only at one end
• Is an ordered collection of entries that can only be accessed at
one end
• Is a data structure of ordered entries such that entries can
only be inserted and removed at one end (called TOP)
• When we say a stack is ordered we mean that there is one
that can be accessed first (the one on the top), that can be
accessed second (just below the top), the third one and so
forth
4

Stack
• It is a data structure that has access to its data only at the end
(LL) or TOP (array)
• It operates on Last-In First-Out (LIFO) basis
– The last item to be added to a stack is the first item to be
removed
– This means, elements are removed from the stack in the
reverse order of that in which they were inserted into the
stack
5

Stack
• Every day examples of such structures
– Stack of dishes
– Stack of pennies
– Stack of folded towels
– Stack of books
• An item may be added or removed from the top of any of the
above stacks
6

Stack
• Other names used for stacks
– Piles
– Push down lists
• A stack has many applications in computer science
– To be addressed later
8

Stack
• A stack has two basic operations, push and pop
• push (or put)
– Is a term used for inserting or adding data (or element) at
the top of the stack or into the stack
– Could be written as Is a function to add an entry at the top
of the stack
– Needs at least one parameter
9

Stack
• pop (get)
– Is a term used to delete or remove an element or data
from the top of the stack
– Could be written as a function to remove the top entry
– No need to define the element to be removed in case of
pop because there is only one element to be removed, the
element at the top
– No need to use parameter for pop operation
– By default pop operation takes (or removes) the top
element
– Is the synonym for delete when it comes to stack
10

Stack
• There are no functions that allow a program to access entries
other than the top entry
• To access any entry other than the top one, the program must
remove entries one at a time form the top until the desired
entry is reached
• Include examples of stacks
– Include example 1, 2, and 3
11

Implementation of stacks
• Stacks can be implemented both as
– An array (contiguous list) and
– Linked lists
• There are a set of operations that will work with either type of
implementation (push, pop, peek, isEmpty, isFull, sizeOfStack)
12

Array representation of Stacks
• The array acts as a stack
• Let the array be called STACK
• Let MAXSTK
– be the maximum number of elements that can be held by
the stack
– Capacity of the stack
– Array implementation requires this constant because it
uses a fixed size array to hold the items of the stack
– The linked list implementation will use dynamic memory
instead and such a constant is not required
13

• Suppose the stack has the following data structure (or
definition)
int STACK[MAXSTK];
• We should have a an integer variable ( or pointer variable)
that points to the top element in the stack
• Call this variable TOP
• This implies TOP will contain the index of the top element of
the stack
14

• At the beginning the TOP variable (or pointer) must be
initialized with -1
• That is, we have no element in the stack at the beginning
int TOP = -1;
• TOP also tells us the total number of elements in the STACK
• How do you calculate the total number of elements in the stack
using TOP?
– Total number of elements in the stack = TOP + 1;
– That is, TOP + 1 is the total number of elements in the stack
15

Stack Errors
• Stack overflow
– This is the condition resulting from trying to push (or add)
an item to a full stack
– To avoid stack overflow, compare the capacity constant
MAXSTK to the stack’s current size obtained with the size
function
16

Stack Errors
• Stack Underflow
– The condition resulting from trying to POP (remove) an
item from an empty stack
– If a program attempts to POP an item of empty stack, then
it is called stack underflow
– In order to avoid a stack underflow, it is possible to write a
function that tests whether the stack is empty or not
17

To push or store an element into the stack
• do the following
– Check if there is enough space
• In order to store data
– TOP must be <= MAXSTK – 2 or
– TOP must be < MAXSTK – 1
• because to store on the top, the top must be empty
– If yes increment TOP and then store the element in
STACK[TOP]
• (incrementing must be done before storing/pushing the element)
– If no, No space in the stack and send an overflow message
(stack overflow)
• We use the operation push
18

push Operation
• Is a function to push or store an element into the stack
• There is nothing (i.e., data) to be returned by this function
because it does only push element
• Thus, it must be of type void
19

push Operation
• Algorithm
– Step 1 - increment the stack TOP by 1
i. Check whether stack is always less than upper limit of
the stack
ii. If it is less than the upper limit go to step 2 else
report “Stack Overflow”
− Step 2 - put the new element at the position pointed by
TOP
20

push Operation
/* This procedure pushes an ITEM on to a stack*/
void push(STACK, TOP, MAXSTK, ITEM)
{
// Is stack already filled?
If(TOP==MAXSTK-1)
cout<<“nOVERFLOW” //print overflow and return
else
{
TOP=TOP + 1; // TOP is set to increase by 1. The value of TOP, in case
//of push, is changed before the insertion in push
STACK[TOP]=ITEM //insert ITEM in a new TOP position
}
return;
}
21

push Operation
• Remark
– Frequently, TOP and MAXSTK are global variables and
hence procedure push can be called using only
push(STACK, ITEM)
– If the array STACK is also declared globally, the procedure
push can be called using only push(ITEM)
22

Push Operation
• C++ implementation
– Include F
– See stack2.cpp
– See stackz.cpp
23

To POP or remove an element from the stack
• do the following
– Check if there is data in the stack (how?)
• Check if TOP > = 0
– If yes
• There is at least one element
• Copy/remove the element in STACK[TOP]
• Then decrement the value of the TOP
– If NO
• No element in the stack
• Send stack underflow message
• We use the operation pop()
24

pop Operation
• The function pop has to return a data, top element, because
the element may be used somewhere else
• The data type of the pop function is the same as the type of
the STACK itself, in this case the type of the array
25

pop Operation
• Algorithm for pop() function
– Step 1
if the stack is empty
give the alert “Stack Underflow!” and quit
else
go to step 2
− Step 2
i. Hold the value for the element pointed by the TOP
ii. Put a NULL value instead
iii. Decrement the TOP by 1
26

pop Operation
/* This procedure deletes the top element of the stack and assigns it
to the variable ITEM*/
DataType pop(STACK, TOP, ITEM)
{
// Stack has item to be removed?
If(TOP==-1)
cout<<“nUNDERFLOW” ; //print UNDERFLOW and return (TOP);
else
{
ITEM=STACK[TOP]; // Assign (set) top element to ITEM
STACK[TOP]=NULL;
TOP=TOP – 1;//TOP is decreased by 1. The value of TOP, in case
//of pop, is changed after the deletion in pop
}
return (ITEM);
}
27

pop Operation
• Remark
– Frequently, TOP and MAXSTK are global variables and
hence procedure pop can be called using only pop(STACK,
ITEM)
– If the array STACK is also declared globally, procedure pop
can be called using only pop(ITEM) or simple pop( )
28

pop Operation
• C++ implementation
– Include G
– Include stack2.cpp (second version)
29

isEmpty() & isFULL()
• We could have other operations, such as an explicit boolean
function
– isEmpty()- that tests for empty stack
– isFull()- that tests for a full stack
togeter with the pop and push functions
• Include definitions of isEmpty() and isFull( )
– Include D
30

peek()
• Sometimes we need information about the last element
without removing the element
• For this we use the function peek( ) or also called topData( )
• Include the definition for topData( )
– Include E
31

sizeOfStack()
• Function to know the current size of the stack
//Returns the number of elements in the stack
int sizeOfStack()
{
/*post condition: return value is total number of items
in the stack */
return (TOP + 1);
}
32

Linked List Implementation of Stack
• A linked list is a natural way to implement a stack as a
dynamic structure whose size can grow and shrink during
execution, without a predefined limit that is determined at
compilation
• The items in the stack are stored in a linked list, with the TOP
of the stack stored at the head node, down to the bottom of
the stack at the final node
• In case of array implementation, we have a variable TOP to
indicate the top element
33

Linked List Implementation of Stack
• In this case, we need two pointers, one pointing to the
beginning node (because we need to have a head pointer) of
the list and the other the top of the list or node (because
operation is performed at the top)
• At the beginning, these two pointers must be initialized to
NULL
node * headPtr=NULL;
node *topPtr=NULL;
34

Defining the Structure
struct number {
int num;
number *next;
}
number * bottomPtr =NULL;
number * topPtr=NULL;
35

To push an element
• Check if there is a free space
– How is this done?
– We need to have a pointer newNumPtr that points to the
newly created node
number * newNumPtr=new number;
– This is used to reserve new space for our data.
– How do we know that this space is reserved?
– Is newNumPt != NULL?
– If != NULL, it means we have memeory, memory is
allocated and this memeory is pointed by newNumPtr and
its address is stored in newNumPtr
– If the value stored at newNumPtr is NULL we have no space
36

To push an element
• If newNumPtr != NULL is true, then
– Store data in the newly created node (How?)
– Create a link between the last node and the newly created
node.
• That is, the next pointer of the last node in the list
should point to the newly created node
– Make the top pointer to point to the last node
• If newNumPtr != NULL is false
– Print “Memeory overflow “
37

Linked list implementation of stacks:
push operation
• Algorithm
– Step 1- if the stack is empty, go to step 2 or else go to step
3
– Step 2- create the new element and make bottomPtr and
topPtr point to the new element and quit
– Step 3- create the new element and make the last (top
most) element of the stack to point to it
– Step 4- make the new element your top most element by
making the topPtr points to it
38

push operation
• Include the C++ code
– Include H
– See stack5.cpp
39

push operation
• Remarks
– The push operation is similar to the insertion operation in
a dynamic singly linked list
– The only difference here is that you can add the new
element only at the end of the list, which means additions
can happen only from the TOP
– Since dynamic list is used for the stack, the stack is also
dynamic, means it has no prior upper limit
– So there is no need to check for the overflow condition at
all!
40

To pop an element
• Check if there is data in the stack
• How do you check whether there is an element in the list or
stack?
– Is bottomPtr != NULL; (ask this question)
41

To pop an element
• If bottomPtr != NULL is true
– bottomPtr is different form NULL
– That is, There is data in the list (at least one)
• If bottomPtr != NULL is false
– bottomPtr=NULL
– Stack is empty
– There is no element in the stack to pop
– Print “Stack Underflow”
• Note: 1. topPtr always points to the last node
• 2. We need a pointer prevNumPtr to point to the previous node of
the last node
number * prevNumPtr;
42

pop operation
• Algorithm
1. If the stack is empty then give an alert message “stack
Underflow” and quite; or else proceed
2. If there is only one element left go to step 3 or else step
4
3. Free that element and make bottomPtr and topPtr to
point to NULL and quit
4. Make “targetPtr” point to just one element before top;;
free the top most element ; make targetPtr to point to
the top most element
43

pop operation
• Include the C++ code
– Include I
– See stack5.cpp
44

pop operation
• The pop operation is similar to the deletion operation in any
linked list but you can only delete from the end of the list and
only one at time
• Here, we need a list pointer, targetPtr, which will be pointing
to the last but one element in the list (stack)
• Every time we pop, the TOP most element will be deleted and
targetPtr will be made as the top most element
• Supposing you have only one element left in the stack, then
we won’t make use of targetPtr. Rather we will take help of
bottomPtr
45

Stack Applications
• Surprisingly, stacks have many application
• Most compilers use stacks to analyze the syntax of a program
• Stacks are used
– To keep track of local variables when a program is run
– To search a maze or family tree or other types of branching
trees
46

Applications of Stacks
Infix to Postfix (RPN) Conversion
Types of Expression
• Notations for Arithmetic Expression
• There are three notations to represent an
arithmetic expression:
• Infix Notation
• Prefix Notation
• Postfix Notation
• The normal (or human) way of expressing mathematical
expressions is called infix form, e.g. 4+5*5.
47

Example : Infix : 4+5*5
• the prefix form: + * 4 5 5
• When the operators come after their
operands, it is called postfix form
Prefix Notation(Polish notation)
• The prefix notation places the operator before the
operands. This notation was introduced by the Polish
mathematician and hence often referred to as polish
notation.
48

Postfix Notation(Reverse Polish notation)
• The postfix notation places the operator after
the operands.
• This notation is just the reverse of Polish
notation and also known as Reverse Polish
notation.
e.g. 4 5 5 * +
49

50
Transforming Infix to Postfix
 Goal is show you how to evaluate infix
algebraic expression, postfix expressions are
easier to evaluate.
 So first look at how to represent an infix
expression by using postfix notation.
Infix Postfix
a+b ab+
(a+b)*c ab+c*
a + b*c abc*+

51
Pencil and paper scheme
 We write the infix expression with a fully
parenthesized infix expression.
 For example: (a+b)*c as ((a+b)*c). Each operator is
associated with a pair of parentheses.
 Now we move each operator to the right, so it
appears immediately before its associated close
parenthesis:
 ((ab+)c*)
 Finally, we remove the parentheses to obtain the
postfix expression. ab+c*

infix Notation Prefix Notation Postfix Notation
A * B * A B AB*
(A+B)/C /+ ABC AB+C/
(A*B) + (D-C) +*AB - DC AB*DC-+
52
Exercise
a + b * c
a * b / (c - d)
a / b + (c - d)
a / b + c - d

Answers
abc*+
ab*cd-/
ab/cd-+
ab/c+d-

• Scan the infix expression from left to right.
• Encounter an operand, place it at the end of new
expression.
• Encounter an operator, save the operator until the right
time to pop up
– Depends on the relative precedence of the next operator.
• Higher precedence operator can be pushed
• Same precedence, operator gets popped.
• Reach the end of input expression, we pop each operator
from the stack and append it to the end of the output

Transforming Infix to Postfix
Converting the infix expression
a + b * c to postfix form

Converting infix expression to postfix form:
a – b + c

Algorithm
initialize stack and postfix output to
empty;
while(not end of infix expression) {
get next infix item
if(item is operand) {
append item to postfix output
}
else if(item == ‘(‘ ) {
push item onto stack
}
else if(item == ‘)’ ) {
pop stack to x
while(x != ‘(‘ ) {
app.x to postfix output &
pop stack to x 57

else {
while(precedence(stack top) >= precedence(item)){
pop stack to x ;
app.x to postfix output
}
push item onto stack
}
}
while(stack not empty){
pop stack to x ;
append x to postfix output
}
58

Operator Precedence (for this algorithm):
4 : ‘(‘ - only popped if a matching ‘)’ is found
3 : All unary operators
2 : / *
1 : + -
59

Converting infix expression to postfix form:
a ^ b ^ c
An exception: if an operator is ^,
push it onto the stack regardless of
what is at the top of the stack

• Infix-to-Postfix Algorithm
Symbol in
Infix
Action
Operand Append to end of output expression
Operator ^ Push ^ onto stack
Operator +,-,
*, or /
Pop operators from stack, append to output
expression until stack empty or top has
lower precedence than new operator. Then
push new operator onto stack
Open
parenthesis
Push ( onto stack, treat it as an operator
with the lowest precedence
Close
parenthesis
Pop operators from stack, append to output
expression until we pop an open
parenthesis. Discard both parentheses.

Steps to convert the infix expression
a / b * ( c + ( d – e ) ) to postfix form.

 (a + b)/(c - d)
 a / (b - c) * d
 a-(b / (c-d) * e + f) ^g
 (a – b * c) / (d * e ^ f * g + h)

Answer
ab+cd-/
abc-/d*
abcd-/e*f + g^-
abc* -def^*g*h+/

Example:
Convert to postfix expression:
a) 4 + 3*(6*3-12)
b) A + B * C + D
c) (A + B) * (C + D)
d) A+(B*C+D)/E
e) 10 + 3 * 5 / (16 - 4)
65

Postfix Evaluation
Consider the postfix expression :
6 5 2 3 + 8 * + 3 + *
66

Evaluating Postfix Expression
• Save operands until we find the operators that apply to
them.
• When see operator, is second operand is the most
recently save value. The value saved before it – is the
operator’s first operand.
• Push the result into the stack. If we are at the end of
expression, the value remains in the stack is the value of
the expression.

Algorithm
initialise stack to empty;
while (not end of postfix expression) {
get next postfix item;
if(item is value)
push it onto the stack;
else if(item is binary operator) {
pop the stack to x;
pop the stack to y;
perform y operator x;
push the results onto the stack;
}
else if (item is unary operator) {
pop the stack to x;
perform operator(x);
push the results onto the stack
}
}
The single value on the stack is the desired result.
68

• The stack during the evaluation of the postfix
expression a b / when a is 2 and b is 4
Evaluating Postfix Expression

Binary operators: +, -, *, /, etc.,
Unary operators: unary minus, square root, sin, cos, exp, etc.,
Example:
Evaluate:
a) 6 5 2 3 + 8 * + 3 + *
b) 4 5 + 7 2 - *
c) 4 2 3 5 1 - + * +
70

The stack during the evaluation of the postfix expression
a b + c / when a is 2, b is 4 and c is 3

Exercise:
 Evaluate the following postfix expression.
Assume that a=2, b=3, c=4, d=5, and e=6
 ae+bd-/
 abc*d*-
 abc-/d*
 ebca^*+d-

73
Application of stack:
Balanced parentheses using Stack
Here are some of the balanced and unbalanced expressions:

74
Steps to find whether a given expression is balanced or unbalanced
1. Input the expression and put it in a character stack.
2. Scan the characters from the expression one by one.
3. If the scanned character is a starting bracket ( ‘ ( ‘ or ‘ , ‘ or ‘ *
‘), then push it to the stack.
4. If the scanned character is a closing bracket ( ‘ ) ’ or ‘ - ’ or ‘ + ’
), then pop from the stack and if the popped character is the
equivalent starting bracket, then proceed. Else, the expression
is unbalanced.
5. After scanning all the characters from the expression, if there
is any parenthesis found in the stack or if the stack is not
empty, then the expression is unbalanced.

Stack in function calls
• Program Stack: Program Stack is the stack which holds
all the function calls, with bottom elements as the main
function.
• Stack Frame: Stack Frame is actually a buffer memory
that is an element of program stack and has data of the
called function ie
– Return Address
– Input Parameter
– Local Variables
• whenever a function is called a new stack frame is
created with all the function’s data and this stack
frame is pushed in the program stack

Series of operations when we call a function:
1. Stack Frame is pushed into stack.
2. Sub-routine instructions are executed.
3. Stack Frame is popped from the stack.
4. Now Program Counter is holding the return
address.

79
Queue
• Is an ordered collection of items from which items may be
deleted at one end (called front of the queue) and into which
items may be inserted at the other end (called the rear of the
queue)
• That is, it is a data structure that has access to its data at the
front and rear

80
Queue
• It is a First-In-First-Out (FIFO) data structure
– That is, the first element inserted into a queue is the first
element to be removed
– The order in which elements enter a queue is the order in
which they leave
– This is in contrast with the queue

81
Queue
• Examples of a queue in the real world
– people waiting in line at the bank form a queue where the
first person in line is the first person to be waited on
– A line at the bus stop
– Automobiles waiting to pass through an intersection (form
a queue) in which the first car in line is the first car through
– A group of cars waiting at a tool booth

82
Queue
• Has three primitive operations applied to it
– enqueue(q,x) also called insert(q,x)
• Inserts item (data) x at the rear of the queue q
– X=dequeue(q) also called remove(q)
• Deletes the front element (data) from the queue q and sets X to its
content
– empty(q)
• Returns false or true depending on whether or not the queue
contains any elements

83
Queue
• Example
Operation Content of queue
Enqueue(q, B)
Enqueue(q, C)
Enqueue(q)
Enqueue(q, G)
Enqueue(q, F)
Enqueue(q)
Enqueue(q, A)
Enqueue(q)

84
Queue
• Two ways to represent (or implement) a queue
– As an array
• Simple array representation of queue
• Circular array representation of queue
– As a linked list

85
Simple array representation of queue
• Array is used to hold the elements
• We need two integer variables, FRONT and REAR, to hold the
positions with in the array of the first and last elements of the
queue
– FRONT tells (or holds) the index of the front element
– REAR tells (or holds) the index of the rear element
• We also need the following integer variables
– QUEUESIZE
• tells (or holds) the total number of data in the queue
– MAX_SIZE
• tells (or stores) the capacity of the array (i.e. queue)

86
• Initially FRONT and REAR must be initialized to -1
int FRONT = -1, REAR = -1;
• Initially QUEUESIZE must be initialized to 0
int QUEUESIZE = 0
• Remark
– Using an array to hold a queue introduces the
possibility of overflow if the queue should grow larger
than the size of the array

87
• Declaring a queue q of integers
#define MAXQUEUE 100
Struct queue{
int items[MAXQUEUE];
int front, rear;
}q;

88
• Algorithm to enqueue data to the queue
– Check if there is space in the queue
– To do this ask this question
• Is REAR < MAX_SIZE - 1 ?
– If (REAR < MAX_SIZE - 1 ) is true
• Increment REAR by 1
• Store the data in queue[REAR]
• Increment QUEUESIZE by 1
• If (FRONT = = -1)
Increment FRONT by 1
– If (REAR < MAX_SIZE - 1) is false
• Display the message “Queue Overflow”

89
• Algorithm to dequeue data from the queue
– Check if there is data in the queue
• Is QUEUESIZE > 0 ?
– If (QUEUESIZE > 0) is true
• Copy the data in queue[FRONT]
• Increment FRONT by 1
• Decrement QUEUESIZE by 1
– If (QUEUESIZE > 0) is false
• Display the message “Queue Underflow”

90
C++ implementation of enqueue and equeue
const int MAX_SIZE=100;
int FRONT =-1, REAR =-1;
int QUEUESIZE = 0;

91
void enqueue(char queue[], char x)
{
if(REAR < MAX_SIZE - 1)
{
REAR++;
queue[REAR] = x;
QUEUESIZE++;
if(FRONT = = -1)
FRONT++;
}
else
cout<<"Queue Overflow";
}

92
int dequeue(char queue[])
{
char x;
if(QUEUESIZE>0)
{
x=queue[FRONT];
FRONT++;
QUEUESIZE--;
}
else
cout<<"Queue Underflow";
return (x);
}

93
Array representation of queue
• Example 1
– Consider a queue with QUEUESIZE = 4 and write a C++
program that performs the operations shown in the table
given in the next page
– Answer
• See queue3.cpp

94

95
Limitation of simple array representation of queue
• The array representation outlined earlier has a limitation
– The variable rear is incremented but never decremented
– Hence, it will quickly reach the end of the array
– At that point, we will not be able to add any more elements
to the queue, yet there is likely to be room in the array
– In fact, it is possible to reach the absurd situation where the
queue is empty, yet no new element can be inserted
– In normal application, the variable front would also be
incremented from time to time (when entries are removed
from the queue)
– This will free up the array application with index values less
than front
• Different solutions to overcome this problem

96
Solution 1
• For using the freed array locations
– Maintain all the queue entries so that front is always equal
to 0 (the first index of the array)
– When queue[0] is removed, we move all the entries in the
array down one locations, so that the value of queue[1] is
moved to queue[0], and then all other entries are also
moved
• How is the above method implemented (see next slide)

97
Solution 1
• Modify the dequeue operation so that when an item is deleted,
the entire queue is shifted to the beginning of the array
• Ignoring the possibility of overflow, the operation
x=dequeue(queue) would be modified as follows
x=queue(FRONT);
for(int i = 0; i < REAR; i++)
{
queue[i] = queue[i+1];
}
REAR--;

98
Solution 1 …
• The queue need no longer contain a front field
• This is because, the element at position 0 of the array is always
at the front of the queue
• The empty queue is represented by the queue in which rear is
-1

99
Solution 1 …
• Is this method efficient?
– The method will work
– But, it is not efficient, in fact too inefficient
– Each deletion involves moving every remaining element of
the queue
– Every time we remove an entry from the queue, we must
move every entry in the queue
– If a queue contains 500 or 100 elements, this is high a price
to pay
– Further, the operation of removing an element from a queue
logically involves manipulation of only one element
– The implementation of that operation should reflect this and
should not involve a host of extraneous operations

100
Solution 1 …
• Exercise
–Include exercise 4.1.3

101
Solution 2
• Is a better approach
• We do not need to move all the array elements
• When the rear index reaches the end of the array, we can
simply start using the variable locations at the front
• One way to think of this arrangement is to think of the array
as being bent into a circle so that the first component of the
array is immediately after the last component of the array
• That is, the successor of the last array index is the first array
index

102
Solution 2…
• In this arrangement (circular queue), the free index positions are
always to the “right after” queue[REAR]
• Example
– Consider a queue of characters with a capacity of 4
– Perform the operations given in the table in the next slide
considering the queue is represented as circular array rather
than simple array
– See also D

104
Solution 2 …
• With this view, the queue’s entries start at queue[FRONT] and
continue forward
• If you reach the end of the array, then come back at queue[0],
and keep going until you find the REAR
• It may help to actually view the array as bent circle, with the
final array element attached back to the front as shown below
• 104
• Incude diagram and the statement just above it on kinfe
• See also mu not
• An array used in this way is called a circular array (if we have
free slot we can use it)

105
Solution 2 …
• The circular array implementation of a queue with capacity
QUEUESIZ can be simulated as follows:
• An array used in this way is called a circular array (if we have
free slot we can use it)
• In any case, queue[rear] is the last entry in the queue
0
1
2
3 4
5
6
7
8
9
10
11
12
13
MAX_SIZE - 1

106
Circular array implementation of queue
• Consider the structure queue[QUEUESIZE]
• We need to have three integer variables, FRONT, REAR and
QUEUESIZE and MAX_SIZE
– REAR tells (stores) the index of the front element
– FRONT tells (stores) the index of the rear element
– QUEUESIZE tells (stores) the total number of data in the queue
– MAX_SIZE tells (stores) the capacity of the queue
• Initially FRONT and REAR must be initialized to -1
int FRONT = -1, REAR = -1;
• Initially QUEUESIZE must be initialized to 0
int QUEUESIZE = 0;

107
• Algorithm to enqueue data to the queue
– Check if there is space in the queue
Is QUEUESIZE < MAX_SIZE ?
– If(QUEUESIZE < MAX_SIZE ) is true
• Increment REAR by 1
• If(REAR = = MAX_SIZE)
REAR = 0
• Store the data in queue[REAR]
• Increment QUEUESIZE by 1
• If(FRONT = = -1)
Increment FRONT by 1
– If(QUEUESIZE < MAX_SIZE ) is false
display the message “Queue Overflow”

108
• Algorithm to dequeue data from the queue
– Check if there is data in the queue
Is QUEUESIZE > 0 ?
– If(QUEUESIZE > 0 ) is true
• Copy the data in queue[FRONT]
• Increment FRONT by 1
• If(FRONT == QUEUESIZE)
Set FRONT = 0
• Decrement QUEUESIZE by 1
– If(QUEUESIZE > 0 ) is false
Display the message “Queue Overflow”

109
(circular array of queue)
const int MAX_SIZE=100;
int FRONT =-1, REAR =-1;
int QUEUESIZE = 0;

110
(Circular array of queue)
void enqueue(int x)
{
if(QUEUESIZE<MAX_SIZE)
{
REAR++;
if(REAR = = MAX_SIZE)
REAR=0;
Num[REAR]=x;
QUEUESIZE++;
if(FRONT = = -1)
FRONT++;
}
else
cout<<"Queue Overflow";
}

111
int dequeue()
{
int x;
if(QUEUESIZE>0)
{
x=Num[FRONT];
FRONT++;
if(FRONT = = MAX_SIZE)
FRONT = 0;
QUEUESIZE--;
}
else
cout<<"Queue Underflow";
return(x);
}

112
• Example 3
– Assume a queue is implemented as circular array
MAX_SIZE = 4 and write a C++ program that
performs the operations shown in the table given
in the next page
– Answer
• See queue5.cpp

113

114
• Example 4
– This is another example on a circular array
implementation queue
– See queue6.cpp

115
Linked list implementation of enqueue and dequeue
operations
• Enqueue- is inserting a node at the end of a linked list
• Dequeue- is deleting the first node in the list
• Define the structure to hold references to queue nodes for the
linked queue implementation
struct queue{
char item;
queue next
};
queue *frontPtr=NULL; //references to the front of the queue
queue *rearPtr=NULL; //references to the rear of the queue

116
Linked list implementation of enqueue operation
/* Adds item to the rear of this queue */
void enqueue(char x)
{
queue * temp = new queue;
if(temp == NULL)
cout<<“nQueue is FULL”;
temp->item = x;
temp -> next = NULL;
to be checked
if(rearPtr == NULL)
frontPtr=temp;
else
rearPtr -> next = temp;
rearPtr = temp;
}

117
Linked list implementation of dequeue operation
/* Removes front element form this queue and return it */
void dequeue()
{
char data;
data = frontPtr -> item;
frontPtr = frontPtr -> next;
if(frontPtr == NULL)
rearPtrr = NULL;
return data;
}

118
Linked list implementation of enqueue and dequeue
operations
• Example
– queue7.cpp
– Queue8.cpp

119
Deque (pronounced as Deck)
• Is a Double Ended Queue
• A stack allows insertion and deletion of elements at only one
end
• A queue allows insertion at one end and deletion at the other
end
• A deque allows insertion and deletion at both ends
• insertion and deletion can occur at either end
DequeueFront EnqueueRear
DequeueRear
EnqueueFront
Front Rear

120
• Has the following basic operations
– EnqueueFront – inserts data at the front of the list
– DequeueFront – deletes data at the front of the list
– EnqueueRear – inserts data at the end of the list
– DequeueRear – deletes data at the end of the list
• Implementation is similar to that of queue
• Is best implemented using doubly linked list

121
• Example
– Queue9.cpp
• Exercise
– How can a queue be represented as a C++ array?
– Write four O(1)-time procedures EnqueueFront,
DequeueFront, EnqueueRear and DequeueRear to insert
elements into and delete elements from both ends of a
deque constructed from an array. Make sure that the
functions work properly for the empty deque and that they
detect overflow and underflow

122
Priority Queue
• Using a queue ensures that customers are served in the exact
order in which they arrive
• However, we often want to assign priorities to customers and
serve the higher priority customers before those of lower
priority
• In fact, in many situations, simple queues are inadequate since
first-in-first-out scheduling has to be overruled using some
priority criteria
• Examples
– A hospital emergency room will handle most severely
injured patients first, even if they are not “first in line”
• In situations like this, a modified queue, or priority queue, is
needed

123
Priority Queue
• Is a data structure that stores entries along with a priority for
each entry
• Is a queue where each element has an associated key which
indicates priority of the elements
• This key is provided at the time of insertion
• Entries are removed in order of priorities
• The highest priority entry is removed first
• If there are several entries with equal high priorities, then the
one that was placed in the priority queue first is the one
removed first

124
Priority Queue
• That is, in priority queues, elements are dequeued according to
– Their priority and
– Their current queue position
• Elements arrive randomly to the priority queue
• Thus, there is no guarantee that the front elements will be the
most likely to be dequeued and the elements put at the end will
be the last candidates for dequeueing

125
Priority Queue
• A wide spectrum of possible criteria can be used to prioritize
elements in different cases
– Frquency of use
– Birth date
– Salary
– Postion
– Status
– gender

126
Priority queue enqueue and dequeue operations
• Dequeue operation deletes data having highest priority in
the list
• One of the previously used dequeue or enqueue operations
has to be modified
• Example:
– Consider the following queue of persons where females
have higher priority than males (gender is the key to give
priority).
Abebe Alemu Aster Belay Kedir Meron Yonas
Male Male Female Male Male Female Male

127
• Dequeue() deletes Aster
• Dequeue() deletes Meron
• Now the queue has data having equal priority and dequeue
operation deletes the front element like in the case of ordinary
queues.
Abebe Alemu Belay Kedir Meron Yonas
Male Male Male Male Female Male
Abebe Alemu Belay Kedir Yonas
Male Male Male Male Male

128
• Dequeue() deletes Abebe
• Dequeue() deletes Alemu
• Thus, in the above example the implementation of the
dequeue operation need to be modified.
Alemu Belay Kedir Yonas
Male Male Male Male
Belay Kedir Yonas
Male Male Male

129
Types of Priority queues
• There are two types of priority queues, ascending priority
queue and descending priority queue

130
Ascending Priority Queue
• Is a collection of items into which items can be inserted
arbitrarily and from which only the smallest item can be
removed
– The operation enqueue(pq,x) insrtes element x into the pq
and
– The operation dequeue(pq) removes the minimum element
from pq and returns its value the queue

131
Descending Priority Queue
• Is a collection of items into which items can be inserted
arbitrarily and from which only the largest item can be
removed
– The operation enqueue(pq,x) inserts element x into the pq
and is logically identical with the enqueue(pq,x) operation
for an ascending priority queue
– The operation dequeue(pq) removes the maximum element
from pq and returns its value

132
Priority queue
• Examples
– Queue10.cpp (array implementation)
– Queue11.cpp (ascending priority)
• Linked list implementation
– Queue12.cpp (descending priority)
– Queue13.cpp
– Queue14.cpp
– Queue15.cpp

133
Reading assignment
• Demerging queues
• Merging queues
• Application of queues
– Page 61

Chapter 5 Stack and Queue.pdf

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Chapter 5 Stack and Queue.pdf