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Queues

Briana B. Morrison

Adapted from Alan Eugenio

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Topics Define Queue

APIs

Applications Radix Sort

Simulation

Implementation Array based

Circular

Empty, one value, full

Linked list based Deques

Priority Queues

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A QUEUEIS A CONTAINER IN WHICH:

. INSERTIONS ARE MADE ONLY AT

THE BACK;

. DELETIONS, RETRIEVALS, AND

MODIFICATIONS ARE MADE ONLY

AT THE FRONT.

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The Queue

A Queue is a FIFO(First in First Out)

Data Structure.

Elements are inserted

in the Rear of the

queue and are

removed at the Front.

C

B C

A B C

A

backfront

push A

A Bfront back

push B

front back push C

front back

pop A

frontback

pop B

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PUSH(ALSO CALLED ENQUEUE) -- TO

INSERT AN ITEM AT THE BACK

POP(ALSO CALLED DEQUEUE) -- TO

DELETE THE FRONT ITEM

IN A QUEUE, THE FIRST ITEMINSERTED WILL BE THE FIRST ITEM

DELETED: FIFO (FIRST-IN, FIRST-OUT)

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METHOD INTERFACES

FOR THE

queueCLASS

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CLASS queue Constructor

queue();

Create an empty queue.

CLASS queue Operations

boolempty() const;

Check whether the queue is empty. Return true if it isempty and false otherwise.

T&front();

Return a reference to the value of the item at the font

of the queue.Precondition: The queue is not empty.

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const T&front() const;

Constant version of front().

voidpop();

Remove the item from the front of the queue.

Precondition: The queue is not empty.

Postcondition: The element at the front of the queueis the element that was addedimmediately after the element justpopped or the queue is empty.

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voidpush(const T& item);

Insert the argument item at the back of the queue.

Postcondition: The queue has a new item at the back

intsize() const;

Return the number of elements in the queue.

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THERE ARE NO ITERATORS!

THE ONLY ACCESSIBLE ITEM IS THEFRONT ITEM.

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DETERMINE THE OUTPUT FROM THE

FOLLOWING:

queue my_queue;

for(inti = 0; i < 10; i++)

my_queue.push (i * i);

while(!my_queue.empty())

{cout

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THE queueCLASS IS TEMPLATED:

template

TIS THE TEMPLATE PARAMETER

FOR THE ITEM TYPE. ContainerIS THE

TEMPLATE PARAMETER FOR THE

CLASS THAT WILL HOLD THE ITEMS,

WITH THE dequeCLASS THE DEFAULT.

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THE queueCLASS IS A CONTAINER

ADAPTOR. A CONTAINER ADAPTORC

CALLS THE METHODS FROM SOME

OTHER CLASS TO DEFINE THE

METHODS IN C.

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SPECIFICALLY, THE queueCLASS

ADAPTS ANY CONTAINER CLASS

THAT HAS push_back, pop_front, front,

size, AND emptyMETHODS.

deque?

list?

vector?

OK

OK

NOT OK: NO pop_frontMETHOD

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Implementing Queue: adapter of std::list

This is a simple adapter class, with following mappings:

Queuepushmaps topush_back

Queue frontmaps front

Queuepop

maps topop_front

...

This is the approach taken by the C++ standard library.

Any sequential container that supportspush_back,

front, andpop_frontcan be used. The list

The deque

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THE STANDARD C++ REQUIRES THAT

THE DEFINITION OF THE queueCLASS INCLUDE

protected:Container c;

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ALSO, THE METHOD DEFINITIONS

ARE PRESCRIBED. FOR EXAMPLE,

public:

voidpush (constvalue_type& x) { c.push_back (x)); }

voidpop( ) { c.pop_front( ); }

constT& front( ) const { returnc.front( ); }

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miniQ.push(10)

miniQ.push(25)

miniQ.push(50)

n = miniQ.front() // n = 10

miniQ.pop()

Queue Statement ListList StatementQueue

10

backfrontqlist.push_back(10)

10

backfront

10 25

backfront

qlist.push_back(25) 10 25

backfront

5010 25

backfront

qlist.push_back(50)10 25

front

50

back

return qlist.front() // return 10

25 50

backfront

qlist.pop_front() 25 50

miniQueue miniQ; // declare an empty queue

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IF EITHER A DEQUE OR LIST IS THE

UNDERLYING CONTAINER,

worstTime(n) IS CONSTANT FOR EACH

METHOD.

EXCEPTION: FOR THE pushMETHOD,IF A DEQUE IS THE UNDERLYING

CONTAINER, worstTime(n) IS LINEAR IN

n, AND averageTime(n) IS CONSTANT

(AND amortizedTime(n) IS CONSTANT).

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Applications of Queues

Direct applications

Waiting lists, bureaucracy

Access to shared resources (e.g., printer)

Multiprogramming

Indirect applications

Auxiliary data structure for algorithms

Component of other data structures

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APPLICATION OF QUEUES:

RADIX SORT

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The Radix SortOrder ten 2 digit numbers in 10 bins from smallestnumber to largest number. Requires 2 calls to the sortAlgorithm.

Initial Sequence: 91 6 85 15 92 35 30 22 39

Pass 0: Distribute the cards into bins according

to the 1's digit (100

).

9130

0

39

987

6

6

3515

85

543

22

92

21

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The Radix SortAfter Collection: 30 91 92 22 85 15 35 6 39

Pass 1: Take the new sequence and distribute thecards into bins determined by the 10's digit (101).

Final Sequence: 6 15 22 30 35 39 85 91 92

6

0

9291

9

85

87654

39

3530

3

22

2

15

1

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Radix Sort Use an array of queues (or vector of queues) as the buckets

void radixSort (vector& v, int d){

int i;

int power = 1;

queue digitQueue[10];

for (i=0;i < d;i++)

{

distribute(v, digitQueue, power);

collect(digitQueue, v);

power *= 10;

}

}

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// support function for radixSort()

// distribute vector elements into one of 10 queues

// using the digit corresponding to power

// power = 1 ==> 1's digit// power = 10 ==> 10's digit

// power = 100 ==> 100's digit

// ...

void distribute(const vector& v, queue digitQueue[],

int power){

int i;

// loop through the vector, inserting each element into

// the queue (v[i] / power) % 10for (i = 0; i < v.size(); i++)

digitQueue[(v[i] / power) % 10].push(v[i]);

}

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// support function for radixSort()

// gather elements from the queues and copy back to the vector

void collect(queue digitQueue[], vector& v)

{

int i = 0, digit;

// scan the vector of queues using indices 0, 1, 2, etc.

for (digit = 0; digit < 10; digit++)

// collect items until queue empty and copy items back

// to the vector

while (!digitQueue[digit].empty())

{

v[i] = digitQueue[digit].front();

digitQueue[digit].pop();

i++;

}

}

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APPLICATION OF QUEUES:

COMPUTER SIMULATION

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PHYSICAL MODEL : DIFFERS FROM

THE SYSTEM ONLY IN SCALE ORINTENSITY.

EXAMPLES: WAR GAMES, PRE-SEASON

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MATHEMATICAL MODEL : A SET OF

EQUATIONS, VARIABLES, ANDASSUMPTIONS

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A

500

D

? 200

500 C

B 400 E

ASSUMPTIONS: ANGLE ADC = ANGLE BCD

BEC FORMS A RIGHT TRIANGLE

DCE FORMS A STRAIGHT LINE

LINE SEGMENT AB PARALLEL TO DC

DISTANCE FROM A TO B?

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IF IT IS INFEASIBLE TO SOLVE THE

MATH MODEL (THAT IS, THE

EQUATIONS) BY HAND, A PROGRAM

IS DEVELOPED.

COMPUTER SIMULATION: THE

DEVELOPMENT OF COMPUTER

PROGRAMS TO SOLVE MATH MODELS

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DEVELOPSystem Computer

Model

VERIFY RUN

Interpretation Output DECIPHER

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IF THE INTERPRETATION DOES NOT

CORRESPOND TO THE BEHAVIOR OF

THE SYSTEM, CHANGE THE MODEL!

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Simulating Waiting Lines Using Queues

Simulation is used to study the performance:

Of a physical (real) system

By using a physical, mathematical, or computer model of

the system

Simulation allows designers to estimate

performance

Beforebuilding a system Simulation can lead to design improvements

Giving betterexpected performance of the system

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Simulating Waiting Lines Using Queues

Simulation is particular useful when:

Building/changing the system is expensive

Changing the system later may be dangerous

Often use computer models to simulate realsystems

Airline check-in counter, for example

Special branch of mathematics for these problems:Queuing Theory

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Simulate Strategies for Airline Check-In

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Simulate Airline Check-In We will maintain a simulated clock

Counts in integer ticks, from 0 At each tick, one or more events can

happen:

1.

Frequent flyer (FF) passenger arrives in line2. Regular (R) passenger arrives in line

3. Agent finishes, then serves next FFpassenger

4. Agent finishes, then serves next Rpassenger

5. Agent is idle (both lines empty)

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Simulate Airline Check-In

Simulation uses someparameters:

Max # FF served between regular passengers

Arrival rate of FF passengers

Arrival rate of R passengers

Service time

Desired output:

Statistics on waiting times, agent idle time, etc. Optionally, a detailed trace

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Design approach:

Agentdata type models airline agent

Passengerdata type models passengers

2 queue, 1 for FF, 1 for R OverallAirline_Checkin_Simclass

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QUEUE APPLICATION

A SIMULATED CAR WASH

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PROBLEM:

GIVEN THE ARRIVAL TIMES AT

SPEEDYS CAR WASH, CALCULATE THE

AVERAGE WAITING TIME PER CAR.

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ANALYSIS:

ONE WASH STATION

10 MINUTES FOR EACH CAR TO GET

WASHED

AT ANY TIME, AT MOST 5 CARS

WAITING TO BE WASHED; ANY OTHERS

TURNED AWAY AND NOT COUNTED

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AVERAGE WAITING TIME = SUM OF

WAITING TIMES / NUMBER OF CARS

IN A GIVEN MINUTE, A DEPARTURE IS

PROCESSED BEFORE AN ARRIVAL.

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IF A CAR ARRIVES WHEN NO CAR IS

BEING WASHED AND NO CAR IS

WAITING, THE CAR IMMEDIATELY

ENTERS THE WASH STATION.

A CAR STOPS WAITING WHEN IT

ENTERS THE WASH STATION.

SENTINEL IS 999.

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SYSTEM TEST 1:8

111113

14161620

999

TIME EVENT WAITING TIME

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TIME EVENT WAITING TIME8 ARRIVAL11 ARRIVAL11 ARRIVAL13 ARRIVAL14 ARRIVAL16 ARRIVAL16 ARRIVAL (OVERFLOW)

18 DEPARTURE 020 ARRIVAL28 DEPARTURE 7

38 DEPARTURE 1748 DEPARTURE 2558 DEPARTURE 3468 DEPARTURE 42

78 DEPARTURE 48

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AVERAGE WAITING TIME

= 173 / 7.0 MINUTES

= 24.7 MINUTES

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CAR WASH APPLET

http://www.cs.lafayette.edu/~collinsw/carwash/car.html

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Implementing a Queue Array based

Where is front? Where is top? Array suffers from rightward drift

To solve, use circular array

How are elements added, removed?

Using circular array, a new problem arises

What does empty look like?

What does single element look like?

What does full look like?

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Array-based Queue

Use an array of size Nin a circular fashion

Two variables keep track of the front and rear

f index of the front element

r index immediately past the rear element

Array location ris kept empty

Q0 1 2 rf

normal configuration

Q0 1 2 fr

wrapped-around configuration

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Implementing queueWith a Circular Array

Basic idea:Maintain two integer indices into an array

front: index of first element in the queue

rear: index of the last element in the queue

Elements thus fall at front through rearKey innovation:

If you hit the end of the array wrap aroundto slot 0

This prevents our needing to shift elements around

Still have to deal with overflow of space

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Implementing Queue With Circular Array

l Q h l A

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Implementing Queue With Circular Array

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Th d d

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The Bounded queue

A

BC

qfrontqback

Insert elements A,B, C

BC qfront

qback

Remove element A

D

C

qfront

qback

Insert element D,

Cqfront

qback

Remove element B

D

Cqfront

qback

Insert element D,

Cqfront qback

Insert element E

Circular ViewArray View

D E

Insert element D, Insert element E

C D

qfrontqback

E C D

qfrontqback

h d l

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Methods to Implement

i = (( i + 1) == max) ? 0 : (i + 1);

if (( i + 1) == max) i = 0; else i = i + 1;

i = ( i + 1) % max;

Q O i

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Queue Operations

We use the

modulo operator

(remainder of

division)

Algorithmsize()

return(N-f+r) mod N

AlgorithmisEmpty()

return(f=r)

Q0 1 2 r

f

Q0 1 2 fr

Q O i ( )

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Queue Operations (cont.)

Algorithmenqueue(o)

ifsize()=N-1then

throw FullQueueException

else

Q[r] or(r + 1) mod N

Operation enqueue throwsan exception if the array isfull

This exception isimplementation-dependent

Q0 1 2 r

f

Q0 1 2 fr

Q O i ( )

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Queue Operations (cont.)

Operation dequeuethrows an exception ifthe queue is empty

This exception isspecified in the queue

ADT

Algorithmdequeue()

ifisEmpty()then

throw EmptyQueueException

else

oQ[f]f(f + 1) mod N

returno

Q0 1 2 rfQ

0 1 2 fr

B d C di i

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Boundary Conditions

I l i C id i

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Implementation Considerations Thephysical model: a linear array with the front

always in the first position and all entries moved upthe array whenever the front is deleted.

A linear array with two indices always increasing.

A circular array with front and rear indices and one

position left vacant. A circular array with front and rear indices and a

Boolean flag to indicate fullness (or emptiness).

A circular array with front and rear indices and an

integer counter of entries. A circular array with front and rear indices taking

special values to indicate emptiness.

G bl A b d Q

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Growable Array-based Queue

In an enqueue operation, when the array is

full, instead of throwing an exception, we can

replace the array with a larger one

Similar to what we did for an array-basedstack

The enqueue operation has amortized running

time

O(n)with the incremental strategy O(1)with the doubling strategy

I l i Q

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Implementing a Queue

Linked List based

Where is front? Where is back?

How are elements added, removed?

Efficiency of operations

Q i h Si l Li k d Li

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Queue with a Singly Linked List

We can implement a queue with a singly linked list

The front element is stored at the first node

The rear element is stored at the last node

The space used is O(n)and each operation of the Queue

ADT takes O(1) time

f

r

nodes

elements

I l ti Q Si l Li k d Li t

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Implementing Queue: Singly-Linked List

This requires frontand rearNodepointers:

template

class queue {

. . .

private:// Insert implementation-specific data fields

// Insert definition of Node here

#include "Node.h"

// Data fields

Node* front_of_queue;Node* back_of_queue;

size_t num_items;

};

U i Si l Li k d Li t t

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Using a Single-Linked List to

Implement a Queue (continued)

Impl m ti Q Si l Li k d Li t

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Implementing Queue: Singly-Linked List

Insert at tail, usingback_of_queuefor speed

Remove using front_of_queue

Adjust sizewhen adding/removing

No need to iterate through to determine size

A l i f th Sp /Ti I

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Analysis of the Space/Time Issues

Time efficiency of singly- or doubly-linked list good:

O(1) for all Queueoperations

Space cost: ~3 extra words per item

vector uses 1 word per item when fully packed

2 words per item when just grown

On average ~1.5 words per item, for larger lists

Comparing the Three Implementations

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Comparing the Three Implementations

All three are comparable in time: O(1) operations

Linked-lists require more storage

Singly-linked list: ~3 extra words / element

Doubly-linked list: ~4 extra words / element

Circular array: 0-1 extra word / element

On average, ~0.5 extra word / element

Anal sis of the Space/Time Iss es

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Analysis of the Space/Time Issues

vectorImplementation Insertion at end of vector is O(1), on average

Removal from the front is linear time: O(n)

Removal from rear of vector is O(1) Insertion at the front is linear time: O(n)

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A DEQUE IS A FINITE SEQUENCE OF

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A DEQUEIS A FINITE SEQUENCE OF

ITEMS SUCH THAT

1.GIVEN ANY INDEX, THE ITEM INTHE SEQUENCE AT THAT INDEXCAN BE ACCESSED OR MODIFIEDIN CONSTANT TIME.

2.AN INSERTION OR DELETION AT

THE FRONTOR BACK OF THESEQUENCE TAKES ONLYCONSTANT TIME, ON AVERAGE.

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THE dequeCLASS IS VERY SIMILAR TO

THE vectorCLASS.

The deque

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The deque

The deque is an abstract data type thatcombines the features of a stack and a

queue.

The name deque is an abbreviation fordouble-ended queue.

The C++ standard defines the deque as a

full-fledged sequential container that supportsrandom access.

The deque class

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The deque class

Member Function Behaviorconst Item_Type&

operator[](size_t index)const;

Item_Type&

operator[](size_t index)

Returns a reference to the element at positionindex.

const Item_Type&

at(size_t index) const;

Item_Type&

at(size_t index)

Returns a reference to the element at positionindex. If indexis not valid, theout_of_rangeexception is thrown.

iterator insert(iterator pos,

const Item_Type& item)

Inserts a copy of iteminto the dequeatposition pos. Returns an iteratorthatreferences the newly inserted item.

iterator erase(iterator pos) Removes the item from the dequeat

position pos. Returns an iteratorthatreferences the item following the one erased.

void remove(const Item_Type&

item)

Removes all occurrences of itemfrom thedeque.

The deque class (2)

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The deque class (2)

Member Function Behaviorvoid push_front(const Item_Type&

item)

Inserts a copy of itemas the first element ofthe deque

void push_back(const Item_Type&

item)

Inserts a copy of itemas the last element ofthe deque.

void pop_front() Removes the first item from the deque.void pop_back() Removes the last item from the deque.

Item_Type& front();

const Item_Type& front() const

Returns a reference to the first item in thedeque.

Item_Type& back();

const Item_Type& back() const

Returns a reference to the last item in thedeque.

iterator begin() Returns an iteratorthat references thefirst item of the deque.

const_iterator begin() const Returns a const_iteratorthat referencesthe first item of the deque.

The deque class (3)

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The deque class (3)

Member Function Behavioriterator end() Returns an iteratorthat references one

past the last item of the deque.

const_iterator end() const Returns a const_iteratorthat referencesone past the last item of the deque.

void swap(deque

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TO TAKE ADVANTAGE OF INSERTIONSAND DELETIONS AT THE FRONT OF ADEQUE, THERE ARE TWO METHODS:

// Postcondition: A copy of x has been inserted at the front// of this deque. The averageTime(n) is// constant, and worstTime (n) is O(n) but// for n consecutive insertions,

// worstTime(n) is only O(n). That is,// amortizedTime(n) is constant.voidpush_front (constT& x);

// Postcondition: The item at the front of this deque has// been deleted.voidpop_front( );

THE deque CLASS ALSO HAS ALL OF

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THE dequeCLASS ALSO HAS ALL OF

THE METHODS FROM THE vectorCLASS

(EXCEPT FOR capacityAND reserve),

AND THE INTERFACES ARE THE SAME!

deq e ords

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dequewords; string word;

for(inti = 0; i < 5; i++) { cin >> word; words.push_back (word);

} // for words.pop_front( ); words.pop_back( ); for(unsignedi = 0; i < words.size(); i++)

cout

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ON PAPER, THAT IS, LOOKING AT BIG-

O TIME ESTIMATES ONLY, A DEQUE IS

SOMETIMES FASTER AND NEVER

SLOWER THAN A VECTOR.

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The Standard Library Implementation

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The Standard Library Implementation

The standard library uses a randomlyaccessible circular array.

Each item in the circular array points to a

fixed size, dynamically allocated array thatcontains the data.

The advantage of this implementation is that when

reallocation is required, only the pointers need to

be copied into the new circular array.

The Standard Library Implementation (2)

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The Standard Library Implementation (2)

IN THE HEWLETT-PACKARD DESIGN

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IN THE HEWLETT PACKARD DESIGN

AND IMPLEMENTATION OF THE deque

CLASS, THERE IS A mapFIELD: A

POINTER TO AN ARRAY. THE ARRAY

ITSELF CONTAINS POINTERS TOBLOCKS THAT HOLD THE ITEMS. THE

startAND finishFIELDS ARE

ITERATORS.

map start

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finish

yes

true now

good

loveclear

right

FOR CONTAINER CLASSES IN

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FOR CONTAINER CLASSES IN

GENERAL, ITERATORS ARE SMART

POINTERS. FOR EXAMPLE operator++

ADVANCES TO THE NEXT ITEM,

WHICH MAY BE AT A LOCATION FAR

AWAY FROM THE CURRENT ITEM.

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DEQUE ITERATORS ARE GENIUSES!

THE iterator CLASS EMBEDDED IN THE

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THE iteratorCLASS EMBEDDED IN THEdequeCLASS HAS FOUR FIELDS:

1.first, A POINTER TO THEBEGINNING OF THE BLOCK;

2.current, A POINTER TO AN ITEM;3.last, A POINTER TO ONE BEYOND

THE END OF THE BLOCK;

4.node, A POINTER TO THELOCATION IN THE MAP ARRAYTHAT POINTS TO THE BLOCK.

map start

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finish

yes

true

now

good

love

clearright

SUPPOSE THE DEQUE SHOWN IN THE

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SUPPOSE THE DEQUE SHOWN IN THE

PREVIOUS SLIDE IS words. HOW DOESRANDOM ACCESS WORK? FOR

EXAMPLE, HOW IS words [5]

ACCESSED?

1.BLOCK NUMBER

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= (index + offset of first item in first block) / block size

= (5 +start.current start.first

) / 4

= (5 + 2) / 4

= 1

2.OFFSET WITHIN BLOCK

= (index+offset of first item in first block) % block size

= 7 % 4

= 3

THE ITEM AT words [5] AT AN OFFSET

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THE ITEM AT words [5], AT AN OFFSET

OF 3 FROM THE START OF BLOCK 1, ISclear.

FOR AN INSERTION OR REMOVAL AT

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FOR AN INSERTION OR REMOVAL AT

SOME INDEX iIN THE INTERIOR OF A

DEQUE, THE NUMBER OF ITEMS

MOVED IS THE MINIMUM OF iAND

length i. THE lengthFIELD CONTAINS

THE NUMBER OF ITEMS.

FOR EXAMPLE TO INSERT AT words [5]

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FOR EXAMPLE, TO INSERT AT words [5],

THE NUMBER OF ITEMS MOVED IS 7 5

= 2.

TO DELETE THE ITEM AT INDEX 1, THE

NUMBER OF ITEMS MOVED IS 1.

EXERCISE: SUPPOSE FOR SOME deque

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EXERCISE: SUPPOSE, FOR SOME deque

CONTAINER, BLOCK SIZE = 10 AND

THE FIRST ITEM IS AT INDEX 7 IN

BLOCK 0. DETERMINE THE BLOCK

AND OFFSET FOR INDEX 31.

Priority Queue

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y Q

Job # 3

Clerk

Job # 4

Supervisor

Job # 2

President

Job # 1

Manager

A Special form of queue from which items are

removed according to their designated priority and

not the order in which they entered.

Items entered the queue in sequential order but will be

removed in the order #2, #1, #4, #3.

CLASS priority_queue Constructor

i it ()

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priority_queue();

Create an empty priority queue. Type T must

implement the operator

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voidpush(const T& item);

Insert the argument item into the priority queue.

Postcondition: The priority queue contains a newelement.

intsize() const;

Return the number of items in the priority queue.

T&top();

Return a reference to the item having the highestpriority.

Precondition: The priority queue is not empty.const T&top();

Constant version of top().

PQ Implementation

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Q p

How would you implement a priority queue?

Summary Slide 1

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10

3

y- Queue

- A first-come-first-served data structure.- Insertion operations (push()) occur at the back of the

sequence

- deletion operations (pop()) occur at the front of the

sequence.

Summary Slide 2

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10

4

y- The radix sort algorithm

- Orders an integer vector by using queues (bins).- This sorting technique has running time O(n) but

has only specialized applications.

- The more general in-place O(n log2n) sortingalgorithms are preferable in most cases.

Summary Slide 3

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10

5

y- Implementing a queue with a fixed-size

array- Indices qfront and qback move circularly around

the array.

- Gives O(1) time push() and pop() operations withno wasted space in the array.

Summary Slide 4

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y- Priority queue

- Pop() returns the highest priority item (largest orsmallest).

- Normally implemented by a heap, which isdiscussed later in the class.

- The push() and pop() operations have runningtime O(log2n)

Queues 2

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