Priority Queue Erick, Eka, Reddy © Sekolah Tinggi Teknik Surabaya 1.

Data Structure

Priority QueueErick, Eka, Reddy

© Sekolah Tinggi Teknik Surabaya

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Outlines» Priority Queue ADT

+ Keys, Priorities, and Total order Relations

+ Sorting with a Priority Queue

» Priority Queue implementation+ Implementation with an unsorted

sequence+ Implementation with a sorted sequence

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Priority Queue Concept

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» A priority queue stores a collection of entries / node

» Each node is a pair(key, value)

» Main methods of the Priority Queue ADT˃ insert(k, x)

inserts an node with key k and value x

˃ removeMin()removes and returns the node with smallest key

» Additional methods˃ min()

returns, but does not remove, an node with smallest key

˃ size(), isEmpty()

» Applications:˃ Standby flyers˃ Auctions˃ Stock market

Keys (1)» Suppose that you have a few assignments

from different courses. Which assignment will you want to work on first?

» You set your priority based on due days. Due days are called keys. 4


Course Priority Due dayDatabase Systems 2 October 3Introduction to C++ 4 October 10Data Structure & Algorithm 1 September 29Structured Systems Analysis

3 October 7

Example: Student records

» Any of the attributes, Student Name, Student Number, or Final Score can be used as keys.

» Note: Keys may not be unique (Final Score). 5


Student Name Student Number Final ScoreBill Scott 110102 65Bob Jones 110140 76Alan Smith 110243 86Susan Kane 110176 80

Example (2)

» Suppose we are looking for tires for a passenger car. How can we weight the tires so we can select the tires?

» The key may consist of not only one attribute such as price. In fact, we want to consider factors such as brands and warranty as well.

» So the key may be more complex than just one value.6


Brand Price Warranty (km)Motormaster $61.49 110,000Goodyear $98.99 220,000Michelin $101.99 150,000

Total Order Relations

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» Keys in a priority queue can be arbitrary objects on which an order is defined

» Two distinct entries in a priority queue can have the same key

» Mathematical concept of total order relation ˃ Reflexive property:

x x˃ Antisymmetric property:

x y y x x = y˃ Transitive property:

x y y z x z

Total Order Relations (2)» Not everything can have a total order relation.» Non-example: » For 2-D vectors v1 = (x1, x2) and v2 = (x3, x4), define the following

ordering rule:

» v1 v2 if x2 - x1 == x4 - x3

» Then we have » 4 - 1 = 7 - 4

» Therefore, (1, 4) (4, 7) and (4, 7) (1, 4).

» But (1,4) (7, 4), namely, the relation does not satisfy the» antisymmetric property.

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Comparison» If a comparison rule defines a total order relation,

it will never lead to a comparison contradiction.

» the smallest key: If we have a finite number of elements with a total order relation, then the smallest key, denoted by kmin, is well-defined: kmin is the key that satisfies kmin k for any other key k.

» Being able to find the smallest key is very important because in many cases, we want to have the element with the smallest key.

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PQ Summary» priority queue: A container of

elements, each having an associated key that is provided at the time the element is inserted.

» The two fundamental methods of a priority queue P:

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insertItem(k,e): Insert an element e with key k into P.

removeMin(): Return and remove from P an element with the smallest key.

Priority Queue Sorting

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» We can use a priority queue to sort a set of comparable elements1. Insert the elements one

by one with a series of insert operations

2. Remove the elements in sorted order with a series of removeMin operations

» The running time of this sorting method depends on the priority queue implementation

Algorithm PQ-Sort(S, C)Input sequence S, comparator C for the elements of SOutput sequence S sorted in increasing order according to CP priority queue with comparator Cwhile not S.isEmpty ()

e S.removeFirst ()P.insert (e, 0)

while not P.isEmpty()e P.removeMin().key()

S.insertLast(e)

Sorting Example (1)» We have a sequence S with 6 integers

(elements). Also we have an empty priority queue P.

» While S is not empty insert to P 12


4 0 7 8 2 1

S

P

Sorting Example : Step 1» After the first step, all the elements are

now in the priority queue, with themselves as keys.

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S

4,40,0 7,7 8,82,21,1

P

Sorting Example : Step 2» The first three elements have been

extracted from P and inserted into S in order. The element with the smallest key in P now is 4,4, which will be extracted next time.

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0 21

S

4,4 7,7 8,8

P

Sorting Example : Final» After the second step: Now the

elements are sorted in S.

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40 7 821

S

P

Methods of a Priority Queue» The priority queue abstract data type

supports the following methods:

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size(): Return the number of elements in P.Input: None; Output: Integer

isEmpty(): Test whether P is empty.Input: None; Output: Boolean

inserItem(k,e): Insert a new element e with key k into P.Input: Objects k (key) and e (element); Output: None

minElement(): Return (but do not remove) an element of P with the smallest key; an error condition occurs if the priority queue is empty.Input: None; Output: Object (element)

minKey(): Return a smallest key in P; an error condition occurs if the priority queue is empty.Input: None; Output: Object (key)

removeMin(): Remove from P and return an element with the smallest key; an error condition occurs if the priority queue is empty.Input: None; Output: Object (element)

Example» The following table shows a series of operations and

their effects on an initially empty priority queue P.

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Operation Output Priority QueueinsertItem(5,A) - {(5,A)}insertItem(9,C) - {(5,A),(9,C)}insertItem(3,B) - {(3,B),(5,A),(9,C)}insertItem(7,D) - {(3,B),(5,A),(7,D),(9,C)}minElement() B {(3,B),(5,A),(7,D),(9,C)}minKey() 3 {(3,B),(5,A),(7,D),(9,C)}removeMin() B {(5,A),(7,D),(9,C)}size() 3 {(5,A),(7,D),(9,C)}removeMin() A {(7,D),(9,C)}removeMin() D {(9,C)}removeMin() C {}removeMin() “error” {}isEmpty() true {}

Implementing Priority Queue» Node Class» Icomparable ADT

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Node Class

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» A node in a priority queue is simply a key-value pair

» Priority queues store nodes to allow for efficient insertion and removal based on keys

» Methods:˃ key(): returns the key for

this node˃ value(): returns the value

associated with this node

» As C# class:public class Node { public int key; public T value;}

Comparator ADT

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» A comparator encapsulates the action of comparing two objects according to a given total order relation

» A generic priority queue uses an auxiliary comparator

» The comparator is external to the keys being compared

» When the priority queue needs to compare two keys, it uses its comparator

» The primary method of the Comparator ADT:˃ compare(x, y): Returns

an integer i such that i < 0 if a < b, i = 0 if a = b, and i > 0 if a > b; an error occurs if a and b cannot be compared.

PQ with a unsorted sequence» Let S be a sequence. A pair of key and

element is denoted as p=(k,e). With an unsorted sequence, we use the method insertLast(p) of S to implement insertItem(k,e) of the priority queue P.

» To perform operations including minElement, minKey, and removeMin, we have to inspect all the elements of the sequence S to find the element with the smallest key.

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PQ Sorted Example» Assume we have the elements stored in an

unsorted sequence show here. » To perform the removeMin() operation, we

have to inspect all elements to find the element (0,0) that has the smallest key.

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4,4 0,0 7,7 8,82,21,1

P

PQ with a sorted sequence» Let S be a sequence. A pair of key and

element is denoted as p=(k,e). With an sorted sequence, we can easily extract the element with the smallest key with the combination of methods remove() and first() of S.

» However, to perform operation insertItem, we need to scan through the sequence S to find the apropriate position to insert the new element and key. 23


PQ Sorted Example» To insert the pair (6,6), we have to scan

through the sequence until we find the right place (between (4,4) and (7,7)).

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4,40,0 7,7 8,82,21,1

P

6,6

Comparing the Two Implementations

Method Unsorted S Sorted Ssize, isEmpty fast fastinsertItem fast O(n)minElement, minKey, removeMin O(n) fast

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» Assume that the size of the sequence is n.

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Inheritance

• Use in the small, when a derived class "is-a" base class– enables code reuse– enables design reuse & polymorphic programming

• Example:– a Student is-a Person

Undergraduate

Person

Student Employee

Graduate Staff Faculty

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Implementation

• C# supports single inheritance– public inheritance only (C++ parlance)– base keyword gives you access to base class's members

public class Student : Person{ private int m_ID;

public Student(string name, int age, int id) // constructor :base(name, age) { this.m_ID = id; }}

Student

Person

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Binding

• C# supports both static and dynamic binding– determined by absence or presence of virtual keyword– derived class must acknowledge with new or override

public class Person{ . . . // statically-bound public string HomeAddress() { … }

// dynamically-bound public virtual decimal Salary() { … }}

public class Student : Person{ . . . public new string HomeAddress() { … }

public override decimal Salary() { … }}

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All classes inherit from System.Object

St r i ng Ar r ay Val ueType Except i on Del egat e Cl ass1

Mul t i castDel egat e

Cl ass2

Cl ass3

Obj ect

Enum1

St r uct ur e1EnumPr i mi t i ve t ypes

Bool ean

Byt e

I nt 16

I nt 32

I nt 64

Char

Si ngl e

Doubl e

Deci mal

Dat eTi me

System-defined types

User-defined types

Del egat e1

Ti meSpan

Gui d

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Part 4

• Interfaces…

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Interfaces

• An interface represents a design

• Example:– the design of an object for iterating across a data structure– interface = method signatures only, no implementation details!– this is how foreach loop works…

public interface IEnumerator{ void Reset(); // reset iterator to beginning bool MoveNext(); // advance to next element object Current { get; } // retrieve current element}

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Why use interfaces?

• Formalize system design before implementation– especially helpful for PITL (programming in the large)

• Design by contract– interface represents contract between client and object

• Decoupling– interface specifies interaction between class A and B– by decoupling A from B, A can easily interact with C, D, …

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.NET is heavily influenced by interfaces

• IComparable• ICloneable• IDisposable• IEnumerable & IEnumerator• IList• ISerializable• IDBConnection, IDBCommand, IDataReader• etc.

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Example

• Sorting– FCL contains methods that sort for you– sort any kind of object– object must implement IComparable

object[] students;

students = new object[n];students[0] = new Student(…);students[1] = new Student(…);...

Array.Sort(students);

public interface IComparable{ int CompareTo(object obj);}

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To be a sortable object…

• Sortable objects must implement IComparable• Example:

– Student objects sort by id

public class Student : Person, IComparable{ private int m_ID; . . .

int IComparable.CompareTo(Object obj) { Student other; other = (Student) obj; return this.m_ID – other.m_ID; }}

base class

interface

Student

Person

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Summary

• Object-oriented programming is *the* paradigm of .NET• C# is a fully object-oriented programming language

– fields, properties, indexers, methods, constructors– garbage collection– single inheritance– interfaces

• Inheritance?– consider when class A "is-a" class B– but you only get single-inheritance, so make it count

• Interfaces?– consider when class C interacts with classes D, E, F, …– a class can implement any number of interfaces

Priority Queue Erick, Eka, Reddy © Sekolah Tinggi Teknik Surabaya 1.

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