Syllabus of Data Structure - Sekolah Tinggi Teknik Surabaya

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Priority Queue
Erick, Eka, Reddy
© Sekolah Tinggi Teknik Surabaya
» 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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» A priority queue stores a » Additional methods
collection of entries /
˃ min()
returns, but does not
node
remove, an node with
» Each node is a pair
smallest key
(key, value)
˃ size(), isEmpty()
» 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
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Applications:
˃ Standby flyers
˃ Auctions
˃ Stock market
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» Suppose that you have a few assignments
from different courses. Which assignment
will you want to work on first?
Course
Database Systems
Introduction to C++
Data Structure & Algorithm
Structured Systems
Analysis
Priority
2
4
1
3
Due day
October 3
October 10
September 29
October 7
» You set your priority based on due days.
Due days are called keys.
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Student Name
Bill Scott
Bob Jones
Alan Smith
Susan Kane
Student Number
110102
110140
110243
110176
Final Score
65
76
86
80
» Any of the attributes, Student Name,
Student Number, or Final Score can be
used as keys.
» Note: Keys may not be unique (Final
Score).
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Brand
Motormaster
Goodyear
Michelin
Price
$61.49
$98.99
$101.99
Warranty (km)
110,000
220,000
150,000
» 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.
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» Keys in a priority
» Mathematical concept
queue can be
of total order relation 
˃ Reflexive property:
arbitrary objects on
xx
which an order is
˃ Antisymmetric property:
defined
xyyxx=y
» Two distinct entries in ˃ Transitive property:
xyyzxz
a priority queue can
have the same key
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» 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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» 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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» 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:
insertItem(k,e): Insert an element e with key k into P.
removeMin(): Return and remove from P an element with
the smallest key.
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» We can use a priority
queue to sort a set of
comparable elements
»
Algorithm PQ-Sort(S, C)
Input sequence S, comparator C for
the elements of S
Output sequence S sorted in
1. Insert the elements one
increasing order according to C
by one with a series of
P  priority queue with
insert operations
comparator C
2. Remove the elements
while not S.isEmpty ()
in sorted order with a
e  S.removeFirst ()
series of removeMin
P.insert (e, 0)
operations
while not P.isEmpty()
e  P.removeMin().key()
The running time of
S.insertLast(e)
this sorting method
depends on the priority
queue implementation
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» We have a sequence S with 6 integers
(elements). Also we have an empty
priority queue P.
S
4
0
7
8
2 1
P
» While S is not empty insert to P
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» After the first step, all the elements are
now in the priority queue, with
themselves as keys.
S
P
0,0
1,1
2,2
4,4
7,7
8,8
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» 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.
S
0 1
2
4,4
7,7
P
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8,8
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» After the second step: Now the
elements are sorted in S.
S
0 1
2 4
7
8
P
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» The priority queue abstract data type
supports the following methods:
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)
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» The following table shows a series of operations and
their effects on an initially empty priority queue P.
Operation
insertItem(5,A)
insertItem(9,C)
insertItem(3,B)
insertItem(7,D)
minElement()
minKey()
removeMin()
size()
removeMin()
removeMin()
removeMin()
removeMin()
isEmpty()
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Output
B
3
B
3
A
D
C
“error”
true
Priority Queue
{(5,A)}
{(5,A),(9,C)}
{(3,B),(5,A),(9,C)}
{(3,B),(5,A),(7,D),(9,C)}
{(3,B),(5,A),(7,D),(9,C)}
{(3,B),(5,A),(7,D),(9,C)}
{(5,A),(7,D),(9,C)}
{(5,A),(7,D),(9,C)}
{(7,D),(9,C)}
{(9,C)}
{}
{}
{}
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» Node Class
» Icomparable ADT
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» A node in a priority
queue is simply a keyvalue 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
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» As C# class:
public class Node {
public int key;
public T value;
}
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» A comparator encapsulates
the action of comparing
two objects according to a » The primary method of
given total order relation
the Comparator ADT:
» A generic priority queue
˃ compare(x, y): Returns
uses an auxiliary
an integer i such that i <
0 if a < b, i = 0 if a = b,
comparator
and i > 0 if a > b; an error
» The comparator is external
occurs if a and b cannot
to the keys being compared
be compared.
» When the priority queue
needs to compare two keys,
it uses its comparator
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» 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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» 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.
P
1,1
4,4
0,0
2,2
7,7
8,8
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» 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.
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» 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)).
P
0,0
1,1
2,2
4,4
7,7
8,8
6,6
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© Sekolah Tinggi Teknik Surabaya
» Assume that the size of the sequence is n.
Method
Unsorted S Sorted S
size, isEmpty
fast
fast
O(n)
insertItem
fast
minElement, minKey, removeMin O(n)
fast
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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
Person
Student
Undergraduate
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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
Person
public class Student : Person
{
private int m_ID;
public Student(string name, int age, int id)
:base(name, age)
{
this.m_ID = id;
}
Student
// constructor
}
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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()
{ … }
public class Student : Person
{
.
.
.
// dynamically-bound
public virtual decimal Salary()
{ … }
public new string HomeAddress()
{ … }
}
public override decimal Salary()
{ … }
}
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All classes inherit from System.Object
System-defined types
Object
User-defined types
String
Array
Primitive types
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Boolean
Single
Byte
Double
Int16
Decimal
Int32
DateTime
Int64
TimeSpan
Char
Guid
ValueType
Exception
Delegate
Class1
Enum
Structure1
Multicast
Delegate
Class2
Delegate1
Class3
Enum1
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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(…);
.
.
.
public interface IComparable
{
int CompareTo(object obj);
}
Array.Sort(students);
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To be a sortable object…
• Sortable objects must implement IComparable
• Example:
– Student objects sort by id
base class
interface
public class Student : Person, IComparable
{
private int m_ID;
.
.
.
Person
Student
int IComparable.CompareTo(Object obj)
{
Student other;
other = (Student) obj;
return this.m_ID – other.m_ID;
}
}
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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
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