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Basic Data Structures
Page 1
BFH-TI: Softwareschule Schweiz
Algorithms and Data Structures
Basic Data Structures
Dr. Rolf Haenni
CAS SD01
Berner Fachhochschule
Technik und Informatik
Rolf Haenni
Algorithms and Data Structures
Basic Data Structures
Page 2
Outline
Data Structures and Abstract Data Types
Linear Data Structures
Implementing Linear Data Structures
Implementation with Arrays
Implementation with Growable Arrays
Implementation with Linked Lists
Iterators
Berner Fachhochschule
Technik und Informatik
Rolf Haenni
Algorithms and Data Structures
Data Structures and Abstract Data Types
Basic Data Structures
Page 3
Outline
Data Structures and Abstract Data Types
Linear Data Structures
Implementing Linear Data Structures
Implementation with Arrays
Implementation with Growable Arrays
Implementation with Linked Lists
Iterators
Berner Fachhochschule
Technik und Informatik
Rolf Haenni
Algorithms and Data Structures
Basic Data Structures
Data Structures and Abstract Data Types
Page 4
Data Structures
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A data structure is a way of storing complex data in a
computer so that it can be used efficiently
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Carefully chosen data structures are crucial for building
efficient algorithms
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Therefore, the quality and performance of large systems
depends heavily on choosing the best data structure
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Different data structures are suited to different kinds of
applications, and some are highly specialized to certain tasks
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Many basic data structures are included in standard libraries
of modern programming languages (e.g. Java Collection API)
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The fundamental building blocks of most data structures are
arrays, records, and references
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Algorithms and Data Structures
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Data Structures and Abstract Data Types
Page 5
Abstract Data Types
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An abstract data type (ADT) is an abstraction of a data
structure
An ADT specifies
Ý Data stored
Ý Operations on the data
Ý Error conditions associated with the data
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The object-oriented programming paradigm supports the
creation of complex ADTs
Ý ADTs are specified as interfaces
Ý ADTs are implemented as classes (which themselves
implement the ADT interface)
Ý Concrete data structures are wrapped into objects of
corresponding classes
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Algorithms and Data Structures
Basic Data Structures
Data Structures and Abstract Data Types
Page 6
Properties of Well-Designed ADTs
Universality The same ADT can be used in different programs
Encapsulation The interface provides an impenetrable barrier
Simplicity The implementation details are entirely hidden
Integrity Internal data is protected against improper use or
bugs
Flexibility The internal implementation can be changed
without affecting the main application(s)
Modularity Important sub-problems are solved independently
Berner Fachhochschule
Technik und Informatik
Rolf Haenni
Algorithms and Data Structures
Linear Data Structures
Basic Data Structures
Page 7
Outline
Data Structures and Abstract Data Types
Linear Data Structures
Implementing Linear Data Structures
Implementation with Arrays
Implementation with Growable Arrays
Implementation with Linked Lists
Iterators
Berner Fachhochschule
Technik und Informatik
Rolf Haenni
Algorithms and Data Structures
Linear Data Structures
Basic Data Structures
Page 8
Linear Data Structures
A linear data structure is a collection of linearly arranged elements
with various ways to access its elements
Stack Insert/remove elements at one end of the collection
Queue Insert/remove elements at different ends of the
collection
Vector Access elements w.r.t. the rank within the collection
List Access elements w.r.t. the position within the collection
Sequence Access elements w.r.t. both ranks and positions
The elements of a linear data structure have all the same “rights”
(no priorities, no hierarchy)
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Algorithms and Data Structures
Linear Data Structures
Basic Data Structures
Page 9
The Stack ADT
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The stack ADT is a linear data structure that stores arbitrary
elements according to the last-in-first-out (LIFO) scheme
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Thus insertion and deletions take place at the same “end” of
the data structure
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Think of a spring-loaded plate dispenser
Applications of stacks
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Ý
Ý
Ý
Ý
Ý
Visited-page history in a web browser
Undo sequence in a text editor
Towers of Hanoi problem
Chain of method calls in the Java Virtual Machine (JVM)
Parsing of arithmetic expressions
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Page 10
The Queue ADT
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The queue ADT is a linear data structure that stores arbitrary
elements according to the first-in-first-out (FIFO) scheme
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Thus insertion and deletions take place at the opposite “ends”
of the data structure
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Think of the queue at the airport security check
Applications of queues
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Ý Waiting lists
Ý Access to shared resources (e.g. a printer)
Ý Multiprogramming
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The Vector ADT
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The vector ADT extends the notion of an array by storing a
sequence of arbitrary elements
Ý An element can be accessed, inserted, or removed by specifying
its rank ∈ {0, 1, 2, . . .} = number of elements preceding it
Ý The size (number of stored elements) of a vector changes
when elements are inserted or deleted
Ý Proper vectors have no fixed maximal size
Ý The ranks of some elements may change when elements are
inserted or deleted
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Think of a ranking in ski racing events
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An exception is thrown if an incorrect rank is specified (e.g. a
negative rank)
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The List ADT
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The list ADT models a linear sequence of positions
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Each position stores an arbitrary element
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List manipulations are always performed relative to some
given positions
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Special positions are the first and the last position in the list
To be as general as possible, we need a position ADT with
two simple operations:
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Ý getElement(): returns the element stored at the position
Ý setElement(e): sets the stored element to e
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The position ADT gives a unified view of diverse ways of
storing data (cell of an array, node of a linked list, etc.)
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A list establishes a before/after relation between positions
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Page 13
The Sequence ADT
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The sequence ADT is the union of the vector and the list ADT
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Elements can be accessed by their rank and/or their position
To transform ranks into positions and vice versa, two “bridge”
operators are needed
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Ý atRank(r): returns the position at rank r
Ý rankOf(p): returns the rank of a position p
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The sequence ADT is thus a general-purpose data structure
for storing linearly ordered collections of elements
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Stacks, queues, vectors, and lists are included as special cases
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Algorithms and Data Structures
Linear Data Structures
Basic Data Structures
Page 14
Operations for Linear Structures I
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General Operations
All: isEmpty(), size()
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Accessing Elements/Positions
Stack: top()
Queue: front()
Vector: elemAtRank(r)
List: first(), last(), before(p), after(p)
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Inserting Elements
Stack: push(e)
Queue: enqueue(e)
Vector: insertAtRank(r,e)
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Page 15
Operations for Linear Structures II
List: insertFirst(e), insertLast(e),
insertBefore(p,e), insertAfter(p,e)
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Removing Elements
Stack: pop()
Queue: dequeue()
Vector: removeAtRank(r)
List: removeElement(p)
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Replacing/Swaping Elements
Vector: replaceAtRank(r,e), swapAtRanks(r,q)
List: replaceElement(p,e), swapElements(p,q)
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Rank/position conversion
Sequence: atRank(r), rankOf(p)
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Page 16
UML Diagram
<<interface>>
<<interface>>
BasicSequence
size()
isEmpty()
<<interface>>
Stack
push(e)
pop()
top()
<<interface>>
Queue
enqueue(e)
dequeue()
front()
<<interface>>
Vector
elemAtRank(r)
insertAtRank(r,e)
removeAtRank(r)
replaceAtRank(r)
swapAtRanks(r,q)
<<interface>>
Sequence
atRank(r)
rankOf(p)
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Position
getElement()
setElement(e)
<<interface>>
List
first()
last()
before(p)
after(p)
isFirst(p)
isLast(p)
insertFirst(e)
insertLast(e)
insertBefore(p,e)
insertAfter(p,e)
removeElement(p)
replaceElement(p,e)
swapElements(p,q)
Rolf Haenni
Algorithms and Data Structures
Linear Data Structures
Basic Data Structures
Page 17
BasicSequence Interface in Java
public interface BasicSequence {
public int size();
public boolean isEmpty();
}
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For more information on Java interfaces, see §6.10 in “Java
ist auch eine Insel”
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Page 18
Stack Interface in Java
public interface Stack extends BasicSequence {
public Object top()
throws EmptyStackException;
public void push(Object e);
public Object pop()
throws EmptyStackException;
}
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I
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Stack inherits general methods from BasicSequence
Requires the definition of a class EmptyStackException
Generic stacks of a particular type can be defined using a
technique called Java Generics (see §6.12 in “Java ist auch
eine Insel”)
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Page 19
Sequence Interface in Java
public interface Sequence extends Vector, List {
public Position atRank(int r);
public Position rankOf(Position p);
}
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Example of multiple inheritance of interfaces (not allowed for
classes)
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Page 20
The Collection Interface
The java.util package provides some predefined interfaces and
classes
<<interface>>
<<interface>>
Iterable
Collection
<<interface>>
<<interface>>
Queue
List
AbstractCollection
AbstractList
AbstractSequentialList
LinkedList
Stack
Vector
ArrayList
AttributeList
RoleList
RoleUnsolvedList
See §12 in “Java ist auch eine Insel”
Berner Fachhochschule
Technik und Informatik
Rolf Haenni
Algorithms and Data Structures
Implementing Linear Data Structures
Basic Data Structures
Page 21
Outline
Data Structures and Abstract Data Types
Linear Data Structures
Implementing Linear Data Structures
Implementation with Arrays
Implementation with Growable Arrays
Implementation with Linked Lists
Iterators
Berner Fachhochschule
Technik und Informatik
Rolf Haenni
Algorithms and Data Structures
Implementing Linear Data Structures
Basic Data Structures
Page 22
Implementing Linear Data Structures
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Linear data structures can be implemented in multiple ways:
Ý Arrays
Ý Records and references (singly/doubly-linked lists)
Ý Combinations of arrays and linked lists
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The choice of the implementation determines the running
times and space requirements of the basic operations
Choosing the “right” implementation . . .
Ý depends on the intended application
Ý is a trade-off between running time, memory space, simplicity
Berner Fachhochschule
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Rolf Haenni
Algorithms and Data Structures
Implementing Linear Data Structures
Basic Data Structures
Page 23
Implementing Linear Data Structures in Java
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In Java, implementing a ADT means to write a class which
implements the interface
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You may have several implementations of the same interface
<<interface>>
BasicSequence
size()
isEmpty()
ArrayStack
n
S
size()
isEmpty()
push(e)
pop()
top()
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<<interface>>
Stack
push(e)
pop()
top()
LinkedListStack
n
top
size()
isEmpty()
push(e)
pop()
top()
Rolf Haenni
Algorithms and Data Structures
Implementing Linear Data Structures
Basic Data Structures
Page 24
Arrays
Most programming languages provide arrays as a simple linear data
structure
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Arrays have a fixed size N
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The elements are usually indexed by i ∈ {0, . . . , N − 1}
A[0] = first element
A[i] = i+1-th element
A[N−1] = last element
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The running time for accessing elements is usually O(1)
→ Random access
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Arrays are similar to vectors (but not identical)
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Algorithms and Data Structures
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Page 25
Linked Lists
A linked list is another fundamental data structure which is easy to
implement in most programming languages
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It consists of a linked sequence of nodes
Each node is a record (or object) which contains:
Ý A data field to store an element (number, string, object, etc.)
Ý One or two references (links, pointers) pointing to the next
and/or the previous nodes
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Other than arrays, a linked list detaches the order of the list
elements from the one used to store them in memory or disk
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It allows only sequential (no random) access to its elements
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Nodes should implement the interface of the position ADT
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Page 26
Singly Linked Lists
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A singly linked list is the simplest form of a linked list
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Each node contains a reference to the next node
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A single reference to the first node is kept in memory
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Sometimes it is useful to keep a reference to the last node
Main Reference
Optional Reference
Node
Record
next
element
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A
B
C
Rolf Haenni
Algorithms and Data Structures
Implementing Linear Data Structures
Basic Data Structures
Page 27
Doubly-Linked Lists
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A doubly-linked list is another simple form of a linked list
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Each node contains two references, one to the next node and
one to the previous node
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Usually references to both ends are kept
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For simplicity, special header and trailer nodes are often added
Reference 1
Reference 2
Node
Record
prev
next
element
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A
B
C
Rolf Haenni
Algorithms and Data Structures
Implementation with Arrays
Basic Data Structures
Page 28
Outline
Data Structures and Abstract Data Types
Linear Data Structures
Implementing Linear Data Structures
Implementation with Arrays
Implementation with Growable Arrays
Implementation with Linked Lists
Iterators
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Rolf Haenni
Algorithms and Data Structures
Implementation with Arrays
Basic Data Structures
Page 29
Array-Based Stacks
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The elements are added from “left” to “right” and removed
from “right” to “left”
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A variable n keeps track of the stack size (= next available
location in the array)
S
0
1
2
3
4
n
N –1
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When the array becomes full, i.e. for n = N, a push operation
will throw a FullStackException
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This exception is implementation-specific (not intrinsic to
stack ADT)
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Implementing Array-Based Stacks I
Algorithm size() // runs in O(1) time
return n
Algorithm isEmpty() // runs in O(1) time
return (n = 0)
Algorithm top() // runs in O(1) time
if isEmpty() then
throw EmptyStackException
else
return S[n − 1]
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Implementing Array-Based Stacks II
Algorithm push(e) // runs in O(1) time
if n = N then
throw FullStackException
else
S[n] ← e
n ←n+1
Algorithm pop() // runs in O(1) time
if isEmpty() then
throw EmptyStackException
else
n ←n−1
return S[n]
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Page 32
Array-Based Queues
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To implement the queue ADT, the array should be used in a
circular fashion
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The elements are added and removed from “left” to “right”
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Two variables f and r keep track of the front and rear
element’s indices (the location r is kept empty)
Q
0
1
f
0
1
2
r
N –1
f
N –1
Q
I
r
When the array becomes full, a enqueue operation will throw
a FullQueueException (implementation-specific exception)
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Implementing Array-Based Queues I
Algorithm size() // runs in O(1) time
return (N + r − f ) mod N
Algorithm isEmpty() // runs in O(1) time
return (f = r )
Algorithm front() // runs in O(1) time
if isEmpty() then
throw EmptyQueueException
else
return Q[f ]
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Implementing Array-Based Queues II
Algorithm enqueue(e) // runs in O(1) time
if size() = N − 1 then // the capacity is only N−1!
throw FullQueueException
else
Q[r ] ← e
r ← (r + 1) mod N
Algorithm dequeue() // runs in O(1) time
if isEmpty() then
throw EmptyQueueException
else
e ← Q[f ]
f ← (f + 1) mod N
return e
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Page 35
Array-Based Vectors
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Vectors are most naturally implemented with arrays (i.e. ranks
= array indices)
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A variable n keeps track of the size of the vector (number of
elements stored)
V
0
1
2
3
4
r
n
N –1
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Operation elemAtRank(r) is implemented in O(1) time by
returning V [r ]
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When the array becomes full, a insertAtRank operation will
throw a FullVectorException (implementation-specific
exception)
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Inserting Elements
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In the operation insertAtRank(r,e), we need to make room
for the new element
Ý shift forward the n − r elements V [r ], . . . , V [n − 1]
V
0
1
2
3
4
r
n
N –1
0
1
2
3
4
r
n
N –1
0
1
2
3
4
V
V
I
e
r
n
N –1
In the worst case, i.e. for r = 0, this takes O(n) time
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Page 37
Removing Elements
I
In the operation removeAtRank(r), we need to fill the hole
left by the removed element
Ý shift backward the n − r − 1 elements V [r + 1], . . . , V [n − 1]
V
0
1
2
3
4
r
n
N –1
0
1
2
3
4
r
n
N –1
0
1
2
3
4
r
V
V
I
n
N –1
In the worst case, i.e. for r = 0, this takes O(n) time
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Basic Data Structures
Page 38
Outline
Data Structures and Abstract Data Types
Linear Data Structures
Implementing Linear Data Structures
Implementation with Arrays
Implementation with Growable Arrays
Implementation with Linked Lists
Iterators
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Technik und Informatik
Rolf Haenni
Algorithms and Data Structures
Implementation with Growable Arrays
Basic Data Structures
Page 39
Growable Array-Based Stack
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To solve the FullStackException problem, replace the array
with a larger one when necessary
Ý Incremental strategy: increase the size by a constant c
Ý Doubling strategy: double the size
Algorithm push(e)
if size() = N then
A ← new array of size ...
for i ← 0 to n − 1 do
A[i] ← S[i]
S ←A
S[n] ← e
n ←n+1
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Page 40
Running Times of Strategies
Doubling Strategy
running time of push(e)
running time of push(e)
Incremental Strategy
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
current number of elements
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0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
current number of elements
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Algorithms and Data Structures
Implementation with Growable Arrays
Basic Data Structures
Page 41
Comparison of the Strategies
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Consider the the total time T (n) needed to perform a series
of n push operations
We assume that we start with an empty stack represented by
an array of size
Ý c, for the incremental strategy
Ý 1, for the doubling strategy
I
We call amortized running time T (n)/n the average time
taken by a push operation over the series of operations
I
We only consider one primitive operation: storing an object in
the array
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Incremental Strategy
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Let n be a multiple of c, i.e. n = kc
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The array needs to be replaced k − 1 times
I
For the total running time T (n) we get
T (n) = n + c + 2c + 3c + . . . + (k − 1)c
= n + c(1 + 2 + 3 . . . + (k − 1))
1 2 1
(k − 1)k
= ··· =
n + n
=n+c
2
2c
2
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The total running time for n push operations is O(n2 )
I
The amortized running time for a push operation is O(n)
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Doubling Strategy
I
Let n be a power of 2, i.e. n = 2k
I
The array needs to be replaced k = log n times
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For the total running time T (n) we get
T (n) = n + 1 + 2 + 4 + . . . + 2k−1 = n + 2k − 1 = 2n − 1
I
The total running time for n push operations is O(n)
I
The amortized running time for a push operation is O(1)
I
As a consequence, the doubling strategy outperforms the
incremental strategy for large stacks
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Page 44
Performance
Growable arrays can also be used to implement queues and vectors
(and lists, but this is not very natural)
Operation
size
isEmpty
top, front, elemAtRank
push, enqueue
insertAtRank
pop, dequeue
removeAtRank
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Doubling
Strategy
O(1)
O(1)
O(1)
O(1)
O(n)
O(1)
O(n)
Incremental
Strategy
O(1)
O(1)
O(1)
O(n)
O(n)
O(1)
O(n)
Rolf Haenni
Algorithms and Data Structures
Implementation with Linked Lists
Basic Data Structures
Page 45
Outline
Data Structures and Abstract Data Types
Linear Data Structures
Implementing Linear Data Structures
Implementation with Arrays
Implementation with Growable Arrays
Implementation with Linked Lists
Iterators
Berner Fachhochschule
Technik und Informatik
Rolf Haenni
Algorithms and Data Structures
Implementation with Linked Lists
Basic Data Structures
Page 46
Stacks and Queues with Singly Linked Lists
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I
Stacks and queues are often implemented as singly linked lists
Nodes implement the position ADT by storing:
Ý Element
Ý Reference next to the next node
I
The top/front element is stored at the first node
I
For stacks, only a reference to the first node needs to be kept
(called top)
I
For queues, references to both ends of the list needs to be
kept (called front and rear )
I
Keep track of the current size n of the stack/queue
I
All operations run in O(1) time, memory grows in O(n)
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Page 47
Implementing Stack with Singly Linked Lists
Algorithm push(e) // runs in O(1) time
new ← new Node(e)
new .next ← top
top ← new
n ←n+1
Algorithm pop() // runs in O(1) time
if isEmpty() then
throw EmptyStackException
else
e ← top.getElement()
top ← top.next
n ←n−1
return e
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Basic Data Structures
Page 48
Doubly-Linked List Implementation
I
I
A doubly-linked list provides a natural implementation of the
list ADT
Nodes implement the position ADT by storing
Ý Element
Ý Reference prev to the previous node
Ý Reference next to the next node
I
The list itself stores two references first and last
first
A
last
B
Berner Fachhochschule
Technik und Informatik
C
D
E
Rolf Haenni
Algorithms and Data Structures
Implementation with Linked Lists
Basic Data Structures
Page 49
Inserting Elements
I
All four insertion operations need to create some new and
redirect some existing links, i.e. they run in O(1) time
p
A
B
C
A
B
C
D
E
q
D
E
D
E
X
A
B
Berner Fachhochschule
Technik und Informatik
C
X
Rolf Haenni
Algorithms and Data Structures
Implementation with Linked Lists
Basic Data Structures
Page 50
Removing Elements
I
The operation removeElement(p), which needs to redirect
some existing links, runs in O(1) time
p
A
B
C
A
B
C
X
D
E
D
E
p
X
A
B
Berner Fachhochschule
Technik und Informatik
C
D
E
Rolf Haenni
Algorithms and Data Structures
Implementation with Linked Lists
Basic Data Structures
Page 51
Running Times Overview
Sequence Operation
size, isEmpty
insertFirst, insertLast
replaceElement, swapElements
first, last, isFirst, isLast, after
before
insertAfter
insertBefore, removeElement
atRank, rankOf, elemAtRank
replaceAtRank, swapAtRanks
insertAtRank, removeAtRank
Berner Fachhochschule
Technik und Informatik
Array
(circular,
growable)
1
1
1
1
1
n
n
1
1
n
Doubly
Linked
List
1
1
1
1
1
1
1
n
n
n
Singly
Linked
List
1
1
1
1
n
1
n
n
n
n
Rolf Haenni
Algorithms and Data Structures
Iterators
Basic Data Structures
Page 52
Outline
Data Structures and Abstract Data Types
Linear Data Structures
Implementing Linear Data Structures
Implementation with Arrays
Implementation with Growable Arrays
Implementation with Linked Lists
Iterators
Berner Fachhochschule
Technik und Informatik
Rolf Haenni
Algorithms and Data Structures
Iterators
Basic Data Structures
Page 53
The Iterator ADT
I
I
An iterator abstracts the process of scanning through a
sequence by keeping a “pointer” to the “current element”
Methods of an iterator ADT
Ý
Ý
Ý
Ý
element(): returns the current element
hasNext(): checks whether the iteration has completed
nextElement(): advances the pointer to the next element
reset(): resets the iterator
I
Can be realized for array or linked-list implementations
I
The order in which the elements are traversed is not
necessarily the “normal” rank-based or position-based order
I
We can use several iterators for the same sequence
I
See §7.1.2 in “Java ist auch eine Insel” (interface iterable)
Berner Fachhochschule
Technik und Informatik
Rolf Haenni
Algorithms and Data Structures