# Lec9

```Vectors
&copy; 2004 Goodrich, Tamassia
Vectors
1
(“Vector” = “Array List” in &sect;6.1)
extends the notion of
array by storing a
sequence of arbitrary
objects
An element can be
accessed, inserted or
removed by specifying
its rank (number of
elements preceding it)
An exception is thrown
if an incorrect rank is
specified (e.g. negative
rank, or larger than
current size)
&copy; 2004 Goodrich, Tamassia
Main vector operations:
 object elemAtRank(integer r):
returns the element at rank r
without removing it, 0≤r≤s-1
 object replaceAtRank(integer r,
object o): replace the element at
rank with o and return the old
element, 0≤r≤s-1
 insertAtRank(integer r, object o):
insert a new element o to have
rank r, 0≤r≤s
 object removeAtRank(integer r):
removes and returns the element
at rank r, 0≤r≤s-1
isEmpty()
Vectors
2
In “Array List” ADT in the book,
these procedures are renamed:
 size() and isEmpty()




 size() and isEmpty()
elemAtRank(integer r)

object replaceAtRank(integer
r, object o)
insertAtRank(integer r, object
o)
object removeAtRank(integer
r)
&copy; 2004 Goodrich, Tamassia
Vectors

get(integer r)
object set(integer r, object
o)


object remove(integer r)
3
Applications of Vectors
Direct applications

Sorted collection of objects (elementary
database)
Indirect applications


Auxiliary data structure for algorithms
Component of other data structures
&copy; 2004 Goodrich, Tamassia
Vectors
4
Array-based Vector
Use an array V of size N
A variable n keeps track of the size of the vector
(number of elements stored)
Operation elemAtRank(r) is implemented in O(1)
time by returning V[r]
V
0 1 2
&copy; 2004 Goodrich, Tamassia
n
r
Vectors
5
Insertion
In operation insertAtRank(r, o), we need to make
room for the new element by shifting forward the
n - r elements V[r], …, V[n - 1]
In the worst case (r = 0), this takes O(n) time
V
0 1 2
r
n
0 1 2
r
n
0 1 2
o
r
V
V
&copy; 2004 Goodrich, Tamassia
Vectors
n
6
Deletion
In operation removeAtRank(r), we need to fill the
hole left by the removed element by shifting
backward the n - r - 1 elements V[r + 1], …, V[n - 1]
In the worst case (r = 0), this takes O(n) time
V
0 1 2
o
r
n
0 1 2
r
n
0 1 2
r
V
V
&copy; 2004 Goodrich, Tamassia
Vectors
n
7
Performance
In the array based implementation of a Vector



The space used by the data structure is O(n)
size, isEmpty, elemAtRank and replaceAtRank run in
O(1) time
insertAtRank and removeAtRank run in O(n) time
If we use the array in a circular fashion,
insertAtRank(0) and removeAtRank(0) run in
O(1) time
In an insertAtRank operation, when the array
is full, instead of throwing an exception, we
can replace the array with a larger one
&copy; 2004 Goodrich, Tamassia
Vectors
8
Lists
&copy; 2004 Goodrich, Tamassia
Vectors
9
The Position ADT models the notion of
place within a data structure where a
single object is stored
It gives a unified view of diverse ways
of storing data, such as


a cell of an array
a node of a linked list
Just one method:

object element(): returns the element
stored at the position
&copy; 2004 Goodrich, Tamassia
Vectors
10
(“List” = “Node List” in &sect; 6.2.3)
sequence of positions
storing arbitrary objects
It establishes a
before/after relation
between positions
Generic methods:

size(), isEmpty()
Accessor methods (all
return type Position):


Update methods:






&copy; 2004 Goodrich, Tamassia
Vectors
first(), last()
prev(p), next(p)
replace(p, e)
remove(p)
insertBefore(p, e)
insertAfter(p, e)
insertFirst(e)
insertLast(e)
11
In the Node List ADT in the book,
these methods are renamed:
 size(), isEmpty()








 size(), isEmpty()
Position first(), last()
Position prev(p), next(p)


Object replace(p, e)
Object remove(p)
Position insertBefore(p, e)
Position insertAfter(p, e)
Position insertFirst(e)
Position insertLast(e)
&copy; 2004 Goodrich, Tamassia
Vectors






Position first(), last()
Position prev(p), next(p)
Object set(p, e)
Object remove(p)
12
A doubly linked list provides a natural
Nodes implement Position and store:



element
prev
next
elem
node
nodes/positions
trailer
elements
&copy; 2004 Goodrich, Tamassia
Vectors
13
Insertion
We visualize operation insertAfter(p, X), which returns position q
p
A
B
C
p
A
q
B
C
X
p
A
&copy; 2004 Goodrich, Tamassia
q
B
Vectors
X
C
14
Insertion Algorithm
Algorithm insertAfter(p,e):
Create a new node v
v.setElement(e)
v.setPrev(p)
v.setNext(p.getNext()) {link v to its successor}
(p.getNext()).setPrev(v) {link p’s old successor to v}
p.setNext(v)
{link p to its new successor, v}
size ++
return v {the position for the element e}
&copy; 2004 Goodrich, Tamassia
Vectors
15
Deletion
We visualize remove(p), where p = last()
p
A
B
C
A
B
C
D
p
D
A
&copy; 2004 Goodrich, Tamassia
B
Vectors
C
16
Deletion Algorithm
Algorithm remove(p):
t = p.element
{a temporary variable to hold the
return value}
(p.getPrev()).setNext(p.getNext())
(p.getNext()).setPrev(p.getPrev())
p.setPrev(null) {invalidating the position p}
p.setNext(null)
size -return t
&copy; 2004 Goodrich, Tamassia
Vectors
17
Performance
In the implementation of the List ADT
by means of a doubly linked list




The space used by a list with n elements is
O(n)
The space used by each position of the list
is O(1)
All the operations of the List ADT run in
O(1) time
Operation element() of the
Position ADT runs in O(1) time
&copy; 2004 Goodrich, Tamassia
Vectors
18
Sequences and
Iterators
&copy; 2004 Goodrich, Tamassia
Vectors
19
union of the Vector and
Elements accessed by


List-based methods:

Rank, or
Position
Generic methods:

size(), isEmpty()
Vector-based methods:

elemAtRank(r),
replaceAtRank(r, o),
insertAtRank(r, o),
removeAtRank(r)
&copy; 2004 Goodrich, Tamassia
Vectors
first(), last(), prev(p),
next(p), replace(p, o),
insertBefore(p, o),
insertAfter(p, o),
insertFirst(o),
insertLast(o),
remove(p)
Bridge methods:

atRank(r), rankOf(p)
20
Applications of Sequences
The Sequence ADT is a basic, generalpurpose, data structure for storing an ordered
collection of elements
Direct applications:


Generic replacement for stack, queue, vector, or
list
Indirect applications:

Building block of more complex data structures
&copy; 2004 Goodrich, Tamassia
Vectors
21
A doubly linked list provides a
reasonable implementation of the
Nodes implement Position and store:



element
Position-based methods
run in constant time
Rank-based methods
require searching from
keeping track of ranks;
hence, run in linear time
nodes/positions
trailer
elements
&copy; 2004 Goodrich, Tamassia
Vectors
22
Array-based Implementation
elements
We use a
circular array
storing
positions
A position
object stores:


Element
Rank
Indices f and l
keep track of
first and last
positions
0
1
3
positions
S
f
&copy; 2004 Goodrich, Tamassia
2
Vectors
l
23
Sequence Implementations
Operation
size, isEmpty
atRank, rankOf, elemAtRank
Array
1
1
List
1
n
first, last, prev, next
replace
replaceAtRank
1
1
1
1
1
n
insertAtRank, removeAtRank
insertFirst, insertLast
n
1
n
1
insertAfter, insertBefore
n
1
remove
n
1
&copy; 2004 Goodrich, Tamassia
Vectors
24
Iterators (&sect; 5.4)
An iterator abstracts the
process of scanning through
a collection of elements
Methods of the ObjectIterator




object object()
boolean hasNext()
object nextObject()
reset()

ObjectIterator elements()
Two notions of iterator:
Extends the concept of
capability
Implementation with an array
&copy; 2004 Goodrich, Tamassia
An iterator is typically
associated with an another
data structure
We can augment the Stack,
Queue, Vector, List and