Transcript N - Texas A&M University

Chapter 6:
Vectors, Lists and Sequences
Nancy Amato
Parasol Lab, Dept. CSE, Texas A&M University
Acknowledgement: These slides are adapted from slides provided with Data Structures and
Algorithms in C++, Goodrich, Tamassia and Mount (Wiley 2004. 2011)
http://parasol.tamu.edu
Vectors: Outline and Reading
• The Vector ADT (§6.1.1)
• Array-based implementation (§6.1.2)
Vectors
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The Vector ADT
• Main vector operations:
• The Vector ADT
• elemAtRank(int r): returns the
extends the notion of
element at rank r without removing
array by storing a
it
sequence of arbitrary
• replaceAtRank(int r, Object o):
objects
replace the element at rank r with o
• An element can be
• insertAtRank(int r, Object o): insert
accessed, inserted or
a new element o to have rank r
removed by specifying
• removeAtRank(int r): removes the
its rank (number of
element at rank r
elements preceding it) • Additional operations size() and
isEmpty()
• An exception is thrown
if an incorrect rank is
specified (e.g., a
negative rank)
Vectors
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Applications of Vectors
• Direct applications
• Sorted collection of objects (elementary database)
• Indirect applications
• Auxiliary data structure for algorithms
• Component of other data structures
Vectors
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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]
N-1
0
V
0 1 2
r
Vectors
n
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Array based Vector: 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
Vectors
n
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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
Vectors
n
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Performance
• In the array based implementation of a Vector
• The space used by the data structure is O(n)
• Size(), isEmpty(), elemAtRank(r) and
replaceAtRank(r,o) run in O(1) time
• insertAtRank(r,o) and removeAtRank(r) run in O(n)
time
• If we use the array in a circular fashion,
insertAtRank(0,o) and removeAtRank(0) run in
O(1) time
• In an insertAtRank(r,o) operation, when the
array is full, instead of throwing an exception,
we can replace the array with a larger one
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Vectors
8
Exercise:
• Implement the Deque ADT using Vector
functions
• Deque functions:
• first(), last(), insertFirst(e), insertLast(e),
removeFirst(), removeLast(), size(), isEmpty()
• Vector functions:
• elemAtRank( r), replaceAtRank(r,e),
insertAtRank(r,e), removeAtRank(r ), size(), isEmpty()
Vectors
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Exercise Solution:
• Implement the Deque ADT using Vector functions
• Deque functions: first(), last(), insertFirst(e), insertLast(e), removeFirst(), removeLast(),
size(), isEmpty()
• Vector functions: elemAtRank( r), replaceAtRank(r,e), insertAtRank(r,e), removeAtRank(r
), size(), isEmpty()
•
•
•
•
•
•
•
•
Deque function : Realization using Vector Functions
size() and isEmpty() fcns can simply call Vector fcns directly
first()
=> elemAtRank(0)
last()
=> elemAtRank(size()-1)
insertFirst(e) => insertAtRank(0,e)
insertLast(e) => insertAtRank(size(), e)
removeFirst() => removeAtRank(0)
removeLast() => removeAtRank(size()-1)
Vectors
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STL vector class
• Functions in the STL vector class (incomplete)
•
•
•
•
•
•
•
•
Size(), capacity() - return #elts in vector, #elts vector can hold
empty() - boolean
Operator[r] - returns reference to elt at rank r (no index check)
At( r) - returns reference to elt at rank r (index checked)
Front(), back() - return references to first/last elts
push_back(e) - insert e at end of vector
pop_back() - remove last elt
vector(n) - creates a vector of size n
• Similarities & Differences with book’s Vector ADT
• STL assignment v[r]=e is equivalent to v.replaceAtRank(r,e)
• No direct STL counerparts of insertAtRank( r) & removeAtRank( r)
• STL also provides more generatl fcns for inserting & removing from
arbitrary positions in the vector - these use iterators
Vectors
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Iterators
• An iterator abstracts the
• An iterator is typically
process of scanning through a
associated with an another
collection of elements
data structure
• Methods of the ObjectIterator • We can augment the Stack,
ADT:
Queue, Vector, and other
container ADTs with method:
• boolean hasNext()
• ObjectIterator elements()
• object next()
• reset()
• Two notions of iterator:
• Extends the concept of
position by adding a traversal
capability
• May be implemented with an
array or singly linked list
Vectors
• snapshot: freezes the
contents of the data structure
at a given time
• dynamic: follows changes to
the data structure
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Iterators
• Some functions supported by STL containers
• Begin(), end() - return iterators to beginning or end of container
• Insert(I,e) - insert e just before the position indicated by iterator
I (analogous to our insertBefore(p))
• Erase(I) - removes the elt at the position indicated by I
(analogous to our remove(p))
• The functions can be used to insert/remove elts from
arbitrary positions in the STL vector and list
Vectors
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Vector Summary
• Vector Operation Complexity for Different Implementations
Array
Fixed-Size or
Expandable
List
Singly or
Doubly
Linked
RemoveAtRank(r),
InsertAtRank(r,o)
O(1) Best Case (r=0,n)
O(n) Worst Case
O(n) Average Case
?
elemAtRank(r),
ReplaceAtRank(r,o)
O(1)
?
Size(), isEmpty()
O(1)
?
Vectors
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Lists and Sequences
Vectors
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Outline and Reading
•
•
•
•
•
•
Singly linked list
Position ADT (§6.2.1)
List ADT (§6.2.2)
Doubly linked list (§ 6.2.3)
Sequence ADT (§6.3.1)
Implementations of the sequence ADT (§6.3.23)
• Iterators (§6.2.5)
Vectors
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Position ADT
• The Position ADT models the notion of place within a
data structure where a single object is stored
• A special null position refers to no object.
• Positions provide a unified view of diverse ways of
storing data, such as
• a cell of an array
• a node of a linked list
• Member functions:
• Object& element(): returns the element stored at this
position
• bool isNull(): returns true if this is a null position
Vectors
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List ADT (§5.2.2)
• The List ADT models a
sequence of positions
storing arbitrary objects
• establishes a before/after
relation between positions
• It allows for insertion
and removal in the
“middle”
• Query methods:
• isFirst(p), isLast(p)
• Accessor methods:
• first(), last()
• before(p), after(p)
• Update methods:
• replaceElement(p, o),
swapElements(p, q)
• insertBefore(p, o),
insertAfter(p, o),
• insertFirst(o), insertLast(o)
• remove(p)
• Generic methods:
• size(), isEmpty()
18
List ADT
• Query methods:
• isFirst(p), isLast(p) :
• return boolean indicating if the given position is the first or last,
resp.
• Accessor methods
• first(), last():
• return the position of the first or last, resp., element of S
• an error occurs if S is empty
• before(p), after(p):
• return the position of the element of S preceding or following,
resp, the one at position p
• an error occurs if S is empty, or p is the first or last, resp.,
position
Vectors
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List ADT
• Update Methods
• replaceElement(p, o)
• Replace the element at position p with e
• swapElements(p, q)
• Swap the elements stored at positions p & q
• insertBefore(p, o), insertAfter(p, o),
• Insert a new element o into S before or after, resp., position p
• Output: position of the newly inserted element
• insertFirst(o), insertLast(o)
• Insert a new element o into S as the first or last, resp., element
• Output: position of the newly inserted element
• remove(p)
• Remove the element at position p from S
Vectors
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Exercise:
• Describe how to implement the following list
ADT operations using a singly-linked list
• list ADT operations: first(), last(), before(p), after(p)
• For each operation, explain how it is implemented
and provide the running time
next
• A singly linked list concrete
data structure consists of a
sequence of nodes
• Each node stores
elem
• element
• link to the next node
node

A
Vectors
B
C
D
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Exercise:
• Describe how to implement the following list ADT
operations using a doubly-linked list
• list ADT operations: first(), last(), before(p), after(p)
• For each operation, explain how it is implemented and provide
the running time
prev
• Doubly-Linked List Nodes
implement Position and
store:
• element
• link to previous node
• link to next node
next
node
elem
header
trailer
• Special trailer and header
nodes
Vectors
elements
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Insertion
• We visualize operation insertAfter(p, X) which returns position q
p
A
B
C
p
A
q
B
C
X
p
A
q
B
Vectors
X
C
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Deletion
• We visualize remove(p), where p = last()
p
A
B
C
A
B
C
D
p
D
A
B
Vectors
C
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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
Vectors
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STL list class
• Functions in the STL list class
•
•
•
•
•
Size() - return #elts in list, empty() - boolean
Front(), back() - return references to first/last elts
Push_front(e), push_back(e) - insert e at front/end
Pop_front(), pop_back() - remove first/last elt
List() - creates an empty list
• Similarities & Differences with book’s List ADT
• STL front() & back() correspond to first() & last() except the STL
functions return the element & not its position
• STL push() & pop() are equiv to List ADT insert and remove when
applied to the beginning & end of the list
• STL also provides fcns for inserting & removing from arbitrary
positions in the list - these use iterators
Vectors
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List Summary
• List Operation Complexity for different implementations
Vectors
List Singly-Linked
List
DoublyLinked
first(), last(), after(p)
insertAfter(p,o),
replaceElement(p,o),
swapElements(p,q)
O(1)
O(1)
before(p), insertBefore(p,o),
remove(p)
O(n) WC & AC
O(1) BC
O(1)
Size(), isEmpty()
O(1)
O(1)
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Sequence ADT
• The Sequence ADT is the union • List-based methods:
of the Vector and List ADTs
• first(), last(), before(p),
• Elements accessed by
after(p),
• Rank, or
replaceElement(p, o),
• Position
swapElements(p, q),
insertBefore(p, o),
• Generic methods:
insertAfter(p, o),
• size(), isEmpty()
insertFirst(o),
• Vector-based methods:
insertLast(o),
• elemAtRank(r), replaceAtRank(r,
remove(p)
o), insertAtRank(r, o),
removeAtRank(r)
• Bridge methods:
• atRank(r), rankOf(p)
Vectors
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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
• small database (e.g., address book)
• Indirect applications:
• Building block of more complex data structures
Vectors
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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
2
3
positions
S
f
Vectors
l
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Sequence Implementations
Operation
size, isEmpty
atRank, rankOf, elemAtRank
first, last, before, after
replaceElement, swapElements
replaceAtRank
insertAtRank, removeAtRank
insertFirst, insertLast
insertAfter, insertBefore
remove
Vectors
Array
1
1
1
1
1
n
1
n
n
List
1
n
1
1
n
n
1
1
1
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