Vectors, Lists, and Sequences
Vectors: Outline and Reading
• The Vector ADT (§6.1.1)
• Array-based implementation (§6.1.2)
The Vector ADT
• The Vector ADT extends • Main vector operations:
the notion of array by
– elemAtRank(int r): returns the
storing a sequence of
element at rank r without removing
it
arbitrary objects
– replaceAtRank(int r, Object o):
• An element can be
replace the element at rank r with o
accessed, inserted or
– insertAtRank(int r, Object o): insert a
removed by specifying
new element o to have rank r
its rank (number of
– removeAtRank(int r): removes the
elements preceding it)
element at rank r
• An exception is thrown
• Additional operations size() and
if an incorrect rank is
isEmpty()
specified (e.g., a
negative rank)
Applications of Vectors
• Direct applications
– Sorted collection of objects (simple database)
• Indirect applications
– Auxiliary data structure for algorithms
– Component of other data structures
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
n
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
n
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
n
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
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()
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)
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 counterparts of insertAtRank( r) & removeAtRank( r)
– STL also provides more general fcns for inserting & removing from
arbitrary positions in the vector - these use iterators
Iterators
• An iterator abstracts the process • An iterator is typically
of scanning through a collection
associated with an another data
of elements
structure
• Methods of the ObjectIterator
• We can augment the Stack,
ADT:
Queue, Vector, and other
container ADTs with method:
– boolean hasNext()
– object next()
– reset()
• Extends the concept of position
by adding a traversal capability
– ObjectIterator elements()
• Two notions of iterator:
– snapshot: freezes the contents
of the data structure at a given
time
– dynamic: follows changes to
the data structure
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 element at the position
indicated by I (analogous to our remove(p))
• The functions can be used to insert/remove
elements from arbitrary positions in the STL
vector and list
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)
?
Lists and Sequences
Outline and Reading
•
•
•
•
•
•
Singly linked list
Position ADT (§6.2.1)
List ADT (§6.2.2)
Sequence ADT (§6.3.1)
Implementations of the sequence ADT (§6.3.2-3)
Iterators (§6.2.5)
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
List ADT (§6.2.2)
• 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)
• Generic methods:
– size(), isEmpty()
• 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)
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
List ADT
• Update Methods
– replaceElement(p, o)
• Replace the element at position p with o
– 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
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 consists of
a sequence of nodes
• Each node stores
• element
• link to the next node
elem
node
head
tail

Leonard
Sheldon
Howard
Raj
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
• Doubly-Linked List Nodes
implement Position and store:
next
prev
• element
• link to previous node
• link to next node
elem
• Special head/tail nodes
node
tail
head
Leonard
Sheldon
Howard
Raj
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
STL list class
• Functions in the STL list class
–
–
–
–
–
size() - return #elements in list, empty() - boolean
front(), back() - return references to first/last elements
push_front(e), push_back(e) - insert e at front/end
pop_front(), pop_back() - remove first/last element
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 functions for inserting & removing from
arbitrary positions in the list - these use iterators
List Summary
• List Operation Complexity for different implementations
List Singly-Linked
List Doubly- Linked
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)
O(1)
Size(), isEmpty()
O(1)
O(1)
Sequence ADT
• The Sequence ADT is the union of
the Vector and List ADTs
• Elements accessed by
– Rank, or
– Position
• Generic methods:
– size(), isEmpty()
• Vector-based methods:
– elemAtRank(r), replaceAtRank(r,
o), insertAtRank(r, o),
removeAtRank(r)
• List-based methods:
– first(), last(), before(p),
after(p),
replaceElement(p, o),
swapElements(p, q),
insertBefore(p, o),
insertAfter(p, o),
insertFirst(o),
insertLast(o), remove(p)
• Bridge methods:
– atRank(r), rankOf(p)
Applications of Sequences
• The Sequence ADT is a basic, general-purpose,
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
Sequence Implementations
Operation
size, isEmpty
atRank, rankOf, elemAtRank
first, last, before, after
replaceElement, swapElements
replaceAtRank
insertAtRank, removeAtRank
insertFirst, insertLast
insertAfter, insertBefore
remove
Array
1
1
1
1
1
n
1
n
n
List
1
n
1
1
n
n
1
1
1