Singly Linked Lists

1
Singly Linked Lists

• What is a singly-linked list?
• Why linked lists?
• Singly-linked lists vs. 1D-arrays
• Representation
• Space Analysis
• Creation, Append and Prepend
• Traversal
• Search
• Insertion after and before an element
• Deletion
• Time Complexity Singly-linked lists vs. 1D-arrays

2
What is a Singly-linked list?

• A singly linked list is a dynamic data structure
  consisting of a sequence of nodes, forming a
  linear ordering.
• Each node stores
• Element (data object)
• Reference (i.e., address) to the next node

Node
Singly-linked list
3
Why linked lists?

• Linked lists are used to implement many important
  data structures such as stacks, queues, graphs,
  hash tables, etc.
• Linked lists are used as components of some data
  structures. Examples B trees, skip lists, etc.
• LISP An important programming language in
  artificial intelligence makes extensive use of
  linked lists in performing symbolic processing.
• Memory management An important role of operating
  systems. An operating system must decide how to
  allocate and reclaim storage for processes
  running on the system. A linked list can be used
  to keep track of portions of memory that are
  available for allocation.
• Scrolled lists, components found in graphical
  user interfaces, can be implemented using linked
  lists.

4
Singly-linked lists vs. 1D-arrays
ID-array Singly-linked list
Fixed size Resizing is expensive Dynamic size
Insertions and Deletions are inefficient Elements are usually shifted Insertions and Deletions are efficient No shifting
Random access i.e., efficient indexing No random access ? Not suitable for operations requiring accessing elements by index such as sorting
No memory waste if the array is full or almost full otherwise may result in much memory waste. Extra storage needed for references however uses exactly as much memory as it needs
Sequential access is faster because of greater locality of references Reason Elements in contiguous memory locations Sequential access is slow because of low locality of references Reason Elements not in contiguous memory locations
5
Representation

• We are using a representation in which a linked
  list has both head and tail references

public class MyLinkedList protected Element
head protected Element tail public final
class Element Object data Element
next Element(Object obj, Element element)
data obj next element public
Object getData()return data public Element
getNext()return next // . . .
6
Representation Space Analysis

• Assume
• There are n nodes in the list
• All data references are null
• Number of references in the list and space
  required

Required space Total Reference
SizeOfSinglyLinkedListElementReference 1 head
SizeOfSinglyLinkedListElementReference 1 tail
nSizeOfSinglyLinkedListElementReference n next
nSizeOfObjectReference n data

• Total space (n 2)SizeOfSinglyLinkedListEleme
  ntReference
• nSizeOfObjectReference
• Hence space complexity is O(n)

7
List Creation

• An empty list is created as follows
• Once created, elements can be inserted into the
  list using either the append or prepend methods
• Also if we have a reference to a node (an
  element), we can use the insertAfter or
  InsertBefore methods of the Element class

MyLinkedList list new MyLinkedList()
for (int k 0 k lt 10 k) list.append(new
Integer(k))
8
Insertion at the end (Append)
public void append(Object obj) Element
element new Element(obj, null) if(head
null) head element else
tail.next element tail element
Complexity is
O(1)
9
Insertion at the beginning (Prepend)
public void prepend(Object obj) Element
element new Element(obj, head) if(head
null) tail element head element
Complexity is
O(1)
10
Traversal

• Begin at the first node, then follow each next
  reference until the
• traversal condition is satisfied or until you
  come to the end.
• To move an Element reference e from one node to
  the next use
• Example Count the number of nodes in a linked
  list.

e e.next
public int countNodes() int count 0
Element e head while(e ! null)
count e e.next return count
Complexity is
O(n)
11
Searching

• To search for an element, we traverse from head
  until we locate the object or we reach the end of
  the list.
• Example Count the number of nodes with data
  field equal to a given object.

public int countNodes(Object obj) int count
0 Element e head while(e ! null)
if(e.data.equals(obj)) count e
e.next return count
Complexity is .

• The following reference relationships are useful
  in searching

However, it is important to ensure that next is
not null in such expressions
12
Insertion after an element

• To insert an object y after a node x
• Move a reference e from the beginning of the
  list to node x
• Element e head
• if(e null) throw new IllegalArgumentException
  (not found)
• while(e ! null !e.data.equals(x))
• e e.next
• if(e null) throw new IllegalArgumentExceptio
  n(not found)
• Create a new node containing y as data and let
  its next reference refer to the
• node after node x
• Element element new Element(y, e.next)
• Make the next reference of node x refer to node
  y
• e.next element
• If the new node was inserted at the end of the
  list, update the tail reference
• if(element.next null) tail element

13
Insertion after an element

• The insertAfter method of the Element class is
  invoked as

MyLinkedList.Element e list.find(obj1) if(e
! null) e.insertAfter(obj2) // insert obj2
after obj1 else System.out.println("Element to
insert before not found")
Complexity is
O(n)

• Within the insertAfter method this refers to
  obj1 node

public void insertAfter(Object obj) // create
a new node for obj2 and make it refer to the
node // after obj1 node Element element
new Element(obj, this.next) // make obj1 node
refer to the new node this.next element
// update tail if the new node was inserted
at the end if(this tail) tail
next
Complexity is
O(1)

• Note The total complexity of the insert after
  operation is O(n) because find is O(n)

14
Insertion before an element

• To insert an object y before a node x
• Move a reference previous from the beginning of
  the list to the node before
• node x
• Element e head, previous
• if(e null) throw new IllegalArgumentExcepti
  on(not found)
• while(e ! null ! e.data.equals(x))
• previous e
• e e.next
• if(e null) throw new IllegalArgumentExcepti
  on(not found)
• Create a new node containing y as data and let
  its next reference refer to the
• node x
• Element element new Element(y, e)
• Make the next reference of the node before node
  x refer to node y
• if(e head)
• head element
• else
• previous.next element

15
Insertion before an element

• The insertBefore method of the Element class is
  invoked as

MyLinkedList.Element e list.find(obj1) if(e
! null) e.insertBefore(obj2) // insert
obj2 before obj1 else System.out.println("Eleme
nt to insert before not found")
Complexity is
O(n)

• Within the insertBefore method this refers to
  obj1 node

public void insertBefore(Object obj) //
create a new node for obj2, make this node point
to obj1 node Element element new
Element(obj, this) if(this head)
head element return Element
previous head // move previous to node
before obj1 node while(previous.next ! this)
previous previous.next
previous.next element // insert
Complexity is
O(n)
16
Deletion

• To delete a node x
• Move a reference previous from the beginning of
  the list to the node before
• node x
• Element e head, previous
• if(e null) throw new IllegalArgumentExcepti
  on(not found)
• while(e ! null ! e.data.equals(x))
• previous e
• e e.next
• if(e null) throw new IllegalArgumentExcepti
  on(not found)
• Bypass the node to be deleted
• if(e head)
• if(head.next null)
• head tail e null
• else
• head head.next
• else
• previous.next e.next

17
Deletion Deleting First and Last Element
public void extractFirst() if(head null)
throw new IllegalArgumentException("item not
found") head head.next if(head
null) tail null
Complexity is
public void extractLast() if(tail null)
throw new IllegalArgumentException("item not
found") if (head tail) head tail
null else Element previous head
while(previous.next ! tail) previous
previous.next previous.next null
tail previous
Complexity is
18
Deletion of an arbitrary element

• To delete an element, we use either the extract
  method of MyLinkedList or that of the Element
  inner class.
• The MyLinkedList extract method (code similar to
  that in slide 16)

public void extract(Object obj) Element
element head Element previous null
while(element ! null ! element.data.equals(obj
)) previous element element
element.next if(element null)
throw new IllegalArgumentException("item not
found") if(element head) head
element.next else previous.next
element.next if(element tail) tail
previous
Complexity is

• The method is invoked as

try list.extract(obj1) catch(IllegalArgumen
tException e) System.out.println("Element not
found")
19
Deletion of an arbitrary element

• The Element extract method invocation and
  implementation

MyLinkedList.Element e list.find(obj1) if(e !
null) e.extract() else System.out.println("
Element not found")
public void extract() Element element
null if(this head) head
next else element head
while(element ! null element.next !
this) element element.next
if(element null)
throw new InvalidOperationException(Not
found) element.next next
if(this tail) tail
element
Complexity is
20
Time Complexity Singly-linked lists vs. 1D-arrays
Operation ID-Array Complexity Singly-linked list Complexity
Insert at beginning O(n) O(1)
Insert at end O(1) O(1) if the list has tail reference O(n) if the list has no tail reference
Insert at middle O(n) O(n)
Delete at beginning O(n) O(1)
Delete at end O(1) O(n)
Delete at middle O(n) O(1) access followed by O(n) shift O(n) O(n) search, followed by O(1) delete
Search O(n) linear search O(log n) Binary search O(n)
Indexing What is the element at a given position k? O(1) O(n)
middle neither at the beginning nor at the end
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Exercises

• Using the Element extract method is less
  efficient than using the
• MyLinkedList extract method. Why?
• For the MyLinkedList class, Implement each of the
  following methods
• String toString()
• Element find(Object obj)
• void insertAt(int n) //counting the nodes from 1.
• void deleteBefore(Object obj) // delete node
  before obj node
• State the complexity of each method.
• Which methods are affected if we do not use the
  tail reference in MyLinkedList class.

22
Doubly Linked Lists

• What is a doubly-linked list?
• Representation
• Space Analysis
• Doubly-linked lists vs. Singly-linked lists
• Creation, Append and Prepend
• Traversal
• Insertion before an element
• Deletion

23
What is a Doubly-linked list?

• A doubly linked list is a dynamic data structure
  consisting of a sequence of nodes, forming a
  linear ordering.
• Each node stores
• Element (data object)
• Reference (i.e., address) to the next node
• Reference (i.e., address) to the previous node

Node
Doubly-linked list
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Representation
public class DoublyLinkedList protected
Element head, tail //. . . public
class Element Object data
Element next, previous Element(Object
obj, Element next, Element previous)
data obj this.next next
this.previous previous public
Object getData()return data public
Element getNext()return next public
Element getPrevious()return previous //
. . .
25
Doubly-Linked Lists Space Analysis

• Assume
• There are n nodes in the list
• All data references are null
• Number of references in the list and space
  required

Required space Total Reference
SizeOfDoublyLinkedListElementReference 1 head
SizeOfDoublyLinkedListElementReference 1 tail
nSizeOfDoublyLinkedListElementReference n next
nSizeOfDoublyLinkedListElementReference n previous
nSizeOfSizeOfObjectReference n data

• Total space (2n 2)SizeOfDoublyLinkedListElem
  entReference
• nSizeOfObjectReference
• Hence space complexity is O(n)

26
Doubly-Linked Lists vs. Singly-linked lists

• A doubly-linked list allows traversing the list
  in either direction.
• Modifying a doubly-linked list usually requires
  changing more references, but is sometimes
  simpler because there is no need to keep track of
  the address of the previous node. In
  singly-linked list, this is required in delete
  and insert before operations.
• ? The extractLast operation is O(1) in
  doubly-linked list whereas it is
• O(n) is singly-linked list
• Doubly-linked lists are used to implement
  dequeues (double-ended queues that support insert
  or delete operations at either end).
• A singly-linked list uses less memory than an
  equivalent doubly-linked list.

27
List Creation and Insertion

• An empty doubly-linked list is created as
  follows
• DoublyLinkedList list new DoublyLinkedList()
• Like a singly-linked list, once created, elements
  can be inserted into the list using either the
  append or prepend methods
• for (int k 0 k lt 10 k)
• list.append(new Int(k))
• Also if we have a reference to a node (an
  element), we can use the insertAfter or
  InsertBefore methods of the Element class.

28
Insertion at the end (append)

• public void append(Object obj)
• Element element new Element(obj, null,
  tail)
• if(head null)
• head tail element
• else
• tail.next element
• tail element

Complexity is
29
Insertion at the beginning (prepend)

• public void prepend(Object obj)
• Element element new Element(obj, head,
  null)
• if(head null)
• head tail element
• else
• head.previous element
• head element

Complexity is
30
Traversal

• For DoublyLinked list, traversal can be done in
  either direction. Forward, starting from the
  head, or backward starting from the tail.
• Example Count the number of nodes in a linked
  list.

Element e head while (e ! null) //do
something e e.next
Element e tail while (e ! null) //do
something e e.previous
public int countNodes() int count 0
Element e head while(e ! null)
count e e.next return
count
Complexity is
31
Traversal (contd)

• Example The following computes the sum of the
  last n nodes

public int sumLastNnodes(int n) if(n lt 0)
throw new IllegalArgumentException("Wrong "
n) if(head null) throw new
ListEmptyException() int count 0,
sum 0 Element e tail while(e ! null
count lt n) sum (Integer)e.data
count e e.previous if(count
lt n) throw new IllegalArgumentException(No.
of nodes lt "n) return sum
Complexity is
32
Insertion before an element

• Inserting before the current node (this) that is
  neither the first nor the last node

Element element new Element(obj, this,
this.previous) this.previous.next
element this.previous element
Complexity is
33
Deletion

• To delete an element, we use either the extract
  method of DoublyLinkedList or that of the Element
  inner class.

public void extract(Object obj) Element
element head while((element ! null)
(!element.data.equals(obj))) element
element.next if(element null)
throw new IllegalArgumentException("item not
found") if(element head) head
element.next if(element.next !
null) element.next.previous
null else element.previous.next
element.next if(element.next !
null) element.next.previous
element.previous if(element tail)
tail element.previous
Complexity is
34
Exercises

• For the DoublyLinkedList class, Implement each of
  the following methods and state its complexity.
• String toString()
• Element find(Object obj)
• void ExtractLast()
• void ExtractFirst()
• void ExtractLastN(int n)
• For the DoublyLinkedList.Element inner class,
  implement each of the following methods and state
  its complexity.
• void insertBefore()
• void insertAfter()
• void extract()
• What are the methods of DoublyLinkedList and its
  Element inner class that are more efficient than
  those of MyLinkedList class?