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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

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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:

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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.

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Singly-linked lists vs. 1D-arrays

ID-arraySingly-linked list

Fixed size: Resizing is expensiveDynamic size

Insertions and Deletions are inefficient: Elements are usually shifted

Insertions and Deletions are efficient: No shifting

Random access i.e., efficient indexingNo 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]

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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;}

} // . . .}

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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:

ReferenceTotalRequired space

head1SizeOfSinglyLinkedListElementReference

tail1SizeOfSinglyLinkedListElementReference

nextnn*SizeOfSinglyLinkedListElementReference

datann*SizeOfObjectReference

• Total space = (n + 2)*SizeOfSinglyLinkedListElementReference

+ n*SizeOfObjectReference

• Hence space complexity is O(n)

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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 < 10; k++) list.append(new Integer(k));

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public void append(Object obj){ Element element = new Element(obj, null); if(head == null) head = element; else tail.next = element; tail = element;}

Insertion at the end (Append)

Complexity is O(1)

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public void prepend(Object obj) { Element element = new Element(obj, head); if(head == null) tail = element; head = element;}

Insertion at the beginning (Prepend)

Complexity is O(1)

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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.public int countNodes(){ int count = 0; Element e = head; while(e != null){ count++;

e = e.next; } return count;}

e = e.next;

Complexity is O(n)

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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:

e .ne xt e .ne xt .ne xt e .ne xt .ne xt .ne xt

e .ne xt .data e .ne xt .ne xt .data

e

e .data

However, it is important to ensure that next is not null in such expressions

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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 IllegalArgumentException(“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;

e

e .ne xtx

e

e .ne xtx

ye l e m e nt

ee .ne xt

x

ye l e m e nt

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Insertion after an element

Complexity is

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)

O(n)

• 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 obj1else System.out.println("Element to insert before not found");

• Within the insertAfter method this refers to obj1 node:

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

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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 IllegalArgumentException(“not found”); while(e != null && ! e.data.equals(x)){ previous = e; e = e.next; } if(e == null) throw new IllegalArgumentException(“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;

e

x

p re v io u s

ye l e m e nt

e

x

p re v io u s

ye l e m e nt

e

x

p re v io u s

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Insertion before an element

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

MyLinkedList.Element e = list.find(obj1); if(e != null) e.insertBefore(obj2); // insert obj2 before obj1else System.out.println("Element to insert before not found");

• The insertBefore method of the Element class is invoked as:

• Within the insertBefore method this refers to obj1 node:

Complexity is O(n)

O(n)

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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 IllegalArgumentException(“not found”); while(e != null && ! e.data.equals(x)){ previous = e; e = e.next; } if(e == null) throw new IllegalArgumentException(“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; if(tail == e) tail = previous; } e = null;

e

x

p re v io u s

e

x

p re v io u s

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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;}

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…

Complexity is…

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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;}

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):

Complexity is…

try{ list.extract(obj1);} catch(IllegalArgumentException e){ System.out.println("Element not found");

}

• The method is invoked as:

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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;}

Deletion of an arbitrary element• The Element extract method invocation and implementation:

Complexity is…

MyLinkedList.Element e = list.find(obj1);if(e != null) e.extract();else System.out.println("Element not found");

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Time Complexity: Singly-linked lists vs. 1D-arrays

OperationID-Array ComplexitySingly-linked list Complexity

Insert at beginningO(n)O(1)

Insert at endO(1)O(1) if the list has tail referenceO(n) if the list has no tail reference

Insert at middle*O(n)O(n)

Delete at beginningO(n)O(1)

Delete at endO(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

SearchO(n) linear searchO(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.

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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

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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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Representationpublic 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;} // . . . }}

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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:

ReferenceTotalRequired space

head1SizeOfDoublyLinkedListElementReference

tail1SizeOfDoublyLinkedListElementReference

nextnn*SizeOfDoublyLinkedListElementReference

previousnn*SizeOfDoublyLinkedListElementReference

datann*SizeOfSizeOfObjectReference

• Total space = (2n + 2)*SizeOfDoublyLinkedListElementReference

+ n*SizeOfObjectReference

• Hence space complexity is O(n)

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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.

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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 < 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.

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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…

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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…

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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…

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public int sumLastNnodes(int n){ if(n <= 0)

throw new IllegalArgumentException("Wrong: " + n); if(head == null)

throw new ListEmptyException();

int count = 0, sum = 0; Element e = tail; while(e != null && count < n){

sum += (Integer)e.data; count++; e = e.previous;

} if(count < n)

throw new IllegalArgumentException(“No. of nodes < "+n); return sum; }

Traversal (cont’d)

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

Complexity is…

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Insertion before an element

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

Complexity is…

Element element = new Element(obj, this, this.previous);this.previous.next = element;this.previous = element;

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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…

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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?