What is the average case time complexity to insert a node in a singly linked list?

Introduction to Singly Linked List

Singly Linked List is a variant of Linked List which allows only forward traversal of linked lists. This is a simple form, yet it is effective for several problems such as Big Integer calculations.

A singly linked list is made up of nodes where each node has two parts:

The first part contains the actual data of the node
The second part contains a link that points to the next node of the list that is the address of the next node.

The beginning of the node marked by a special pointer named START. The pointer points to the fist node of the list but the link part of the last node has no next node to point to.

The main difference from an array is:

Elements are not stored in contiguous memory locations.
Size of Linked List need not be known in advance. It can increase at runtime depending on number of elements dynamically without any overhead.

In Singly Linked List, only the pointer to the first node is stored. The other nodes are accessed one by one.

To get the address of ith node, we need to traverse all nodes before it because the address of ith node is stored with i-1th node and so on.

Linked List | Set 2 (Inserting a node)

We have introduced Linked Lists in the previous post. We also created a simple linked list with 3 nodes and discussed linked list traversal.
All programs discussed in this post consider the following representations of linked list.

C++

// A linked list node 

class Node
{ 

public:

int data;

Node *next;
}; 
// This code is contributed by rathbhupendra

// A linked list node

struct Node
{

int data;

struct Node *next;
};

Java

// Linked List Class

class LinkedList
{

Node head; // head of list

/* Node Class */

class Node

{

int data;

Node next;

// Constructor to create a new node

Node(int d) {data = d; next = null; }

}
}

Python

# Node class

class Node:

# Function to initialize the node object

def __init__(self, data):

self.data = data # Assign data

self.next = None # Initialize next as null
# Linked List class

class LinkedList:

# Function to initialize the Linked List object

def __init__(self):

self.head = None

/* Linked list Node*/

public class Node
{

public int data;

public Node next;

public Node(int d) {data = d; next = null; }
}

Javascript

In this post, methods to insert a new node in linked list are discussed. A node can be added in three ways
1) At the front of the linked list
2) After a given node.
3) At the end of the linked list.

Insert N elements in a Linked List one after other at middle position

Given an array of N elements. The task is to insert the given elements at the middle position in the linked list one after another. Each insert operation should take O(1) time complexity.
Examples:

Input: arr[] = {1, 2, 3, 4, 5}
Output: 1 -> 3 -> 5 -> 4 -> 2 -> NULL
1 -> NULL
1 -> 2 -> NULL
1 -> 3 -> 2 -> NULL
1 -> 3 -> 4 -> 2 -> NULL
1 -> 3 -> 5 -> 4 -> 2 -> NULL
Input: arr[] = {5, 4, 1, 2}
Output: 5 -> 1 -> 2 -> 4 -> NULL

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: There are two cases:

Number of elements present in the list are less than 2.
Number of elements present in the list are more than 2.
- The number of elements already present are even say N then the new element is inserted in the middle position that is (N / 2) + 1.
- The number of elements already present are odd then the new element is inserted next to the current middle element that is (N / 2) + 2.

We take one additional pointer ‘middle’ which stores the address of current middle element and a counter which counts the total number of elements.
If the elements already present in the linked list are less than 2 then middle will always point to the first position and we insert the new node after the current middle.
If the elements already present in the linked list are more than 2 then we insert the new node next to the current middle and increment the counter.
If there are an odd number of elements after insertion then the middle points to the newly inserted node else there is no change in the middle pointer.
Below is the implementation of the above approach:

C++

// C++ implementation of the approach
#include 

using namespace std;
// Node structure

struct Node {

int value;

struct Node* next;
};
// Class to represent a node
// of the linked list

class LinkedList {

private:

struct Node *head, *mid;

int count;

public:

LinkedList();

void insertAtMiddle(int);

void show();
};
LinkedList::LinkedList()
{

head = NULL;

mid = NULL;

count = 0;
}
// Function to insert a node in
// the middle of the linked list

void LinkedList::insertAtMiddle(int n)
{

struct Node* temp = new struct Node();

struct Node* temp1;

temp->next = NULL;

temp->value = n;

// If the number of elements

// already present are less than 2

if (count < 2) {

if (head == NULL) {

head = temp;

}

else {

temp1 = head;

temp1->next = temp;

}

count++;

// mid points to first element

mid = head;

}

// If the number of elements already present

// are greater than 2

else {

temp->next = mid->next;

mid->next = temp;

count++;

// If number of elements after insertion

// are odd

if (count % 2 != 0) {

// mid points to the newly

// inserted node

mid = mid->next;

}

}
}
// Function to print the nodes
// of the linked list

void LinkedList::show()
{

struct Node* temp;

temp = head;

// Initializing temp to head

// Iterating and printing till

// The end of linked list

// That is, till temp is null

while (temp != NULL) {

cout << temp->value << " -> ";

temp = temp->next;

}

cout << "NULL";

cout << endl;
}
// Driver code

int main()
{

// Elements to be inserted one after another

int arr[] = { 1, 2, 3, 4, 5 };

int n = sizeof(arr) / sizeof(arr[0]);

LinkedList L1;

// Insert the elements

for (int i = 0; i < n; i++)

L1.insertAtMiddle(arr[i]);

// Print the nodes of the linked list

L1.show();

return 0;
}

Java

// Java implementation of the approach 

class GFG
{
// Node ure 

static class Node
{ 

int value;

Node next;
}; 
// Class to represent a node 
// of the linked list 

static class LinkedList
{ 

Node head, mid;

int count;
LinkedList() 
{ 

head = null;

mid = null;

count = 0;
} 
// Function to insert a node in 
// the middle of the linked list 

void insertAtMiddle(int n)
{ 

Node temp = new Node();

Node temp1;

temp.next = null;

temp.value = n;

// If the number of elements

// already present are less than 2

if (count < 2)

{

if (head == null)

{

head = temp;

}

else

{

temp1 = head;

temp1.next = temp;

}

count++;

// mid points to first element

mid = head;

}

// If the number of elements already present

// are greater than 2

else

{

temp.next = mid.next;

mid.next = temp;

count++;

// If number of elements after insertion

// are odd

if (count % 2 != 0)

{

// mid points to the newly

// inserted node

mid = mid.next;

}

}
} 
// Function to print the nodes 
// of the linked list 

void show()
{ 

Node temp;

temp = head;

// Initializing temp to head

// Iterating and printing till

// The end of linked list

// That is, till temp is null

while (temp != null)

{

System.out.print( temp.value + " -> ");

temp = temp.next;

}

System.out.print( "null");

System.out.println();
} 
}
// Driver code 

public static void main(String args[])
{ 

// Elements to be inserted one after another

int arr[] = { 1, 2, 3, 4, 5 };

int n = arr.length;

LinkedList L1=new LinkedList();

// Insert the elements

for (int i = 0; i < n; i++)

L1.insertAtMiddle(arr[i]);

// Print the nodes of the linked list

L1.show();
}
}
// This code is contributed by Arnab Kundu

Python3

# Python3 implementation of the approach 
# Node ure 

class Node:

def __init__(self):

self.value = 0

self.next = None
# Class to represent a node 
# of the linked list 

class LinkedList:

def __init__(self) :

self.head = None

self.mid = None

self.count = 0

# Function to insert a node in

# the middle of the linked list

def insertAtMiddle(self , n):

temp = Node()

temp1 = None

temp.next = None

temp.value = n

# If the number of elements

# already present are less than 2

if (self.count < 2):

if (self.head == None) :

self.head = temp

else:

temp1 = self.head

temp1.next = temp

self.count = self.count + 1

# mid points to first element

self.mid = self.head

# If the number of elements already present

# are greater than 2

else:

temp.next = self.mid.next

self.mid.next = temp

self.count = self.count + 1

# If number of elements after insertion

# are odd

if (self.count % 2 != 0):

# mid points to the newly

# inserted node

self.mid = self.mid.next

# Function to print the nodes

# of the linked list

def show(self):

temp = None

temp = self.head

# Initializing temp to self.head

# Iterating and printing till

# The end of linked list

# That is, till temp is None

while (temp != None) :

print( temp.value, end = " -> ")

temp = temp.next

print( "None")
# Driver code 
# Elements to be inserted one after another 

arr = [ 1, 2, 3, 4, 5]

n = len(arr)

L1 = LinkedList()
# Insert the elements 

for i in range(n):

L1.insertAtMiddle(arr[i])
# Print the nodes of the linked list 
L1.show() 
# This code is contributed by Arnab Kundu

// C# implementation of the approach 

using System;

class GFG
{
// Node ure 

public class Node
{ 

public int value;

public Node next;
}; 
// Class to represent a node 
// of the linked list 

public class LinkedList
{ 

public Node head, mid;

public int count;

public LinkedList()
{ 

head = null;

mid = null;

count = 0;
} 
// Function to insert a node in 
// the middle of the linked list 

public void insertAtMiddle(int n)
{ 

Node temp = new Node();

Node temp1;

temp.next = null;

temp.value = n;

// If the number of elements

// already present are less than 2

if (count < 2)

{

if (head == null)

{

head = temp;

}

else

{

temp1 = head;

temp1.next = temp;

}

count++;

// mid points to first element

mid = head;

}

// If the number of elements already present

// are greater than 2

else

{

temp.next = mid.next;

mid.next = temp;

count++;

// If number of elements after insertion

// are odd

if (count % 2 != 0)

{

// mid points to the newly

// inserted node

mid = mid.next;

}

}
} 
// Function to print the nodes 
// of the linked list 

public void show()
{ 

Node temp;

temp = head;

// Initializing temp to head

// Iterating and printing till

// The end of linked list

// That is, till temp is null

while (temp != null)

{

Console.Write( temp.value + " -> ");

temp = temp.next;

}

Console.Write( "null");

Console.WriteLine();
} 
}
// Driver code 

public static void Main(String []args)
{ 

// Elements to be inserted one after another

int []arr = { 1, 2, 3, 4, 5 };

int n = arr.Length;

LinkedList L1=new LinkedList();

// Insert the elements

for (int i = 0; i < n; i++)

L1.insertAtMiddle(arr[i]);

// Print the nodes of the linked list

L1.show();
}
}
// This code contributed by Rajput-Ji

Javascript

Output: 1 -> 3 -> 5 -> 4 -> 2 -> NULL

Time Complexity : O(N)
Auxiliary Space: O(1)

What is the average case time complexity to insert a node in a singly linked list?

Article Tags :

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Mathematical

Practice Tags :

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Singly Linked List vs Doubly Linked List

Before looking at the differences between the singly linked list and doubly linked list, we first understand what is singly linked list and doubly linked list separately.

Basic concepts and nomenclatureEdit

Each record of a linked list is often called an 'element' or 'node'.

The field of each node that contains the address of the next node is usually called the 'next link' or 'next pointer'. The remaining fields are known as the 'data', 'information', 'value', 'cargo', or 'payload' fields.

The 'head' of a list is its first node. The 'tail' of a list may refer either to the rest of the list after the head, or to the last node in the list. In Lisp and some derived languages, the next node may be called the 'cdr' (pronounced could-er) of the list, while the payload of the head node may be called the 'car'.

Singly linked listEdit

Singly linked lists contain nodes which have a data field as well as 'next' field, which points to the next node in line of nodes. Operations that can be performed on singly linked lists include insertion, deletion and traversal.

A singly linked list whose nodes contain two fields: an integer value and a link to the next node

The following code demonstrates how to add a new node with data "value" to the end of a singly linked list:

node addNode(node head, int value) { node temp, p; // declare two nodes temp and p temp = createNode(); // assume createNode creates a new node with data = 0 and next pointing to NULL. temp->data = value; // add element's value to data part of node if (head == NULL) { head = temp; // when linked list is empty } else { p = head; // assign head to p while (p->next != NULL) { p = p->next; // traverse the list until p is the last node. The last node always points to NULL. } p->next = temp; // Point the previous last node to the new node created. } return head; }

Doubly linked listEdit

In a 'doubly linked list', each node contains, besides the next-node link, a second link field pointing to the 'previous' node in the sequence. The two links may be called 'forward('s') and 'backwards', or 'next' and 'prev'('previous').

A doubly linked list whose nodes contain three fields: an integer value, the link forward to the next node, and the link backward to the previous node

A technique known as XOR-linking allows a doubly linked list to be implemented using a single link field in each node. However, this technique requires the ability to do bit operations on addresses, and therefore may not be available in some high-level languages.

Many modern operating systems use doubly linked lists to maintain references to active processes, threads, and other dynamic objects.[2] A common strategy for rootkits to evade detection is to unlink themselves from these lists.[3]

Multiply linked listEdit

In a 'multiply linked list', each node contains two or more link fields, each field being used to connect the same set of data records in a different order of same set (e.g., by name, by department, by date of birth, etc.). While doubly linked lists can be seen as special cases of multiply linked list, the fact that the two and more orders are opposite to each other leads to simpler and more efficient algorithms, so they are usually treated as a separate case.

Circular linked listEdit

In the last node of a list, the link field often contains a null reference, a special value is used to indicate the lack of further nodes. A less common convention is to make it point to the first node of the list; in that case, the list is said to be 'circular' or 'circularly linked'; otherwise, it is said to be 'open' or 'linear'. It is a list where the last pointer points to the first node.

In the case of a circular doubly linked list, the first node also points to the last node of the list.

Sentinel nodesEdit

In some implementations an extra 'sentinel' or 'dummy' node may be added before the first data record or after the last one. This convention simplifies and accelerates some list-handling algorithms, by ensuring that all links can be safely dereferenced and that every list (even one that contains no data elements) always has a "first" and "last" node.

Empty listsEdit

An empty list is a list that contains no data records. This is usually the same as saying that it has zero nodes. If sentinel nodes are being used, the list is usually said to be empty when it has only sentinel nodes.

Hash linkingEdit

The link fields need not be physically part of the nodes. If the data records are stored in an array and referenced by their indices, the link field may be stored in a separate array with the same indices as the data records.

List handlesEdit

Since a reference to the first node gives access to the whole list, that reference is often called the 'address', 'pointer', or 'handle' of the list. Algorithms that manipulate linked lists usually get such handles to the input lists and return the handles to the resulting lists. In fact, in the context of such algorithms, the word "list" often means "list handle". In some situations, however, it may be convenient to refer to a list by a handle that consists of two links, pointing to its first and last nodes.

Combining alternativesEdit

The alternatives listed above may be arbitrarily combined in almost every way, so one may have circular doubly linked lists without sentinels, circular singly linked lists with sentinels, etc.

TradeoffsEdit

As with most choices in computer programming and design, no method is well suited to all circumstances. A linked list data structure might work well in one case, but cause problems in another. This is a list of some of the common tradeoffs involving linked list structures.

Linked lists vs. dynamic arraysEdit

A dynamic array is a data structure that allocates all elements contiguously in memory, and keeps a count of the current number of elements. If the space reserved for the dynamic array is exceeded, it is reallocated and (possibly) copied, which is an expensive operation.

Linked lists have several advantages over dynamic arrays. Insertion or deletion of an element at a specific point of a list, assuming that we have indexed a pointer to the node (before the one to be removed, or before the insertion point) already, is a constant-time operation (otherwise without this reference it is O(n)), whereas insertion in a dynamic array at random locations will require moving half of the elements on average, and all the elements in the worst case. While one can "delete" an element from an array in constant time by somehow marking its slot as "vacant", this causes fragmentation that impedes the performance of iteration.

Moreover, arbitrarily many elements may be inserted into a linked list, limited only by the total memory available; while a dynamic array will eventually fill up its underlying array data structure and will have to reallocate—an expensive operation, one that may not even be possible if memory is fragmented, although the cost of reallocation can be averaged over insertions, and the cost of an insertion due to reallocation would still be amortized O(1). This helps with appending elements at the array's end, but inserting into (or removing from) middle positions still carries prohibitive costs due to data moving to maintain contiguity. An array from which many elements are removed may also have to be resized in order to avoid wasting too much space.

On the other hand, dynamic arrays (as well as fixed-size array data structures) allow constant-time random access, while linked lists allow only sequential access to elements. Singly linked lists, in fact, can be easily traversed in only one direction. This makes linked lists unsuitable for applications where it's useful to look up an element by its index quickly, such as heapsort. Sequential access on arrays and dynamic arrays is also faster than on linked lists on many machines, because they have optimal locality of reference and thus make good use of data caching.

Another disadvantage of linked lists is the extra storage needed for references, which often makes them impractical for lists of small data items such as characters or boolean values, because the storage overhead for the links may exceed by a factor of two or more the size of the data. In contrast, a dynamic array requires only the space for the data itself (and a very small amount of control data).[note 1] It can also be slow, and with a naïve allocator, wasteful, to allocate memory separately for each new element, a problem generally solved using memory pools.

Some hybrid solutions try to combine the advantages of the two representations. Unrolled linked lists store several elements in each list node, increasing cache performance while decreasing memory overhead for references. CDR coding does both these as well, by replacing references with the actual data referenced, which extends off the end of the referencing record.

A good example that highlights the pros and cons of using dynamic arrays vs. linked lists is by implementing a program that resolves the Josephus problem. The Josephus problem is an election method that works by having a group of people stand in a circle. Starting at a predetermined person, one may count around the circle n times. Once the nth person is reached, one should remove them from the circle and have the members close the circle. The process is repeated until only one person is left. That person wins the election. This shows the strengths and weaknesses of a linked list vs. a dynamic array, because if the people are viewed as connected nodes in a circular linked list, then it shows how easily the linked list is able to delete nodes (as it only has to rearrange the links to the different nodes). However, the linked list will be poor at finding the next person to remove and will need to search through the list until it finds that person. A dynamic array, on the other hand, will be poor at deleting nodes (or elements) as it cannot remove one node without individually shifting all the elements up the list by one. However, it is exceptionally easy to find the nth person in the circle by directly referencing them by their position in the array.

The list ranking problem concerns the efficient conversion of a linked list representation into an array. Although trivial for a conventional computer, solving this problem by a parallel algorithm is complicated and has been the subject of much research.

A balanced tree has similar memory access patterns and space overhead to a linked list while permitting much more efficient indexing, taking O(log n) time instead of O(n) for a random access. However, insertion and deletion operations are more expensive due to the overhead of tree manipulations to maintain balance. Schemes exist for trees to automatically maintain themselves in a balanced state: AVL trees or red–black trees.

Singly linked linear lists vs. other listsEdit

While doubly linked and circular lists have advantages over singly linked linear lists, linear lists offer some advantages that make them preferable in some situations.

A singly linked linear list is a recursive data structure, because it contains a pointer to a smaller object of the same type. For that reason, many operations on singly linked linear lists (such as merging two lists, or enumerating the elements in reverse order) often have very simple recursive algorithms, much simpler than any solution using iterative commands. While those recursive solutions can be adapted for doubly linked and circularly linked lists, the procedures generally need extra arguments and more complicated base cases.

Linear singly linked lists also allow tail-sharing, the use of a common final portion of sub-list as the terminal portion of two different lists. In particular, if a new node is added at the beginning of a list, the former list remains available as the tail of the new one—a simple example of a persistent data structure. Again, this is not true with the other variants: a node may never belong to two different circular or doubly linked lists.

In particular, end-sentinel nodes can be shared among singly linked non-circular lists. The same end-sentinel node may be used for every such list. In Lisp, for example, every proper list ends with a link to a special node, denoted by nil or (), whose CAR and CDR links point to itself. Thus a Lisp procedure can safely take the CAR or CDR of any list.

The advantages of the fancy variants are often limited to the complexity of the algorithms, not in their efficiency. A circular list, in particular, can usually be emulated by a linear list together with two variables that point to the first and last nodes, at no extra cost.

Doubly linked vs. singly linkedEdit

Double-linked lists require more space per node (unless one uses XOR-linking), and their elementary operations are more expensive; but they are often easier to manipulate because they allow fast and easy sequential access to the list in both directions. In a doubly linked list, one can insert or delete a node in a constant number of operations given only that node's address. To do the same in a singly linked list, one must have the address of the pointer to that node, which is either the handle for the whole list (in case of the first node) or the link field in the previous node. Some algorithms require access in both directions. On the other hand, doubly linked lists do not allow tail-sharing and cannot be used as persistent data structures.

Circularly linked vs. linearly linkedEdit

A circularly linked list may be a natural option to represent arrays that are naturally circular, e.g. the corners of a polygon, a pool of buffers that are used and released in FIFO ("first in, first out") order, or a set of processes that should be time-shared in round-robin order. In these applications, a pointer to any node serves as a handle to the whole list.

With a circular list, a pointer to the last node gives easy access also to the first node, by following one link. Thus, in applications that require access to both ends of the list (e.g., in the implementation of a queue), a circular structure allows one to handle the structure by a single pointer, instead of two.

A circular list can be split into two circular lists, in constant time, by giving the addresses of the last node of each piece. The operation consists in swapping the contents of the link fields of those two nodes. Applying the same operation to any two nodes in two distinct lists joins the two list into one. This property greatly simplifies some algorithms and data structures, such as the quad-edge and face-edge.

The simplest representation for an empty circular list (when such a thing makes sense) is a null pointer, indicating that the list has no nodes. Without this choice, many algorithms have to test for this special case, and handle it separately. By contrast, the use of null to denote an empty linear list is more natural and often creates fewer special cases.

For some applications, it can be useful to use singly linked lists that can vary between being circular and being linear, or even circular with a linear initial segment. Algorithms for searching or otherwise operating on these have to take precautions to avoid accidentally entering an endless loop. One usual method is to have a second pointer walking the list at half or double the speed, and if both pointers meet at the same node, you know you found a cycle.

Using sentinel nodesEdit

Sentinel node may simplify certain list operations, by ensuring that the next or previous nodes exist for every element, and that even empty lists have at least one node. One may also use a sentinel node at the end of the list, with an appropriate data field, to eliminate some end-of-list tests. For example, when scanning the list looking for a node with a given value x, setting the sentinel's data field to x makes it unnecessary to test for end-of-list inside the loop. Another example is the merging two sorted lists: if their sentinels have data fields set to +∞, the choice of the next output node does not need special handling for empty lists.

However, sentinel nodes use up extra space (especially in applications that use many short lists), and they may complicate other operations (such as the creation of a new empty list).

However, if the circular list is used merely to simulate a linear list, one may avoid some of this complexity by adding a single sentinel node to every list, between the last and the first data nodes. With this convention, an empty list consists of the sentinel node alone, pointing to itself via the next-node link. The list handle should then be a pointer to the last data node, before the sentinel, if the list is not empty; or to the sentinel itself, if the list is empty.

The same trick can be used to simplify the handling of a doubly linked linear list, by turning it into a circular doubly linked list with a single sentinel node. However, in this case, the handle should be a single pointer to the dummy node itself.[8]

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