Search an element in a Linked List [Iterative and Recursive]
Write a function that searches a given key ‘x’ in a given singly linked list. The function should return true if x is present in linked list and false otherwise.
For example, if the key to be searched is 15 and linked list is 14->21->11->30->10, then function should return false. If key to be searched is 14, then the function should return true.
Iterative Solution
Following is iterative implementation of above algorithm to search a given key.
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// Iterative C++ program to search
// an element in linked list
#include
using namespace std;
/* Link list node */
class Node
{
public:
int key;
Node* next;
};
/* Given a reference [pointer to pointer] to the head
of a list and an int, push a new node on the front
of the list. */
void push[Node** head_ref, int new_key]
{
/* allocate node */
Node* new_node = new Node[];
/* put in the key */
new_node->key = new_key;
/* link the old list off the new node */
new_node->next = [*head_ref];
/* move the head to point to the new node */
[*head_ref] = new_node;
}
/* Checks whether the value x is present in linked list */
bool search[Node* head, int x]
{
Node* current = head; // Initialize current
while [current != NULL]
{
if [current->key == x]
return true;
current = current->next;
}
return false;
}
/* Driver program to test count function*/
int main[]
{
/* Start with the empty list */
Node* head = NULL;
int x = 21;
/* Use push[] to construct below list
14->21->11->30->10 */
push[&head, 10];
push[&head, 30];
push[&head, 11];
push[&head, 21];
push[&head, 14];
search[head, 21]? coutnext;
}
return false;
}
/* Driver program to test count function*/
int main[]
{
/* Start with the empty list */
struct Node* head = NULL;
int x = 21;
/* Use push[] to construct below list
14->21->11->30->10 */
push[&head, 10];
push[&head, 30];
push[&head, 11];
push[&head, 21];
push[&head, 14];
search[head, 21]? printf["Yes"] : printf["No"];
return 0;
}
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// Iterative Java program to search an element
// in linked list
//Node class
class Node
{
int data;
Node next;
Node[int d]
{
data = d;
next = null;
}
}
//Linked list class
class LinkedList
{
Node head; //Head of list
//Inserts a new node at the front of the list
public void push[int new_data]
{
//Allocate new node and putting data
Node new_node = new Node[new_data];
//Make next of new node as head
new_node.next = head;
//Move the head to point to new Node
head = new_node;
}
//Checks whether the value x is present in linked list
public boolean search[Node head, int x]
{
Node current = head; //Initialize current
while [current != null]
{
if [current.data == x]
return true; //data found
current = current.next;
}
return false; //data not found
}
//Driver function to test the above functions
public static void main[String args[]]
{
//Start with the empty list
LinkedList llist = new LinkedList[];
/*Use push[] to construct below list
14->21->11->30->10 */
llist.push[10];
llist.push[30];
llist.push[11];
llist.push[21];
llist.push[14];
if [llist.search[llist.head, 21]]
System.out.println["Yes"];
else
System.out.println["No"];
}
}
// This code is contributed by Pratik Agarwal
|
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# Iterative Python program to search an element
# in linked list
# Node class
class Node:
# Function to initialise the node object
def __init__[self, data]:
self.data = data # Assign data
self.next = None # Initialize next as null
# Linked List class
class LinkedList:
def __init__[self]:
self.head = None # Initialize head as None
# This function insert a new node at the
# beginning of the linked list
def push[self, new_data]:
# Create a new Node
new_node = Node[new_data]
# 3. Make next of new Node as head
new_node.next = self.head
# 4. Move the head to point to new Node
self.head = new_node
# This Function checks whether the value
# x present in the linked list
def search[self, x]:
# Initialize current to head
current = self.head
# loop till current not equal to None
while current != None:
if current.data == x:
return True # data found
current = current.next
return False # Data Not found
# Code execution starts here
if __name__ == '__main__':
# Start with the empty list
llist = LinkedList[]
''' Use push[] to construct below list
14->21->11->30->10 '''
llist.push[10];
llist.push[30];
llist.push[11];
llist.push[21];
llist.push[14];
if llist.search[21]:
print["Yes"]
else:
print["No"]
# This code is contributed by Ravi Shankar
|
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// Iterative C# program to search an element
// in linked list
using System;
// Node class
public class Node
{
public int data;
public Node next;
public Node[int d]
{
data = d;
next = null;
}
}
// Linked list class
public class LinkedList
{
Node head; // Head of list
// Inserts a new node at the front of the list
public void push[int new_data]
{
// Allocate new node and putting data
Node new_node = new Node[new_data];
// Make next of new node as head
new_node.next = head;
// Move the head to point to new Node
head = new_node;
}
// Checks whether the value x is present in linked list
public bool search[Node head, int x]
{
Node current = head; // Initialize current
while [current != null]
{
if [current.data == x]
return true; // data found
current = current.next;
}
return false; // data not found
}
// Driver code
public static void Main[String []args]
{
// Start with the empty list
LinkedList llist = new LinkedList[];
/*Use push[] to construct below list
14->21->11->30->10 */
llist.push[10];
llist.push[30];
llist.push[11];
llist.push[21];
llist.push[14];
if [llist.search[llist.head, 21]]
Console.WriteLine["Yes"];
else
Console.WriteLine["No"];
}
}
// This code contributed by Rajput-Ji
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// Iterative javascript program
// to search an element
// in linked list
//Node class
class Node {
constructor[d] {
this.data = d;
this.next = null;
}
}
// Linked list class
var head; // Head of list
// Inserts a new node at the front of the list
function push[new_data]
{
// Allocate new node and putting data
var new_node = new Node[new_data];
// Make next of new node as head
new_node.next = head;
// Move the head to point to new Node
head = new_node;
}
// Checks whether the value
// x is present in linked list
function search[ head , x]
{
var current = head; // Initialize current
while [current != null] {
if [current.data == x]
return true; // data found
current = current.next;
}
return false; // data not found
}
// Driver function to test
// the above functions
// Start with the empty list
/*
Use push[] to construct below
list 14->21->11->30->10
*/
push[10];
push[30];
push[11];
push[21];
push[14];
if [search[head, 21]]
document.write["Yes"];
else
document.write["No"];
// This code contributed by aashish2995
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Output:
Yes
Recursive Solution
bool search[head, x] 1] If head is NULL, return false. 2] If head's key is same as x, return true; 3] Else return search[head->next, x]Following is the recursive implementation of the above algorithm to search a given key.
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// Recursive C++ program to search
// an element in linked list
#include
using namespace std;
/* Link list node */
struct Node
{
int key;
struct Node* next;
};
/* Given a reference [pointer to pointer] to the head
of a list and an int, push a new node on the front
of the list. */
void push[struct Node** head_ref, int new_key]
{
/* allocate node */
struct Node* new_node =
[struct Node*] malloc[sizeof[struct Node]];
/* put in the key */
new_node->key = new_key;
/* link the old list off the new node */
new_node->next = [*head_ref];
/* move the head to point to the new node */
[*head_ref] = new_node;
}
/* Checks whether the value x is present in linked list */
bool search[struct Node* head, int x]
{
// Base case
if [head == NULL]
return false;
// If key is present in current node, return true
if [head->key == x]
return true;
// Recur for remaining list
return search[head->next, x];
}
/* Driver code*/
int main[]
{
/* Start with the empty list */
struct Node* head = NULL;
int x = 21;
/* Use push[] to construct below list
14->21->11->30->10 */
push[&head, 10];
push[&head, 30];
push[&head, 11];
push[&head, 21];
push[&head, 14];
search[head, 21]? cout next = [*head_ref];
/* move the head to point to the new node */
[*head_ref] = new_node;
}
/* Checks whether the value x is present in linked list */
bool search[struct Node* head, int x]
{
// Base case
if [head == NULL]
return false;
// If key is present in current node, return true
if [head->key == x]
return true;
// Recur for remaining list
return search[head->next, x];
}
/* Driver program to test count function*/
int main[]
{
/* Start with the empty list */
struct Node* head = NULL;
int x = 21;
/* Use push[] to construct below list
14->21->11->30->10 */
push[&head, 10];
push[&head, 30];
push[&head, 11];
push[&head, 21];
push[&head, 14];
search[head, 21]? printf["Yes"] : printf["No"];
return 0;
}
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|
// Recursive Java program to search an element
// in linked list
// Node class
class Node
{
int data;
Node next;
Node[int d]
{
data = d;
next = null;
}
}
// Linked list class
class LinkedList
{
Node head; //Head of list
//Inserts a new node at the front of the list
public void push[int new_data]
{
//Allocate new node and putting data
Node new_node = new Node[new_data];
//Make next of new node as head
new_node.next = head;
//Move the head to point to new Node
head = new_node;
}
// Checks whether the value x is present
// in linked list
public boolean search[Node head, int x]
{
// Base case
if [head == null]
return false;
// If key is present in current node,
// return true
if [head.data == x]
return true;
// Recur for remaining list
return search[head.next, x];
}
// Driver function to test the above functions
public static void main[String args[]]
{
// Start with the empty list
LinkedList llist = new LinkedList[];
/* Use push[] to construct below list
14->21->11->30->10 */
llist.push[10];
llist.push[30];
llist.push[11];
llist.push[21];
llist.push[14];
if [llist.search[llist.head, 21]]
System.out.println["Yes"];
else
System.out.println["No"];
}
}
// This code is contributed by Pratik Agarwal
|
|
# Recursive Python program to
# search an element in linked list
# Node class
class Node:
# Function to initialise
# the node object
def __init__[self, data]:
self.data = data # Assign data
self.next = None # Initialize next as null
class LinkedList:
def __init__[self]:
self.head = None # Initialize head as None
# This function insert a new node at
# the beginning of the linked list
def push[self, new_data]:
# Create a new Node
new_node = Node[new_data]
# Make next of new Node as head
new_node.next = self.head
# Move the head to
# point to new Node
self.head = new_node
# Checks whether the value key
# is present in linked list
def search[self, li, key]:
# Base case
if[not li]:
return False
# If key is present in
# current node, return true
if[li.data == key]:
return True
# Recur for remaining list
return self.search[li.next, key]
# Driver Code
if __name__=='__main__':
li = LinkedList[]
li.push[1]
li.push[2]
li.push[3]
li.push[4]
key = 4
if li.search[li.head,key]:
print["Yes"]
else:
print["No"]
# This code is contributed
# by Manoj Sharma
|
|
// Recursive C# program to search
// an element in linked list
using System;
// Node class
public class Node
{
public int data;
public Node next;
public Node[int d]
{
data = d;
next = null;
}
}
// Linked list class
public class LinkedList
{
Node head; //Head of list
//Inserts a new node at the front of the list
public void push[int new_data]
{
//Allocate new node and putting data
Node new_node = new Node[new_data];
//Make next of new node as head
new_node.next = head;
//Move the head to point to new Node
head = new_node;
}
// Checks whether the value x is present
// in linked list
public bool search[Node head, int x]
{
// Base case
if [head == null]
return false;
// If key is present in current node,
// return true
if [head.data == x]
return true;
// Recur for remaining list
return search[head.next, x];
}
// Driver code
public static void Main[]
{
// Start with the empty list
LinkedList llist = new LinkedList[];
/* Use push[] to construct below list
14->21->11->30->10 */
llist.push[10];
llist.push[30];
llist.push[11];
llist.push[21];
llist.push[14];
if [llist.search[llist.head, 21]]
Console.WriteLine["Yes"];
else
Console.WriteLine["No"];
}
}
// This code is contributed by PrinciRaj1992
|
|
// Recursive javascript program to search an element
// in linked list
// Node class
class Node {
constructor[val] {
this.data = val;
this.next = null;
}
}
// Linked list class
var head; // Head of list
// Inserts a new node at the front of the list
function push[new_data] {
// Allocate new node and putting data
var new_node = new Node[new_data];
// Make next of new node as head
new_node.next = head;
// Move the head to point to new Node
head = new_node;
}
// Checks whether the value x is present
// in linked list
function search[head , x] {
// Base case
if [head == null]
return false;
// If key is present in current node,
// return true
if [head.data == x]
return true;
// Recur for remaining list
return search[head.next, x];
}
// Driver function to test the above functions
// Start with the empty list
/*
* Use push[] to construct below list 14->21->11->30->10
*/
push[10];
push[30];
push[11];
push[21];
push[14];
if [search[head, 21]]
document.write["Yes"];
else
document.write["No"];
// This code contributed by gauravrajput1
|
Output:
YesThis article is contributed by Ravi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
LinkedList Class
- Reference
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Linked List Operations: Traverse, Insert and Delete
In this tutorial, you will learn different operations on a linked list. Also, you will find implementation of linked list operations in C/C++, Python and Java.
There are various linked list operations that allow us to perform different actions on linked lists. For example, the insertion operation adds a new element to the linked list.
Here's a list of basic linked list operations that we will cover in this article.
- Traversal - access each element of the linked list
- Insertion - adds a new element to the linked list
- Deletion - removes the existing elements
- Search - find a node in the linked list
- Sort - sort the nodes of the linked list
Before you learn about linked list operations in detail, make sure to know about Linked List first.
Things to Remember about Linked List
- head points to the first node of the linked list
- next pointer of the last node is NULL, so if the next current node is NULL, we have reached the end of the linked list.
In all of the examples, we will assume that the linked list has three nodes 1 --->2 --->3 with node structure as below:
struct node { int data; struct node *next; };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:
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]