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IB DP Computer Science Study Notes

5.4.3 Sketching Various Linked Lists

Linked lists are a versatile and fundamental data structure used in computer science. They are particularly useful for applications where the size of the data set can change dynamically. Understanding how to sketch and manipulate these structures is essential for grasping their practical use in various algorithms and applications.

Introduction to Linked Lists

Linked lists are composed of a series of nodes, each holding data and a reference (or pointer) to the next node in the sequence. Unlike arrays, they offer dynamic memory allocation and can efficiently handle operations like insertion and deletion.

Single Linked Lists

A single linked list is a sequence of nodes with each node pointing to the next. It begins with a 'head' node and ends when a node points to null.

Diagrammatic Representation

  • Nodes: Each node contains data and a single pointer. They are typically represented as boxes or circles.
  • Head: The starting node, often represented with an entry arrow.
  • Null Termination: The end of the list, indicating no further nodes.


  • Addition: New nodes can be added anywhere; commonly at the beginning, end, or between two existing nodes. This involves changing the pointer of the previous node and setting the new node’s pointer to the subsequent node.
  • Deletion: To remove a node, redirect the pointer of its predecessor to its successor. Special cases occur when deleting the head or the last node.
  • Modification: Involves only changing the data within a node; the list's structure remains unaffected.
  • Searching: A sequential process, starting from the head and moving through each node until the target data is found or the end is reached.

Double Linked Lists

Each node in a double linked list points to both its next and previous nodes, offering greater flexibility in traversal.

Diagrammatic Representation

  • Nodes: Each contains two pointers – one for the next node and one for the previous.
  • Head and Tail: The first and last nodes respectively. The tail node's next pointer is null, and the head's previous pointer is also null.


  • Addition: Similar to a single linked list, but requires updating the previous pointer of the next node and the next pointer of the previous node.
  • Deletion: Involves changing both the next pointer of the predecessor and the previous pointer of the successor.
  • Modification and Searching: More efficient as traversal can be from either end or can proceed in either direction.

Circular Linked Lists

In a circular linked list, the sequence forms a loop, where the last node links back to the first, facilitating continuous traversal.

Diagrammatic Representation

  • Nodes: Similar to those in single or double linked lists, but without any null pointers.
  • Circular Link: A unique characteristic where the last node points back to the head.


  • Addition and Deletion: These operations require careful adjustment of pointers to maintain the circular structure.
  • Modification and Searching: Enhanced due to the ability to traverse the list indefinitely, without needing to restart from the head.

Practical Examples and Scenarios

Real-world scenarios help illustrate the functionality of linked lists:

Scenario 1: Playlist Management

Consider managing a music playlist as a circular double linked list, where each song is a node. Users can easily navigate back and forth (thanks to the double links) and loop around the playlist endlessly (due to the circular nature).

Scenario 2: Undo Functionality in Applications

An application's undo and redo functionality is aptly represented by a double linked list. Each action by the user creates a new node, allowing navigation through their history of actions in both directions.

Scenario 3: Online Gaming

Online gaming environments, especially those involving multiple players, often use single linked lists to track players. New players are added at the end of the list; players leaving the game result in node deletions, and frequent searches are conducted to update scores or statuses.

Advanced Considerations

While the basic operations on linked lists are straightforward, several advanced considerations can arise:

Memory Management

Efficient handling of memory is crucial. In languages like C++, careful memory management must be practised to prevent memory leaks during deletion.

Algorithm Complexity

Operations in linked lists can have varying complexities. For example, searching is generally O(n) as it may require traversing the entire list. Understanding these complexities is key to choosing the right data structure for a given problem.

Pointer Errors

In languages that use pointers, incorrect handling can lead to errors like lost updates, where a node is accidentally detached from a list, or dangling pointers, where a pointer references a deallocated memory.

Use in Advanced Data Structures

Linked lists are not just standalone structures but are also components in more complex structures like hash tables, adjacency lists for graphs, and more.


Mastering linked lists involves understanding their structure, visualising their operations, and applying these concepts in practical scenarios. The ability to manipulate these dynamic data structures is integral to proficient algorithm design and effective problem-solving in computer science.


To detect a loop in a linked list, the Floyd’s Cycle-Finding Algorithm (often termed the "tortoise and the hare" algorithm) is commonly used. This algorithm involves two pointers, a slow pointer and a fast pointer. Initially, both are set to point to the head of the list. Then, in each step of the algorithm, the slow pointer advances by one node, and the fast pointer advances by two nodes. If the linked list contains a loop, the fast pointer will eventually meet the slow pointer within the loop. If the fast pointer reaches the end of the list (null), then the list doesn't have a loop. This method is efficient because it only requires O(n) time and O(1) additional space.

Time complexity in linked lists varies based on the operation:

  • Searching: O(n) since it might require traversing the entire list to find an element.
  • Insertion: Generally O(1) if you're inserting at the beginning or if you already have a reference to the node after which you wish to insert. However, if you need to find a specific position first, it becomes O(n).
  • Deletion: Similar to insertion, it's O(1) if you have direct access to the node you want to delete. But if you need to search for a node first, the complexity becomes O(n).
  • Traversal: O(n) as each node must be visited. These complexities are largely due to the non-contiguous storage of data in linked lists and the need to potentially traverse through nodes sequentially to reach a particular node.

Sentinel nodes are special nodes in a linked list that are used to simplify the implementation of a linked list, particularly the operations of insertion and deletion. They are not used to store meaningful data but instead act as placeholders or markers. Typically, a sentinel node can be used as a "dummy" head or tail in a linked list. By doing so, certain edge cases in list operations, like inserting or deleting nodes at the beginning or end of the list, are simplified. This is because these operations can be performed uniformly without needing to check for null pointers. For example, in a list with a dummy head, insertion at the beginning of the list does not require checking if the head is null – you can simply insert the new node after the dummy head. This uniformity often leads to simpler, more error-free implementation.

Linked lists are versatile and can indeed be used as the foundational building block for various other data structures. For example:

  • Stacks and Queues: Both of these can be easily implemented using either single or double linked lists. The stack follows the Last In, First Out (LIFO) principle, while the queue uses the First In, First Out (FIFO) principle. The dynamic nature of linked lists allows for the constant time addition and removal of elements, which are key operations in both stacks and queues.
  • Graphs: Linked lists can represent adjacency lists in graphs. Each node's list contains its adjacent nodes, allowing the graph to dynamically add or remove nodes and edges.
  • Hash Tables: Linked lists can resolve collisions in hash tables using chaining. Each bucket of a hash table, which may experience a collision, can store a linked list of all elements hashing to the same bucket.

Linked lists allocate memory dynamically and hence do not require a contiguous memory block, unlike arrays. Each node in a linked list is stored independently in memory, with the node itself containing a pointer to the next (and possibly previous) node. This allocation method allows for efficient memory utilisation, especially in scenarios where the exact number of required elements is not known beforehand or can change. Arrays, in contrast, typically require the size to be specified at the time of declaration, leading to potential memory wastage or a need for resizing if the array becomes too small or large. Furthermore, resizing an array is a costly operation involving creating a new array and copying all elements to it. Linked lists avoid these inefficiencies by allowing nodes to be added or removed without affecting the rest of the structure.

Practice Questions

Explain how the addition of a new node in a double linked list differs from that in a single linked list. Illustrate your answer with a diagram.

The addition of a new node in a double linked list is more complex than in a single linked list. In a double linked list, the new node must be linked to both the preceding and succeeding nodes. This requires updating the next pointer of the previous node to point to the new node, and the previous pointer of the new node to point to the previous node. Similarly, the next pointer of the new node should point to the subsequent node, and the previous pointer of the subsequent node should be updated to point to the new node. In contrast, in a single linked list, only the next pointer of the previous node and the new node need to be updated. The diagram for each would show these pointer updates, highlighting the bidirectional linking in double linked lists versus the unidirectional linking in single linked lists.

Consider a scenario where you need to implement a history feature in a web browser. Would you use a single, double, or circular linked list for this feature? Justify your answer.

For implementing a history feature in a web browser, a double linked list would be most suitable. This is because a double linked list allows traversal in both directions: backwards (previous web pages) and forwards (forward navigation after going back). This mirrors the typical use case in a browser history, where users often go back to a previously visited page and then move forward again. A double linked list provides an efficient way to navigate through the history in this manner. In contrast, a single linked list would only allow movement in one direction, making it cumbersome to navigate forwards again. A circular linked list, while allowing continuous traversal, lacks the intuitive back and forth navigation inherent in a double linked list and would complicate the user experience by looping the history.

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Written by: Alfie
Cambridge University - BA Maths

A Cambridge alumnus, Alfie is a qualified teacher, and specialises creating educational materials for Computer Science for high school students.

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