Data Structures & AlgorithmsCircular Linked Lists

Circular Linked Lists

LevelIntermediate

Duration55 mins

TopicCircular Linked Lists

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What Is a Circular Linked List

When the End Meets the Beginning

In every linked list we've explored so far, there has been a clear beginning and a definitive end. We start at the head, traverse through the chain of nodes, and eventually reach a node whose next pointer is null—the unmistakable signal that our journey through the list has concluded.

But what if there were no end?

Imagine a structure where the last node, instead of pointing to nothing, points back to the first node. The chain becomes a ring. The sequence becomes a cycle. There is no terminal point, no null to signal completion—only an endless loop of nodes, each connected to the next, with the final connection completing the circle.

This is the circular linked list: a variation of the linked list where the notion of "ending" transforms into the concept of "returning to the start." This seemingly simple modification opens entirely new possibilities and solves problems that linear linked lists cannot address elegantly.

What You Will Master

By the end of this page, you will deeply understand what a circular linked list is, why it exists, how it differs structurally from linear linked lists, and the conceptual model for reasoning about cyclic data structures. This foundation is essential for understanding the variants, operations, and real-world applications covered in subsequent pages.

The Conceptual Foundation

To understand circular linked lists, we must first examine the fundamental question: What does it mean for a data structure to be circular?

In mathematics and computer science, a cycle is a path that begins and ends at the same point. When we apply this concept to linked lists, we create a structure where traversal from any node will eventually return to that same node if we keep following the next pointers.

The Linear Model (Traditional Linked List):

head → [A] → [B] → [C] → [D] → null

In this model, traversal terminates. Starting from the head, we can visit A, then B, then C, then D, and then we stop—there is nowhere else to go. The null pointer is the wall at the end of the corridor.

The Circular Model:

head → [A] → [B] → [C] → [D] ─┐
        ↑                      │
        └──────────────────────┘

In the circular model, D's next pointer doesn't point to null—it points back to A. Now, starting from any node and following next pointers will cycle through all nodes indefinitely. There is no termination unless we explicitly introduce a stopping condition.

The Nature of Cycles

A cycle doesn't mean confusion or chaos—it means continuity. Just as a clock's hands cycle through hours without ever 'ending,' a circular linked list cycles through its elements in a well-defined, predictable order. The structure is deterministic; only our traversal strategy changes.

The Philosophical Shift:

Traditional linked lists answer the question: "What comes next, and when do we stop?"

Circular linked lists answer a different question: "What comes next, and how do we know when we've completed a full cycle?"

This shift from termination-based thinking to cycle-completion thinking is fundamental. It affects how we design traversal algorithms, how we insert and delete nodes, and how we reason about the structure's invariants.

Formal Definition and Structure

Let us establish a precise definition:

A circular linked list is a linked list in which the last node's next pointer references the first node (or a designated entry node), forming a closed loop.

This definition applies to the most common form—the singly circular linked list. The concept extends naturally to doubly linked lists, creating doubly circular linked lists where bidirectional cycles exist.

Structural Components:

A circular linked list, like its linear counterpart, consists of nodes. Each node in a singly circular linked list contains:

Data — The value stored in the node
Next Pointer — A reference to the next node in the sequence

The critical difference is in what the last node's next pointer references:

List Type	Last Node's `next` Points To
Linear Singly Linked List	`null`
Circular Singly Linked List	First node (head)

Circular Linked List Node Structure
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// Node structure is identical to a standard linked list node
class CircularListNode<T> {
    data: T;
    next: CircularListNode<T> | null;
    
    constructor(data: T) {
        this.data = data;
        // Initially points to itself (single-node circular list)
        // or null until properly linked
        this.next = null;
    }
}
 
// The difference lies in how we connect the nodes
class CircularLinkedList<T> {
    // We can use 'head' or 'tail' as the entry point
    // Using 'tail' can be advantageous (explained later)
    private head: CircularListNode<T> | null;
    private size: number;
    
    constructor() {
        this.head = null;
        this.size = 0;
    }
    
    isEmpty(): boolean {
        return this.head === null;
    }
    
    getSize(): number {
        return this.size;
    }
}

Node Reusability

Notice that the node class itself is structurally identical to a standard linked list node. The 'circularity' is not a property of individual nodes—it's a property of how the nodes are linked together. The same node class can be used in linear or circular configurations depending on how we manage the connections.

Visualizing the Circular Structure

Visualization is crucial for developing intuition about circular linked lists. Unlike linear lists that we naturally draw as a straight line, circular lists require a different mental model.

Linear Representation with Loop-Back:

The simplest visualization shows the nodes in a line, with an arrow looping back from the last node to the first:

head → [10] → [20] → [30] → [40] ─┐
         ↑                         │
         └─────────────────────────┘

Ring Representation:

A more accurate visualization shows the nodes arranged in a circle, emphasizing that no node is truly "first" or "last" in the logical sense—they are distinguished only by our choice of entry point:

            ┌────────┐
            │        │
            ↓        │
         [head]      │
          [10]       │
         ↙    ↘      │
       [40]    [20]  │
         ↖    ↙      │
          [30]       │
            │        │
            └────────┘

Memory Layout Perspective:

Just like linear linked lists, circular linked lists have nodes scattered throughout memory. The addresses are arbitrary; only the pointer values create the circular structure:

Address       Contents
─────────────────────────────────────────────────
0x1000        Node { data: 10, next: 0x3000 }   ← head points here
0x2000        (other data)
0x3000        Node { data: 20, next: 0x5000 }
0x4000        (other data)
0x5000        Node { data: 30, next: 0x7000 }
0x6000        (other data)
0x7000        Node { data: 40, next: 0x1000 }   ← points back to head!

Following pointers: 0x1000 → 0x3000 → 0x5000 → 0x7000 → 0x1000 → ...

The circular nature is entirely in the pointer connections, not in the physical memory layout.

The Single-Node Case

A circular linked list with a single node is a valid structure. In this case, the node's next pointer points to itself: head → [A] ⟲. This self-referential single node forms the smallest possible circular linked list.

Why Circular? Motivation and Use Cases

Every data structure exists to solve specific problems more elegantly than alternatives. Circular linked lists shine in scenarios involving cyclic or repetitive processing, where the concept of "starting over" is inherent to the problem.

Motivating Question:

Imagine you're managing a game where multiple players take turns in a fixed order. After the last player's turn, play returns to the first player. With a linear linked list, after reaching the last player, you'd need special logic to jump back to the head. With a circular linked list, you simply follow the next pointer—it naturally brings you back to the first player.

This seamless wraparound is the core value proposition of circular linked lists.

Scenarios Where Circular Lists Excel

•Round-Robin Scheduling — Processes in an operating system are given CPU time in rotation. After the last process, scheduling returns to the first. A circular list naturally models this.
•Circular Buffers — Fixed-size buffers where new data overwrites the oldest data. Circular lists provide the wraparound semantics inherently.
•Turn-Based Games — Players take turns in sequence, cycling indefinitely. Circular lists eliminate edge-case handling for returning to the first player.
•Repeating Playlists — Music or video playlists that loop continuously. The last track's "next" is naturally the first track.
•Token Ring Networks — In some networking protocols, a token circulates around nodes in a ring topology. Circular lists model this exactly.
•Josephus Problem — A classic algorithmic problem where people in a circle are eliminated at regular intervals. Circular lists enable direct simulation.

The Common Pattern:

All these scenarios share a common characteristic: there is no natural 'end' to the sequence. The data or entities form a logical cycle, and operations repeatedly traverse or process elements in order, wrapping around when reaching what would otherwise be the end.

When your problem domain has this cyclic nature, using a linear list with explicit wraparound logic is fighting against the grain of the data. A circular list aligns the data structure with the problem's inherent structure.

Design Intuition

When choosing between linear and circular linked lists, ask: 'Does my problem have a definitive start and end, or does it cycle through elements repeatedly?' If the latter, circular linked lists may provide cleaner, more intuitive code.

Comparison with Linear Linked Lists

Understanding the differences between circular and linear linked lists is essential for making informed design decisions. The table below provides a detailed comparison:

Circular vs. Linear Linked Lists: Detailed Comparison
Aspect	Linear Linked List	Circular Linked List
Last Node's next	Points to null	Points to head (first node)
End Detection	Check for next === null	Check if current === head (after starting from head)
Traversal Nature	Terminates naturally	Infinite unless manually stopped
Full Traversal Logic	while (current !== null)	do-while with head comparison
Edge Case: Empty List	head === null	head === null (same)
Edge Case: Single Node	head.next === null	head.next === head (self-loop)
Access to First Node	Directly via head	Directly via head
Access to Last Node	O(n) traversal (or O(1) with tail pointer)	O(n) traversal, or O(1) if using tail-referenced design
Insertion at Head	O(1)	O(1), but must update last node's next
Deletion at Head	O(1)	O(1), but must update last node's next
Natural Wraparound	Requires explicit logic	Built-in
Memory Overhead	One pointer per node	One pointer per node (same)

When to Use Circular

•Problem inherently involves cyclic processing
•Need to repeatedly cycle through elements
•Wraparound logic would clutter linear list code
•Modeling systems with ring-like structures
•Algorithms that eliminate elements in rotation

When to Use Linear

•Data has a clear beginning and end
•Traversal should terminate naturally
•Simpler termination condition is preferable
•Accidental infinite loops would be catastrophic
•Standard LIFO/FIFO patterns (stacks, queues)

The Entry Point Dilemma: Head or Tail?

In a circular linked list, every node is connected. There is no true "first" or "last" in the cycle—only a node we designate as our entry point. This raises an interesting design question: Should we maintain a reference to the head, the tail, or something else?

Option 1: Head Reference (Most Common)

We keep a reference to the node we consider "first." This is the traditional approach and feels natural because it mirrors linear linked lists.

head → [A] → [B] → [C] → [D] ─┐
        ↑                      │
        └──────────────────────┘

Option 2: Tail Reference (Often More Efficient)

Alternatively, we keep a reference to the node we consider "last." Since tail.next gives us the head in a circular list, we effectively have O(1) access to both ends:

        ┌──────────────────────┐
        │                      ↓
       [D] → [A] → [B] → [C] ─┘
        ↑
       tail (and tail.next is head, which is [A])

Tail-Referenced Circular List Advantage
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// With a tail reference, we get O(1) access to both ends
 
class TailReferencedCircularList<T> {
    private tail: CircularListNode<T> | null;
    private size: number;
    
    constructor() {
        this.tail = null;
        this.size = 0;
    }
    
    // Get the head in O(1) time
    getHead(): CircularListNode<T> | null {
        if (this.tail === null) return null;
        return this.tail.next; // In a circular list, tail.next is head
    }
    
    // Get the tail in O(1) time
    getTail(): CircularListNode<T> | null {
        return this.tail;
    }
    
    // Insert at the beginning in O(1) time
    insertAtHead(data: T): void {
        const newNode = new CircularListNode(data);
        
        if (this.tail === null) {
            // First node: points to itself
            newNode.next = newNode;
            this.tail = newNode;
        } else {
            // New node points to current head
            newNode.next = this.tail.next;
            // Tail now points to new node (making it the new head)
            this.tail.next = newNode;
        }
        this.size++;
    }
    
    // Insert at the end in O(1) time
    insertAtTail(data: T): void {
        const newNode = new CircularListNode(data);
        
        if (this.tail === null) {
            // First node: points to itself
            newNode.next = newNode;
            this.tail = newNode;
        } else {
            // New node points to head
            newNode.next = this.tail.next;
            // Current tail points to new node
            this.tail.next = newNode;
            // Update tail reference
            this.tail = newNode;
        }
        this.size++;
    }
}

The Tail-Reference Advantage

With a head-referenced design, inserting at the tail requires O(n) traversal to find the last node. With a tail-referenced design, inserting at either end is O(1). Since tail.next gives us the head, we lose nothing. This is why many circular linked list implementations prefer a tail reference.

Invariants of a Valid Circular Linked List

Just as linear linked lists have structural invariants, circular linked lists have their own set of conditions that must always hold true. Understanding and maintaining these invariants is critical for writing correct code.

Core Invariants:

•Closure — Following next pointers from any node will eventually return to that node. The structure forms a closed loop with no null termination (except when the list is empty).
•Full Reachability — Every node in the list is reachable from the entry point (head or tail) by following next pointers.
•Unique Path Length — Starting from the entry point, following exactly size links returns to the entry point (completing exactly one cycle).
•Single Cycle — All nodes belong to the same cycle. There are no separate loops or disconnected nodes within the structure.
•Entry Point Validity — The entry point (head or tail) is either null (empty list) or references a valid node within the cycle.
•Self-Reference Consistency — A single-node circular list has node.next === node (the node points to itself).

What Breaks These Invariants?

Breaking the Cycle — If any node's next accidentally becomes null (except in an empty list), the structure is broken. Traversal will hit null instead of cycling.
Multiple Entry Points — If head and tail become inconsistent (e.g., tail.next ≠ head after an operation), the list's invariants are violated.
Orphaned Nodes — Nodes that are removed from the cycle but not properly unlinked can cause memory leaks and logical inconsistencies.
Size Mismatch — If the size counter doesn't match the actual number of nodes in the cycle, operations relying on size will behave incorrectly.

The Infinite Loop Danger

The most common bug when working with circular linked lists is the infinite loop. If your termination condition is 'while (current !== null)' as with linear lists, you'll loop forever because current never becomes null. Always use an entry-point comparison or count-based termination.

Traversal in Circular Linked Lists

Traversing a circular linked list requires a fundamentally different approach than traversing a linear list. Since there is no null terminator, we must use alternative methods to know when we've visited all nodes.

Method 1: Entry Point Comparison (Do-While Pattern)

Start from the entry point and continue until we return to it:

Circular List Traversal
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// Method 1: Do-While with Entry Point Comparison
function traverseCircular<T>(head: CircularListNode<T> | null): void {
    if (head === null) {
        console.log("List is empty");
        return;
    }
    
    let current = head;
    
    // Use do-while because we need to visit head before checking
    do {
        console.log(current.data);
        current = current.next!;
    } while (current !== head);
    // We stop when we're back at the starting point
}
 
// Method 2: Count-Based Traversal
function traverseByCount<T>(
    head: CircularListNode<T> | null, 
    size: number
): void {
    if (head === null || size === 0) {
        console.log("List is empty");
        return;
    }
    
    let current = head;
    
    for (let i = 0; i < size; i++) {
        console.log(current.data);
        current = current.next!;
    }
}
 
// Method 3: Using a "visited once" flag (for complex traversals)
function traverseWithFlag<T>(head: CircularListNode<T> | null): void {
    if (head === null) return;
    
    let current: CircularListNode<T> | null = head;
    let isFirst = true;
    
    while (isFirst || current !== head) {
        isFirst = false;
        console.log(current!.data);
        current = current!.next;
    }
}

Choosing a Traversal Method

The do-while pattern (Method 1) is most common because it doesn't require knowing the size. The count-based approach (Method 2) is useful when you have a size field and want to avoid the comparison overhead on each iteration. Both are O(n) for a complete traversal.

Summary: What Is a Circular Linked List

We've thoroughly explored the concept and structure of circular linked lists. Let's consolidate the key understandings:

Key Takeaways

•Definition — A circular linked list is a linked list where the last node's next pointer references the first node, forming a closed loop with no null termination.
•No True End — Unlike linear lists, circular lists have no terminal null pointer. Every node has a valid successor, and traversal can continue indefinitely.
•Same Node Structure — The node class is identical to linear linked lists. Circularity is a property of the connections, not the nodes themselves.
•Entry Point Choice — We can maintain a reference to the head, the tail, or both. Tail references often provide O(1) access to both ends.
•Natural Cyclic Semantics — Circular lists excel in problems that are inherently cyclic: round-robin processing, repeating sequences, and ring-topology simulations.
•Traversal Requires Care — The standard while (current !== null) pattern causes infinite loops. Use entry-point comparison or count-based termination.
•Invariant: Closure — A valid circular list forms exactly one closed cycle containing all nodes, with every node reachable from the entry point.

Your Mental Model:

Think of a circular linked list as a merry-go-round. Each horse (node) is connected to the next, and after the last horse comes the first again. There is no "end" to the ride—you simply keep going round and round until you decide to stop. The entry point (where you get on) is just a reference point; the structure itself is symmetric and continuous.

What's Next:

Now that you understand the fundamental concept of circular linked lists, we'll explore the two main variants in depth: singly circular linked lists and doubly circular linked lists. Understanding both variants is essential for choosing the right tool for your specific use case.

Page Complete

You now have a deep understanding of what circular linked lists are, why they exist, and how they differ structurally from linear linked lists. This foundation prepares you for exploring the singly and doubly circular variants in the next page.

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Data Structures & AlgorithmsCircular Linked Lists

Circular Linked Lists

LevelIntermediate

Duration55 mins

TopicCircular Linked Lists

1 / 4

What Is a Circular Linked List

When the End Meets the Beginning

But what if there were no end?

What You Will Master

The Conceptual Foundation

To understand circular linked lists, we must first examine the fundamental question: What does it mean for a data structure to be circular?

The Linear Model (Traditional Linked List):

head → [A] → [B] → [C] → [D] → null

The Circular Model:

head → [A] → [B] → [C] → [D] ─┐
        ↑                      │
        └──────────────────────┘

The Nature of Cycles

The Philosophical Shift:

Traditional linked lists answer the question: "What comes next, and when do we stop?"

Circular linked lists answer a different question: "What comes next, and how do we know when we've completed a full cycle?"

Formal Definition and Structure

Let us establish a precise definition:

A circular linked list is a linked list in which the last node's next pointer references the first node (or a designated entry node), forming a closed loop.

Structural Components:

A circular linked list, like its linear counterpart, consists of nodes. Each node in a singly circular linked list contains:

Data — The value stored in the node
Next Pointer — A reference to the next node in the sequence

The critical difference is in what the last node's next pointer references:

List Type	Last Node's `next` Points To
Linear Singly Linked List	`null`
Circular Singly Linked List	First node (head)

Circular Linked List Node Structure
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// Node structure is identical to a standard linked list node
class CircularListNode<T> {
    data: T;
    next: CircularListNode<T> | null;
    
    constructor(data: T) {
        this.data = data;
        // Initially points to itself (single-node circular list)
        // or null until properly linked
        this.next = null;
    }
}
 
// The difference lies in how we connect the nodes
class CircularLinkedList<T> {
    // We can use 'head' or 'tail' as the entry point
    // Using 'tail' can be advantageous (explained later)
    private head: CircularListNode<T> | null;
    private size: number;
    
    constructor() {
        this.head = null;
        this.size = 0;
    }
    
    isEmpty(): boolean {
        return this.head === null;
    }
    
    getSize(): number {
        return this.size;
    }
}

Node Reusability

Visualizing the Circular Structure

Visualization is crucial for developing intuition about circular linked lists. Unlike linear lists that we naturally draw as a straight line, circular lists require a different mental model.

Linear Representation with Loop-Back:

The simplest visualization shows the nodes in a line, with an arrow looping back from the last node to the first:

head → [10] → [20] → [30] → [40] ─┐
         ↑                         │
         └─────────────────────────┘

Ring Representation:

            ┌────────┐
            │        │
            ↓        │
         [head]      │
          [10]       │
         ↙    ↘      │
       [40]    [20]  │
         ↖    ↙      │
          [30]       │
            │        │
            └────────┘

Memory Layout Perspective:

Just like linear linked lists, circular linked lists have nodes scattered throughout memory. The addresses are arbitrary; only the pointer values create the circular structure:

Address       Contents
─────────────────────────────────────────────────
0x1000        Node { data: 10, next: 0x3000 }   ← head points here
0x2000        (other data)
0x3000        Node { data: 20, next: 0x5000 }
0x4000        (other data)
0x5000        Node { data: 30, next: 0x7000 }
0x6000        (other data)
0x7000        Node { data: 40, next: 0x1000 }   ← points back to head!

Following pointers: 0x1000 → 0x3000 → 0x5000 → 0x7000 → 0x1000 → ...

The circular nature is entirely in the pointer connections, not in the physical memory layout.

The Single-Node Case

Why Circular? Motivation and Use Cases

Motivating Question:

This seamless wraparound is the core value proposition of circular linked lists.

Scenarios Where Circular Lists Excel

•Round-Robin Scheduling — Processes in an operating system are given CPU time in rotation. After the last process, scheduling returns to the first. A circular list naturally models this.
•Circular Buffers — Fixed-size buffers where new data overwrites the oldest data. Circular lists provide the wraparound semantics inherently.
•Turn-Based Games — Players take turns in sequence, cycling indefinitely. Circular lists eliminate edge-case handling for returning to the first player.
•Repeating Playlists — Music or video playlists that loop continuously. The last track's "next" is naturally the first track.
•Token Ring Networks — In some networking protocols, a token circulates around nodes in a ring topology. Circular lists model this exactly.
•Josephus Problem — A classic algorithmic problem where people in a circle are eliminated at regular intervals. Circular lists enable direct simulation.

The Common Pattern:

Design Intuition

Comparison with Linear Linked Lists

Understanding the differences between circular and linear linked lists is essential for making informed design decisions. The table below provides a detailed comparison:

Circular vs. Linear Linked Lists: Detailed Comparison
Aspect	Linear Linked List	Circular Linked List
Last Node's next	Points to null	Points to head (first node)
End Detection	Check for next === null	Check if current === head (after starting from head)
Traversal Nature	Terminates naturally	Infinite unless manually stopped
Full Traversal Logic	while (current !== null)	do-while with head comparison
Edge Case: Empty List	head === null	head === null (same)
Edge Case: Single Node	head.next === null	head.next === head (self-loop)
Access to First Node	Directly via head	Directly via head
Access to Last Node	O(n) traversal (or O(1) with tail pointer)	O(n) traversal, or O(1) if using tail-referenced design
Insertion at Head	O(1)	O(1), but must update last node's next
Deletion at Head	O(1)	O(1), but must update last node's next
Natural Wraparound	Requires explicit logic	Built-in
Memory Overhead	One pointer per node	One pointer per node (same)

When to Use Circular

•Problem inherently involves cyclic processing
•Need to repeatedly cycle through elements
•Wraparound logic would clutter linear list code
•Modeling systems with ring-like structures
•Algorithms that eliminate elements in rotation

When to Use Linear

•Data has a clear beginning and end
•Traversal should terminate naturally
•Simpler termination condition is preferable
•Accidental infinite loops would be catastrophic
•Standard LIFO/FIFO patterns (stacks, queues)

The Entry Point Dilemma: Head or Tail?

Option 1: Head Reference (Most Common)

We keep a reference to the node we consider "first." This is the traditional approach and feels natural because it mirrors linear linked lists.

head → [A] → [B] → [C] → [D] ─┐
        ↑                      │
        └──────────────────────┘

Option 2: Tail Reference (Often More Efficient)

Alternatively, we keep a reference to the node we consider "last." Since tail.next gives us the head in a circular list, we effectively have O(1) access to both ends:

        ┌──────────────────────┐
        │                      ↓
       [D] → [A] → [B] → [C] ─┘
        ↑
       tail (and tail.next is head, which is [A])

Tail-Referenced Circular List Advantage
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// With a tail reference, we get O(1) access to both ends
 
class TailReferencedCircularList<T> {
    private tail: CircularListNode<T> | null;
    private size: number;
    
    constructor() {
        this.tail = null;
        this.size = 0;
    }
    
    // Get the head in O(1) time
    getHead(): CircularListNode<T> | null {
        if (this.tail === null) return null;
        return this.tail.next; // In a circular list, tail.next is head
    }
    
    // Get the tail in O(1) time
    getTail(): CircularListNode<T> | null {
        return this.tail;
    }
    
    // Insert at the beginning in O(1) time
    insertAtHead(data: T): void {
        const newNode = new CircularListNode(data);
        
        if (this.tail === null) {
            // First node: points to itself
            newNode.next = newNode;
            this.tail = newNode;
        } else {
            // New node points to current head
            newNode.next = this.tail.next;
            // Tail now points to new node (making it the new head)
            this.tail.next = newNode;
        }
        this.size++;
    }
    
    // Insert at the end in O(1) time
    insertAtTail(data: T): void {
        const newNode = new CircularListNode(data);
        
        if (this.tail === null) {
            // First node: points to itself
            newNode.next = newNode;
            this.tail = newNode;
        } else {
            // New node points to head
            newNode.next = this.tail.next;
            // Current tail points to new node
            this.tail.next = newNode;
            // Update tail reference
            this.tail = newNode;
        }
        this.size++;
    }
}

The Tail-Reference Advantage

Invariants of a Valid Circular Linked List

Core Invariants:

•Closure — Following next pointers from any node will eventually return to that node. The structure forms a closed loop with no null termination (except when the list is empty).
•Full Reachability — Every node in the list is reachable from the entry point (head or tail) by following next pointers.
•Unique Path Length — Starting from the entry point, following exactly size links returns to the entry point (completing exactly one cycle).
•Single Cycle — All nodes belong to the same cycle. There are no separate loops or disconnected nodes within the structure.
•Entry Point Validity — The entry point (head or tail) is either null (empty list) or references a valid node within the cycle.
•Self-Reference Consistency — A single-node circular list has node.next === node (the node points to itself).

What Breaks These Invariants?

Breaking the Cycle — If any node's next accidentally becomes null (except in an empty list), the structure is broken. Traversal will hit null instead of cycling.
Multiple Entry Points — If head and tail become inconsistent (e.g., tail.next ≠ head after an operation), the list's invariants are violated.
Orphaned Nodes — Nodes that are removed from the cycle but not properly unlinked can cause memory leaks and logical inconsistencies.
Size Mismatch — If the size counter doesn't match the actual number of nodes in the cycle, operations relying on size will behave incorrectly.

The Infinite Loop Danger

Traversal in Circular Linked Lists

Method 1: Entry Point Comparison (Do-While Pattern)

Start from the entry point and continue until we return to it:

Circular List Traversal
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// Method 1: Do-While with Entry Point Comparison
function traverseCircular<T>(head: CircularListNode<T> | null): void {
    if (head === null) {
        console.log("List is empty");
        return;
    }
    
    let current = head;
    
    // Use do-while because we need to visit head before checking
    do {
        console.log(current.data);
        current = current.next!;
    } while (current !== head);
    // We stop when we're back at the starting point
}
 
// Method 2: Count-Based Traversal
function traverseByCount<T>(
    head: CircularListNode<T> | null, 
    size: number
): void {
    if (head === null || size === 0) {
        console.log("List is empty");
        return;
    }
    
    let current = head;
    
    for (let i = 0; i < size; i++) {
        console.log(current.data);
        current = current.next!;
    }
}
 
// Method 3: Using a "visited once" flag (for complex traversals)
function traverseWithFlag<T>(head: CircularListNode<T> | null): void {
    if (head === null) return;
    
    let current: CircularListNode<T> | null = head;
    let isFirst = true;
    
    while (isFirst || current !== head) {
        isFirst = false;
        console.log(current!.data);
        current = current!.next;
    }
}

Choosing a Traversal Method

Summary: What Is a Circular Linked List

We've thoroughly explored the concept and structure of circular linked lists. Let's consolidate the key understandings:

Key Takeaways

•Definition — A circular linked list is a linked list where the last node's next pointer references the first node, forming a closed loop with no null termination.
•No True End — Unlike linear lists, circular lists have no terminal null pointer. Every node has a valid successor, and traversal can continue indefinitely.
•Same Node Structure — The node class is identical to linear linked lists. Circularity is a property of the connections, not the nodes themselves.
•Entry Point Choice — We can maintain a reference to the head, the tail, or both. Tail references often provide O(1) access to both ends.
•Natural Cyclic Semantics — Circular lists excel in problems that are inherently cyclic: round-robin processing, repeating sequences, and ring-topology simulations.
•Traversal Requires Care — The standard while (current !== null) pattern causes infinite loops. Use entry-point comparison or count-based termination.
•Invariant: Closure — A valid circular list forms exactly one closed cycle containing all nodes, with every node reachable from the entry point.

Your Mental Model:

What's Next:

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