Detect if a Linked List is Circular in DSA C++ - Time & Space Complexity

Time Complexity: Detect if a Linked List is Circular

O(n)

Understanding Time Complexity

We want to know how long it takes to check if a linked list loops back on itself.

The question is: how does the time needed grow as the list gets longer?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

bool isCircular(Node* head) {
  if (!head) return false;
  Node* slow = head;
  Node* fast = head->next;
  while (fast && fast->next) {
    if (slow == fast) return true;
    slow = slow->next;
    fast = fast->next->next;
  }
  return false;
}
    

This code checks if a linked list has a loop by moving two pointers at different speeds.

Identify Repeating Operations

• Primary operation: The while loop that moves two pointers through the list.
• How many times: At most, it runs once per node until pointers meet or list ends.

How Execution Grows With Input

As the list grows, the number of steps grows roughly in a straight line with the number of nodes.

Input Size (n)	Approx. Operations
10	About 10 steps
100	About 100 steps
1000	About 1000 steps

Pattern observation: The steps increase directly with the list size, not faster or slower.

Final Time Complexity

Time Complexity: O(n)

This means the time to detect a loop grows linearly with the number of nodes in the list.

Common Mistake

[X] Wrong: "The fast pointer moves twice as fast, so the time is O(n/2), which is constant time."

[OK] Correct: Even though the fast pointer moves faster, the time still grows with list size. O(n/2) simplifies to O(n), so it is not constant time.

Interview Connect

Understanding this helps you explain how to find loops efficiently, a common real-world problem in linked list questions.

Self-Check

"What if we used only one pointer moving one step at a time? How would the time complexity change?"

Practice

1. What does it mean if a linked list is circular?

easy

A. The linked list nodes are sorted in ascending order

B. The linked list has no nodes

C. The last node points back to one of the previous nodes, forming a loop

D. The linked list has only one node

Solution

1. Step 1: Understand linked list structure

   A linked list is a chain of nodes where each node points to the next node.

2. Step 2: Define circular linked list

   If the last node points back to any previous node, it creates a loop, making the list circular.

3. Final Answer:

   The last node points back to one of the previous nodes, forming a loop -> Option C

4. Quick Check:

   Circular linked list = last node points back [OK]

Hint: Circular means last node links back to earlier node [OK]

Common Mistakes:

• Thinking circular means sorted list
• Confusing empty list with circular list
• Assuming single node list is always circular

2. Which of the following C++ code snippets correctly checks if a linked list is circular using a slow and fast pointer approach?

easy

A. while(slow != nullptr) { slow = slow->next; if(slow == fast) return true; } return false;

B. while(fast != nullptr && fast->next != nullptr) { slow = slow->next; fast = fast->next->next; if(slow == fast) return true; } return false;

C. for(int i=0; i<10; i++) { if(slow == fast) return true; slow = slow->next; fast = fast->next; } return false;

D. if(head == nullptr) return true; else return false;

Solution

1. Step 1: Understand slow and fast pointer method

   Two pointers move at different speeds; if they meet, a loop exists.

2. Step 2: Analyze code correctness

   while(fast != nullptr && fast->next != nullptr) { slow = slow->next; fast = fast->next->next; if(slow == fast) return true; } return false; correctly moves slow by one and fast by two nodes, checking for equality inside the loop.

3. Final Answer:

   while(fast != nullptr && fast->next != nullptr) { slow = slow->next; fast = fast->next->next; if(slow == fast) return true; } return false; -> Option B

4. Quick Check:

   Slow-fast pointer meeting means circular [OK]

Hint: Fast pointer moves twice as fast; meeting means loop [OK]

Common Mistakes:

• Not checking fast and fast->next for nullptr
• Moving pointers at same speed
• Returning true without pointer meeting

3. What will be the output of the following C++ code snippet?

struct Node {
  int data;
  Node* next;
};

Node* head = new Node{1, nullptr};
Node* second = new Node{2, nullptr};
Node* third = new Node{3, nullptr};
head->next = second;
second->next = third;
third->next = second; // creates a loop

bool isCircular(Node* head) {
  Node* slow = head;
  Node* fast = head;
  while(fast != nullptr && fast->next != nullptr) {
    slow = slow->next;
    fast = fast->next->next;
    if(slow == fast) return true;
  }
  return false;
}

std::cout << (isCircular(head) ? "Circular" : "Not Circular") << std::endl;

medium

A. Circular

B. Not Circular

C. Compilation Error

D. Runtime Error

Solution

1. Step 1: Analyze linked list structure

   third->next points to second, creating a loop between second and third nodes.

2. Step 2: Trace slow and fast pointers

   Slow moves one step, fast moves two steps; they will meet inside the loop, returning true.

3. Final Answer:

   Circular -> Option A

4. Quick Check:

   Loop detected by slow-fast pointers = Circular [OK]

Hint: Loop causes slow and fast pointers to meet [OK]

Common Mistakes:

• Ignoring the loop created by third->next
• Assuming no loop means Not Circular
• Confusing pointer assignments

4. Identify the error in the following code that attempts to detect if a linked list is circular:

bool isCircular(Node* head) {
  Node* slow = head;
  Node* fast = head;
  while(fast != nullptr) {
    slow = slow->next;
    fast = fast->next->next;
    if(slow == fast) return true;
  }
  return false;
}

medium

A. The loop condition should check fast->next != nullptr to avoid null pointer dereference

B. The slow pointer should move two steps instead of one

C. The function should return false inside the loop

D. The function should initialize slow and fast to nullptr

Solution

1. Step 1: Check loop condition safety

   fast->next->next is accessed, so fast->next must not be nullptr to avoid crash.

2. Step 2: Correct loop condition

   Change while(fast != nullptr) to while(fast != nullptr && fast->next != nullptr) to prevent null pointer dereference.

3. Final Answer:

   The loop condition should check fast->next != nullptr to avoid null pointer dereference -> Option A

4. Quick Check:

   Check fast and fast->next to avoid crash [OK]

Hint: Always check fast->next before fast->next->next [OK]

Common Mistakes:

• Not checking fast->next leads to runtime error
• Moving slow pointer incorrectly
• Returning false too early

5. You have a singly linked list where the last node might point back to any node, making it circular. How would you modify the list to remove the circular link and make it linear again?

hard

A. Set head to nullptr to clear the list

B. Delete all nodes after the head node

C. Reverse the linked list to break the loop

D. Detect the loop using slow and fast pointers, then find the node before the loop start and set its next to nullptr

Solution

1. Step 1: Detect loop using slow and fast pointers

   Use two pointers moving at different speeds; if they meet, a loop exists.

2. Step 2: Find the node before loop start and break the loop

   Reset one pointer to head, move both one step until they meet; the node before meeting point is loop end. Set its next to nullptr.

3. Final Answer:

   Detect the loop using slow and fast pointers, then find the node before the loop start and set its next to nullptr -> Option D

4. Quick Check:

   Find loop start, break link to remove circular [OK]

Hint: Find loop start, set previous node's next to nullptr [OK]

Common Mistakes:

• Deleting nodes unnecessarily
• Reversing list does not remove loop
• Setting head to nullptr loses entire list