Detect if a Linked List is Circular in DSA Python - 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.

How does the time needed grow as the list gets longer?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

class Node:
    def __init__(self, val):
        self.val = val
        self.next = None

def is_circular(head):
    slow = head
    fast = head
    while fast and fast.next:
        slow = slow.next
        fast = fast.next.next
        if slow == fast:
            return True
    return False
    

This code uses two pointers moving at different speeds to detect a loop in the linked list.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• 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	Up to 10 steps
100	Up to 100 steps
1000	Up to 1000 steps

Pattern observation: The time grows linearly as the list gets longer.

Final Time Complexity

Time Complexity: O(n)

This means the time to detect a loop grows in direct proportion to 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 faster than O(n)."

[OK] Correct: We ignore constants in Big-O, so O(n/2) is still O(n). The time still grows linearly with input size.

Interview Connect

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

Self-Check

"What if we used only one pointer to check for a loop by visiting nodes and marking them? 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 linked list has only one node

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

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 D

4. Quick Check:

   Circular linked list means loop exists [OK]

Hint: Circular means last node links back inside list [OK]

Common Mistakes:

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

2. Which of the following Python code snippets correctly checks if a linked list is circular?

easy

A. def is_circular(head):
if not head:
return False
slow = head
fast = head.next
while fast and fast.next:
if slow == fast:
return True
slow = slow.next
fast = fast.next.next
return False

B. def is_circular(head):
current = head
while current:
current = current.next
return True

C. def is_circular(head):
return head.next == None

D. def is_circular(head):
return head == None

Solution

1. Step 1: Understand Floyd's cycle detection

   Using two pointers (slow and fast) moving at different speeds detects loops if they meet.

2. Step 2: Analyze code correctness

   def is_circular(head): if not head: return False slow = head fast = head.next while fast and fast.next: if slow == fast: return True slow = slow.next fast = fast.next.next return False uses slow and fast pointers correctly and returns True if they meet, else False.

3. Final Answer:

   Code using slow and fast pointers to detect loop -> Option A

4. Quick Check:

   Floyd's cycle detection = def is_circular(head): if not head: return False slow = head fast = head.next while fast and fast.next: if slow == fast: return True slow = slow.next fast = fast.next.next return False [OK]

Hint: Use two pointers moving at different speeds [OK]

Common Mistakes:

• Not checking fast pointer's next before accessing
• Assuming reaching None means circular
• Returning True without loop detection

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

class Node:
    def __init__(self, val):
        self.val = val
        self.next = None

head = Node(1)
second = Node(2)
third = Node(3)
head.next = second
second.next = third
third.next = head  # Creates a circular linked list

def is_circular(head):
    slow = head
    fast = head
    while fast and fast.next:
        slow = slow.next
        fast = fast.next.next
        if slow == fast:
            return True
    return False

print(is_circular(head))

medium

A. True

B. False

C. None

D. Runtime Error

Solution

1. Step 1: Understand list structure

   Nodes 1->2->3, and third points back to head, making a loop.

2. Step 2: Trace Floyd's cycle detection

   Slow and fast pointers will eventually meet inside the loop, so function returns True.

3. Final Answer:

   True -> Option A

4. Quick Check:

   Loop detected = True [OK]

Hint: Loop means slow and fast pointers meet [OK]

Common Mistakes:

• Ignoring the loop created by third.next = head
• Assuming function returns False by default
• Confusing None return with False

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

def is_circular(head):
    slow = head
    fast = head
    while fast.next and fast.next.next:
        slow = slow.next
        fast = fast.next.next
        if slow == fast:
            return True
    return False

medium

A. The function should return True outside the loop unconditionally

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

C. The while loop condition should check if fast and fast.next exist

D. The fast pointer should start at head.next

Solution

1. Step 1: Analyze while loop condition

   Current condition checks fast.next and fast.next.next but does not check if fast itself is None.

2. Step 2: Correct condition to avoid runtime error

   Should check if fast and fast.next exist before accessing fast.next.next to prevent errors.

3. Final Answer:

   The while loop condition should check if fast and fast.next exist -> Option C

4. Quick Check:

   Check fast and fast.next to avoid errors [OK]

Hint: Always check pointer exists before accessing next [OK]

Common Mistakes:

• Not checking if fast is None before fast.next
• Moving slow pointer incorrectly
• Returning True too early

5. You have a linked list where the last node points to the third node, making it circular. How would you modify the detection function to also return the node where the loop starts?

hard

A. Return the node where slow and fast first meet during detection

B. Use Floyd's algorithm: after detecting loop, reset slow to head and move slow and fast one step until they meet; that node is loop start

C. Return the head node always as loop start

D. Use a hash set to store visited nodes and return the first repeated node

Solution

1. Step 1: Detect loop using Floyd's cycle detection

   Use slow and fast pointers to find if loop exists by meeting inside the loop.

2. Step 2: Find loop start node

   Reset slow to head, then move slow and fast one step at a time; the node where they meet is the loop start.

3. Final Answer:

   Use Floyd's algorithm with reset slow pointer to find loop start -> Option B

4. Quick Check:

   Reset slow and move both pointers to find loop start [OK]

Hint: After loop detection, reset slow to head and move both one step [OK]

Common Mistakes:

• Returning meeting point as loop start without reset
• Always returning head as loop start
• Not handling loop detection before finding start