Detect Cycle in Linked List Floyd's Algorithm in DSA C++ - Time & Space Complexity

Time Complexity: Detect Cycle in Linked List Floyd's Algorithm

O(n)

Understanding Time Complexity

We want to understand how long it takes to check if a linked list has a cycle using Floyd's Algorithm.

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

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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

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

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop that moves slow by 1 step and fast by 2 steps.
• How many times: At most, the loop runs proportional to the number of nodes before a cycle is detected or the list ends.

How Execution Grows With Input

As the linked list length n grows, the number of steps to detect a cycle grows roughly linearly.

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

Pattern observation: The steps increase roughly in direct proportion to the list size.

Final Time Complexity

Time Complexity: O(n)

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

Common Mistake

[X] Wrong: "Because fast moves twice as fast, the time complexity is O(n/2), so it is faster than O(n)."

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

Interview Connect

Understanding Floyd's cycle detection helps you show how to use pointers smartly to solve problems efficiently.

Self-Check

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

Practice

1. What is the main purpose of Floyd's Cycle Detection Algorithm in a linked list?

easy

A. To find the middle element of the linked list

B. To sort the linked list in ascending order

C. To reverse the linked list

D. To detect if there is a cycle (loop) in the linked list

Solution

1. Step 1: Understand Floyd's Algorithm goal

   Floyd's Cycle Detection Algorithm is designed to check if a linked list contains a cycle, meaning a node points back to a previous node.

2. Step 2: Compare with other linked list operations

   Sorting, reversing, or finding the middle element are different operations and not the purpose of this algorithm.

3. Final Answer:

   To detect if there is a cycle (loop) in the linked list -> Option D

4. Quick Check:

   Cycle detection = Detect cycle [OK]

Hint: Floyd's algorithm finds loops, not sorting or reversing [OK]

Common Mistakes:

• Confusing cycle detection with sorting
• Thinking it finds the middle node
• Assuming it reverses the list

2. Which of the following correctly initializes the two pointers used in Floyd's Cycle Detection Algorithm in C++?

easy

A. ListNode* slow = head; ListNode* fast = head;

B. ListNode* slow = head->next; ListNode* fast = head;

C. ListNode* slow = nullptr; ListNode* fast = head;

D. ListNode* slow = head; ListNode* fast = head->next;

Solution

1. Step 1: Understand pointer initialization in Floyd's algorithm

   Both slow and fast pointers start at the head node to begin traversal.

2. Step 2: Check each option

   ListNode* slow = head; ListNode* fast = head; initializes both pointers at head, which is the standard approach. Other options start pointers differently, which can cause errors or miss cycles.

3. Final Answer:

   ListNode* slow = head; ListNode* fast = head; -> Option A

4. Quick Check:

   Both pointers start at head [OK]

Hint: Start both pointers at head for Floyd's algorithm [OK]

Common Mistakes:

• Starting fast pointer at head->next causes skipping
• Initializing slow as nullptr causes null errors
• Swapping slow and fast initialization

3. What will be the output of the following code snippet detecting a cycle in a linked list?

#include <iostream>

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

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

int main() {
  Node a{1, nullptr};
  Node b{2, nullptr};
  Node c{3, nullptr};
  a.next = &b;
  b.next = &c;
  c.next = &b; // cycle here
  bool result = hasCycle(&a);
  std::cout << (result ? "Cycle detected" : "No cycle") << std::endl;
  return 0;
}

medium

A. No cycle

B. Cycle detected

C. Compilation error

D. Runtime error

Solution

1. Step 1: Analyze linked list structure

   Nodes a, b, c are linked as a->b->c, and c points back to b, creating a cycle.

2. Step 2: Trace Floyd's algorithm behavior

   Slow and fast pointers will eventually meet inside the cycle, so hasCycle returns true.

3. Final Answer:

   Cycle detected -> Option B

4. Quick Check:

   Cycle exists, so output is "Cycle detected" [OK]

Hint: If a node points back, Floyd detects cycle [OK]

Common Mistakes:

• Ignoring the cycle created by c.next = &b
• Assuming no cycle because list looks linear
• Confusing output with compilation or runtime errors

4. Identify the bug in this Floyd's Cycle Detection code snippet:

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

medium

A. The return type should be void, not bool

B. Pointers slow and fast should start at different nodes

C. The while condition should check if fast is not nullptr before fast->next

D. The comparison if (slow == fast) should be outside the loop

Solution

1. Step 1: Check while loop condition safety

   The code checks fast->next and fast->next->next but does not check if fast itself is nullptr, risking null pointer access.

2. Step 2: Correct condition order

   Proper check is to ensure fast is not nullptr before accessing fast->next to avoid runtime errors.

3. Final Answer:

   The while condition should check if fast is not nullptr before fast->next -> Option C

4. Quick Check:

   Check fast != nullptr before fast->next [OK]

Hint: Always check pointer not null before accessing members [OK]

Common Mistakes:

• Skipping null check on fast pointer
• Starting pointers at different nodes unnecessarily
• Moving comparison outside loop causing infinite loop

5. Given a linked list where a cycle exists, how can Floyd's algorithm be extended to find the exact node where the cycle begins?

hard

A. After detecting the cycle, reset one pointer to head and move both pointers one step at a time until they meet at the cycle start.

B. Count the number of nodes in the list and divide by two to find the cycle start.

C. Use a hash set to store visited nodes and find the first repeated node.

D. Reverse the linked list and check where the reversal breaks to find the cycle start.

Solution

1. Step 1: Detect cycle using Floyd's algorithm

   First, detect the cycle by moving slow and fast pointers until they meet inside the cycle.

2. Step 2: Find cycle start node

   Reset one pointer to head, then move both pointers one step at a time; their meeting point is the cycle start node.

3. Final Answer:

   After detecting the cycle, reset one pointer to head and move both pointers one step at a time until they meet at the cycle start. -> Option A

4. Quick Check:

   Reset pointer + move stepwise = cycle start [OK]

Hint: Reset pointer to head after detection, move both stepwise [OK]

Common Mistakes:

• Trying to find cycle start by counting nodes
• Using extra memory like hash sets (not Floyd's method)
• Reversing list which breaks structure