Detect Cycle in Linked List Floyd's Algorithm in DSA Go - 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 needed grow as the list gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

func hasCycle(head *ListNode) bool {
    slow, fast := head, head
    for fast != nil && fast.Next != nil {
        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 cycle in a linked list.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The loop where slow moves one step and fast moves two steps through the list.
• How many times: At most, the loop runs once per node until pointers meet or list ends.

How Execution Grows With Input

As the list size grows, the number of steps to detect a cycle grows roughly in a straight line.

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

Pattern observation: The time grows directly with the number of nodes, so doubling nodes roughly doubles steps.

Final Time Complexity

Time Complexity: O(n)

This means the time to detect a cycle grows linearly with the list size.

Common Mistake

[X] Wrong: "Because fast moves two steps, the time complexity is O(n/2) or constant time."

[OK] Correct: Even though fast moves faster, the loop still depends on the number of nodes, so time grows linearly, not constant.

Interview Connect

Understanding this linear time check helps you explain efficient cycle detection clearly and confidently in interviews.

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 idea behind Floyd's Cycle Detection Algorithm in a linked list?

easy

A. Reverse the linked list and check if it matches the original

B. Count the number of nodes and check if it exceeds a limit

C. Use two pointers moving at different speeds to detect a cycle

D. Use a hash map to store visited nodes

Solution

1. Step 1: Understand Floyd's Algorithm concept

   It uses two pointers, one moving one step at a time (slow), and another moving two steps at a time (fast).

2. Step 2: Detect cycle by pointer meeting

   If the fast pointer meets the slow pointer, it means there is a cycle in the linked list.

3. Final Answer:

   Use two pointers moving at different speeds to detect a cycle -> Option C

4. Quick Check:

   Two pointers at different speeds = cycle detection [OK]

Hint: Two pointers at different speeds detect cycle fast [OK]

Common Mistakes:

• Thinking counting nodes detects cycle
• Assuming reversing list helps detect cycle
• Using extra memory like hash map is not Floyd's method

2. Which of the following Go code snippets correctly initializes the two pointers for Floyd's cycle detection in a linked list named head?

easy

A. slow, fast := head, head

B. slow, fast := head.next, head.next.next

C. slow, fast := head, head.next

D. slow, fast := nil, head

Solution

1. Step 1: Initialize both pointers at head

   Floyd's algorithm starts both slow and fast pointers at the head node.

2. Step 2: Move pointers inside loop, not at initialization

   Moving fast or slow pointer ahead at initialization can skip nodes or cause errors.

3. Final Answer:

   slow, fast := head, head -> Option A

4. Quick Check:

   Both pointers start at head [OK]

Hint: Start both pointers at head before loop [OK]

Common Mistakes:

• Starting fast pointer ahead causing nil pointer errors
• Initializing slow pointer ahead skipping first node
• Setting slow pointer to nil initially

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

type Node struct {
  val  int
  next *Node
}

func hasCycle(head *Node) bool {
  slow, fast := head, head
  for fast != nil && fast.next != nil {
    slow = slow.next
    fast = fast.next.next
    if slow == fast {
      return true
    }
  }
  return false
}

// Linked list setup
n1 := &Node{val: 1}
n2 := &Node{val: 2}
n3 := &Node{val: 3}
n4 := &Node{val: 4}
n1.next = n2
n2.next = n3
n3.next = n4
n4.next = n2 // cycle here

fmt.Println(hasCycle(n1))

medium

A. true

B. false

C. runtime error

D. infinite loop

Solution

1. Step 1: Analyze linked list structure

   The list nodes are connected 1->2->3->4, and 4 points back to 2, creating a cycle.

2. Step 2: Trace Floyd's algorithm behavior

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

3. Final Answer:

   true -> Option A

4. Quick Check:

   Cycle exists, so output is true [OK]

Hint: Cycle means pointers meet, so return true [OK]

Common Mistakes:

• Assuming no cycle because list looks linear
• Expecting false without tracing pointers
• Confusing cycle detection with list length

4. Identify the bug in this Go code snippet for detecting a cycle in a linked list:

func hasCycle(head *Node) bool {
  slow, fast := head, head
  for fast != nil {
    slow = slow.next
    fast = fast.next.next
    if slow == fast {
      return true
    }
  }
  return false
}

medium

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

B. The loop condition should check fast.next != nil to avoid nil pointer dereference

C. The fast pointer should move one step instead of two

D. The function should return false inside the loop when pointers don't meet

Solution

1. Step 1: Check loop condition safety

   The loop only checks fast != nil, but fast.next could be nil causing fast.next.next to panic.

2. Step 2: Correct loop condition

   Loop should check both fast != nil and fast.next != nil to safely move fast pointer two steps.

3. Final Answer:

   The loop condition should check fast.next != nil to avoid nil pointer dereference -> Option B

4. Quick Check:

   Check fast and fast.next != nil before moving fast [OK]

Hint: Always check fast.next != nil before fast.next.next [OK]

Common Mistakes:

• Ignoring fast.next nil check causing runtime panic
• Moving slow pointer two steps incorrectly
• Returning false too early inside loop

5. Given a linked list where the cycle starts at node with value 3, how can Floyd's algorithm be extended to find the exact node where the cycle begins?

hard

A. Count nodes until pointers meet and return the count as cycle start position

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

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

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

Solution

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

   First, detect if a cycle exists by moving slow and fast pointers until they meet.

2. Step 2: Find cycle start node

   Reset one pointer to head, then move both pointers one step at a time; the node where they meet is the cycle start.

3. Final Answer:

   After detecting the cycle, reset one pointer to head and move both one step until they meet at cycle start -> Option D

4. Quick Check:

   Reset pointer + move one step = cycle start [OK]

Hint: Reset pointer to head after cycle detection to find start [OK]

Common Mistakes:

• Trying to count nodes instead of resetting pointer
• Using extra memory instead of pointer method
• Reversing list which breaks structure