name: Fast & Slow Pointers
slug: fast-slow-pointers
difficulty_level: 2

description: >
  Use two pointers moving at different speeds to detect cycles, find midpoints,
  or identify patterns in sequences. The fast pointer advances twice as quickly,
  allowing detection of structural properties without extra space.

when_to_use: |
  - Detecting cycles in linked lists or sequences
  - Finding the middle of a linked list
  - Finding the start of a cycle
  - Happy number problem
  - Palindrome linked list verification

metaphor: |
  Imagine two runners on a circular track. If one runs twice as fast as the other,
  the fast runner will eventually lap the slow runner—they'll meet at some point
  on the track. This proves the track is circular (has a cycle).

  Another analogy: finding the middle of a line of people. Have two people start
  at the front—one takes one step at a time, the other takes two. When the fast
  person reaches the end, the slow person is at the middle.

core_concept: |
  The **fast & slow pointers** technique (also called Floyd's cycle detection or
  "tortoise and hare") uses two pointers moving at different speeds:

  - **Slow pointer**: Moves 1 step at a time
  - **Fast pointer**: Moves 2 steps at a time

  Key insights:

  1. **Cycle detection**: In a cyclic structure, fast will eventually catch up to
     slow (they'll meet inside the cycle). In a non-cyclic structure, fast will
     reach the end.

  2. **Finding middle**: When fast reaches the end, slow is at the middle (fast
     traveled 2x the distance).

  3. **Finding cycle start**: After detecting a cycle, reset one pointer to start.
     Move both at the same speed—they meet at the cycle start. (Mathematical proof:
     the distances work out perfectly.)

visualization: |
  **Cycle Detection:**

  ```
  List: 1 → 2 → 3 → 4 → 5
                    ↑   ↓
                    7 ← 6

  Step 1: slow=1, fast=1
  Step 2: slow=2, fast=3
  Step 3: slow=3, fast=5
  Step 4: slow=4, fast=7
  Step 5: slow=5, fast=4
  Step 6: slow=6, fast=6 ← They meet! Cycle exists.
  ```

  **Finding Middle:**

  ```
  List: 1 → 2 → 3 → 4 → 5 → null

  Step 1: slow=1, fast=1
  Step 2: slow=2, fast=3
  Step 3: slow=3, fast=5
  Step 4: fast reaches null

  slow is at middle (3)
  ```

  **Finding Cycle Start:**

  ```
  After detecting cycle at node X:

  1. Reset slow to head, keep fast at meeting point
  2. Move both at same speed (1 step each)
  3. They meet at cycle start

  Why? Math: Let's say:
  - Distance from head to cycle start = A
  - Distance from cycle start to meeting point = B
  - Cycle length = C

  At meeting: slow traveled A + B
              fast traveled A + B + nC (some complete cycles)

  Since fast travels 2x: 2(A + B) = A + B + nC
  Therefore: A + B = nC, so A = nC - B = (n-1)C + (C-B)

  This means: distance from head to cycle start
            = distance from meeting point to cycle start (going forward)
  ```

code_template: |
  class ListNode:
      def __init__(self, val=0, next=None):
          self.val = val
          self.next = next


  def has_cycle(head: ListNode) -> bool:
      """Detect if linked list has a cycle."""
      if not head or not head.next:
          return False

      slow = head
      fast = head

      while fast and fast.next:
          slow = slow.next
          fast = fast.next.next

          if slow == fast:
              return True

      return False


  def find_cycle_start(head: ListNode) -> ListNode:
      """Find the node where the cycle begins."""
      if not head or not head.next:
          return None

      # Phase 1: Detect cycle
      slow = fast = head
      while fast and fast.next:
          slow = slow.next
          fast = fast.next.next
          if slow == fast:
              break
      else:
          return None  # No cycle

      # Phase 2: Find cycle start
      slow = head
      while slow != fast:
          slow = slow.next
          fast = fast.next

      return slow


  def find_middle(head: ListNode) -> ListNode:
      """Find the middle node of linked list."""
      if not head:
          return None

      slow = fast = head

      while fast and fast.next:
          slow = slow.next
          fast = fast.next.next

      return slow  # Middle (or second middle if even length)


  def is_happy_number(n: int) -> bool:
      """Check if number is happy (sum of squared digits eventually = 1)."""
      def get_next(num: int) -> int:
          total = 0
          while num > 0:
              digit = num % 10
              total += digit * digit
              num //= 10
          return total

      slow = n
      fast = get_next(n)

      while fast != 1 and slow != fast:
          slow = get_next(slow)
          fast = get_next(get_next(fast))

      return fast == 1


  def is_palindrome_linked_list(head: ListNode) -> bool:
      """Check if linked list is a palindrome."""
      if not head or not head.next:
          return True

      # Find middle
      slow = fast = head
      while fast and fast.next:
          slow = slow.next
          fast = fast.next.next

      # Reverse second half
      prev = None
      while slow:
          next_node = slow.next
          slow.next = prev
          prev = slow
          slow = next_node

      # Compare halves
      left, right = head, prev
      while right:
          if left.val != right.val:
              return False
          left = left.next
          right = right.next

      return True

recognition_signals:
  - "linked list cycle"
  - "detect cycle"
  - "find middle"
  - "happy number"
  - "palindrome linked list"
  - "circular array"
  - "Floyd's"
  - "tortoise and hare"
  - "meeting point"
  - "cycle start"

common_mistakes:
  - title: Not checking fast.next before advancing
    description: |
      Accessing `fast.next.next` without first checking `fast.next` causes
      null pointer errors when the list has even length.
    fix: |
      Always check both `fast` and `fast.next`:
      ```python
      while fast and fast.next:
          fast = fast.next.next
      ```

  - title: Wrong initialization for cycle detection
    description: |
      Starting slow and fast at different positions (e.g., slow=head, fast=head.next)
      changes the math for finding the cycle start.
    fix: |
      Start both at head for consistency. The algorithms are designed assuming
      both start at the same position.

  - title: Forgetting to handle empty or single-node lists
    description: |
      Accessing head.next or head.next.next on empty or single-node lists
      causes errors.
    fix: |
      Add early returns:
      ```python
      if not head or not head.next:
          return False  # or appropriate value
      ```

  - title: Confusing meeting point with cycle start
    description: |
      Returning the meeting point instead of finding the actual cycle start
      gives the wrong answer.
    fix: |
      After detecting a cycle (meeting point), reset one pointer to head and
      advance both at the same speed to find the cycle start.

variations:
  - name: Cycle detection
    description: |
      Determine if a linked list or sequence has a cycle. If fast catches slow,
      there's a cycle.
    example: "Linked List Cycle, Circular Array Loop"

  - name: Finding cycle start
    description: |
      After detecting a cycle, find the node where the cycle begins using the
      two-phase approach.
    example: "Linked List Cycle II"

  - name: Finding middle
    description: |
      When fast reaches the end, slow is at the middle. Useful for divide and
      conquer on linked lists.
    example: "Middle of Linked List, Sort List (merge sort needs middle)"

  - name: Happy number
    description: |
      Treat the sequence of digit-square sums as a linked list. Either reaches 1
      (happy) or cycles (unhappy).
    example: "Happy Number"

  - name: Palindrome check
    description: |
      Find middle, reverse second half, compare. Combines finding middle with
      linked list reversal.
    example: "Palindrome Linked List"

related_patterns:
  - two-pointers
  - linkedlist-reversal

prerequisite_patterns:
  - two-pointers