Floyd’s Cycle Finding Algorithm

Last Updated : 10 May, 2023

Floyd’s cycle finding algorithm or Hare-Tortoise algorithm is a pointer algorithm that uses only two pointers, moving through the sequence at different speeds. This algorithm is used to find a loop in a linked list. It uses two pointers one moving twice as fast as the other one. The faster one is called the fast pointer and the other one is called the slow pointer.

How Does Floyd’s Cycle Finding Algorithm Works?

While traversing the linked list one of these things will occur-

The Fast pointer may reach the end (NULL) this shows that there is no loop in the linked list.
The Fast pointer again catches the slow pointer at some time therefore a loop exists in the linked list.

Example:

Loop exists

Pseudocode:

Initialize two-pointers and start traversing the linked list.
Move the slow pointer by one position.
Move the fast pointer by two positions.
If both pointers meet at some point then a loop exists and if the fast pointer meets the end position then no loop exists.

Below is the C++ program to implement the above approach:

C++

// C++ program to implement
// the above approach
#include <bits/stdc++.h>
using namespace std;
 
class Node {
public:
    int data;
    Node* next;
 
    Node(int data)
    {
        this->data = data;
        next = NULL;
    }
};
 
// initialize a new head for the linked list
Node* head = NULL;
class Linkedlist {
public:
    // insert new value at the start
    void insert(int value)
    {
        Node* newNode = new Node(value);
        if (head == NULL)
            head = newNode;
        else {
            newNode->next = head;
            head = newNode;
        }
    }
 
    // detect if there is a loop
    // in the linked list
    bool detectLoop()
    {
        Node *slowPointer = head,
             *fastPointer = head;
 
        while (slowPointer != NULL
               && fastPointer != NULL
               && fastPointer->next != NULL) {
            slowPointer = slowPointer->next;
            fastPointer = fastPointer->next->next;
            if (slowPointer == fastPointer)
                return 1;
        }
 
        return 0;
    }
};
 
int main()
{
    Linkedlist l1;
    // inserting new values
    l1.insert(10);
    l1.insert(20);
    l1.insert(30);
    l1.insert(40);
    l1.insert(50);
 
    // adding a loop for the sake
    // of this example
    Node* temp = head;
    while (temp->next != NULL)
        temp = temp->next;
 
    temp->next = head;
 
    // loop added;
 
    if (l1.detectLoop())
        cout << "Loop exists in the"
             << " Linked List" << endl;
    else
        cout << "Loop does not exists "
             << "in the Linked List" << endl;
 
    return 0;
}

Java

// Java program to implement
// the above approach
 
import java.util.*;
 
class GFG{
 
static class Node {
    int data;
    Node next;
 
    Node(int data)
    {
        this.data = data;
        next = null;
    }
};
 
// initialize a new head for the linked list
static Node head = null;
static class Linkedlist {
    // insert new value at the start
    void insert(int value)
    {
        Node newNode = new Node(value);
        if (head == null)
            head = newNode;
        else {
            newNode.next = head;
            head = newNode;
        }
    }
 
    // detect if there is a loop
    // in the linked list
    boolean detectLoop()
    {
        Node slowPointer = head,
             fastPointer = head;
 
        while (slowPointer != null
               && fastPointer != null
               && fastPointer.next != null) {
            slowPointer = slowPointer.next;
            fastPointer = fastPointer.next.next;
            if (slowPointer == fastPointer)
                return true;
        }
 
    return false;
    }
}
 
public static void main(String[] args)
{
    Linkedlist l1 = new Linkedlist();
    // inserting new values
    l1.insert(10);
    l1.insert(20);
    l1.insert(30);
    l1.insert(40);
    l1.insert(50);
 
    // adding a loop for the sake
    // of this example
    Node temp = head;
    while (temp.next != null)
        temp = temp.next;
 
    temp.next = head;
 
    // loop added;
 
    if (l1.detectLoop())
        System.out.print("Loop exists in the"
            + " Linked List" +"\n");
    else
        System.out.print("Loop does not exists "
            + "in the Linked List" +"\n");
 
}
}
 
// This code is contributed by 29AjayKumar

Python3

# Python code for the above approach
class Node:
    def __init__(self, d):
        self.data = d
        self.next = None
 
# initialize a new head for the linked list
head = None
 
# detect if there is a loop
# in the linked list
def detectLoop(head):
    slowPointer = head
    fastPointer = head
 
    while (slowPointer != None
           and fastPointer != None
           and fastPointer.next != None):
        slowPointer = slowPointer.next
        fastPointer = fastPointer.next.next
        if (slowPointer == fastPointer):
            return 1
 
    return 0
 
# inserting new values
head = Node(10)
head.next = Node(20)
head.next.next = Node(30)
head.next.next.next = Node(40)
head.next.next.next.next = Node(50)
 
# adding a loop for the sake
# of this example
temp = head
while (temp.next != None):
    temp = temp.next
 
temp.next = head
 
# loop added;
if (detectLoop(head)):
    print("Loop exists in the Linked List")
else:
    print("Loop does not exists in the Linked List")
 
# This code is contributed by Saurabh Jaiswal

C#

// C# program to implement
// the above approach
using System;
using System.Collections.Generic;
public class GFG {
 
  public class Node {
    public int data;
    public Node next;
 
    public Node(int data) {
      this.data = data;
      next = null;
    }
  };
 
  // initialize a new head for the linked list
  static Node head = null;
 
  public class Linkedlist
  {
 
    // insert new value at the start
    public void insert(int value) {
      Node newNode = new Node(value);
      if (head == null)
        head = newNode;
      else {
        newNode.next = head;
        head = newNode;
      }
    }
 
    // detect if there is a loop
    // in the linked list
    public bool detectLoop() {
      Node slowPointer = head, fastPointer = head;
 
      while (slowPointer != null && fastPointer != null && 
             fastPointer.next != null)
      {
        slowPointer = slowPointer.next;
        fastPointer = fastPointer.next.next;
        if (slowPointer == fastPointer)
          return true;
      }
 
      return false;
    }
  }
 
  public static void Main(String[] args) {
    Linkedlist l1 = new Linkedlist();
    // inserting new values
    l1.insert(10);
    l1.insert(20);
    l1.insert(30);
    l1.insert(40);
    l1.insert(50);
 
    // adding a loop for the sake
    // of this example
    Node temp = head;
    while (temp.next != null)
      temp = temp.next;
 
    temp.next = head;
 
    // loop added;
    if (l1.detectLoop())
      Console.Write("Loop exists in the" + " Linked List" + "\n");
    else
      Console.Write("Loop does not exists " + "in the Linked List" + "\n");
 
  }
}
 
// This code is contributed by umadevi9616

Javascript

<script>
        // JavaScript code for the above approach 
        class Node
        {
            constructor(d) 
            {
                this.data = d;
                this.next = null;
            }
        }
 
        // initialize a new head for the linked list
        let head = null;
 
        // detect if there is a loop
        // in the linked list
        function detectLoop(head) {
            let slowPointer = head;
            let fastPointer = head;
 
            while (slowPointer != null
                && fastPointer != null
                && fastPointer.next != null) {
                slowPointer = slowPointer.next;
                fastPointer = fastPointer.next.next;
                if (slowPointer == fastPointer)
                    return 1;
            }
 
            return 0;
        }
 
        // inserting new values
        head = new Node(10);
        head.next = new Node(20);
        head.next.next = new Node(30);
        head.next.next.next = new Node(40);
        head.next.next.next.next = new Node(50);
 
        // adding a loop for the sake
        // of this example
        let temp = head;
        while (temp.next != null)
            temp = temp.next;
 
        temp.next = head;
 
        // loop added;
        if (detectLoop(head))
            document.write("Loop exists in the Linked List" + "<br>");
        else
            document.write("Loop does not exists in the Linked List" + "<br>");
 
       // This code is contributed by Potta Lokesh
    </script>

Output

Loop exists in the Linked List

Time complexity: O(n), as the loop is traversed once.
Auxiliary Space: O(1), only two pointers are used therefore constant space complexity.

Why Does Floyd’s Algorithm Works?

Let us consider an example:

Why floyd algorithm work

Let,

X = Distance between the head(starting) to the loop starting point.

Y = Distance between the loop starting point and the first meeting point of both the pointers.

C = The distance of the loop

So before both the pointer meets-

The slow pointer has traveled X + Y + s * C distance, where s is any positive constant number.

The fast pointer has traveled X + Y + f * C distance, where f is any positive constant number.

Since the fast pointer is moving twice as fast as the slow pointer, we can say that the fast pointer covered twice the distance the slow pointer covered. Therefore-

X + Y + f * C = 2 * (X + Y + s * C)

X + Y = f * C – 2 * s * C

We can say that,

f * C – 2 * s * C = (some integer) * C

= K * C

Thus,

X + Y = K * C – ( 1 )

X = K * C – Y – ( 2 )

Where K is some positive constant.

Now if reset the slow pointer to the head(starting position) and move both fast and slow pointer by one unit at a time, one can observe from 1st and 2nd equation that both of them will meet after traveling X distance at the starting of the loop because after resetting the slow pointer and moving it X distance, at the same time from loop meeting point the fast pointer will also travel K * C – Y distance(because it already has traveled Y distance).
From equation (2) one can say that X = K * C – Y therefore, both the pointers will travel the distance X i.e. same distance after the pink node at some point to meet at the starting point of the cycle.
Here, by some point, it means that the fast pointer can complete the K * C distance out of which it has already covered the Y distance.
Conclusively, we find that slow pointer moves 1 step at a time while fast pointer moves 2 steps at a time so the fast pointer catches or surpasses slow pointer if there is a loop, just like start <end in Binary Search. Else if there exists no loop then fast pointer reaches the NULL before than slow (and wins the race: tortoise ->slow pointer and hare/rabbit->fast pointer).

Below is the C++ program to implement the above approach-

C++

// C++ program to implement
// the above approach
#include <bits/stdc++.h>
using namespace std;
 
class Node {
public:
    int data;
    Node* next;
 
    Node(int data)
    {
        this->data = data;
        next = NULL;
    }
};
 
// initialize a new head
// for the linked list
Node* head = NULL;
class Linkedlist {
public:
    // insert new value at the start
    void insert(int value)
    {
        Node* newNode = new Node(value);
        if (head == NULL)
            head = newNode;
        else {
            newNode->next = head;
            head = newNode;
        }
    }
 
    // detect if there is a loop
    // in the linked list
    Node* detectLoop()
    {
        //If link list is empty, there are no loop.
        //If only one element is found in link list, there are no loop.
        if(head==NULL || head->next==NULL){ 
            return NULL; 
        }
       
        Node *slowPointer = head, *fastPointer = head;
 
        while (slowPointer != NULL && fastPointer != NULL
               && fastPointer->next != NULL) {
            slowPointer = slowPointer->next;
            fastPointer = fastPointer->next->next;
            if (slowPointer == fastPointer)
                break;
        }
 
        // if no loop exists
        if (slowPointer != fastPointer)
            return NULL;
 
        // reset slow pointer to head
        // and traverse again
        slowPointer = head;
        while (slowPointer != fastPointer) {
            slowPointer = slowPointer->next;
            fastPointer = fastPointer->next;
        }
 
        return slowPointer;
    }
};
 
int main()
{
    Linkedlist l1;
    // inserting new values
    l1.insert(10);
    l1.insert(20);
    l1.insert(30);
    l1.insert(40);
    l1.insert(50);
 
    // adding a loop for the sake
    // of this example
    Node* temp = head;
    while (temp->next != NULL)
        temp = temp->next;
    // loop added;
    temp->next = head;
 
    Node* loopStart = l1.detectLoop();
    if (loopStart == NULL)
        cout << "Loop does not exists" << endl;
    else {
        cout << "Loop does exists and starts from "
             << loopStart->data << endl;
    }
 
    return 0;
}

Java

// Java program to implement
// the above approach
import java.util.*;
public class GFG {
 
    static class Node {
        int data;
        Node next;
 
        Node(int data)
        {
            this.data = data;
            next = null;
        }
    }
 
    // initialize a new head for the linked list
    static Node head = null;
    static class Linkedlist {
        // insert new value at the start
        void insert(int value)
        {
            Node newNode = new Node(value);
            if (head == null)
                head = newNode;
            else {
                newNode.next = head;
                head = newNode;
            }
        }
 
        // detect if there is a loop
        // in the linked list
        public Node detectLoop()
        {
            Node slowPointer = head, fastPointer = head;
 
            while (slowPointer != null
                   && fastPointer != null
                   && fastPointer.next != null) {
                slowPointer = slowPointer.next;
                fastPointer = fastPointer.next.next;
                if (slowPointer == fastPointer)
                    break;
            }
 
            // if no loop exists
            if (slowPointer != fastPointer)
                return null;
 
            // reset slow pointer to head
            // and traverse again
            slowPointer = head;
            while (slowPointer != fastPointer) {
                slowPointer = slowPointer.next;
                fastPointer = fastPointer.next;
            }
 
            return slowPointer;
        }
    }
 
    public static void main(String[] args)
    {
        Linkedlist l1 = new Linkedlist();
        // inserting new values
        l1.insert(10);
        l1.insert(20);
        l1.insert(30);
        l1.insert(40);
        l1.insert(50);
 
        // adding a loop for the sake
        // of this example
 
        Node temp = head;
        while (temp.next != null)
            temp = temp.next;
 
        temp.next = head;
 
        // loop added;
 
        Node loopStart = l1.detectLoop();
        if (loopStart == null)
            System.out.println("Loop does not exists");
        else {
            System.out.println(
                "Loop does exists and starts from "
                + loopStart.data);
        }
    }
}
 
// This code is contributed by jana_sayantan.

Python3

# Python code for the above approach
class Node:
    def __init__(self, d):
        self.data = d
        self.next = None
  
# initialize a new head for the linked list
head = None
  
# detect if there is a loop
# in the linked list
def detectLoop(head):
    slowPointer = head
    fastPointer = head
  
    while (slowPointer != None
           and fastPointer != None
           and fastPointer.next != None):
        slowPointer = slowPointer.next
        fastPointer = fastPointer.next.next
        if (slowPointer == fastPointer):
            break
 
    # if no loop exists
    if (slowPointer != fastPointer):
            return None
     
    # reset slow pointer to head
    # and traverse again
    slowPointer = head
    while (slowPointer != fastPointer):
      slowPointer = slowPointer.next
      fastPointer = fastPointer.next
     
    return slowPointer
   
  
# inserting new values
head = Node(50)
head.next = Node(40)
head.next.next = Node(30)
head.next.next.next = Node(20)
head.next.next.next.next = Node(10)
  
# adding a loop for the sake
# of this example
temp = head
while (temp.next != None):
    temp = temp.next
# loop added
temp.next = head
 
loopStart = detectLoop(head)
if (loopStart == None):
    print("Loop does not exists")
else:
    print(f"Loop does exists and starts from {loopStart.data}")
  
# This code is contributed by Priyank Namdeo

C#

// C# program to implement the above approach
 
using System;
public class GFG
{
 
    public class Node
    {
        public int data;
        public Node next;
 
        public Node(int data)
        {
            this.data = data;
            next = null;
        }
    }
 
    // initialize a new head for the linked list
    static Node head = null;
    public class Linkedlist
    {
        // insert new value at the start
        public void insert(int value)
        {
            Node newNode = new Node(value);
            if (head == null)
                head = newNode;
            else
            {
                newNode.next = head;
                head = newNode;
            }
        }
 
        // detect if there is a loop
        // in the linked list
        public Node detectLoop()
        {
            Node slowPointer = head, fastPointer = head;
 
            while (slowPointer != null
                   && fastPointer != null
                   && fastPointer.next != null)
            {
                slowPointer = slowPointer.next;
                fastPointer = fastPointer.next.next;
                if (slowPointer == fastPointer)
                    break;
            }
 
            // if no loop exists
            if (slowPointer != fastPointer)
                return null;
 
            // reset slow pointer to head
            // and traverse again
            slowPointer = head;
            while (slowPointer != fastPointer)
            {
                slowPointer = slowPointer.next;
                fastPointer = fastPointer.next;
            }
 
            return slowPointer;
        }
    }
 
    public static void Main()
    {
        Linkedlist l1 = new Linkedlist();
        // inserting new values
        l1.insert(10);
        l1.insert(20);
        l1.insert(30);
        l1.insert(40);
        l1.insert(50);
 
        // adding a loop for the sake
        // of this example
 
        Node temp = head;
        while (temp.next != null)
            temp = temp.next;
 
        temp.next = head;
 
        // loop added;
 
        Node loopStart = l1.detectLoop();
        if (loopStart == null)
            Console.WriteLine("Loop does not exists");
        else
        {
            Console.WriteLine(
                "Loop does exists and starts from "
                + loopStart.data);
        }
    }
}
 
// This code is contributed by Saurabh Jaiswal

Javascript

// JavaScript code for the above approach
 
class Node
{
    constructor(d)
    {
        this.data = d;
        this.next = null;
    }
}
 
// initialize a new head for the linked list
let head = null;
 
function detectLoop(head) {
    let slowPointer = head, fastPointer = head;
     
    while(slowPointer!=null && fastPointer!=null && fastPointer.next!=null){
        slowPointer = slowPointer.next;
        fastPointer = fastPointer.next.next;
        if(slowPointer==fastPointer){
            break;
        }
    }
     
    // if no loop exists
    if(slowPointer!=fastPointer){
        return null;
    }
     
    // reset slow pointer to head and traverse again
    slowPointer = head;
    while(slowPointer!=fastPointer){
        slowPointer = slowPointer.next;
        fastPointer = fastPointer.next;
    }
     
    return slowPointer;
}
 
// inserting new values
head = new Node(50);
head.next = new Node(40);
head.next.next = new Node(30);
head.next.next.next = new Node(20);
head.next.next.next.next = new Node(10);
 
// adding a loop for the sake
// of this example
let temp = head;
while (temp.next != null)
    temp = temp.next;
 
temp.next = head;
 
// loop added;
let loopStart = detectLoop(head);
if (loopStart==null)
console.log("Loop does not exists" + "<br>");
else
    console.log("Loop does exists and starts from " + loopStart.data);
 
// This code is contributed by lokeshmvs21.

Output

Loop does exists and starts from 50

Time complexity: O(n), as we have traversed the loop once and then traveled X distance.
Auxiliary space: O(1), as only pointers are used therefore constant space complexity.

Suggest improvement

Brent's Cycle Detection Algorithm

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Floyd’s Cycle Finding Algorithm

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