# Floyd's Cycle Finding Algorithm

#### 2022-06-14 22:10

___
##### Algorithms:
#algorithm #Floyd_s_cycle_finding_algorithm 
##### Data structures:
#linked_list 
##### Difficulty:
#CS_analysis #difficulty-easy
##### Related problems:
```expander
tag:#coding_problem tag:#Floyd_s_cycle_finding_algorithm  -tag:#template_remove_me 
```
 
- [[Leetcode Linked-List-Cycle]]
 

##### Links:
- [g4g](https://www.geeksforgeeks.org/floyds-cycle-finding-algorithm/)
___

### What is Floyd's Cycle Finding Algorithm?
[Floyd’s cycle finding algorithm](https://www.geeksforgeeks.org/detect-loop-in-a-linked-list/) or Hare-Tortoise algorithm is a **pointer algorithm** that uses only **two pointers**, moving through the sequence at different speeds.

It uses two pointers one moving twice as fast as the other one. The faster one is called the faster pointer and the other one is called the slow pointer.

### How does it work?

#### Part 1. **Verify** if there is a loop
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 n the linked list.
-   The Fast pointer again **catches the slow pointer at some** time therefore a loop exists in the linked list.

**Pseudo-code:**

-   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.
#### Part 2. **Locating the start** of the loop

Let us consider an example:

![Why floyd algorithm work](https://media.geeksforgeeks.org/wp-content/uploads/20220105124053/6-300x139.jpg)

-   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).
- Because X = K * C - Y, while fast pointer was at Y from start of loop, running X will place it at the start of loop, meeting the slow pointer.

**Pseudo-code**
- Place slow pointer at the head.
- Move one step at a time until they met.
- The start of the loop is where they met

#### Example
```cpp
// 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() {
    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;
}

```
### When to use?

This algorithm is used ==to find a loop in a linked list==, Also it ==can locate where the loop starts.==

It takes O(n) time complexity, O(1)space complexity.