Find first node of loop in a linked list

Write a function findFirstLoopNode() that checks whether a given Linked List contains loop. If loop is present then it returns point to first node of loop. Else it returns NULL.

Example :

Input : Head of bellow linked list

Output : Pointer to node 2

We have discussed Floyd’s loop detection algorithm. Below are steps to find first node of loop.

1. If a loop is found, initialize slow pointer to head, let fast pointer be at its position.
2. Move both slow and fast pointers one node at a time.
3. The point at which they meet is the start of the loop.





// C++ program to return first node of loop.
#include <bits/stdc++.h>
using namespace std;
struct Node {
    int key;
    struct Node* next;
Node* newNode(int key)
    Node* temp = new Node;
    temp->key = key;
    temp->next = NULL;
    return temp;
// A utility function to print a linked list
void printList(Node* head)
    while (head != NULL) {
        cout << head->key << " ";
        head = head->next;
    cout << endl;
// Function to detect and remove loop
// in a linked list that may contain loop
Node* detectAndRemoveLoop(Node* head)
    // If list is empty or has only one node
    // without loop
    if (head == NULL || head->next == NULL)
        return NULL;
    Node *slow = head, *fast = head;
    // Move slow and fast 1 and 2 steps
    // ahead respectively.
    slow = slow->next;
    fast = fast->next->next;
    // Search for loop using slow and
    // fast pointers
    while (fast && fast->next) {
        if (slow == fast)
        slow = slow->next;
        fast = fast->next->next;
    // If loop does not exist  
    if (slow != fast)
        return NULL;
    // If loop exists. Start slow from
    // head and fast from meeting point.
    slow = head;
    while (slow != fast) {
        slow = slow->next;
        fast = fast->next;
    return slow;
/* Driver program to test above function*/
int main()
    Node* head = newNode(50);
    head->next = newNode(20);
    head->next->next = newNode(15);
    head->next->next->next = newNode(4);
    head->next->next->next->next = newNode(10);
    /* Create a loop for testing */
    head->next->next->next->next->next = head->next->next;
    Node* res = detectAndRemoveLoop(head);
    if (res == NULL)
        cout << "Loop does not exist";
        cout << "Loop starting node is " << res->key;
    return 0;



Loop starting node is 15

How does this approach work?
Let slow and fast meet at some point after Floyd’s Cycle finding algorithm. Below diagram shows the situation when cycle is found.


We can conclude below from above diagram

Distance traveled by fast pointer = 2 * (Distance traveled 
                                         by slow pointer)

(m + n*x + k) = 2*(m + n*y + k)

Note that before meeting the point shown above, fast
was moving at twice speed.

x -->  Number of complete cyclic rounds made by 
       fast pointer before they meet first time

y -->  Number of complete cyclic rounds made by 
       slow pointer before they meet first time

From above equation, we can conclude below

    m + k = (x-2y)*n

Which means m+k is a multiple of n. 

So if we start moving both pointers again at same speed such that one pointer (say slow) begins from head node of linked list and other pointer (say fast) begins from meeting point. When slow pointer reaches beginning of loop (has made m steps), fast pointer would have made also moved m steps as they are now moving same pace. Since m+k is a multiple of n and fast starts from k, they would meet at the beginning. Can they meet before also? No because slow pointer enters the cycle first time after m steps.

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