Find cycle in a linked list. (Method 2)

By | March 5, 2018

Question: Given a linked list, return the node where the cycle begins. If there is no cycle, return null.

Output: 0

The easier method to find the cycle has been discussed in this post.

This method is based upon the idea of a circular race track. If there is a slow runner and a fast runner, eventually the fast runner will catch up the slow runner from behind. A similar analogy can be applied to a tortoise and a hare and hence the method is also called Tortoise and Hare method or the Floyd-Warshal algorithm.

The method is divided into 2 steps.

STEP 1: Determine if there is a loop.
The first phase/step of this method is to determine if there is actually any loop in the Linked List. If no loop is found, then we need to directly return null.

STEP 2: Find the loop.

  • We initialize 2 pointers. Fast(hare) and slow(tortoise).
  • Advance slow pointer by 1 step. Advance fast pointer by 2 steps.
  • Keep performing the above step in a loop until both the pointers meet.
  • As soon as they meet, start one more pointer from the beginning of the list.
  • Move the slow pointer and the new pointer one step at a time.
  • The place where they meet is the point where the loop starts in the linked list.

To have a better understanding of why this method works, have a look at the following image.

  • Both hare and tortoise start from POINT 1.
  • The hare covers distance ‘F’ enters the loop and starts moving in the loop at POINT 2.
  • The hare runs for a while as he is fast and moves on path ‘a’.
  • The tortoise now enters the loop at POINT 2 after covering ‘F’ and keeps on moving on path ‘a’. While the hare is now on path ‘b’.
  • The tortoise keeps on moving on path ‘a’ and the hare eventually catches up with the tortoise at POINT 3.
  • Start a new pointer from POINT 1 and it runs at the same speed of tortoise.

We know that speed of hare is double the speed of tortoise.
Let us look at a little math now:-

​2 * distance(tortoise)​ ​​​​= distance(hare)​
2 * (F + a) = F + a + b + a​
​2F + 2a = F + 2a + b
​F​ = b​​

Hence POINT 2 is the point where loop starts.

The source-code for the above algorithm is given below.

// Template for a linked list node
class ListNode {
    int value;
    ListNode next;

public class Solution {

    private ListNode getIntersect(ListNode head) {
        ListNode tortoise = head;
        ListNode hare = head;

        // A fast pointer will either loop around a cycle and meet the slow
        // pointer or reach the `null` at the end of a non-cyclic list.
        while (hare != null && != null) {
            tortoise =;
            hare =;
            if (tortoise == hare) {
                return tortoise;

        return null;

    public ListNode detectCycle(ListNode head) {
        if (head == null) {
            return null;

        // If there is a cycle, the fast/slow pointers will intersect at some
        // node. Otherwise, there is no cycle, so we cannot find an entrance to
        // a cycle.
        ListNode intersect = getIntersect(head);
        if (intersect == null) {
            return null;

        // To find the entrance to the cycle, we have two pointers traverse at
        // the same speed -- one from the front of the list, and the other from
        // the point of intersection.
        ListNode ptr1 = head;
        ListNode ptr2 = intersect;
        while (ptr1 != ptr2) {
            ptr1 =;
            ptr2 =;

        return ptr1;

Time Complexity: O(n)
Space Complexity: O(1)

A sample problem can be found here.

Author: nikoo28

a tech-savvy guy and a design buff...

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