# Finding loop in a singly linked-list

You can detect it by simply running two pointers through the list.
Start the first pointer p1 on the first node and the second pointer p2 on the second node.

Advance the first pointer by one every time through the loop, advance the second pointer by two. If there is a loop, they will eventually point to the same node. If there’s no loop, you’ll eventually hit the end with the advance-by-two pointer.

Consider the following loop:

``````head -> 1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 8
^                        |
|                        |
+------------------------+``````

Starting A at 1 and B at 2, they take on the following values:

``````p1   p2
=    =
1    2
2    4
3    6
4    8
5    4
6    6
``````

Because they’re equal, and P2 should always be beyond `p1` in a non-looping list (because it’s advancing by two as opposed to the advance-by-one behavior of `p1)`, it means you’ve discovered a loop.

The pseudo-code will go something like this:

• Start with hn(Head Node )of the linked list
• If hn==null; return false; //empty list
• if hn.next!=null
• p1=hn; p2=hn;
• While p2!=null loop
• p1=p1.next//Advancing p1 by 1
• if p2.next!=null
• p2=p2.next.next//Advancing p2 by 2
• else return false
• if p1==p2
• return true// Loop was found
• return false// till now no loop was found

Once you know a node within the loop, there’s an `O(n)` guaranteed method to find the start of the loop.

Let’s return to the original position after you’ve found an element somewhere in the loop but you’re not sure where the start of the loop is.

``````                                 p1,p2 (this is where p1 and p2
|               first met).
v
head -> 1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 8
^                        |
|                        |
+------------------------+
``````

This is the process to follow:

• First, advance `p2` and set the `loopsize` to `1`.
• Second, while `p1` and `p2` are not equal, continue to advance `p2`, increasing the `loopsize` each time. That gives the size of the loop, six in this case.
If the `loopsize` ends up as `1`, you know that you must already be at the start of the loop, so simply return `A` as the start, and skip the rest of the steps below.
• Third, simply set both `p1` and `p2` to the first element then advance `p2` exactly `loopsize` times (to the `7` in this case). This gives two pointers that are different by the size of the loop.
• Lastly, while `p1` and `p2` are not equal, you advance them together. Since they remain exactly `loopsize` elements apart from each other at all times, `p1` will enter the loop at exactly the same time as `p2` returns to the start of the loop. You can see that with the following walk through:
• `loopsize` is evaluated as `6`
• set both `p1` and `p2` to `1`
• advance `p2` by `loopsize` elements to `7`
• `1` and `7` aren’t equal so advance both
• `2` and `8` aren’t equal so advance both
• `3` and `3` are equal so that is your loop start

Now, since each those operations are `O(n)` and performed sequentially, the whole thing is `O(n)`.

Source

Algorithm Name: Floyd–Warshall algorithm

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