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Question: To find starting node of loop if cycle exists in linked list

Approach:

(1)Using Hare-Tortoise algorithm, find if cycle exists(No issues with this step)

(2)Let P be the node where hare and tortoise meets.Let H be head pointer on linked list.Traverse one node at a time from H and P until they meet.

Doubt: Logic behind (2). How does traversing one step at a time from H and P ensure that they will meet at the start of loop?

References:

Problem and Solution

Partially Explained Logic(Refer second approach in this blog)

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Let T be the number of links traversed by the tortoise, and let S be the number of links traversed before reaching the cycle.

We know that traveling T links along the cycle puts you back where you started, since the tortoise and the hare reach the same spot P after traveling T and 2T links respectively. If you travel S links from P, this is equivalent to traveling S links from the start and then traveling T links, which is equivalent to just traveling S links from the start.

Thus, when the pointer traveling from the start of the list has traveled S links, it reaches the start of the cycle, and it meets up with the pointer traveling from P.

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2 Comments

"We know that traveling T links along the cycle puts you back where you started, since the tortoise and the hare reach the same spot P after traveling T and 2T links respectively". This is my doubt. How do we know? Please elaborate
@user3409814: Seriously? That's the part that confuses you? Okay, suppose you travel T links from the start. You reach the tortoise's location, which is P. You keep traveling for another T links. You reach the hare's location, which is also P. Therefore, traveling T links from P puts you back at P. By the symmetry of the cycle's structure, we can tell that traveling T links from any point in the cycle will put you back where you started.

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