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You have a singly-linked list and want to check if it contains a cycle.
A singly-linked list is built with nodes, where each node has:
For example:
A cycle occurs when a node’s next points back to a previous node in the list. The linked list is no longer linear with a beginning and end—instead, it cycles through a loop of nodes.
Write a function contains_cycle that takes the first node in a singly-linked list and returns a boolean indicating whether the list contains a cycle.
Careful—a cycle can occur in the middle of a list, or it can simply mean the last node links back to the first node. Does your function work for both?
We can do this in time and space!
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time and space.
The runtime analysis is a little tricky. The worst case is when we do have a cycle, so we don't return until fast_runner equals slow_runner. But how long will that take?
First, we notice that when both runners are circling around the cycle fast_runner can never skip over slow_runner. Why is this true?
Suppose fast_runner had just skipped over slow_runner. fast_runner would only be 1 node ahead of slow_runner, since their speeds differ by only 1. So we would have something like this:
What would the step right before this "skipping step" look like? fast_runner would be 2 nodes back, and slow_runner would be 1 node back. But wait, that means they would be at the same node! So fast_runner didn't skip over slow_runner! (This is a proof by contradiction.)
Since fast_runner can't skip over slow_runner, at most slow_runner will run around the cycle once and fast_runner will run around twice. This gives us a runtime of .
For space, we store two variables no matter how long the linked list is, which gives us a space cost of .
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