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:

• node.next—the next node in the list.
• node.value—the data held in the node. For example, if our linked list stores people in line at the movies, node.value might be the person's name.

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.

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 .

1. How would you detect the first node in the cycle? Define the first node of the cycle as the one closest to the head of the list.
2. Would the program always work if the fast runner moves three steps every time the slow runner moves one step?
3. What if instead of a simple linked list, you had a structure where each node could have several "next" nodes? This data structure is called a "directed graph." How would you test if your directed graph had a cycle?