What's the worst case complexity of Robert W. Floyd's cycle detection algorithm?
In this answer, I show that $x_i = x_{2i}$ iff $i \geq \mu$ and $i$ is a multiple of $\lambda$. In particular, this will happen for some index $i$ in the range $\{\mu,\ldots,\mu+\lambda-1\}$. This shows that the algorithm runs in time $O(\mu + \lambda)$.