Diameter of the thick part of moduli space and simultaneous Whitehead moves

Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the $ϵ$-thick part of moduli space of S equipped with the Teichm"uller or Thurston’s Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order $\log \frac{g+p}{\epsilon}$\mathrm{.}Thesameresultalsoholdsforthe$\epsilon$-thick part of the moduli space of metric graphs of rank n equipped with the Lipschitz metric. The proof involves a sorting algorithm that sorts an arbitrary labeled tree with n labels with simultaneous Whitehead moves, where the number of steps is of order log(n).