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Let S be a surface of genus g with p punctures with negative Eulercharacteristic. We study the diameter of the $\epsilon$-thick part of modulispace of S equipped with the Teichm\"uller or Thurston's Lipschitz metric. Weshow that the asymptotic behaviors in both metrics are of order $\log\frac{g+p}{\epsilon}$. The same result also holds for the $\epsilon$-thick partof the moduli space of metric graphs of rank n equipped with the Lipschitzmetric. The proof involves a sorting algorithm that sorts an arbitrary labeledtree with n labels with simultaneous Whitehead moves, where the number of stepsis of order log(n).

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