strategy in the BST model, in a near-optimal manner. In particular, we prove the following tight result.

\[thm: LF simulation\] ${\mbox{\sf OPT}}(X) \leq O(\log k) \cdot {\mbox{\sf F}}^{k}(X)$.

The proof of Theorem \[thm: LF simulation\] is a refinement of an earlier argument [@CombineBST], improving the overhead factor from $O(k)$ to $O(\log{k})$. The logarithmic dependence on $k$ is, in general, the best possible. To see this, consider a sequence $S$ of length $m$, over $k$ distinct items with average cost $\Omega{(\log{k})}$ (e.g. a random sequence from $[k]^m$ does the job). While ${\mbox{\sf OPT}}{(S)} = \Theta(m\log{k})$, clearly ${\mbox{\sf F}}^{k}(X)=O(m)$, as each of the $k$ items can be served with its own private finger.

In the definition of ${\mbox{\sf F}}^k(X)$ we assume a *static* reference tree $R$ for the $k$-finger execution. The offline BST simulation in the proof of Theorem \[thm: LF simulation\] works in fact (with the same overhead) even if $R$ is *dynamic*, i.e. if the multi-finger adversary can perform rotations at any of the fingers. In this case, however, the $k$-finger bound is too strong to be useful; already the $k=1$ case captures the dynamic BST optimum. Our next result is the online counterpart of Theorem \[thm: LF simulation\]. In 