algorithms.

The strength of the $k$-finger model lies in the $k$-server abstraction. In order to establish a BST property of the form ${\mbox{\sf OPT}}(X) \leq \beta(\ell) \cdot O(g(X))$, it is now sufficient to prove ${\mbox{\sf F}}^{\ell}(X) \leq \left(\beta(\ell)/\log{\ell}\right) \cdot O(g(X))$. In other words, our technique reduces the task of bounding the cost in the BST model to designing $k$-server strategies, which typically admits much cleaner combinatorial arguments. We illustrate this approach by showing that the unified property with a fixed time-window holds in the BST model.

\[thm: strategy for min-dist\] For some fixed functions $\alpha(\cdot),\gamma(\cdot)$, we have:  ${\mbox{\sf F}}^{\alpha(\ell)}(X) \leq \gamma(\ell) \cdot {\mbox{\sf UB}}^{\ell} $.

Theorems \[thm: strategy for min-dist\] and \[thm: LF simulation\] together imply Theorem \[thm: weak UB\]. Moreover, Theorem \[thm: online simulation\] implies that the property holds for *online* BST algorithms (we later specify the involved functions). The $k$-finger approach can be used to show further BST properties. For example, we connect *decomposability* (refer to 4 for definitions) and finger properties by showing that even one finger is enough to obtain the *traversal* property in significantly generalized form.

\[thm:decomp\] Let $X$ be a $d$-decomposable sequence. Then ${\mbox{\sf F}}^{1}(X) = O(\log d) \cdot |X| $.

As a corollary, 