in turn imply static finger. Our main tool in proving Theorem \[thm: weak UB\] is a generalization of the lazy finger property allowing multiple fingers. The model is motivated by the famous $k$-server problem. For an input sequence $X \in [n]^m$ and a static BST $R$ with nodes associated with the keys in $[n]$, we have $k$ servers located initially at arbitrary nodes in $R$. At time $t=1,\ldots, m$, the request $x_t$ arrives, and we move a server of our choice to the node of $R$ that stores $x_t$. The cost for serving a sequence $X$ is equal to the total movement in $R$ to serve the sequence $X$.

Denote by ${\mbox{\sf F}}^{k}_R(X)$ the cost of the optimal (offline) strategy that serves sequence $X$ in $R$ with $k$ servers, minimized over all possible initial server locations. Let ${\mbox{\sf F}}^{k}(X) = \min_R {\mbox{\sf F}}^{k}_R(X)$. We call ${\mbox{\sf F}}^{k}(X)$ the [*$k$-finger cost*]{} of $X$. We remark that the value of ${\mbox{\sf F}}^{k}_R(X)$ is polynomial-time computable for each $R$, $k\in {\mathbb N}$, and $X \in [n]^m$ by dynamic programming. Clearly, ${\mbox{\sf F}}^{1}(X) \geq {\mbox{\sf F}}^{2}(X) \geq \cdots \geq {\mbox{\sf F}}^{n}(X)$ holds for all $X$.

We first show that one can simulate any $k$-finger 