Key-independent optimality

There are many binary search tree algorithms that can look up a sequence of ${\displaystyle m}$ keys ${\displaystyle X=x_{1},x_{2},\cdots ,x_{m}}$, where each ${\displaystyle x_{i}}$ is a number between ${\displaystyle 1}$ and ${\displaystyle n}$. For each sequence ${\displaystyle X}$, let ${\displaystyle {\textit {OPT}}(X)}$ be the fastest binary search tree algorithm that looks up the elements in ${\displaystyle X}$ in order. Let ${\displaystyle b}$ be one of the ${\displaystyle n!}$ possible permutation of the sequence ${\displaystyle 1,2,\cdots ,n}$, chosen at random, where ${\displaystyle b(i)}$ is the ${\displaystyle i}$th entry of ${\displaystyle b}$. Let ${\displaystyle b(X)=b(x_{1}),b(x_{2}),\cdots ,b(x_{m})}$. Iacono defined, for a sequence ${\displaystyle X}$, that ${\displaystyle {\textit {KIOPT}}(X)=E[{\textit {OPT}}(b(X))]}$.

A data structure has key-independent optimality if it can lookup the elements in ${\displaystyle X}$ in time ${\displaystyle O({\textit {KIOPT}}(X))}$.

Key-independent optimality has been proved to be asymptotically equivalent to the working set theorem. Splay trees are known to have key-independent optimality.