Key-independent optimality

Key-independent optimality is a property of some binary search tree data structures in computer science proposed by John Iacono. Suppose that key-value pairs are stored in a data structure, and that the keys have no relation to their paired values. A data structure has key-independent optimality if, when randomly assigning the keys, the expected performance of the data structure is within a constant factor of the optimal data structure. Key-independent optimality is related to dynamic optimality.

Definitions
There are many binary search tree algorithms that can look up a sequence of $$m$$ keys $$X = x_1, x_2, \cdots, x_m$$, where each $$x_i$$ is a number between $$1$$ and $$n$$. For each sequence $$X$$, let $$\textit{OPT}(X)$$ be the fastest binary search tree algorithm that looks up the elements in $$X$$ in order. Let $$b$$ be one of the $$n!$$ possible permutation of the sequence $$1, 2, \cdots, n$$, chosen at random, where $$b(i)$$ is the $$i$$th entry of $$b$$. Let $$b(X) = b(x_1), b(x_2), \cdots ,b(x_m)$$. Iacono defined, for a sequence $$X$$, that $$\textit{KIOPT}(X) = E[\textit{OPT}(b(X))]$$.

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

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