Key-independent optimality

Key-independent optimality is a property of some binary search tree data structures in computer science proposed by John Iacono.^[1] 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 ${\displaystyle m}$ 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 ${\displaystyle O({\textit {KIOPT}}(X))}$.

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.

References