key-independent optimality

Key-independent optimality is a property of some binary search tree data structures in computer science

proposed by John Iacono.{{cite web|title=John Iacono. Key independent optimality. Algorithmica, 42(1):3-10, 2005.|url=http://john2.poly.edu/papers/algo04/paper.pdf|url-status=dead|archiveurl=https://web.archive.org/web/20100613054302/http://john2.poly.edu/papers/algo04/paper.pdf|archivedate=2010-06-13}}

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.

References

Category:Trees (data structures)