What Does Dynamic Optimality Mean in External Memory?

TL;DR

This paper redefines dynamic optimality for external-memory data structures, introducing buffered-propagation trees and the llotree, which adapt to workload locality and achieve optimality in insert/update/delete-heavy scenarios.

Contribution

It introduces buffered-propagation trees and the llotree, a novel data structure that achieves dynamic optimality in external memory for specific input classes.

Findings

Buffered-propagation trees adapt to input locality.

The llotree is statically optimal.

The llotree achieves dynamic optimality for certain input classes.

Abstract

In this paper, we revisit the question of how the dynamic optimality of search trees should be defined in external memory. A defining characteristic of external-memory data structures is that there is a stark asymmetry between queries and inserts/updates/deletes: by making the former slightly asymptotically slower, one can make the latter significantly asymptotically faster (even allowing for operations with sub-constant amortized I/Os). This asymmetry makes it so that rotation-based search trees are not optimal (or even close to optimal) in insert/update/delete-heavy external-memory workloads. To study dynamic optimality for such workloads, one must consider a different class of data structures. The natural class of data structures to consider are what we call buffered-propagation trees. Such trees can adapt dynamically to the locality properties of an input sequence in order to optimize the interactions between different inserts/updates/deletes and queries. We also present a new form of beyond-worst-case analysis that allows for us to formally study a continuum between static and dynamic optimality. Finally, we give a novel data structure, called the \jellotree, that is statically optimal and that achieves dynamic optimality for a large natural class of inputs defined by our beyond-worst-case analysis.