Dynamic External Hashing: The Limit of Buffering

TL;DR

This paper investigates the fundamental tradeoff between query and insertion costs in external hash tables with internal memory buffers, establishing limits on how buffering can improve performance.

Contribution

It proves that for certain query costs, internal buffers do not significantly reduce insertion costs, answering an open question in external memory data structures.

Findings

High query accuracy limits insertion efficiency when buffering is constrained.

Buffers are ineffective for reducing insertion costs if query costs are very close to optimal.

The results clarify the inherent tradeoffs in external hash table design.

Abstract

Hash tables are one of the most fundamental data structures in computer science, in both theory and practice. They are especially useful in external memory, where their query performance approaches the ideal cost of just one disk access. Knuth gave an elegant analysis showing that with some simple collision resolution strategies such as linear probing or chaining, the expected average number of disk I/Os of a lookup is merely $1+1/2^{\Omega(b)}$, where each I/O can read a disk block containing $b$ items. Inserting a new item into the hash table also costs $1+1/2^{\Omega(b)}$ I/Os, which is again almost the best one can do if the hash table is entirely stored on disk. However, this assumption is unrealistic since any algorithm operating on an external hash table must have some internal memory (at least $\Omega(1)$ blocks) to work with. The availability of a small internal memory buffer can dramatically reduce the amortized insertion cost to $o(1)$ I/Os for many external memory data structures. In this paper we study the inherent query-insertion tradeoff of external hash tables in the presence of a memory buffer. In particular, we show that for any constant $c>1$, if the query cost is targeted at $1+O(1/b^{c})$ I/Os, then it is not possible to support insertions in less than $1-O(1/b^{\frac{c-1}{4}})$ I/Os amortized, which means that the memory buffer is essentially useless. While if the query cost is relaxed to $1+O(1/b^{c})$ I/Os for any constant $c<1$, there is a simple dynamic hash table with $o(1)$ insertion cost. These results also answer the open question recently posed by Jensen and Pagh.