On the Optimal Time/Space Tradeoff for Hash Tables

TL;DR

This paper explores the fundamental tradeoff between time and space in hash tables, demonstrating that increasing insertion/deletion time can exponentially decrease wasted space, surpassing previous bounds.

Contribution

It introduces a new tradeoff curve showing that for any k in [log* n], hash tables can achieve O(k)-time operations with exponentially decreasing wasted bits, improving upon prior bounds.

Findings

Achieves O(k)-time insertions/deletions with O(log^{(k)} n) wasted bits per key.

Shows the tradeoff curve is optimal within a broad class of hash tables.

Surpasses the previously known O(loglog n) wasted bits bound.

Abstract

For nearly six decades, the central open question in the study of hash tables has been to determine the optimal achievable tradeoff curve between time and space. State-of-the-art hash tables offer the following guarantee: If keys/values are Theta(log n) bits each, then it is possible to achieve constant-time insertions/deletions/queries while wasting only O(loglog n) bits of space per key when compared to the information-theoretic optimum. Even prior to this bound being achieved, the target of O(loglog n) wasted bits per key was known to be a natural end goal, and was proven to be optimal for a number of closely related problems (e.g., stable hashing, dynamic retrieval, and dynamically-resized filters). This paper shows that O(loglog n) wasted bits per key is not the end of the line for hashing. In fact, for any k \in [log* n], it is possible to achieve O(k)-time insertions/deletions, O(1)-time queries, and O(\log^{(k)} n) wasted bits per key (all with high probability in n). This means that, each time we increase insertion/deletion time by an \emph{additive constant}, we reduce the wasted bits per key \emph{exponentially}. We further show that this tradeoff curve is the best achievable by any of a large class of hash tables, including any hash table designed using the current framework for making constant-time hash tables succinct.