Optimal Non-Adaptive Cell Probe Dictionaries and Hashing

TL;DR

This paper introduces a simple, optimal non-adaptive data structure for static dictionaries, matching lower bounds and highlighting a complexity gap with adaptive solutions, also providing an efficient hash family implementation.

Contribution

It presents a provably optimal non-adaptive dictionary data structure with matching lower bounds and introduces an efficient hash family in the cell probe model.

Findings

Optimal non-adaptive dictionary data structure with O(lg(u/n)/lg(s/n)) query time.

Matching lower bounds establish the optimality of the data structure.

First implementation of an n-wise independent hash family with optimal evaluation time.

Abstract

We present a simple and provably optimal non-adaptive cell probe data structure for the static dictionary problem. Our data structure supports storing a set of n key-value pairs from [u]x[u] using s words of space and answering key lookup queries in t = O(lg(u/n)/ lg(s/n)) nonadaptive probes. This generalizes a solution to the membership problem (i.e., where no values are associated with keys) due to Buhrman et al. We also present matching lower bounds for the non-adaptive static membership problem in the deterministic setting. Our lower bound implies that both our dictionary algorithm and the preceding membership algorithm are optimal, and in particular that there is an inherent complexity gap in these problems between no adaptivity and one round of adaptivity (with which hashing-based algorithms solve these problems in constant time). Using the ideas underlying our data structure, we also obtain the first implementation of a n-wise independent family of hash functions with optimal evaluation time in the cell probe model.