De Dictionariis Dynamicis Pauco Spatio Utentibus

TL;DR

This paper introduces dynamic dictionaries on the word RAM that achieve asymptotically optimal space and constant-time operations for insertions, deletions, and queries, improving upon previous bounds.

Contribution

It provides new upper bounds and lower bounds for dynamic dictionaries supporting perfect-hashing and membership queries with optimal space efficiency.

Findings

Achieves optimal space bounds for membership queries: Theta(n lg(u/n)) bits.

Provides new upper bounds for perfect-hashing queries depending on hashcode range.

Establishes a lower bound of Omega(n lg(n/(t+1))) bits for space complexity.

Abstract

We develop dynamic dictionaries on the word RAM that use asymptotically optimal space, up to constant factors, subject to insertions and deletions, and subject to supporting perfect-hashing queries and/or membership queries, each operation in constant time with high probability. When supporting only membership queries, we attain the optimal space bound of Theta(n lg(u/n)) bits, where n and u are the sizes of the dictionary and the universe, respectively. Previous dictionaries either did not achieve this space bound or had time bounds that were only expected and amortized. When supporting perfect-hashing queries, the optimal space bound depends on the range {1,2,...,n+t} of hashcodes allowed as output. We prove that the optimal space bound is Theta(n lglg(u/n) + n lg(n/(t+1))) bits when supporting only perfect-hashing queries, and it is Theta(n lg(u/n) + n lg(n/(t+1))) bits when also supporting membership queries. All upper bounds are new, as is the Omega(n lg(n/(t+1))) lower bound.