Optimal Static Dictionary with Worst-Case Constant Query Time

TL;DR

This paper introduces a new static dictionary data structure that achieves worst-case constant query time using near-optimal space, improving upon previous solutions by reducing the extra space needed for constant-time queries.

Contribution

It presents the first static dictionary with worst-case constant query time using space close to the information-theoretic optimum, with most extra space dedicated to a universal lookup table.

Findings

Achieves worst-case constant query time with near-optimal space.

Uses only $ ext{OPT} + n^{ ext{ extepsilon}}$ bits of space.

Most extra space is for a universal lookup table and hash functions.

Abstract

In this paper, we design a new succinct static dictionary with worst-case constant query time. A dictionary data structure stores a set of key-value pairs with distinct keys in $[U]$ and values in $[\sigma]$, such that given a query $x\in [U]$, it quickly returns if $x$ is one of the input keys, and if so, also returns its associated value. The textbook solution to dictionaries is hash tables. On the other hand, the (information-theoretical) optimal space to encode such a set of key-value pairs is only $\text{OPT} := \log\binom{U}{n}+n\log \sigma$. We construct a dictionary that uses $\text{OPT} + n^{\epsilon}$ bits of space, and answers queries in constant time in worst case. Previously, constant-time dictionaries are only known with $\text{OPT} + n/\text{poly}\log n$ space [P\v{a}tra\c{s}cu 2008], or with $\text{OPT}+n^{\epsilon}$ space but expected constant query time [Yu 2020]. We emphasize that most of the extra $n^{\epsilon}$ bits are used to store a lookup table that does not depend on the input, and random bits for hash functions. The "main" data structure only occupies $\text{OPT}+\text{poly}\log n$ bits.