Small Uncolored and Colored Choice Dictionaries

TL;DR

This paper introduces highly space-efficient choice dictionaries that operate in constant time, significantly improving previous data structures by reducing space and time complexity for managing subsets and their colors.

Contribution

The authors present new choice and colored choice dictionaries with constant operation time and near-optimal space usage, advancing the efficiency of space-conscious data structures.

Findings

Constant-time operations for choice dictionaries

Space usage is optimal or near-optimal

Applications to graph algorithms like BFS

Abstract

A choice dictionary can be initialized with a parameter $n\in\mathbb{N}$ and subsequently maintains an initially empty subset $S$ of $\{1,\ldots,n\}$ under insertion, deletion, membership queries and an operation $\textit{choice}$ that returns an arbitrary element of $S$. The choice dictionary is fundamental in space-efficient computing and has numerous applications. The best previous choice dictionary can be initialized with $n$ and $t\in\mathbb{N}$ and subsequently executes all operations in $O(t)$ time and occupies $n+O(n({t/w})^t+\log n)$ bits on a word RAM with a word length of $w=\Omega(\log n)$ bits. We describe a new choice dictionary that executes all operations in constant time and, in addition to the space needed to store the integer $n$, occupies only $n+1$ bits, which is shown to be optimal if $w=o(n)$. A generalization of the choice dictionary called a colored choice dictionary is initialized with $c\in\mathbb{N}$ in addition to $n$ and subsequently maintains a semipartition $(S_0,\ldots,S_{c-1})$ of $\{1,\ldots,n\}$ under the operations $\textit{setcolor}(j,\ell)$, which moves $\ell$ from its current subset to $S_j$, $\textit{color}(\ell)$, which returns the unique $j\in\{0,\ldots,c-1\}$ with $\ell\in S_j$, and $\textit{choice}(j)$, which returns an arbitrary element of $S_j$. We describe new colored choice dictionaries that, if initialized with constant $c$, execute $\textit{setcolor}$, $\textit{color}$ and $\textit{choice}$ in constant time and occupy $n\log_2\!c+1$ bits plus the space needed to store $n$ if $c$ is a power of 2, and at most $n\log_2\!c+n^\epsilon$ bits in general, for arbitrary fixed $\epsilon>0$. We also study the possibility of iterating over the set $S$ or over $S_j$ for given $j\in\{0,\ldots,c-1\}$ and an application of this to breadth-first search.