Resizable Retrieval

2026-06-14

AI Summary

This work resolves an open problem posed by Demaine et al. (2006) concerning the design of a dynamic retrieval data structure that supports constant-time queries, insertions, and deletions while using space dependent only on the current number of keys $n$, rather than a pre-specified upper bound. We present the first dynamic retrieval structure achieving $O(1)$ worst-case operation time with space usage of $nv + O(n \log\log(U/n))$ bits, matching the information-theoretic lower bound. Our approach combines information-theoretically compact encoding, dynamic hashing, and amortized analysis, complemented by careful memory management. Beyond settling this fundamental retrieval problem, our techniques yield practical applications, including efficient dynamic filters and space-efficient memory allocators.

Abstract

A dynamic retrieval data structure encodes a function $f:K \rightarrow [2^v]$ for a set $K \subseteq [U]$, while supporting queries $f(x)$ for $x\in K$, insertions \texttt{Insert}$(x, f(x))$ for $x \notin K$, and deletions \texttt{Delete}$(x)$ for $x \in K$. Given an upper bound $N$ on $|K|$, it is known how to solve the dynamic retrieval problem with $O(1)$-time operations and space $Nv + O(N \log \log (U/N))$ bits. An open question, first posed by Demaine et al. in 2006, is whether a similar bound can be achieved with a resizable data structure, whose space bound is parameterized by the \emph{current} size $n$ of $K$. We answer this question in the affirmative and prove matching lower bounds for the space-time trade-off achieved by our data structure. We also give corollaries for space-efficient memory allocation and dynamic filters.

Problem

• Resizable Retrieval
• Dynamic Data Structure
• Space Efficiency
• Retrieval Problem
• Memory Allocation

Innovation

• resizable retrieval
• dynamic data structures
• space-efficient
• constant-time operations
• information-theoretic lower bounds