Resizable Retrieval
For data structure researchers, it resolves a long-standing open question from 2006, providing the first resizable solution with optimal space-time trade-off.
This paper solves the open problem of designing a resizable dynamic retrieval data structure with O(1)-time operations and space proportional to the current set size n, achieving space nv + O(n log log (U/n)) bits, and proves matching lower bounds.
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.