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Dynamic Dictionary with Subconstant Wasted Bits per Key (2310.20536v1)

Published 31 Oct 2023 in cs.DS

Abstract: Dictionaries have been one of the central questions in data structures. A dictionary data structure maintains a set of key-value pairs under insertions and deletions such that given a query key, the data structure efficiently returns its value. The state-of-the-art dictionaries [Bender, Farach-Colton, Kuszmaul, Kuszmaul, Liu 2022] store $n$ key-value pairs with only $O(n \log{(k)} n)$ bits of redundancy, and support all operations in $O(k)$ time, for $k \leq \log* n$. It was recently shown to be optimal [Li, Liang, Yu, Zhou 2023b]. In this paper, we study the regime where the redundant bits is $R=o(n)$, and show that when $R$ is at least $n/\text{poly}\log n$, all operations can be supported in $O(\log* n + \log (n/R))$ time, matching the lower bound in this regime [Li, Liang, Yu, Zhou 2023b]. We present two data structures based on which range $R$ is in. The data structure for $R<n/\log{0.1} n$ utilizes a generalization of adapters studied in [Berger, Kuszmaul, Polak, Tidor, Wein 2022] and [Li, Liang, Yu, Zhou 2023a]. The data structure for $R \geq n/\log{0.1} n$ is based on recursively hashing into buckets with logarithmic sizes.

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