Optimal Non-Adaptive Cell Probe Dictionaries and Hashing
arxiv(2023)
摘要
We present a simple and provably optimal non-adaptive cell probe data
structure for the static dictionary problem. Our data structure supports
storing a set of n key-value pairs from [u]x[u] using s words of space and
answering key lookup queries in t = O(lg(u/n)/ lg(s/n)) nonadaptive probes.
This generalizes a solution to the membership problem (i.e., where no values
are associated with keys) due to Buhrman et al. We also present matching lower
bounds for the non-adaptive static membership problem in the deterministic
setting. Our lower bound implies that both our dictionary algorithm and the
preceding membership algorithm are optimal, and in particular that there is an
inherent complexity gap in these problems between no adaptivity and one round
of adaptivity (with which hashing-based algorithms solve these problems in
constant time). Using the ideas underlying our data structure, we also obtain
the first implementation of a n-wise independent family of hash functions with
optimal evaluation time in the cell probe model.
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