Optimal Non-Adaptive Cell Probe Data Structures for Dictionaries and Hashing

Core Concepts

Optimal non-adaptive cell probe data structures for storing dictionaries and evaluating n-wise independent hash functions.

Abstract

The paper presents optimal non-adaptive cell probe data structures for the dictionary problem and for evaluating n-wise independent hash functions. For the dictionary problem, the authors construct a simple and provably optimal non-adaptive cell probe data structure that supports storing a set of n key-value pairs from [u] × [u] using s words of space and answering key lookup queries in t = O(lg(u/n)/ lg(s/n)) non-adaptive probes. This generalizes a previous solution for the membership problem. The authors also present matching lower bounds for the non-adaptive static membership problem in the deterministic setting, showing that their dictionary algorithm and the preceding membership algorithm are optimal. This implies an inherent complexity gap between no adaptivity and one round of adaptivity for these problems. Using the ideas underlying their dictionary data structure, the authors also obtain the first implementation of an n-wise independent family of hash functions with optimal evaluation time in the cell probe model. The key technical ingredients are the use of non-contractive expander graphs and Hall's marriage theorem to construct the dictionary, and the use of a slightly stronger expander property along with a randomized weight assignment to construct the hash functions.

Stats

The data structure stores n key-value pairs from [u] × [u] using s memory cells of w = Θ(lg u) bits. The query time is t = O(lg(u/n)/ lg(s/n)) non-adaptive probes. The hash function evaluation time is also t = O(lg(u/n)/ lg(s/n)) cell probes.

Quotes

None.

Key Insights Distilled From

Optimal Non-Adaptive Cell Probe Dictionaries and Hashing

by Kasper Green... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2308.16042.pdf

Deeper Inquiries

What are the implications of the inherent complexity gap between no adaptivity and one round of adaptivity for the dictionary and membership problems

The inherent complexity gap between no adaptivity and one round of adaptivity for the dictionary and membership problems has significant implications for the design of practical data structures. In the context of non-adaptive data structures, the gap highlights the trade-off between query time and adaptivity. Non-adaptive data structures, while simpler and more efficient in certain computational settings, inherently require more probes to achieve the same query time as adaptive structures. This complexity gap underscores the importance of adaptivity in reducing query time and improving efficiency. Practically, this insight can inform the design of data structures in various computational settings by guiding the choice between adaptivity and query time. In scenarios where adaptivity is limited or costly, non-adaptive data structures may be preferred despite the higher query time. On the other hand, in applications where query time is critical and adaptivity is not a constraint, adaptive data structures can offer significant performance improvements. Understanding this complexity gap allows designers to make informed decisions based on the specific requirements and constraints of the computational environment.

How might this insight inform the design of practical data structures in various computational settings

The techniques used to construct optimal non-adaptive data structures for dictionaries and hash functions may have limitations when extended to other data structure problems beyond these specific domains. While the non-contractive expander approach is effective for addressing the static dictionary and hashing problems, its applicability to different data structure problems depends on the underlying characteristics of those problems. One limitation of the non-contractive expander approach is its reliance on the existence of suitable expanders with specific properties, such as non-contractiveness and expansion factors. These properties may not be readily available or easily constructed for all types of data structures. Additionally, the construction and analysis of non-adaptive data structures using expanders require careful consideration of the problem's structure and requirements, which may not always align with the properties of available expanders. Therefore, while the techniques used in constructing non-adaptive data structures can provide valuable insights and approaches for certain problems, their direct extension to other data structure problems may face challenges due to the unique characteristics and constraints of those problems.

Can the techniques used to construct the optimal non-adaptive data structures be extended to other data structure problems beyond dictionaries and hash functions

The lack of explicit constructions of the required expanders can impact the practical applicability of the presented results in several ways. Firstly, without explicit constructions, the implementation of non-adaptive data structures based on these expanders may rely on probabilistic arguments or non-constructive proofs, which can limit the reproducibility and efficiency of the implementations. Explicit constructions are essential for ensuring the feasibility and reliability of the data structures in real-world applications. Furthermore, the absence of tight explicit constructions hinders the scalability and optimization of the data structures for larger problem instances. Tight explicit constructions would provide precise control over the parameters of the data structures, such as space complexity and query time, leading to more efficient and practical implementations. The prospects for obtaining tight explicit constructions in the future depend on advancements in the theory of expanders and combinatorial structures. Research efforts focused on developing explicit constructions with optimal parameters for non-adaptive data structures could enhance the applicability and impact of the presented results in various computational settings. Collaborative work between researchers in theoretical computer science and data structure design may lead to breakthroughs in this area, paving the way for more efficient and reliable non-adaptive data structures.