Efficient Randomized Streaming Algorithms for Word Problems of Finitely Generated Groups

Conceitos essenciais

Finitely generated linear groups, metabelian groups, and free solvable groups admit randomized streaming algorithms with logarithmic space complexity for their word problems. The class of finitely generated groups with a logspace randomized streaming algorithm for the word problem is closed under several group-theoretic constructions. In contrast, Thompson's group F has only a linear space randomized streaming algorithm for its word problem.

Resumo

The paper studies deterministic and randomized streaming algorithms for word problems of finitely generated groups. The main results are:

For finitely generated linear groups, metabelian groups, and free solvable groups, the authors show the existence of randomized streaming algorithms with logarithmic space complexity for their word problems.

The class of finitely generated groups with a logspace randomized streaming algorithm for the word problem is closed under several group-theoretical constructions: finite extensions, graph products, and wreath products by finitely generated abelian groups.

In contrast, the authors provide an example of a finitely presented group, Thompson's group F, where the word problem has only a linear space randomized streaming algorithm.

The deterministic streaming space complexity of a group's word problem is directly linked to the growth function of the group. The randomized streaming space complexity is lower bounded by the logarithm of the logarithm of the group's growth function.

Randomized streaming algorithms for subgroup membership problems in free groups and direct products of free groups are also studied.

The paper provides a comprehensive analysis of the streaming complexity of group word problems, identifying classes of groups with efficient randomized streaming algorithms as well as examples of groups with inherent limitations.

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Streaming word problems

by Mark... às arxiv.org 04-24-2024

https://arxiv.org/pdf/2202.04060.pdf

Perguntas Mais Profundas

What are some potential applications of the efficient randomized streaming algorithms for group word problems presented in this paper

The efficient randomized streaming algorithms for group word problems presented in this paper have several potential applications in both theoretical and practical settings.

Group Theory Research: These algorithms can be used to efficiently solve word problems in various classes of groups, such as linear groups, metabelian groups, and free solvable groups. This can aid researchers in studying the properties and structures of these groups more effectively.

Complexity Theory: The study of streaming algorithms for word problems in groups contributes to the understanding of the complexity of group theoretical computations. It provides insights into the space complexity of solving fundamental group theory problems.

Cryptography: Efficient algorithms for word problems in groups are essential in cryptographic protocols based on group theory, such as public-key cryptography. The ability to solve these problems efficiently can enhance the security and performance of cryptographic systems.

Data Processing: Streaming algorithms are valuable in processing large datasets or streams of data. In the context of group word problems, these algorithms can be applied to analyze and manipulate group elements in real-time data streams.

Are there any other group-theoretic constructions, beyond the ones considered, that preserve the existence of logspace randomized streaming algorithms for the word problem

While the paper discusses group theoretical constructions like finite extensions, graph products, and wreath products by finitely generated abelian groups that preserve the existence of logspace randomized streaming algorithms for the word problem, there are other constructions that may also exhibit similar properties.

Direct Products: Direct products of groups could potentially preserve the existence of logspace randomized streaming algorithms for the word problem. By analyzing the structure of direct products and their impact on computational complexity, it may be possible to extend the results to this construction.

Quotient Groups: Investigating how quotient groups interact with streaming algorithms could be another avenue to explore. Understanding how the properties of quotient groups affect the efficiency of streaming algorithms for word problems can provide valuable insights.

Automorphism Groups: Exploring the streaming complexity of word problems in automorphism groups of groups could be another interesting direction. By studying the behavior of automorphisms and their impact on computational algorithms, new insights into streaming complexity could be gained.

Can the techniques developed in this paper be extended to study the streaming complexity of other computational problems in group theory, such as the conjugacy problem or the isomorphism problem

The techniques developed in this paper can be extended to study the streaming complexity of other computational problems in group theory, such as the conjugacy problem or the isomorphism problem.

Conjugacy Problem: By adapting the framework of randomized streaming algorithms, researchers can investigate the space complexity of solving the conjugacy problem in groups. This can lead to efficient algorithms for determining whether two elements in a group are conjugate.

Isomorphism Problem: Applying similar techniques to the isomorphism problem can help in understanding the space requirements for determining if two groups are isomorphic. By developing randomized streaming algorithms for this problem, researchers can explore the computational complexity of group isomorphisms in a streaming setting.