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arXiv · 2303.01957

Linear Space Data Structures for Finite Groups with Constant Query-time

Bireswar Das·Anant Kumar·Shivdutt Sharma·Dhara Thakkar
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Abstract

A finite group of order $n$ can be represented by its Cayley table. In the word-RAM model the Cayley table of a group of order $n$ can be stored using $O(n^2)$ words and can be used to answer a multiplication query in constant time. It is interesting to ask if we can design a data structure to store a group of order $n$ that uses $o(n^2)$ space but can still answer a multiplication query in constant time. We design a constant query-time data structure that can store any finite group using $O(n)$ words where $n$ is the order of the group. Farzan and Munro (ISSAC 2006) gave an information theoretic lower bound of $Ω(n)$ on the number of words to store a group of order $n$. Since our data structure achieves this lower bound and answers queries in constant time, it is optimal in both space usage and query-time. A crucial step in the process is essentially to design linear space and constant query-time data structures for nonabelian simple groups. The data structures for nonableian simple groups are designed using a lemma that we prove using the Classification Theorem for Finite Simple Groups (CFSG).

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BibTeXRIS

Bireswar Das, Anant Kumar, Shivdutt Sharma, Dhara Thakkar. 2023-03-03. Linear Space Data Structures for Finite Groups with Constant Query-time. https://arxiv.org/abs/2303.01957

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