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

Authors Bireswar Das, Anant Kumar, Shivdutt Sharma, Dhara Thakkar

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Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Data compression
Keywords
  • Compact Data Structures
  • Space Efficient Representations
  • Finite Groups
  • Simple Groups
  • Classification Theorem for Finite Simple Groups

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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(n²) 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(n²) 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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Bireswar Das, Anant Kumar, Shivdutt Sharma, and Dhara Thakkar. Linear Space Data Structures for Finite Groups with Constant Query-Time. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.STACS.2022.25

Author Details

Bireswar Das
  • Indian Institute of Technology Gandhinagar, India
Anant Kumar
  • Indian Institute of Technology Gandhinagar, India
Shivdutt Sharma
  • Indian Institute of Information Technology, Una, India
Dhara Thakkar
  • Indian Institute of Technology Gandhinagar, India

Funding

  • Das, Bireswar: Funded by Ministry of Education, Govt. of India.
  • Thakkar, Dhara: Funded by CSIR-UGC NET JRF Fellowship.

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