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DOI: 10.4230/LIPIcs.STACS.2022.25
URN: urn:nbn:de:0030-drops-158350
URL: https://drops.dagstuhl.de/opus/volltexte/2022/15835/
Das, Bireswar ; Kumar, Anant ; Sharma, Shivdutt ; Thakkar, Dhara
Linear Space Data Structures for Finite Groups with Constant Query-Time
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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).
BibTeX - Entry
@InProceedings{das_et_al:LIPIcs.STACS.2022.25,
author = {Das, Bireswar and Kumar, Anant and Sharma, Shivdutt and Thakkar, Dhara},
title = {{Linear Space Data Structures for Finite Groups with Constant Query-Time}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {25:1--25:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/15835},
URN = {urn:nbn:de:0030-drops-158350},
doi = {10.4230/LIPIcs.STACS.2022.25},
annote = {Keywords: Compact Data Structures, Space Efficient Representations, Finite Groups, Simple Groups, Classification Theorem for Finite Simple Groups}
}
| Keywords: | Compact Data Structures, Space Efficient Representations, Finite Groups, Simple Groups, Classification Theorem for Finite Simple Groups | |
| Seminar: | 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022) | |
| Issue date: | 2022 | |
| Date of publication: | 09.03.2022 |
