Abstract
The Cayley table representation of a group uses \(O(n^2)\) words for a group of order n and answers multiplication queries in time O(1) in word RAM model. It is interesting to ask if there is a \(o(n^2)\) space representation of groups that still has O(1) query-time. We show that for any \(\delta \), \(\frac{1}{\log n} \le \delta \le 1\), there is an \(\mathcal {O}(\frac{n^{1 +\delta }}{\delta })\) space representation for groups of order n with \(\mathcal {O}(\frac{1}{\delta })\) query-time.
We also show that for Dedekind groups, simple groups and several group classes defined in terms of semidirect product, there are linear space representation to answer multiplication queries in logarithmic time.
Farzan and Munro (ISSAC’06) defined a model for group representation and gave a succinct data structure for abelian groups with constant query-time. They asked if their result can be extended to categorically larger group classes. We show we can construct data structures in their model to represent Hamiltonian groups and extensions of abelian groups by cyclic groups to answer multiplication queries in constant time.
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Notes
- 1.
An elementary abelian 2-group is an abelian group in which every nontrivial element has order 2.
- 2.
If \((\alpha _1,\cdots ,\alpha _k)\notin \text {Image}(F)\), then the value could be arbitrary.
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Das, B., Sharma, S., Vaidyanathan, P.R. (2019). Succinct Representations of Finite Groups. In: Gąsieniec, L., Jansson, J., Levcopoulos, C. (eds) Fundamentals of Computation Theory. FCT 2019. Lecture Notes in Computer Science(), vol 11651. Springer, Cham. https://doi.org/10.1007/978-3-030-25027-0_16
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