Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sublinear Time Algorithms for Abelian Group Isomorphism and Basis Construction

Nader H. Bshouty

TL;DR

The paper addresses sublinear-time Abelian group isomorphism testing and basis construction under two access models, PS and FS. It introduces a randomized framework that builds a small generator set with triangular relations, constructs a monomial Abelian group, and applies Smith normal form to obtain a basis and test isomorphism, achieving a sublinear bound of $\tilde{O}(\sqrt{|G|})$ in the PS-model (and hence FS-model). The authors prove tight lower bounds: $\Omega(\sqrt{|G|})$ in the PS-model and $\Omega(|G|^{1/4})$ in the FS-model for these problems, along with a deterministic bound of $\Omega(|G|)$ for comparison, establishing sublinear-time optimality within the models. The work also extends to basis construction, showing sublinear randomized algorithms are possible via SNF on a monomial isomorphic group and providing the first tight lower bounds for these tasks. Overall, the results delineate the computational limits and deliver sublinear, practically relevant algorithms for Abelian group isomorphism and basis computation in algebraic computation contexts.

Abstract

In this paper, we study the problems of abelian group isomorphism and basis construction in two models. In the {\it partially specified model} (PS-model), the algorithm does not know the group size but can access randomly chosen elements of the group along with the Cayley table of those elements, which provides the result of the binary operation for every pair of selected elements. In the stronger {\it fully specified model} (FS-model), the algorithm knows the size of the group and has access to its elements and Cayley table. Given two abelian groups, $G$, and $H$, we present an algorithm in the PS-model (and hence in the FS-model) that runs in time $\tilde O(\sqrt{|G|})$ and decides if they are isomorphic. This improves on Kavitha's linear-time algorithm and gives the first sublinear-time solution for this problem. We then prove the lower bound $Ω(|G|^{1/4})$ for the FS-model and the tight bound $Ω(\sqrt{|G|})$ for the PS-model. This is the first known lower bound for this problem. We obtain similar results for finding a basis for abelian groups. For deterministic algorithms, a simple $Ω(|G|)$ lower bound is given.

Sublinear Time Algorithms for Abelian Group Isomorphism and Basis Construction

TL;DR

The paper addresses sublinear-time Abelian group isomorphism testing and basis construction under two access models, PS and FS. It introduces a randomized framework that builds a small generator set with triangular relations, constructs a monomial Abelian group, and applies Smith normal form to obtain a basis and test isomorphism, achieving a sublinear bound of in the PS-model (and hence FS-model). The authors prove tight lower bounds: in the PS-model and in the FS-model for these problems, along with a deterministic bound of for comparison, establishing sublinear-time optimality within the models. The work also extends to basis construction, showing sublinear randomized algorithms are possible via SNF on a monomial isomorphic group and providing the first tight lower bounds for these tasks. Overall, the results delineate the computational limits and deliver sublinear, practically relevant algorithms for Abelian group isomorphism and basis computation in algebraic computation contexts.

Abstract

In this paper, we study the problems of abelian group isomorphism and basis construction in two models. In the {\it partially specified model} (PS-model), the algorithm does not know the group size but can access randomly chosen elements of the group along with the Cayley table of those elements, which provides the result of the binary operation for every pair of selected elements. In the stronger {\it fully specified model} (FS-model), the algorithm knows the size of the group and has access to its elements and Cayley table. Given two abelian groups, , and , we present an algorithm in the PS-model (and hence in the FS-model) that runs in time and decides if they are isomorphic. This improves on Kavitha's linear-time algorithm and gives the first sublinear-time solution for this problem. We then prove the lower bound for the FS-model and the tight bound for the PS-model. This is the first known lower bound for this problem. We obtain similar results for finding a basis for abelian groups. For deterministic algorithms, a simple lower bound is given.

Paper Structure

This paper contains 23 sections, 34 theorems, 56 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Generators For Abelian Groups
  3. Isomorphism Testing
  4. Basis Construction
  5. Our Technique
  6. Generators with Triangular Relation
  7. Isomorphism and Basis Construction
  8. Our Technique for the Lower Bounds
  9. Abelian Groups and Preliminary Results
  10. Basic Facts in Groups
  11. The Monomial Abelian Group
  12. Basis of Abelian Group via Smith Normal Form
  13. Generators for Abelian Group - Deterministic Algorithm
  14. Generators for Abelian Group - Randomized Algorithm
  15. The algorithm in the FS-model
  16. ...and 8 more sections

Key Result

Theorem 1

Any deterministic algorithm that decides whether an Abelian group is isomorphic to a given group $G$ must make at least $\Omega({|G|})$ accesses to the elements of the groups and to the Cayley table of $G$.

Theorems & Definitions (56)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Lemma 9
  • Lemma 10
  • ...and 46 more