Sublinear Time Algorithms for Abelian Group Isomorphism and Basis Construction

2025-04-03

AI Summary

This paper studies the group isomorphism testing and basis construction problems for finite abelian groups, designing sublinear-time algorithms under two oracle models: Partially Specified (PS), where only random sampling and local Cayley table queries are allowed and the group order |G| is unknown; and Fully Specified (FS), where |G| is known and the full Cayley table is accessible. Methodologically, the approach integrates randomized sampling, localized Cayley table queries, structural decomposition of abelian groups, and probabilistic analysis. The main contributions are: (i) the first Õ(√|G|)-time algorithm for both isomorphism testing and basis construction in the PS model—breaking the previously believed linear lower bound; and (ii) the first rigorous lower bounds: Ω(√|G|) for PS and Ω(|G|^{1/4}) for FS. These results substantially improve upon Kavitha’s linear-time algorithm and establish new complexity-theoretic benchmarks for abelian group algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

In his 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 $ ilde 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 $Omega(|G|^{1/4})$ for the FS-model and the tight bound $Omega(sqrt{|G|})$ 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 $Omega(|G|)$ lower bound is given.

Problem

• Develops sublinear-time algorithm for abelian group isomorphism
• Proves lower bounds for group isomorphism in two models
• Extends results to basis construction for abelian groups

Innovation

• Sublinear-time algorithm for group isomorphism
• Partially specified model with random element access
• Tight lower bounds for basis construction