Universal partial tori

2024-09-19

Designs, Codes and Cryptography

AI Summary

This paper addresses the insufficient compression capability of two-dimensional De Bruijn tori by introducing the concept of “universal partial tori”, extending classical De Bruijn tori and one-dimensional universal partial cycles to support compact two-dimensional sequence encoding with wildcards. Methodologically, it (i) defines and constructs infinite families of universal partial tori for the first time; (ii) introduces a novel one-dimensional combinatorial structure—the “universal partial family”—as the core building block; and (iii) integrates combinatorial design, Eulerian circuits, and recursive construction techniques. Key contributions include: (i) a theoretical breakthrough in wildcard-based compression for two-dimensional structures; (ii) computational enumeration yielding multiple small-scale concrete instances; (iii) a rigorous existence proof for universal partial tori and an equivalence characterization linking them to universal partial families; and (iv) a parameterized infinite family of explicit constructions.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Data Compression
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

A De Bruijn cycle is a cyclic sequence in which every word of length n over an alphabet $$mathcal {A}$$ A appears exactly once. De Bruijn tori are a two-dimensional analogue. Motivated by recent progress on universal partial cycles and words, which shorten De Bruijn cycles using a wildcard character, we introduce universal partial tori and matrices. We find them computationally and construct infinitely many of them using one-dimensional variants of universal cycles, including a new variant called a universal partial family.

Problem

• Extend De Bruijn cycles to 2D partial tori
• Construct universal partial matrices efficiently
• Introduce new universal partial family variant

Innovation

• Introduces universal partial tori and matrices
• Uses wildcard character to shorten cycles
• Constructs variants with universal partial family