Universal cycle constructions for k-subsets and k-multisets

2026-03-12

AI Summary

This work addresses the long-standing challenge that universal cycles for k-subsets and k-multisets under conventional string representations do not always exist and lack efficient construction methods. To overcome this, the authors introduce a novel representation based on labeled graph structures and develop a unified framework for constructing universal cycles by integrating bounded-weight de Bruijn sequences—generalizing constructions such as the Grandmama sequence—with successor rule algorithms and necklace concatenation techniques. This approach yields the first efficient universal cycle construction for k-multisets that works for all parameters \(n, k \geq 2\), generating each symbol in \(O(1)\) amortized time using only \(O(n)\) space. The result establishes the first practical and general solution for universal cycles of k-multisets.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data

Abstract

A universal cycle for a set S of combinatorial objects is a cyclic sequence of length |S|that contains a representation of each element in S exactly once as a substring. If S is the set of k-subsets of [n] = {1, 2, . . . , n}, it is well-known that universal cycles do not always exists when applying a simple string representation, where 12 or 21 could represent the subset {1, 2}. Similarly, if S is the set of k-multisets of [n], it is also known that universal cycles do not always exist using a similar representation, where 112, 121, or 211 could represent the multiset {1, 1, 2}. By mapping these sets to an appropriate family of labeled graphs, universal cycles are known to exist, but without a known efficient construction. In this paper we consider a new representation for k-subsets and k-multisets that leads to efficient universal cycle constructions for all n, k >=2. We provide successor-rule algorithms to construct such universal cycles in O(n) time per symbol using O(n) space and demonstrate that necklace concatenation algorithms allow the same sequences to be generated in O(1) amortized time per symbol. They are the first known efficient universal cycle constructions for k-multisets. The results are obtained by considering constructions for bounded-weight de Bruijn sequences. In particular, we demonstrate that a bounded-weight generalization of the Grandmama de Bruijn sequence can be constructed in O(1) amortized time per symbol.

Problem

• universal cycle
• k-subsets
• k-multisets
• combinatorial generation
• efficient construction

Innovation

• universal cycle
• k-multisets
• successor-rule algorithm
• necklace concatenation
• bounded-weight de Bruijn sequence