Decoding universal cycles for t-subsets and t-multisets by decoding bounded-weight de Bruijn sequences

2026-03-12

AI Summary

Existing approaches lack efficient decoding algorithms for universal cycles of t-subsets and t-multisets. This work presents the first polynomial-time and polynomial-space decoding method for bounded-weight de Bruijn sequences and applies it to the decoding problem of these universal cycles. By integrating combinatorial mathematics with de Bruijn sequence theory, the proposed algorithm achieves the first decoding scheme that simultaneously satisfies both polynomial time and space complexity. This advancement significantly improves the decoding efficiency for universal cycles of t-subsets and t-multisets, addressing a critical gap in the current literature.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A universal cycle for a set S of combinatorial objects is a cyclic sequence of length |S| that contains a representative of each element in S exactly once as a substring. Despite the many universal cycle constructions known in the literature for various sets including k-ary strings of length n, permutations of order n, t-subsets of an n-set, and t-multisets of an n-set, remarkably few have efficient decoding (ranking/unranking) algorithms. In this paper we develop the first polynomial time/space decoding algorithms for bounded-weight de Bruijn sequences for strings of length nover an alphabet of size k. The results are then applied to decode universal cycles for t-subsets and t-multisets.

Problem

• universal cycles
• decoding
• t-subsets
• t-multisets
• de Bruijn sequences

Innovation

• bounded-weight de Bruijn sequences
• universal cycles
• decoding algorithm
• t-subsets
• t-multisets