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

TL;DR

This paper introduces polynomial time decoding algorithms for bounded-weight de Bruijn sequences, enabling efficient decoding of universal cycles for t-subsets and t-multisets, which were previously lacking such algorithms.

Contribution

It presents the first efficient polynomial time/space decoding algorithms for bounded-weight de Bruijn sequences and applies them to universal cycles of t-subsets and t-multisets.

Findings

Developed polynomial time decoding algorithms for bounded-weight de Bruijn sequences.

Applied decoding algorithms to universal cycles for t-subsets and t-multisets.

Enhanced the efficiency of decoding universal cycles in combinatorial objects.

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.