Universal cycle constructions for k-subsets and k-multisets

TL;DR

This paper introduces efficient universal cycle constructions for k-subsets and k-multisets using a new representation, enabling the first known efficient algorithms for these combinatorial objects.

Contribution

It provides successor-rule algorithms and necklace concatenation methods for universal cycles of k-subsets and k-multisets with optimal time complexity.

Findings

Universal cycles exist for all n, k >= 2 with the new representation.

Algorithms operate in O(n) time per symbol, using O(n) space.

First efficient universal cycle constructions for k-multisets.

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.