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Universal cycle constructions for k-subsets and k-multisets

Colin Campbell, Luke Janik-Jones, Joe Sawada

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.

Universal cycle constructions for k-subsets and 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.
Paper Structure (10 sections, 10 theorems, 14 equations, 2 figures, 2 tables)

This paper contains 10 sections, 10 theorems, 14 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cycle-joining trees and successor rules
  4. Concatenation trees
  5. Bounded-weight de Bruijn sequences
  6. Fixed weight and weight range
  7. Applying the MSR
  8. Application to $k$-subsets and $k$-multisets
  9. Acknowledgment
  10. C implementation of the bounded weight "Grandmama" de Bruijn sequence

Key Result

Theorem 1

Let $T$ be a concatenation tree for a PCR-based cycle-joining tree with the Chain Property for an underlying set $\mathbf{S}$. Then $\mathcal{U}_T$ is a universal cycle for $\mathbf{S}$ with successor rule $h(\alpha)$.

Figures (2)

  • Figure 1: (a) The first non-zero cycle-joining tree for $\Sigma_5(3,4)$ where the nodes are represented by the necklaces in $\mathbf{N}_5(3,4)$. (b) A corresponding concatenation tree where the change index of the root is assigned to the final symbol, as indicated by the small box on top of the node. All children are left children, and the labels on the nodes do not need to change. An RCL traversal of this tree visits the necklaces in colex order.
  • Figure 2: The first non-zero cycle-joining tree based on the MSR for $\Sigma_5(3,4)$ where the nodes are represented by the necklaces in $\mathbf{N}_5(4)$ with weight exactly $w=4$. The labeled pre-order traversal of this tree visits the necklaces in reverse colex order. The tree is drawn in the style of a concatenation tree even though they are not formally defined for MSR-based cycle-joining trees.

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Conjecture 8
  • Theorem 9
  • Theorem 10
  • ...and 1 more