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

Daniel Gabric, Wazed Imam, Lukas 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 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.

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

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.
Paper Structure (14 sections, 9 theorems, 17 equations, 1 algorithm)

This paper contains 14 sections, 9 theorems, 17 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounding the weight
  4. Necklace properties
  5. Ranking the Granddaddy de Bruijn sequence $\mathcal{G}_k(n)$
  6. Decoding the bounded-weight de Bruijn sequence $\mathcal{G}_k(n,w^\uparrow)$
  7. Unranking
  8. An upper bound on weight
  9. Applications to $t$-subsets and $t$-multisets
  10. Proof of Lemma \ref{['lem:T']}: Enumerating $T_k(n,w,\alpha)$
  11. Enumerating preliminary sets
  12. Computing $T_k(n,w,\alpha)$
  13. Analysis
  14. Acknowledgment

Key Result

Lemma 7

The string $\mathop{\mathrm{ap}}\nolimits(\delta_1)$ has suffix $\mathbf{p}$.

Theorems & Definitions (10)

  • Remark 6
  • Lemma 7
  • Lemma 8
  • Corollary 9
  • Lemma 10
  • Theorem 11
  • Theorem 12
  • Corollary 13
  • Theorem 14
  • Theorem 15