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A Connection Between Unbordered Partial Words and Sparse Rulers

Aleksi Saarela, Aleksi Vanhatalo

Abstract

$\textit{Partial words}$ are words that contain, in addition to letters, special symbols $\diamondsuit$ called $\textit{holes}$. Two partial words of $a=a_0 \dots a_n$ and $b=b_0 \dots b_n$ are $\textit{compatible}$ if for all $i$, $a_i = b_i$ or at least one of $a_i, b_i$ is a hole. A partial word is $\textit{unbordered}$ if it does not have a nonempty proper prefix and a suffix that are compatible. Otherwise the partial word is $\textit{bordered}$. A set $R \subseteq \{0, \dots, n\}$ is called a $\textit{complete sparse ruler of length $n$}$ if for all $k \in \{0, \dots, n\}$ there exists $r, s \in R$ such that $k = r - s$. These are also known as $\textit{restricted difference bases}$. From the definitions it follows that the more holes a partial word has, the more likely it is to be bordered. By introducing a connection between unbordered partial words and sparse rulers, we improve bounds on the maximum number of holes an unbordered partial word can have over alphabets of sizes $4$ or greater. We also provide a counterexample for a previously reported theorem. We then study a two-dimensional generalization of these results. We adapt methods from one-dimensional case to solve the correct asymptotic for the number of holes an unbordered two-dimensional binary partial word can have. This generalization might invoke further research questions.

A Connection Between Unbordered Partial Words and Sparse Rulers

Abstract

are words that contain, in addition to letters, special symbols called . Two partial words of and are if for all , or at least one of is a hole. A partial word is if it does not have a nonempty proper prefix and a suffix that are compatible. Otherwise the partial word is . A set is called a n if for all there exists such that . These are also known as . From the definitions it follows that the more holes a partial word has, the more likely it is to be bordered. By introducing a connection between unbordered partial words and sparse rulers, we improve bounds on the maximum number of holes an unbordered partial word can have over alphabets of sizes or greater. We also provide a counterexample for a previously reported theorem. We then study a two-dimensional generalization of these results. We adapt methods from one-dimensional case to solve the correct asymptotic for the number of holes an unbordered two-dimensional binary partial word can have. This generalization might invoke further research questions.
Paper Structure (8 sections, 15 theorems, 37 equations, 4 figures)

This paper contains 8 sections, 15 theorems, 37 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Partial words
  3. The connection with sparse rulers
  4. Applying ruler theory to unbordered partial words
  5. Wichmann-rulers and Wichmann-words
  6. Taking these ideas into 2 dimensions
  7. Quick note on cross-bifix-free codes
  8. Conclusions

Key Result

theorem 1

For all $n> 9$ and $k\geq 4$,

Figures (4)

  • Figure 1: Ruler $\{0, 1, 3, 6, 13, 20, 24, 28, 29\}$ of length $29$ and equivalent unbordered partial word of length $30$ over the alphabet $\{1,2,3,\ldots,9\}$. Here the hole symbol is just an empty space.
  • Figure 2: Word constructed in Theorem \ref{['wichmannword4']} with parameters $r=s=2$. Empty space denotes a hole.
  • Figure 3: One dimensional ruler of length $24$ (so there is $25$ "slots").
  • Figure 5: A 2D ruler turned into 2D unbordered binary partial word. Here the alphabet consists of disks and triangles and an empty cell denotes hole. Note the few extra marks placed on top of the rectangle areas compared to figure \ref{['2Drulerpicture']}.

Theorems & Definitions (38)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • theorem 1: Combinatorics-on-partial-word-borders and blanchet2009many
  • definition 6
  • definition 7
  • theorem 2: Leech and redei1949representation
  • theorem 3
  • ...and 28 more