A Connection Between Unbordered Partial Words and Sparse Rulers

Abstract

Partial words are words that contain, in addition to letters, special symbols $\diamondsuit$ called holes. Two partial words $a=a_0 \dots a_n$ and $b=b_0 \dots b_n$ are 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 unbordered if it does not have a proper nonempty prefix and a suffix that are compatible. Otherwise, the partial word is bordered.

A set $R \subseteq \{0, \dots, n\}$ is called a 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 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 the 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, depending on the reported values of the minimal number of marks in a length-$135$ ruler. We have not verified this value.

We then study a two-dimensional generalization of these results. We adapt methods from the one-dimensional case to solve the correct asymptotic for the number of holes that an unbordered two-dimensional binary partial word can have.