Universal partial words over non-binary alphabets

Abstract

Abstract. Chen, Kitaev, Mütze, and Sun recently introduced the notion of universal partial words, a generalization of universal words and de Bruijn sequences. Universal partial words allow for a wild-card character ⋄, which is a placeholder for any letter in the alphabet. We extend results from the original paper and develop additional proof techniques to study these objects. For non-binary alphabets, we show that universal partial words have periodic ⋄ structure and are cyclic, and we give number-theoretic conditions on the existence of universal partial words. In addition, we provide an explicit construction for an infinite family of universal partial words over non-binary alphabets.

On shortening u-cycles and u-words for permutations

“…This paper opens up a new research direction of shortening u-cycles and u-words for permutations that naturally extends analogous studies conducted for the celebrated de Bruijn sequences [1,5]. We were able to offer two different ways to approach the problem, namely, via linear extensions of posets, and via usage of (restricted) ✸s, and we discussed several existence and non-existence results related to the context.…”

“…The following lemma is an analogue in the case of permutations of Theorem 4.1 in [5] and Lemma 14 in [1] obtained for words. Lemma 9.…”

“…As a straightforward extension of the objects in [1,5] to the case of permutations, our u-cycles and u-words will contain ✸(s), whose meaning needs to be redefined though to avoid factors not orderisomorphic to permutations. In analogy with [1,5], we call u-cycles and u-words for permutations containing at least one ✸ universal partial cycles (u-p-cycles) and universal partial words (u-p-words) for permutations, respectively. Introducing these notions helps us to distinguish between shortening using linear extensions of posets (when the resulting objects are still called by us u-cycles and u-words; see Section 1.1), and shortening using ✸s, in which case the obtained objects are called u-p-cycles and u-p-words.…”

“…• Our second approach is a plain extension of the studies in [1,5] to the case of permutations. However, using the "wildcard" symbol ✸ seems to be inefficient in the context (it is dominated by non-existence results; see Section 3.1), so we consider its refine-…”

Optimized Sequence Library Design for Efficient In Vitro Interaction Mapping

“…Previous theoretical studies have considered the problem of covering all k-mers using joker characters, but with different restrictions and limitations, making them impractical for library design applications (Blanchet-Sadri et al, 2010; H. Z. Q. Chen et al, 2016; Goeckner et al, 2016; Wyatt, 2013). None of these works considered the problem with the restriction that we defined, i.e.…”