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Joe Sawada

Publications and source records attributed to Joe Sawada.

24 records · Page 2Linked to original sources
arXiv2018

Constructing de Bruijn sequences by concatenating smaller universal cycles

We present sufficient conditions for when an ordering of universal cycles $α_1, α_2, \ldots, α_m$ for disjoint sets $\mathbf{S}_1, \mathbf{S}_2, \ldots , \mathbf{S}_m$ can be concatenated together to obtain a universal cycle for $\mathbf{S} = \mathbf{S}_1 \cup \mathbf{S}_2 \cup \cdots \cup \mathbf{S}_m$. When $\mathbf{S}$ is the set of all $k$-ary strings of length $n$, the result of such a successful construction is a de Bruijn sequence. Our conditions are applied to generalize two previously known de Bruijn sequence constructions and then they are applied to develop three new de Bruijn sequence constructions.

math.CO↗
arXiv2016

On Prefix Normal Words and Prefix Normal Forms

A $1$-prefix normal word is a binary word with the property that no factor has more $1$s than the prefix of the same length; a $0$-prefix normal word is defined analogously. These words arise in the context of indexed binary jumbled pattern matching, where the aim is to decide whether a word has a factor with a given number of $1$s and $0$s (a given Parikh vector). Each binary word has an associated set of Parikh vectors of the factors of the word. Using prefix normal words, we provide a characterization of the equivalence class of binary words having the same set of Parikh vectors of their factors. We prove that the language of prefix normal words is not context-free and is strictly contained in the language of pre-necklaces, which are prefixes of powers of Lyndon words. We give enumeration results on $\textit{pnw}(n)$, the number of prefix normal words of length $n$, showing that, for sufficiently large $n$, \[ 2^{n-4 \sqrt{n \lg n}} \le \textit{pnw}(n) \le 2^{n - \lg n + 1}. \] For fixed density (number of $1$s), we show that the ordinary generating function of the number of prefix normal words of length $n$ and density $d$ is a rational function. Finally, we give experimental results on $\textit{pnw}(n)$, discuss further properties, and state open problems.

cs.DM↗
arXiv2015

Charm bracelets and their application to the construction of periodic Golay pairs

A $k$-ary charm bracelet is an equivalence class of length $n$ strings with the action on the indices by the additive group of the ring of integers modulo $n$ extended by the group of units. By applying an $O(n^3)$ amortized time algorithm to generate charm bracelet representatives with a specified content, we construct 29 new periodic Golay pairs of length $68$.

math.CO↗
arXiv2014

Normal, Abby Normal, Prefix Normal

A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled pattern matching. In this paper we present results about the number $pnw(n)$ of prefix normal words of length $n$, showing that $pnw(n) =Ω\left(2^{n - c\sqrt{n\ln n}}\right)$ for some $c$ and $pnw(n) = O \left(\frac{2^n (\ln n)^2}{n}\right)$. We introduce efficient algorithms for testing the prefix normal property and a "mechanical algorithm" for computing prefix normal forms. We also include games which can be played with prefix normal words. In these games Alice wishes to stay normal but Bob wants to drive her "abnormal" -- we discuss which parameter settings allow Alice to succeed.

cs.FL↗
arXiv2014

On Combinatorial Generation of Prefix Normal Words

A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled pattern matching. In this paper we present an efficient algorithm for exhaustively listing the prefix normal words with a fixed length. The algorithm is based on the fact that the language of prefix normal words is a bubble language, a class of binary languages with the property that, for any word w in the language, exchanging the first occurrence of 01 by 10 in w results in another word in the language. We prove that each prefix normal word is produced in O(n) amortized time, and conjecture, based on experimental evidence, that the true amortized running time is O(polylog(n)).

cs.DS↗