Related papers: Constructing de Bruijn sequences by concatenating …
Assembling genomic sequences from a set of overlapping reads is one of the most fundamental problems in computational biology. Algorithms addressing the assembly problem fall into two broad categories -- based on the data structures which…
The flow of contracting systems contracts 1-dimensional parallelotopes, i.e., line segments, at an exponential rate. One reason for the usefulness of contracting systems is that many interconnections of contracting sub-systems yield an…
Given $n$ pairwise disjoint sets $X_1,\ldots, X_n$, we call the elements of $S=X_1\times\ldots\times X_n$ strings. A nonempty set of strings $W\subseteq S$ is said to be well-connected if for every $v\in W$ and for every $i\, (1\le i\le…
The distribution of a given sequence in the set of all sequences with n ones and m = M - n zeros are found by relating the problem to the partitions of a natural number in m natural summands, taking into account the order. The formulas…
Analogously to de Bruijn sequences, orientable sequences have application in automatic position-location applications and, until recently, studies of these sequences focused on the binary case. In recent work by Alhakim et al., a range of…
We introduce the following natural generalization of trace reconstruction, parameterized by a deletion probability $\delta \in (0,1)$ and length $n$: There is a length $n$ string of probabilities, $S=p_1,\ldots,p_n,$ and each "trace" is…
We study an interesting family of cooperating coroutines, which is able to generate all patterns of bits that satisfy certain fairly general ordering constraints, changing only one bit at a time. (More precisely, the directed graph of…
A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian; this baseline result is used as the basis of existence proofs for universal cycles (also known as ucycles or generalized deBruijn cycles or…
The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…
A cycle system of order $n$ is a decomposition of the edges of the complete graph $K_n$ into cycles of a fixed length. A cycle system is said to be $k$-colourable if we can assign $k$ colours to its vertices so that no cycle is…
Square Heffter arrays are $n\times n$ arrays such that each row and each column contains $k$ filled cells, each row and column sum is divisible by $2nk+1$ and either $x$ or $-x$ appears in the array for each integer $1\leq x\leq nk$.…
A number of applications of Steiner triple systems (e.g. disk erasure codes) exist that require a special ordering of its blocks. Universal cycles, introduced by Chung, Diaconis, and Graham in 1992, and Gray codes are examples of listing…
The graph of overlapping permutations is a directed graph that is an analogue to the De Bruijn graph. It consists of vertices that are permutations of length $n$ and edges that are permutations of length $n+1$ in which an edge $a_1\cdots…
The skew of a binary string is the difference between the number of zeroes and the number of ones, while the length of the string is the sum of these two numbers. We consider certain suffixes of the lexicographically-least de Bruijn…
We decide completely the cycle structure of pure summing register (PSR) and complementary summing register (CSR). Based on the state diagram of CSR, we derive an algorithm to generate de Bruijn cycles from CSR inspired by Tuvi Etzion's…
The concept of symbolic sequences play important role in study of complex systems. In the work we are interested in ultrametric structure of the set of cyclic sequences naturally arising in theory of dynamical systems. Aimed at construction…
We consider perfect 1-error correcting codes over a finite field with $q$ elements (briefly $q$-ary 1-perfect codes). In this paper, a generalized concatenation construction for $q$-ary 1-perfect codes is presented that allows us to…
Universal cycle for $k$-permutations is a cyclic arrangement in which each $k$-permutation appears exactly once as $k$ consecutive elements. Enumeration problem of universal cycles for $k$-permutations is discussed and one new enumerating…
Frequent sequence mining methods often make use of constraints to control which subsequences should be mined. A variety of such subsequence constraints has been studied in the literature, including length, gap, span, regular-expression, and…
In this paper we introduce a particular class of Heffter arrays, called globally simple Heffter arrays, whose existence gives at once orthogonal cyclic cycle decompositions of the complete graph and of the cocktail party graph. In…