Related papers: Arbitrary-length analogs to de Bruijn sequences
A special type of cyclic sequences named adjacency-hopping de Bruijn sequences is introduced in this paper. It is theoretically proved the existence of such sequences, and the number of such sequences is derived. These sequences guarantee…
We study algorithms for solving three problems on strings. The first one is the Most Frequently String Search Problem. The problem is the following. Assume that we have a sequence of $n$ strings of length $k$. The problem is finding the…
For every natural number $n\geq 2$ and every finite sequence $L$ of natural numbers, we consider the set $UD_n(L)$ of all uniquely decodable codes over an $n$-letter alphabet with the sequence $L$ as the sequence of code word lengths, as…
Pseudo-random arrays and perfect maps are the two-dimensional analogs of M-sequences and de Bruijn sequences, respectively. We modify the definitions to be applied to codes. These codes are also the two-dimensional analogs of certain…
A universal cycle for a set S of combinatorial objects is a cyclic sequence of length |S|that contains a representation of each element in S exactly once as a substring. If S is the set of k-subsets of [n] = {1, 2, . . . , n}, it is…
We consider string matching with variable length gaps. Given a string $T$ and a pattern $P$ consisting of strings separated by variable length gaps (arbitrary strings of length in a specified range), the problem is to find all ending…
Let $L_{k,\alpha}^{\mathbb{Z}}$ denote the set of all bi-infinite $\alpha$-power free words over an alphabet with $k$ letters, where $\alpha$ is a positive rational number and $k$ is positive integer. We prove that if $\alpha\geq 5$, $k\geq…
An orientable sequence of order $n$ is a cyclic binary sequence such that each length-$n$ substring appears at most once \emph{in either direction}. Maximal length orientable sequences are known only for $n\leq 7$, and a trivial upper bound…
Let $L=(L_d)_{d \in \mathbb N}$ be any ordered probability sequence, i.e., satisfying $0 < L_{d+1} \le L_d$ for each $d \in \mathbb N$ and $\sum_{d \in \mathbb N} L_d =1$. We construct sequences $A = (a_i)_{i \in \mathbb N}$ on the…
When flipping a fair coin, let $W = L_1L_2...L_N$ with $L_i\in\{H,T\}$ be a binary word of length $N=2$ or $N=3$. In this paper, we establish second- and third-order linear recurrence relations and their generating functions to discuss the…
We introduce a novel definition of approximate palindromes in strings, and provide an algorithm to find all maximal approximate palindromes in a string with up to $k$ errors. Our definition is based on the usual edit operations of…
Given an alphabet $S$, we consider the size of the subsets of the full sequence space $S^{\rm {\bf Z}}$ determined by the additional restriction that $x_i\not=x_{i+f(n)},\ i\in {\rm {\bf Z}},\ n\in {\rm {\bf N}}.$ Here $f$ is a positive,…
An M-sequence generated by a primitive polynomial has many interesting and desirable properties. A pseudo-random array is the two-dimensional generalization of an M-sequence. There are non-primitive polynomials all of whose non-zero…
In this paper we define a new problem, motivated by computational biology, $LCSk$ aiming at finding the maximal number of $k$ length $substrings$, matching in both input strings while preserving their order of appearance. The traditional…
When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct…
A problem of reconstructing words from their subwords involves determining the minimum amount of information needed, such as multisets of scattered subwords of a specific length or the frequency of scattered subwords from a given set, in…
This paper investigates the approximability of the Longest Common Subsequence (LCS) problem. The fastest algorithm for solving the LCS problem exactly runs in essentially quadratic time in the length of the input, and it is known that under…
A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique…
We use modular arithmetic to construct a de Bruijn array containing all fillings of an L (a 2 x 2 array with the upper right corner removed) with digits chosen from {0,...,k-1}.
The de Bruijn graph $G_K$ of a set of strings $S$ is a key data structure in genome assembly that represents overlaps between all the $K$-length substrings of $S$. Construction and navigation of the graph is a space and time bottleneck in…