Related papers: De Bruijn Sequences with Minimum Discrepancy
The distance of a binary operation from being associative can be "measured" by its associative spectrum, an appropriate sequence of positive integers. Particular instances and general properties of associative spectra are studied.
The Longest Common Subsequence (LCS) is a fundamental string similarity measure, and computing the LCS of two strings is a classic algorithms question. A textbook dynamic programming algorithm gives an exact algorithm in quadratic time, and…
Mordechay B. Levin has constructed a number $\lambda$ which is normal in base 2, and such that the sequence $(\left\{2^n \lambda\right\})_{n=0,1,2,\ldots}$ has very small discrepancy $D_N$. Indeed we have $N\cdot D_N = \mathcal{O}…
A binary triangle of size $n$ is a triangle of zeroes and ones, with $n$ rows, built with the same local rule as the standard Pascal triangle modulo $2$. A binary triangle is said to be balanced if the absolute difference between the…
The discrepancy BR for an $m \times n$ $0,1$-matrix from Brualdi and Sanderson \cite{Brualdi1998} counts the minimum number of $1$'s which need to be shifted in each row to the left to achieve its Ferrers matrix, i.e. each row consists of…
We consider two independent binary i.i.d. random strings $X$ and $Y$ of equal length $n$ and the optimal alignments according to a symmetric scoring functions only. We decompose the space of scoring functions into five components. Two of…
In this note, we extend the notion of minimal gaps to the higher dimensional sequences. We bound the minimal gap for $(\{\boldsymbol{a}_n\boldsymbol{\alpha}\}),$ $(\{a_n\boldsymbol{\alpha}\})$ and…
We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a…
The aims of this paper are twofold. First, it discusses the Littlewood conjecture and its variants with respect to uniformly distributed sequences. The second aim is to determine the exact order of the discrepancy of the van der…
Results of an exhaustive search for minimum peak sidelobe level binary sequences are presented. Several techniques for efficiency implementation of search algorithm are described. A table of number of non-equivalent optimal binary sequences…
We consider the number of occurrences of subwords (non-consecutive sub-sequences) in a given word. We first define the notion of subword entropy of a given word that measures the maximal number of occurrences among all possible subwords. We…
We design, implement and test a simple algorithm which computes the approximate entropy of a finite binary string of arbitrary length. The algorithm uses a weighted average of the Shannon Entropies of the string and all but the last binary…
Let be the following strategy to construct a walk in a labeled digraph: at each vertex, we follow the unvisited arc of minimum label. In this work we study for which languages, applying the previous strategy over the corresponding de Bruijn…
There are many randomness notions. On the classical account, many of them are about whether a given infinite binary sequence is random for some given probability. If so, this probability turns out to be the same for all these notions, so…
Suppose we know that an object is in a sorted table and we want to determine the index of that object. To achieve this goal we could perform a binary search. However, suppose it is time-consuming to determine the relative position of that…
Distances are fundamental primitives whose choice significantly impacts the performances of algorithms in machine learning and signal processing. However selecting the most appropriate distance for a given task is an endeavor. Instead of…
Orientable sequences of order n are infinite periodic sequences with symbols drawn from a finite alphabet of size k with the property that any particular subsequence of length n occurs at most once in a period in either direction. They were…
Many consensus string problems are based on Hamming distance. We replace Hamming distance by the more flexible (e.g., easily coping with different input string lengths) dynamic time warping distance, best known from applications in time…
Consider an infinite sequence of independent, uniformly chosen points from $[0,1]^d$. After looking at each point in the sequence, an overseer is allowed to either keep it or reject it, and this choice may depend on the locations of all…
A binary string representation of prime occurrences is a sequence of bits, where $1$ entries encode positions of prime numbers. This is a convenient representation for analysis of prime distribution, since it allows for application of a…