Unit Square Research Articles

Combinatorial Properties and Recognition of Unit Square Visibility Graphs

Unit square visibility graphs (USV) are described by axis-parallel visibility between unit squares placed in the plane. If the squares are required to be placed on integer grid coordinates, then USV become unit square grid visibility graphs (USGV), an alternative characterisation of the well-known rectilinear graphs. We extend known combinatorial results for USGV and we show that, in the weak case (i.e., visibilities do not necessarily translate into edges of the represented combinatorial graph), the area minimisation variant of their recognition problem is {{,mathrm{{textsf{N}}{textsf{P}}},}}-hard. We also provide combinatorial insights with respect to USV, and as our main result, we prove their recognition problem to be {{,mathrm{{textsf{N}}{textsf{P}}},}}-hard, which settles an open question.

Geometric dominating-set and set-cover via local-search

In this paper, we study two classic optimization problems: minimum geometric dominating set and set cover. In the dominating-set problem, for a given set of objects in the plane as input, the objective is to choose a minimum number of input objects such that every input object is dominated by the chosen set of objects. Here, we say that one object is dominated by another if their intersection is nonempty. For the second problem, for a given set of points and objects in the plane, the objective is to choose a minimum number of objects to cover all the points. This is a particular version of the set-cover problem.Both problems have been well-studied, subject to various restrictions on the input objects. These problems are APX-hard for object sets consisting of axis-parallel rectangles, ellipses, α-fat objects of constant description complexity, and convex polygons. On the other hand, PTASs (polynomial time approximation schemes) are known for object sets consisting of disks or unit squares. Surprisingly, a PTAS was unknown even for arbitrary squares. For both problems obtaining a PTAS remains open for a large class of objects.For the dominating-set problem, we prove that a popular local-search algorithm leads to a (1+ε) approximation for a family of homothets of a convex object (which includes arbitrary squares, k-regular polygons, translated and scaled copies of a convex set, etc.) in nO(1/ε2) time. On the other hand, the same approach leads to a PTAS for the geometric covering problem when the objects are convex pseudodisks (which include disks, unit height rectangles, homothetic convex objects, etc.). Consequently, we obtain an easy-to-implement approximation algorithm for both problems for a large class of objects, significantly improving the best-known approximation guarantees.

Spanners in randomly weighted graphs: Euclidean case

AbstractGiven a connected graph and a length function we let denote the shortest distance between vertex and vertex . A ‐spanner is a subset such that if denotes shortest distances in the subgraph then for all . We study the size of spanners in the following scenario: we consider a random embedding of into the unit square with Euclidean edge lengths. For constant, we prove the existence w.h.p. of ‐spanners for that have edges. These spanners can be constructed in time. (We will use to indicate that the hidden constant depends on ). There are constraints on preventing it going to zero too quickly.

On the geometric priority set cover problem

We study the priority set cover problem for simple geometric set systems in the plane. For pseudo-halfspaces in the plane we obtain a PTAS via local search by showing that the corresponding set system admits a planar support. We show that the problem is APX-hard even for unit disks in the plane and argue that in this case the standard local search algorithm can output a solution that is arbitrarily bad compared to the optimal solution. We then present an LP-relative constant factor approximation algorithm (which also works in the weighted setting) for unit disks via quasi-uniform sampling. As a consequence we obtain a constant factor approximation for the capacitated set cover problem with unit disks. For arbitrary size disks, we show that the problem is at least as hard as the vertex cover problem in general graphs even when the disks have nearly equal sizes. We also present a few simple results for unit squares and orthants in the plane.

Two-Scale Elastic Shape Optimization for Additive Manufacturing

In this paper, a two-scale approach for elastic shape optimization of fine-scale structures in additive manufacturing is investigated. To this end, a free material optimization is performed on the macro-scale using elasticity tensors in a set of microscopically realizable tensors. A database of these realizable tensors and their cost values is obtained with a shape and topology optimization on microscopic cells, working within a fixed set of elasticity tensors samples. This microscopic optimization takes into account manufacturability constraints via predefined material bridges to neighboring cells at the faces of the microscopic fundamental cell. For the actual additive manufacturing on a chosen fine-scale, a piecewise constant elasticity tensor ansatz on grid cells of a macroscopic mesh is applied. The macroscopic optimization is performed in an efficient online phase, whereas the associated cellwise optimal material patterns are retrieved from the database that was computed offline. For that, the set of admissible realizable elasticity tensors is parametrized using tensor product cubic B-splines over the unit square matching the precomputed samples. This representation is then efficiently used in an interior point method for the free material optimization on the macro-scale.

Covering rectangles by few monotonous polyominoes

A monotonous polyomino is formed by all lattice unit squares met by the graph of some fixed monotonous continuous function $$f:[a,b] \to \mathbb{R}$$ with $$f(k) \notin \mathbb{Z}$$ whenever $$k \in \mathbb{Z}$$ . Our main result says that the least cardinality of a covering of a lattice $$(m \times n)$$ -rectangle by monotonous polyominoes is $$\big\lceil \frac{2}{3}(m+n-\sqrt{m^2+n^2-mn})\big\rceil.$$ The paper is motivated by a problem on arrangements of straight lines on chessboards.

Baxter permuton and Liouville quantum gravity

The Baxter permuton is a random probability measure on the unit square which describes the scaling limit of uniform Baxter permutations. We determine an explicit formula for the density of the expectation of the Baxter permuton. This answers a question of Dokos and Pak (Online J Anal Comb 9:12, 2014). We also prove that all pattern densities of the Baxter permuton are strictly positive, distinguishing it from other permutons arising as scaling limits of pattern-avoiding permutations. Our proofs rely on a recent connection between the Baxter permuton and Liouville quantum gravity (LQG) coupled with the Schramm-Loewner evolution (SLE). The method works equally well for a two-parameter generalization of the Baxter permuton recently introduced by the first author, except that the density is not as explicit. This new family of permutons, called skew Brownian permuton, describes the scaling limit of a number of random constrained permutations. We finally observe that in the LQG/SLE framework, the expected proportion of inversions in a skew Brownian permuton equals $$\frac{\pi -2\theta }{2\pi }$$ where $$\theta $$ is the so-called imaginary geometry angle between a certain pair of SLE curves.

Normalized Weighting Schemes for Image Interpolation Algorithms

Image interpolation algorithms pervade many modern image processing and analysis applications. However, when their weighting schemes inefficiently generate very unrealistic estimates, they may negatively affect the performance of the end-user applications. Therefore, in this work, the author introduced four weighting schemes based on some geometric shapes for digital image interpolation operations. Moreover, the quantity used to express the extent of each shape’s weight was the normalized area, especially when the sums of areas exceeded a unit square size. The introduced four weighting schemes are based on the minimum side-based diameter (MD) of a regular tetragon, hypotenuse-based radius (HR), the virtual pixel length-based height for the area of the triangle (AT), and the virtual pixel length for hypotenuse-based radius for the area of the circle (AC). At the smaller scaling ratio, the image interpolation algorithm based on the HR scheme scored the highest at 66.6% among non-traditional image interpolation algorithms presented. However, at the higher scaling ratio, the AC scheme-based image interpolation algorithm scored the highest at 66.6% among non-traditional algorithms presented, and, here, its image interpolation quality was generally superior or comparable to the quality of images interpolated by both non-traditional and traditional algorithms.

D-overlap functions: Construction, characterization and ordinal sum representation

In this paper, we focus on the so-called D-overlap functions defined over grid of unit square, which are a class of discrete aggregation operators. The core contents are: (i) it proposes the notion of D-overlap functions and shows three construction methods for D-overlap functions in virtue of n-dimensional semi-quasi-overlap functions, convex combination and I-matrixes; (ii) it gives equivalent characterization of D-overlap functions using I-matrixes and, by upper triangular matrix, a representation of the minimum operator min is got; (iii) it provides ordinal sum representation of 1-Lipschitz D-overlap functions. These theoretical results can facilitate the applications of overlap functions related theory in real problems involving qualitative information fusion, especially in expert systems, pattern recognition and statistical data processing.

A Peano-based space-filling surface of fractal dimension three

Although space-filling curves are well known, and have many applications in parallel computing and data mapping, there is a need for a space-filling surface that is a continuous mapping from two-dimensional domain onto the unit cube. This would allow efficient implementation of a 2D problem on parallel processors which are interconnected into a 3D grid. Such a surface is presented in this paper, which uses Hilbert’s geometric approach to generate a mapping from a unit square to a triangular prism. Using two such mappings we can create a mapping from a rectangle to a unit cube. To culminate, we use the mapping to produce a continuous omnichromatic picture, that is, one for which the colors change continuously, and under sufficient resolution, contains every possible RGB value.

COUNTABLE SPACES WITH PEANO PROPERTY

In 1890, Giuseppe Peano published an example of a continuous curve passing through every point of the square $[0,1]^2$. A curve with such properties is called a Peano curve. In fact, Peano constructed a continuous surjective mapping from the unit segment $[0,1]$ to the square $[0,1]^2$. Peano's research was motivated by one result of George Cantor that the set of points of a unit segment has the same cardinality as the set of points of a unit square.In 1890, Giuseppe Peano published an example of a continuous curve passing through every point of the square $[0,1]^2$. A curve with such properties is called a Peano curve. In fact, Peano constructed a continuous surjective mapping from the unit segment $[0,1]$ to the square $[0,1]^2$. Peano's research was motivated by one result of George Cantor that the set of points of a unit segment has the same cardinality as the set of points of a unit square. According to the Hahn-Mazurkevich theorem the Hausdorff topological space $X$ is a continuous image of a unit segment $[0,1]$ if and only if when $X$ is compact, metrizable, connected, locally connected and nonempty. The Hausdorff continuous image of a segment is called {\it Peano space} or {\it Peano continuum}. Sierpinski proved that a connected compact metric space $X$ is a Peano continuum if and only if for every $\varepsilon>0$ the space $X$ can be covered by connected sets of the diameter $\le\varepsilon$. Therefore, naturally arises question about the investigation of disconnected metric spaces $X$ for which there is a continuous surjection between $X$ and $X^2$. Sierpinski characterized rational numbers as a metric countable space without isolated points. Hausdorff described irrational numbers as a metric, separable, completely metrizable, zero-dimensional and nowhere locally compact space. It follows, in particular, that the square $\mathbb Q^2$ is a continuous image of the set $\mathbb Q$ and the square of irrational numbers is a continuous image of the set of irrational numbers. Thus, it would be interesting to find a description of other disconnected subsets of the real line, except those that are homeomorphic to $\mathbb Q$ or $\mathbb R\setminus Q$. In this article we will focus on countable sets such that the set of isolated points of which may not be empty. The main result is the following (see Theorem 2): the square of a countable regular topological space $X$ is its continuous image if and only if $X$ is not compact.

Square-tiled tori

We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})$-action on square-tiled tori and we classify $\mathrm{SL}(2,\mathbf{Z})$-orbits using two numerical invariants that can be easily computed. We deduce the exact size of every $\mathrm{SL}(2,\mathbf{Z})$-orbit. In particular, this answers a question by M. Bolognesi on the number of cyclic covers of the torus, which corresponds to particular $\mathrm{SL}(2,\mathbf{Z})$-orbits of square-tiled tori. We also give the asymptotic behavior of the number of cyclic square-tiled tori.

Directed random geometric graphs: structural and spectral properties

In this work we analyze structural and spectral properties of a model of directed random geometric graphs: given n vertices uniformly and independently distributed on the unit square, a directed edge is set between two vertices if their distance is smaller than the connection radius , which is randomly drawn from a Pareto distribution. This Pareto distribution is characterized by the power-law decay α and the lower bound of its support ; thus the graphs depend on three parameters . By increasing , for fixed , the model transits from isolated vertices () to complete graphs (). We first propose a phenomenological expression for the average degree which works well for α > 3, when k is a self-averaging quantity. Then we numerically demonstrate that , for all α, where is the number of nonisolated vertices of G. Finally, we explore the spectral properties of by the use of adjacency matrices represented by diluted random matrix ensembles; a non-Hermitian and a Hermitian one. We find that is a good scaling parameter of spectral and eigenvector properties of G mainly for large α.

Complexity and Approximation for Discriminating and Identifying Code Problems in Geometric Setups

We study geometric variations of the discriminating code problem. In the discrete version of the problem, a finite set of points P and a finite set of objects S are given in \(\mathbb {R}^d\). The objective is to choose a subset \(S^* \subseteq S\) of minimum cardinality such that for each point \(p_i \in P\), the subset \(S_i^* \subseteq S^*\) covering \(p_i\) satisfies \(S_i^*\ne \emptyset \), and each pair \(p_i,p_j \in P\), \(i \ne j\), we have \(S_i^* \ne S_j^*\). In the continuous version of the problem, the solution set \(S^*\) can be chosen freely among a (potentially infinite) class of allowed geometric objects. In the 1-dimensional case (\(d=1\)), the points in P are placed on a horizontal line L, and the objects in S are finite-length line segments aligned with L (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. This is also in contrast with most geometric covering problems, which are usually polynomial-time solvable in one dimension. Still for the 1-dimensional discrete version, we design a polynomial-time 2-approximation algorithm. We also design a PTAS for both discrete and continuous versions in one dimension, for the restriction where the intervals are all required to have the same length. We then study the 2-dimensional case (\(d=2\)) for axis-parallel unit square objects. We show that both continuous and discrete versions are NP-complete, and design polynomial-time approximation algorithms that produce \((16\cdot OPT+1)\)-approximate and \((64\cdot OPT+1)\)-approximate solutions respectively, using rounding of suitably defined integer linear programming problems. Finally, we apply our techniques to a related variant of the discrete problem, where instead of points and geometric objects we just have a set S of objects. The goal is to select a small subset \(S^*\) of objects so that all objects of S are discriminated by their intersection with the objects of \(S^*\). This problem can be viewed as a graph problem by stating it in terms of the vertices of the geometric intersection graph of S. Under this graph-theoretical form, it is known as the identifying code problem. We show that the identifying code problem for axis-parallel unit square intersection graphs (in \(d=2\)) can be solved in the same manner as for the discrete version of the discriminating code problem for unit square objects described above, and all our positive approximation results still hold in this setting.

The area of empty axis-parallel boxes amidst 2-dimensional lattice points

The dispersion of a point set in the unit square is the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices in the plane, that is, the supremum of the area of the empty axis-parallel boxes amidst the lattice points. We introduce a framework with which to study this based on the continued fractions expansions of the lattice generators. We give necessary and sufficient conditions under which a lattice has finite dispersion. We obtain an exact formula for the dispersion of the lattices associated to subrings of the ring of integers of quadratic fields. We have tight bounds for the dispersion of a lattice based on the largest continued fraction coefficient of the generators, accurate to within one half. We provide an equivalent formulation of Zaremba's conjecture. Using this framework we are able to give very short proofs of previous results.

A bivariate extension of the Omega distribution for two-dimensional proportional data

Abstract When data generating mechanism generates two correlated data sets both defined on the unit interval, a bivariate probabilistic distribution defined on the unit square is needed for modelling the data. For this purpose, we give a Marshall-Olkin type bivariate extension of an omega distribution in this paper. This is in fact a bivariate unit-exponentiated-half-logistic distribution. We study its mathematical properties in detail. The distribution contains neither an exponential term nor any special function which complicates the computations. Maximum likelihood estimation method and its large sample inference are considered for model parameters. Alternatively, we also propose an expectation- maximization algorithm to compute the estimates. To see the performances of the proposed estimators and validate the theoretical results obtained for estimation, we present the results of a simulation study. Data fitting demonstrations show its applicability in modelling random proportions.

Analysis and application of Aczel-Alsina aggregation operators based on bipolar complex fuzzy information in multiple attribute decision making

Bipolar complex fuzzy information is a new idea that can easily depict awkward and vague information in real life problems. The main theory of the bipolar complex fuzzy set is the fusion of two different theories, called bipolar fuzzy set and complex fuzzy set. The bipolar complex fuzzy set contained two different grades, called positive membership grade which is considered a positive aspect of the problem, and negative membership grade which is considered a negative aspect of the problem, both contain in a unit square of complex plane. The major analysis of this theory is to diagnose the interrelationship among any number of attributes. For this, we defined various Aczel-Alsina operational laws and proved their influential results. Further, by using the Aczel-Alsina operational laws and bipolar complex fuzzy information, we diagnosed the theory of bipolar complex fuzzy Aczel-Alsina weighted averaging (BCFAAWA), bipolar complex fuzzy Aczel-Alsina ordered weighted averaging (BCFAAOWA), and bipolar complex fuzzy Aczel-Alsina hybrid averaging (BCFAAHA) operators. To modify the influence of the explored work, we discussed various important results and some properties (Idempotency, Monotonicity, and Boundedness) of the proposed work. Additionally, we computed a multi-attribute decision-making rule under the presence of the initiated approaches and tried to justify it with the help of some examples. Finally, to evaluate the influence and dominancy of the diagnosed work, we compared our deliberated work with various prevailing approaches and also described their geometrical representations.

Analysis of the Threshold for Energy Consumption in Displacement of Random Sensors.

The fundamental problem of energy-efficient reallocation of mobile random sensors to provide full coverage without interference is addressed in this paper. We consider n mobile sensors with the identical sensing range placed randomly on the unit interval and on the unit square. The sensors move from their initial random positions to the final locations so that: (a) every point on the unit interval or on the unit square is within the range of a sensor; (b) each pair of sensors is at a Euclidean distance greater than or equal to s; (c) the energy consumption for the movement of the sensors to the final positions is minimized. As a cost measure for the energy in the movement of sensors, we consider a-total movement defined as the sum ∑i=1ndia, for some constant a>0, provided that the i-th sensor is displaced the distance di. The main contribution is summarized as follows: (1) if the sensors are placed on the unit interval, we explain the sharp increase around the sensing radius equal to 12n and the interference distance equal to 1n for the expected minimal a-total displacement; (2) if the sensors are placed on the unit square, we explain the sharp increase around the square sensing radius equal to 12n and the interference distance equal to 1n for the expected minimal a-total displacement. We designed and analysed three algorithms. The probabilistic analysis of our protocols is based on a novel mathematical theory of the Beta distribution.

The Sierpinski Carpet as a Final Coalgebra

We advance the program of connections between final coalgebras as sources of circularity in mathematics and fractal sets of real numbers. In particular, we are interested in the Sierpinski carpet, taking it as a fractal subset of the unit square. We construct a category of square sets and an endofunctor on it which corresponds to the operation of gluing copies of a square set along segments. We show that the initial algebra and final coalgebra exist for our functor, and that the final coalgebra is bi-Lipschitz equivalent to the Sierpinski carpet. Along the way, we make connections to topics such as the iterative construction of initial algebras as colimits, corecursive algebras, and the classic treatment of fractal sets due to Hutchinson.

Designing Visualisations for Bayesian Problems According to Multimedia Principles

Questions involving Bayesian Reasoning often arise in events of everyday life, such as assessing the results of a breathalyser test or a medical diagnostic test. Bayesian Reasoning is perceived to be difficult, but visualisations are known to support it. However, prior research on visualisations for Bayesian Reasoning has only rarely addressed the issue on how to design such visualisations in the most effective way according to research on multimedia learning. In this article, we present a concise overview on subject-didactical considerations, together with the most fundamental research of both Bayesian Reasoning and multimedia learning. Building on these aspects, we provide a step-by-step development of the design of visualisations which support Bayesian problems, particularly for so-called double-trees and unit squares.