On k-Convex Polygons

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{$2$-convex} objects considered.Comment: 23 pages, 19 figure

An upper bound on the k-modem illumination problem

A variation on the classical polygon illumination problem was introduced in [Aichholzer et. al. EuroCG'09]. In this variant light sources are replaced by wireless devices called k-modems, which can penetrate a fixed number k, of "walls". A point in the interior of a polygon is "illuminated" by a k-modem if the line segment joining them intersects at most k edges of the polygon. It is easy to construct polygons of n vertices where the number of k-modems required to illuminate all interior points is Omega(n/k). However, no non-trivial upper bound is known. In this paper we prove that the number of k-modems required to illuminate any polygon of n vertices is at most O(n/k). For the cases of illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we give a tighter bound of 6n/k + 1. Moreover, we present an O(n log n) time algorithm to achieve this bound.Comment: 9 pages, 4 figure

Geometric optimization on visibility problems: metaheuristic and exact solutions

Doutoramento em MatemáticaOs problemas de visibilidade têm diversas aplicações a situações reais. Entre os mais conhecidos, e exaustivamente estudados, estão os que envolvem os conceitos de vigilância e ocultação em estruturas geométricas (problemas de vigilância e ocultação). Neste trabalho são estudados problemas de visibilidade em estruturas geométricas conhecidas como polígonos, uma vez que estes podem representar, de forma apropriada, muitos dos objectos reais e são de fácil manipulação computacional. O objectivo dos problemas de vigilância é a determinação do número mínimo de posições para a colocação de dispositivos num dado polígono, de modo a que estes dispositivos consigam “ver” a totalidade do polígono. Por outro lado, o objectivo dos problemas de ocultação é a determinação do número máximo de posições num dado polígono, de modo a que quaisquer duas posições não se consigam “ver”. Infelizmente, a maior parte dos problemas de visibilidade em polígonos são NP-difíceis, o que dá origem a duas linhas de investigação: o desenvolvimento de algoritmos que estabelecem soluções aproximadas e a determinação de soluções exactas para classes especiais de polígonos. Atendendo a estas duas linhas de investigação, o trabalho é dividido em duas partes. Na primeira parte são propostos algoritmos aproximados, baseados essencialmente em metaheurísticas e metaheurísticas híbridas, para resolver alguns problemas de visibilidade, tanto em polígonos arbitrários como ortogonais. Os problemas estudados são os seguintes: “Maximum Hidden Vertex Set problem”, “Minimum Vertex Guard Set problem”, “Minimum Vertex Floodlight Set problem” e “Minimum Vertex k-Modem Set problem”. São também desenvolvidos métodos que permitem determinar a razão de aproximação dos algoritmos propostos. Para cada problema são implementados os algoritmos apresentados e é realizado um estudo estatístico para estabelecer qual o algoritmo que obtém as melhores soluções num tempo razoável. Este estudo permite concluir que as metaheurísticas híbridas são, em geral, as melhores estratégias para resolver os problemas de visibilidade estudados. Na segunda parte desta dissertação são abordados os problemas “Minimum Vertex Guard Set”, “Maximum Hidden Set” e “Maximum Hidden Vertex Set”, onde são identificadas e estudadas algumas classes de polígonos para as quais são determinadas soluções exactas e/ou limites combinatórios.Visibility problems have several applications to real-life problems. Among the most distinguished and exhaustively studied visibility problems are the ones involving concepts of guarding and hiding on geometrical structures (guarding and hiding problems). This work deals with visibility problems on geometrical structures known as polygons, since polygons are appropriate representations of many real-world objects and are easily handled by computers. The objective of the guarding problems studied in this thesis is to find a minimum number of device positions on a given polygon such that these devices collectively ''see'' the whole polygon. On the other hand, the goal of the hiding problems is to find a maximum number of positions on a given polygon such that no two of these positions can “see" each other. Unfortunately, most of the visibility problems on polygons are NP-hard, which opens two lines of investigation: the development of algorithms that establish approximate solutions and the determination of exact solutions on special classes of polygons. Accordingly, this work is divided in two parts where these two lines of investigation are considered. The first part of this thesis proposes approximation algorithms, mainly based on metaheuristics and hybrid metaheuristics, to tackle some visibility problems on arbitrary and orthogonal polygons. The addressed problems are the Maximum Hidden Vertex Set problem, the Minimum Vertex Guard Set problem, the Minimum Vertex Floodlight Set problem and the Minimum Vertex k-Modem Set problem. Methods that allow the determination of the performance ratio of the developed algorithms are also proposed. For each problem, the proposed algorithms are implemented and a statistical study is performed to determine which of the developed methods obtains the best solution in a reasonable amount of time. This study allows to conclude that, in general, the hybrid metaheuristics are the best approach to solve the studied visibility problems. The second part of this dissertation addresses the Minimum Vertex Guard Set problem, the Maximum Hidden Set problem and the Maximum Hidden Vertex Set problem, where some classes of polygons are identified and studied and for which are determined exact solutions and/or combinatorial bounds

Conflict-free Chromatic Art Gallery Coverage

We consider a chromatic variant of the art gallery problem, where each guard is assigned one of k distinct colors. A placement of such colored guards is conflict-free if each point of the polygon is seen by some guard whose color appears exactly once among the guards visible to that point. What is the smallest number k(n) of colors that ensure a conflict-free covering of all n-vertex polygons? We call this the conflict-free chromatic art gallery problem. The problem is motivated by applications in distributed robotics and wireless sensor networks where colors indicate the wireless frequencies assigned to a set of covering "landmarks" in the environment so that a mobile robot can always communicate with at least one landmark in its line-of-sight range without interference. Our main result shows that k(n) is O(log n) for orthogonal and for monotone polygons, and O(log^2 n) for arbitrary simple polygons. By contrast, if all guards visible from each point must have distinct colors, then k(n)is Omega(n) for arbitrary simple polygons and Omega(sqrt(n)) for orthogonal polygons, as shown by Erickson and LaValle [Proc. of RSS 2011]

Coverage with k-transmitters in the presence of obstacles

For a fixed integer k ≥ 0, a k-transmitter is an omnidirectional wireless transmitter with an infinite broadcast range that is able to penetrate up to k “walls”, represented as line segments in the plane. We develop lower and upper bounds for the number of k-transmitters that are necessary and sufficient to cover a given collection of line segments, polygonal chains and polygons.Postprint (published version

Conflict-Free Chromatic Art Gallery Coverage

We consider a chromatic variant of the art gallery problem, where each guard is assigned one of k distinct colors. A placement of such colored guards is conflict-free if each point of the polygon is seen by some guard whose color appears exactly once among the guards visible to that point. What is the smallest number k(n) of colors that ensure a conflict-free covering of all n-vertex polygons? We call this the conflict-free chromatic art gallery problem. Our main result shows that k(n) is O(logn) for orthogonal and for monotone polygons, and O(log2 n) for arbitrary simple polygons. By contrast, if all guards visible from each point must have distinct colors, then k(n) is Ω(n) for arbitrary simple polygons, as shown by Erickson and LaValle (Robotics: Science and Systems, vol.VII, pp.81-88, 2012). The problem is motivated by applications in distributed robotics and wireless sensor networks but is also of interest from a theoretical point of view

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version