Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs

AbstractIn this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs.We transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that: (1) ⌊n+13⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊2n+15⌋ edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: (1) a diagonal 2-dominating set of size ⌊n+13⌋ in linear time and space, (2) an edge 2-dominating set of size ⌊2n+15⌋ in O(n2) time and O(n) space, and (3) an edge 2-dominating set of size ⌊3n7⌋ in O(n) time and space.Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: (1) a mobile guard set of size ⌊n+13⌋ in O(nlogn) time, (2) an edge guard set of size ⌊2n+15⌋ in O(n2) time, and (3) an edge guard set of size ⌊3n7⌋ in O(nlogn) time. All space requirements are linear. Finally, we show that ⌊n3⌋ mobile or ⌈n3⌉ edge guards are sometimes necessary.When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ⌈n+14⌉ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ⌈n+14⌉, can be computed in O(n) time and space

Geometric optimization and querying : exact & approximate

This thesis has two main parts. The first part deals with the stage illumination problem. Given a stage represented by a line segment L and a set of lightsources represented by a set of points S in the plane, assign powers to the lightsources such that every point on the stage receives a sufficient amount, e.g. one unit, of light while minimizing the overall power consumption. By assuming that the amount of light arriving from a fixed lightsource decreases rapidly with the distance from the lightsource, this becomes an interesting geometric optimization problem. We present different solutions, based on convex optimization, discretization and linear programming, as well as a purely combinatorial approximation algorithm. Some experimental results are also provided. In the second part of this thesis, we are concerned with two different geometric problems whose solutions are based on the construction of a data structure that would allow for efficient queries. The central idea of our data structures is the well-separated pair decomposition. The first problem we address is the k-hop restricted shortest path under the power-euclidean distance function. Given a set P of n points in the plane and the distance function jpqjd +Cp for some constant d > 1, nonnegative offset cost Cp and p;q 2 P, where jpqj denotes the Euclidean distance between p and q, we consider the problem of finding paths between any pair of points that minimize the lenght of the path and do not use more than some constant number k of hops. Known exact algorithms for this problem required W(nlogn) per query pair (p;q). We relax the exactness requirement and only require approximate (1+e) solutions which allows us to derive schemes which guarantee constant query time using linear space and O(nlogn) preprocessing time. The dependence on e is polynomial in 1=e. We also develop a tool that might be of independent interest: For any pair of points p;q 2 P report in constant time the cluster pair (A;B) representing (p;q) in a well-separated pair decomposition of P. The second problem in this part is so-called cone-restricted nearest neighbor. For a given point set in Euclidean space we consider the problem of finding (approximate) nearest neighbors of a query point but restricting only to points that lie within a fixed cone with apex at the query point. We investigate the structure of the Voronoi diagram induced by this notion of proximity and present approximate and exact data structures for answering cone-restricted nearest neighbor queries. In particular, we develop an approximate Voronoi diagram of size O((n=ed) log(1=e)) that can be used to answer cone-restricted nearest neighbor queries in O(log(n=e)) time.Diese Arbeit besteht aus zwei Teilen. Der erste Teil behandelt das Stage Illumination Problem. Hierbei möchte man eine Bühne, die durch ein Geradenstück repräsentiert ist, durch Lichtquellen, die durch Punkte in der Ebene repräsentiert sind, so beleuchten, dass jeder Punkt der Bühne genügend Licht erhält und dabei möglichst wenig Energie verbrauchen. Wenn man annimmt, dass die Lichtintensität stark mit der Entfernung zur Lichtquelle abnimmt, so stellt dies ein interesanntes geometrisches Optimierungsproblem dar. Wir geben verschiedene Lösungen an, die sowohl auf konvexer Optimierung, Diskretisierung und Linearer Programmierung basieren, als auch einen kombinatorischen Approximationsalgorithmus. Es werden auch experimentelle Resultate angegeben. Im zweiten Teil dieser Arbeit behandeln wir zwei verschiedene geometrische Probleme, deren Lösungen auf einer Datenstruktur basieren, die effiziente Anfragen beantworten kann. Die zentrale Idee unserer Datenstruktur ist die well-separated pair decomposition WSPD. Das erste Problem, das wir ansprechen ist das k-hop restricted shortest path under the power-euclidean distance function. Für n Punkte in der Ebene möchte man den kürzesten Pfad zwischen zwei beliebigen Punkten finden, der nicht mehr als k Kanten benötigt. Bekannte exakte Algorithmen für dieses Problem benötigen W(nlogn) Zeit pro Anfrage (p;q). Wir lockern die Exaktheitsforderung und verlangen nur eine (1+e)-Approximation. Dies erlaubt uns eine Methode zu entwickeln, die konstante Zeit pro Anfrage garaniert und nur linearen Platz benötigt bei einer Vorverarbeitungszeit von O(nlogn). Die Abhängigkeit von e ist polynomiell in 1=e. Außerdem entwickeln wir eine Methode, die davon unabhängig von Interesse ist. Für ein Punktepaar p;q 2 P bestimmen wir in konstanter Zeit das Cluster-paar (A;B), das (p;q) in einer WSPD von P bestimmt. Das zweite Problem in diesem Teil ist das sogenannte cone-restricted nearest neighbor problem. Für eine gegebene Menge von Punkten im Euklidischen Raum betrachten wir das Problem den nächsten Nachbarpunkt zu bestimmen, der in einem Kegel liegt, dessen Spitze ein beliebiger Anfragepunkt ist. Wir untersuchen das dazugehörige Voronoi- Diagramm und entwickeln effiziente Datenstrukturen sowohl für exakte als auch für approximative cone-restricted nearest neighbor-Anfragen. Im speziellen entwickeln wir ein approximatives Voronoi-Diagramm der Größe O((n=ed) log(1=e)), das dazu benutzt werden kann, Anfragen in der Zeit O(log(n=e)) zu beantworten

Geometric optimization and querying : exact & approximate

This thesis has two main parts. The first part deals with the stage illumination problem. Given a stage represented by a line segment L and a set of lightsources represented by a set of points S in the plane, assign powers to the lightsources such that every point on the stage receives a sufficient amount, e.g. one unit, of light while minimizing the overall power consumption. By assuming that the amount of light arriving from a fixed lightsource decreases rapidly with the distance from the lightsource, this becomes an interesting geometric optimization problem. We present different solutions, based on convex optimization, discretization and linear programming, as well as a purely combinatorial approximation algorithm. Some experimental results are also provided. In the second part of this thesis, we are concerned with two different geometric problems whose solutions are based on the construction of a data structure that would allow for efficient queries. The central idea of our data structures is the well-separated pair decomposition. The first problem we address is the k-hop restricted shortest path under the power-euclidean distance function. Given a set P of n points in the plane and the distance function jpqjd +Cp for some constant d > 1, nonnegative offset cost Cp and p;q 2 P, where jpqj denotes the Euclidean distance between p and q, we consider the problem of finding paths between any pair of points that minimize the lenght of the path and do not use more than some constant number k of hops. Known exact algorithms for this problem required W(nlogn) per query pair (p;q). We relax the exactness requirement and only require approximate (1+e) solutions which allows us to derive schemes which guarantee constant query time using linear space and O(nlogn) preprocessing time. The dependence on e is polynomial in 1=e. We also develop a tool that might be of independent interest: For any pair of points p;q 2 P report in constant time the cluster pair (A;B) representing (p;q) in a well-separated pair decomposition of P. The second problem in this part is so-called cone-restricted nearest neighbor. For a given point set in Euclidean space we consider the problem of finding (approximate) nearest neighbors of a query point but restricting only to points that lie within a fixed cone with apex at the query point. We investigate the structure of the Voronoi diagram induced by this notion of proximity and present approximate and exact data structures for answering cone-restricted nearest neighbor queries. In particular, we develop an approximate Voronoi diagram of size O((n=ed) log(1=e)) that can be used to answer cone-restricted nearest neighbor queries in O(log(n=e)) time.Diese Arbeit besteht aus zwei Teilen. Der erste Teil behandelt das Stage Illumination Problem. Hierbei möchte man eine Bühne, die durch ein Geradenstück repräsentiert ist, durch Lichtquellen, die durch Punkte in der Ebene repräsentiert sind, so beleuchten, dass jeder Punkt der Bühne genügend Licht erhält und dabei möglichst wenig Energie verbrauchen. Wenn man annimmt, dass die Lichtintensität stark mit der Entfernung zur Lichtquelle abnimmt, so stellt dies ein interesanntes geometrisches Optimierungsproblem dar. Wir geben verschiedene Lösungen an, die sowohl auf konvexer Optimierung, Diskretisierung und Linearer Programmierung basieren, als auch einen kombinatorischen Approximationsalgorithmus. Es werden auch experimentelle Resultate angegeben. Im zweiten Teil dieser Arbeit behandeln wir zwei verschiedene geometrische Probleme, deren Lösungen auf einer Datenstruktur basieren, die effiziente Anfragen beantworten kann. Die zentrale Idee unserer Datenstruktur ist die well-separated pair decomposition WSPD. Das erste Problem, das wir ansprechen ist das k-hop restricted shortest path under the power-euclidean distance function. Für n Punkte in der Ebene möchte man den kürzesten Pfad zwischen zwei beliebigen Punkten finden, der nicht mehr als k Kanten benötigt. Bekannte exakte Algorithmen für dieses Problem benötigen W(nlogn) Zeit pro Anfrage (p;q). Wir lockern die Exaktheitsforderung und verlangen nur eine (1+e)-Approximation. Dies erlaubt uns eine Methode zu entwickeln, die konstante Zeit pro Anfrage garaniert und nur linearen Platz benötigt bei einer Vorverarbeitungszeit von O(nlogn). Die Abhängigkeit von e ist polynomiell in 1=e. Außerdem entwickeln wir eine Methode, die davon unabhängig von Interesse ist. Für ein Punktepaar p;q 2 P bestimmen wir in konstanter Zeit das Cluster-paar (A;B), das (p;q) in einer WSPD von P bestimmt. Das zweite Problem in diesem Teil ist das sogenannte cone-restricted nearest neighbor problem. Für eine gegebene Menge von Punkten im Euklidischen Raum betrachten wir das Problem den nächsten Nachbarpunkt zu bestimmen, der in einem Kegel liegt, dessen Spitze ein beliebiger Anfragepunkt ist. Wir untersuchen das dazugehörige Voronoi- Diagramm und entwickeln effiziente Datenstrukturen sowohl für exakte als auch für approximative cone-restricted nearest neighbor-Anfragen. Im speziellen entwickeln wir ein approximatives Voronoi-Diagramm der Größe O((n=ed) log(1=e)), das dazu benutzt werden kann, Anfragen in der Zeit O(log(n=e)) zu beantworten

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

Limited range coverage problems

Doutoramento em MatemáticaTal como o título indica, esta tese estuda problemas de cobertura com alcance limitado. Dado um conjunto de antenas (ou qualquer outro dispositivo sem fios capaz de receber ou transmitir sinais), o objectivo deste trabalho é calcular o alcance mínimo das antenas de modo a que estas cubram completamente um caminho entre dois pontos numa região. Um caminho que apresente estas características é um itinerário seguro. A definição de cobertura é variável e depende da aplicação a que se destina. No caso de situações críticas como o controlo de fogos ou cenários militares, a definição de cobertura recorre à utilização de mais do que uma antena para aumentar a eficácia deste tipo de vigilância. No entanto, o alcance das antenas deverá ser minimizado de modo a manter a vigilância activa o maior tempo possível. Consequentemente, esta tese está centrada na resolução deste problema de optimização e na obtenção de uma solução particular para cada caso. Embora este problema de optimização tenha sido investigado como um problema de cobertura, é possível estabelecer um paralelismo entre problemas de cobertura e problemas de iluminação e vigilância, que são habitualmente designados como problemas da Galeria de Arte. Para converter um problema de cobertura num de iluminação basta considerar um conjunto de luzes em vez de um conjunto de antenas e submetê-lo a restrições idênticas. O principal tema do conjunto de problemas da Galeria de Arte abordado nesta tese é a 1-boa iluminação. Diz-se que um objecto está 1-bem iluminado por um conjunto de luzes se o invólucro convexo destas contém o objecto, tornando assim este conceito num tipo de iluminação de qualidade. O objectivo desta parte do trabalho é então minimizar o alcance das luzes de modo a manter uma iluminação de qualidade. São também apresentadas duas variantes da 1-boa iluminação: a iluminação ortogonal e a boa !-iluminação. Esta última tem aplicações em problemas de profundidade e visualização de dados, temas que são frequentemente abordados em estatística. A resolução destes problemas usando o diagrama de Voronoi Envolvente (uma variante do diagrama de Voronoi adaptada a problemas de boa iluminação) é também proposta nesta tese.As the title implies, this thesis studies limited range coverage problems. Given a set of antennas (or any wireless device able to send or receive some sort of signal), the objective of the discussion that follows is to calculate the antennas’ minimum range so that a path between two points within a region is covered by the antennas, a path known as a safe route. The definition of coverage is variable and depends on the applications. In some instances, for example, when monitoring is critical as in the case of fires or military, the definition of coverage necessarily involves the use of multiple antennas to increase the effectiveness of monitoring. However, it is also desirable to extend a network’s lifespan, normally achieved by minimising the antennas’ range. Therefore the focus of this thesis will be the resolution of this dual problem and an affective solution is offered for each case. Although this question has been researched as an issue of coverage, it is also possible to establish a relation between coverage and illumination and visibility, known as Art Gallery problems. To conceptualise coverage problems as Art Gallery problems, all that is needed is to consider a set of lights instead of a set of antennas, which are subject to a similar set of restrictions. The main focus of the Art Gallery problems addressed in this thesis is 1-good illumination. An object is 1-well illuminated if it is fully contained by the convex hull of a set of lights, making this a type of quality illumination. The objective of the discussion that follows is therefore to minimise the lights’ range whilst maintaining a quality illumination. Moreover, two variants of 1-good illumination are also presented: orthogonal good illumination and good ! -illumination. The latter being related to data depth problems and data visualisation that are frequently used in statistics. The resolution of these problems using the Embracing Voronoi diagram (a variant of Voronoi diagrams adapted to good illumination) is also discussed in this thesis

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