Exact Algorithms for Terrain Guarding

Given a 1.5-dimensional terrain T, also known as an x-monotone polygonal chain, the Terrain Guarding problem seeks a set of points of minimum size on T that guards all of the points on T. Here, we say that a point p guards a point q if no point of the line segment pq is strictly below T. The Terrain Guarding problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm [SODA 2005]. However, only in 2010 King and Krohn [SODA 2010] finally showed that Terrain Guarding is NP-hard. In spite of the remarkable developments in approximation algorithms for Terrain Guarding, next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this paper, we answer the first question affirmatively by developing an n^O(sqrt{k})-time algorithm for both Discrete Terrain Guarding and Continuous Terrain Guarding. We also make non-trivial progress with respect to the second question: we show that Discrete Orthogonal Terrain Guarding, a well-studied special case of Terrain Guarding, is fixed-parameter tractable

A finite dominating set of cardinality O(k) and a witness set of cardinality O(n) for 1.5D terrain guarding problem

1.5 dimensional (1.5D) terrain is characterized by a piecewise linear curve. Locating minimum number of guards on the terrain (T) to cover/guard the whole terrain is known as 1.5D terrain guarding problem. Approximation algorithms and a polynomial-time approximation scheme have been presented for the problem. The problem has been shown to be NP-Hard. In the problem, the set of possible guard locations and the set of points to be guarded are uncountable. To solve the problem to optimality, a finite dominating set (FDS) of size O (n2) and a witness set of size O (n3) have been presented, where n is the number of vertices on T. We show that there exists an even smaller FDS of cardinality O (k) and a witness set of cardinality O(n), where k is the number of convex points. Convex points are vertices with the additional property that between any two convex points the piecewise linear curve representing the terrain is convex. Since it is always true that k≤ n for n≥ 2 and since it is possible to construct terrains such that n= 2 k, the existence of an FDS with cardinality O(k) and a witness set of cardinality of O (n) leads to the reduction of decision variables and constraints respectively in the zero-one integer programming formulation of the problem. © 2017, Springer Science+Business Media New York

The Continuous 1.5{D} Terrain Guarding Problem: {D}iscretization, Optimal Solutions, and {PTAS}

In the NP-hard continuous 1.5D Terrain Guarding Problem (TGP) we are given an x-monotone chain of line segments in the plain (the terrain $T$), and ask for the minimum number of guards (located anywhere on $T$) required to guard all of $T$. We construct guard candidate and witness sets $G, W \subset T$ of polynomial size, such that any feasible (optimal) guard cover $G' \subseteq G$ for $W$ is also feasible (optimal) for the continuous TGP. This discretization allows us to: (1) settle NP-completeness for the continuous TGP; (2) provide a Polynomial Time Approximation Scheme (PTAS) for the continuous TGP using the existing PTAS for the discrete TGP by Gibson et al.; (3) formulate the continuous TGP as an Integer Linear Program (IP). Furthermore, we propose several filtering techniques reducing the size of our discretization, allowing us to devise an efficient IP-based algorithm that reliably provides optimal guard placements for terrains with up to 1000000 vertices within minutes on a standard desktop computer

Effect of Boundary Approximation on Visibility

The problem of simplifying a complex shape with simpler ones is an important research area in computer science and engineering. In this thesis, we investigate the effect on the visibility properties of polygons when their boundaries are approximated to make them simpler. We present two algorithms for approximating a restricted class of polygons called 1.5 D terrain. We also present experimental investigations on the performance of reviewed and proposed approximation algorithms

On guarding real terrains: the terrain guarding and the blocking path problems

Locating a minimum number of guards on a terrain such that every point on the terrain is guarded by at least one of the guards is known as the Terrain Guarding Problem (TGP). In this paper, a realistic example of the terrain guarding problem is studied, involving the surveillance of a rugged geographical terrain by means of thermal cameras. A number of issues related to TGP are addressed with integer-programming models proposed to solve the problem. Also, a sensitivity analysis is carried out in which five fictitious terrains are created to see the effect of the resolution of the terrain, and of terrain characteristics, on coverage optimization and the required number of guards. Finally, a new problem, which is called the Blocking Path Problem (BPP), is introduced. BPP is about guarding a path on the terrain with a minimum number of guards such that the path blocks all possible infiltration routes. A discussion is provided about the relation of BPP to the Network Interdiction Problem (NIP), which has been studied extensively by the operations research community, and to the k-Barrier Coverage Problem, which has been studied under the Sensor Deployment Problem. BPP is solved via an integer-programming formulation based on a network paradigm

Resolução do problema da galeria de arte : um método prático e robusto para o posicionamento ótimo de guardas-ponto

Orientadores: Cid Carvalho de Souza, Pedro Jussieu de RezendeDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Nesta dissertação, apresentamos nossa pesquisa sobre o Problema da Galeria de Arte (AGP), um dos problemas mais estudados em Geometria Computacional. O AGP, que é um problema NP-difícil, consiste em encontrar o número mínimo de guardas suficiente para garantir a cobertura visual de uma galeria de arte representada por um polígono. Na versão do problema tratada neste trabalho, usualmente chamada de Problema da Galeria de Arte com Guardas-Ponto, os guardas podem ser posicionados em qualquer lugar do polígono e o objetivo é cobrir toda a região, que pode ou não conter buracos. Nós estudamos como aplicar conceitos e algoritmos de Geometria Computacional, bem como Técnicas de Programação Inteira, com a finalidade de resolver o AGP de forma exata. Este trabalho culminou na criação de um novo algoritmo para o AGP, cuja ideia é gerar, de forma iterativa, limitantes superiores e inferiores para o problema através da resolução de versões discretizadas do AGP, que são reduzidas a instâncias do Problema de Cobertura de Conjuntos. O algoritmo foi implementado e testado em mais de 2800 instâncias, de diferentes tamanhos e classes. A técnica foi capaz de resolver, em minutos, mais de 90% de todas as instâncias consideradas, incluindo polígonos com milhares de vértices, e ampliou em muito o conjunto de casos para os quais são conhecidas soluções exatas. Até onde sabemos, apesar do extensivo estudo do AGP nas últimas quatro décadas, nenhum outro algoritmo demonstrou a capacidade de resolver o AGP de forma tão eficaz como a técnica aqui descritaAbstract: In this dissertation, we present our research on the Art Gallery Problem (AGP), one of the most investigated problems in Computational Geometry. The AGP, which is a known NP-hard problem, consists in finding the minimum number of guards sufficient to ensure the visibility coverage of an art gallery represented as a polygon. In the version of the problem treated in this work, usually called Art Gallery Problem with Point Guards, the guards can be placed anywhere in the polygon and the objective is to cover the whole region, which may or not have holes. We studied how to apply Computational Geometry concepts and algorithms as well as Integer Programming techniques in order to solve the AGP to optimality. This work culminated in the creation of a new algorithm for the AGP, whose idea is to iteratively generate upper and lower bounds for the problem through the resolution of discretized versions of the AGP, which are reduced to instances of the Set Cover Problem. The algorithm was implemented and tested on more than 2800 instances of different sizes and classes of polygons. The technique was able to solve in minutes more than 90% of all instances considered, including polygons with thousands of vertices, greatly increasing the set of instances for which exact solutions are known. To the best of our knowledge, in spite of the extensive study of the AGP in the last four decades, no other algorithm has shown the ability to solve the AGP as effectively as the one described hereMestradoCiência da ComputaçãoMestre em Ciência da Computaçã

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