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104
Art gallery and illumination problems
 In Handbook on Computational Geometry, Elsevier Science Publishers, J.R. Sack and
, 2000
"... How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke pub ..."
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Cited by 86 (3 self)
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How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke published his book Art Gallery Theorems and Algorithms which has further fueled this area of research. The present book is being written almost 10 years since the publication of O’Rourke’s book, and the need for an uptodate manuscript on Art Gallery or Illumination Problems is evident. Some important open problems stated in O’Rourke’s book, such as... have been solved. New directions of research have since been investigated, including: watchman routes, floodlight illumination problems, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this book, we study these results and try to give a complete
VisibilityBased PursuitEvasion in a Polygonal Environment
 International Journal of Computational Geometry and Applications
, 1997
"... This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. This problem was first introduced by Suzuki and Yamashita. Our stud ..."
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Cited by 85 (25 self)
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This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. This problem was first introduced by Suzuki and Yamashita. Our study of this problem is motivated in part by robotics applications, such as surveillance with a mobile robot equipped with a camera that must find a moving target in a cluttered workspace. A few bounds are introduced, and a complete algorithm is presented for computing a successful motion strategy for a single pursuer. For simplyconnected free spaces, it is shown that the minimum number of pursuers required is \Theta(lg n). For multiplyconnected free spaces, the bound is \Theta( p h + lg n) pursuers for a polygon that has n edges and h holes. A set of problems that are solvable by a single pursuer and require a linear number of recontaminations is shown. The complete algorithm searches a f...
How To Learn An Unknown Environment I: The Rectilinear Case
 Journal of the ACM
, 1997
"... We consider the problem faced by a robot that must explore and learn an unknown room with obstacles in it. We seek algorithms that achieve a bounded ratio of the worstcase distance traversed in order to see all visible points of the environment (thus creating a map), divided by the optimum distance ..."
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Cited by 73 (0 self)
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We consider the problem faced by a robot that must explore and learn an unknown room with obstacles in it. We seek algorithms that achieve a bounded ratio of the worstcase distance traversed in order to see all visible points of the environment (thus creating a map), divided by the optimum distance needed to verify the map, if we had it in the beginning. The situation is complicated by the fact that the latter offline problem (the problem of optimally verifying a map) is NPhard. Although we show that there is no such "competitive" algorithm for general obstacle courses, we give a competitive algorithm for the case of a polygonal room with a bounded number of obstacles in it. We restrict ourselves to the rectilinear case, where each side of the obstacles and the room is parallel to one of the coordinates, and the robot must also move either parallel or perpendicular to the sides. (In a subsequent paper, we will discuss the extension to polygons of general shapes.) We also discuss t...
A VisibilityBased PursuitEvasion Problem
 SUBMITTED TO THE INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY AND APPLICATIONS
"... This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. A visibility region is associated witheach pursuer, and the goal is ..."
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Cited by 64 (1 self)
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This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. A visibility region is associated witheach pursuer, and the goal is to guarantee that the evader will ultimately lie in at least one visibility region. The study of this problem is motivated inpart by robotics applications, such as surveillance with a mobile robot equipped withacamera that must nd a moving target in a cluttered workspace. A few bounds are introduced, and a complete algorithm is presented for computing a successful motion strategy. For a simplyconnected free space, a logarithmic bound is established on the minimum of pursuers needed. Loose bounds for multiplyconnected free spaces are also given. A set of problems that are solvable by a single pursuer and require a linear number of recontaminations is shown. The complete algorithm searches a nite cell complex that is constructed onthebasis of critical information changes. This concept can be applied in principle to multiplepursuer problems, and the case of a single pursuer has been implemented. Several solution strategies are shown, most of which were computed in a few seconds on a standard workstation.
Visibilitybased pursuitevasion with limited field of view
 International Journal of Robotics Research
, 2004
"... We study a form of the pursuitevasion problem, in which one or more searchers must move through a given environment so as to guarantee detection of any and all evaders, which can move arbitrarily fast. Our goal is to develop techniques for coordinating teams of robots to execute this task in ap ..."
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Cited by 63 (2 self)
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We study a form of the pursuitevasion problem, in which one or more searchers must move through a given environment so as to guarantee detection of any and all evaders, which can move arbitrarily fast. Our goal is to develop techniques for coordinating teams of robots to execute this task in application domains such as clearing a building, for reasons of security or safety. To this end, we introduce a new class of searcher, the #searcher, which can be readily instantiated as a physical mobile robot. We present a detailed analysis of the pursuitevasion problem using #searchers. We show that computing the minimum number of #searchers required to search a given environment is NPhard, and present the first complete search algorithm for a single #searcher. We show how this algorithm can be extended to handle multiple searchers, and give examples of computed trajectories.
Realtime Combinatorial Tracking of a Target Moving Unpredictably among Obstacles
 In IEEE Int. Conf. Robot. & Autom
, 2002
"... Many applications require continuous monitoring of a moving target by a controllable vision system. Although the goal of tracking objects is not new, traditional techniques usually ignore the presence of obstacles and focus on imaging and target recognition issues. For a target moving among obstacle ..."
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Cited by 35 (3 self)
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Many applications require continuous monitoring of a moving target by a controllable vision system. Although the goal of tracking objects is not new, traditional techniques usually ignore the presence of obstacles and focus on imaging and target recognition issues. For a target moving among obstacles, the goal of tracking involves a complex motion problem: a controllable observer (e.g., a robot) must anticipate that the target may become occluded by an obstacle and move to prevent such an event from occurring. This paper describes a strategy for computing the motions of a mobile robot operating in a 2D workspace without prior knowledge of the target's intention or the distribution of obstacles in the scene. The proposed algorithm governs the motion of the observer based on current measurements of the target's position and the location of the local obstacles. The approach is combinatorial in the sense that the algorithm explicitly computes a description of the geometric arrangement between the target and the observer's visibility region produced by the local obstacles. The algorithm computes a continuous control law based on this description. The new tracking strategy has been implemented in a realtime robotic system.
The floodlight problem
 J. ASSOC. COMPUT. MACH
, 1993
"... Given three angles summing to 2, given n points in the plane and a tripartition k1 + k2 + k3 = n, we can tripartition the plane into three wedges of the given angles so that the ith wedge contains ki of the points. This new result on dissecting point sets is used to prove that lights of speci ed an ..."
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Cited by 27 (7 self)
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Given three angles summing to 2, given n points in the plane and a tripartition k1 + k2 + k3 = n, we can tripartition the plane into three wedges of the given angles so that the ith wedge contains ki of the points. This new result on dissecting point sets is used to prove that lights of speci ed angles not exceeding can be placed at n xed points in the plane to illuminate the entire plane if and only if the angles sum to at least 2pi. We give O(n log n) algorithms for both these problems.
Guarding Polyhedral Terrains
, 1992
"... We prove that b c vertex guards are always sufficient and sometimes necessary to guard the surface of an nvertex polyhedral terrain. We also show that b guards are sometimes necessary to guard the surface of an nvertex polyhedral terrain. ..."
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Cited by 26 (6 self)
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We prove that b c vertex guards are always sufficient and sometimes necessary to guard the surface of an nvertex polyhedral terrain. We also show that b guards are sometimes necessary to guard the surface of an nvertex polyhedral terrain.
Efficient Visibility Queries in Simple Polygons
"... We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n3 log n) preprocessing time and O(n3) space, we can, given a query point q inside or outside an n verte ..."
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Cited by 24 (2 self)
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We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n3 log n) preprocessing time and O(n3) space, we can, given a query point q inside or outside an n vertex polygon, recover the visibility polygon of q in O(log n + k) time, where k is the size of the visibility polygon, and recover the number of vertices visible from q in O(log n) time. The key notion
Distributed deployment of asynchronous guards in art galleries
 in American Control Conference, (Minneapolis, MN
, 2006
"... Abstract — This paper presents deployment algorithms for multiple mobile robots with lineofsight sensing and communication capabilities in a simple nonconvex polygonal environment. The objective of the proposed algorithms is to achieve full visibility of the environment. We solve the problem by co ..."
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Cited by 23 (9 self)
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Abstract — This paper presents deployment algorithms for multiple mobile robots with lineofsight sensing and communication capabilities in a simple nonconvex polygonal environment. The objective of the proposed algorithms is to achieve full visibility of the environment. We solve the problem by constructing a novel data structure called the vertexinduced tree and designing schemes to deploy over the nodes of this tree by means of distributed algorithms. The agents are assumed to have access to a local memory and their operation is partially asynchronous. I.