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33
Improved Approximation Algorithms for Geometric Set Cover
, 2005
"... Given a collection S of subsets of some set U, and M ⊂ U, the set cover problem is to find the smallest subcollection C ⊂ S such that M is a subset of the union of the sets in C. While the general problem is NPhard to solve, even approximately, here we consider some geometric special cases, where u ..."
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Cited by 76 (6 self)
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Given a collection S of subsets of some set U, and M ⊂ U, the set cover problem is to find the smallest subcollection C ⊂ S such that M is a subset of the union of the sets in C. While the general problem is NPhard to solve, even approximately, here we consider some geometric special cases, where usually U = ℜ d. Extending prior results[BG95], we show that approximation algorithms with provable performance exist, under a certain general condition: that for a random subset R ⊂ S and function f(), there is a decomposition of the complement U \ ∪Y∈RY into an expected f(R) regions, each region of a particular simple form. We show that under this condition, a cover of size O(f(C)) can be found. Our proof involves the generalization of shallow cuttings [Mat92] to more general geometric situations. We obtain constantfactor approximation algorithms for covering by unit cubes in ℜ³, for guarding a onedimensional terrain, and for covering by similarsized fat triangles in ℜ². We also obtain improved approximation guarantees for fat triangles, of arbitrary size, and for a class of fat objects.
Locating guards for visibility coverage of polygons
 in Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX
, 2007
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 17 (2 self)
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Terrain Guarding is NPHard
, 2009
"... A set G of points on a 1.5dimensional terrain, also known as an xmonotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in G. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. ..."
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Cited by 8 (0 self)
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A set G of points on a 1.5dimensional terrain, also known as an xmonotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in G. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem asks for a minimum guarding set for the given input terrain. Using a reduction from PLANAR 3SAT we prove that the decision version of this problem is NPhard. This solves a significant open problem and complements recent positive approximability results for the optimization problem. 1
Guarding a terrain by two watchtowers
 In Proc. 21st Annu. ACM Sympos. Computational Geometry
, 2005
"... Given a polyhedral terrain T with n vertices, the twowatchtower problem for T asks to find two vertical segments, called watchtowers, of smallest common height, whose bottom endpoints (bases) lie on T, and whose top endpoints guard T, in the sense that each point on T is visible from at least one o ..."
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Cited by 8 (1 self)
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Given a polyhedral terrain T with n vertices, the twowatchtower problem for T asks to find two vertical segments, called watchtowers, of smallest common height, whose bottom endpoints (bases) lie on T, and whose top endpoints guard T, in the sense that each point on T is visible from at least one of them. There are three versions of the problem, discrete, semidiscrete, and continuous, depending on whether two, one, or none of the two bases are restricted to be among the vertices of T, respectively. In this paper we present the following results for the twowatchtower problem in R 2 and R 3: (1) We show that the discrete twowatchtowers problem in R 2 can be solved in O(n 2 log 4 n) time, significantly improving previous solutions. The algorithm works, without increasing its asymptotic running time, for the semicontinuous version, where one of the towers is allowed to be placed anywhere on T. (2) We show that the continuous twowatchtower problem in R 2 can be solved in O(n 3 α(n)log 3 n) time, again significantly improving previous results. (3) Still in R 2, we show that the continuous version of the problem of guarding a finite set P ⊂ T of m points by two watchtowers of smallest common height can be solved in O(mnlog 4 n) time.
Computing Visibility on Terrains in External Memory
"... We describe a novel application of the distribution sweeping technique to computing visibility on terrains. Given an arbitrary viewpoint v, the basic problem we address is computing the visibility map or viewshed of v, which is the set of points in the terrain that are visible from v. We give the fi ..."
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Cited by 7 (1 self)
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We describe a novel application of the distribution sweeping technique to computing visibility on terrains. Given an arbitrary viewpoint v, the basic problem we address is computing the visibility map or viewshed of v, which is the set of points in the terrain that are visible from v. We give the first I/Oefficient algorithm to compute the viewshed of v on a grid terrain in external memory. Our algorithm is based on Van Kreveld’s O(n lg n) time algorithm for the same problem in internal memory. It uses O(sort(n)) I/Os, where sort(n) is the complexity of sorting n items of data in the I/Omodel. We present an implementation and experimental evaluation of the algorithm. Our implementation clearly outperforms the previous (inmemory) algorithms and can compute visibility for terrains of up to 4 GB in a few hours on a lowcost machine.
Improved Approximations for Guarding 1.5Dimensional Terrains
"... We present a 4approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the currently best approximation factor of 5 (see [14]). Unlike most of the previous techniques, our method is base ..."
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We present a 4approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the currently best approximation factor of 5 (see [14]). Unlike most of the previous techniques, our method is based on rounding the linear programming relaxation of the corresponding covering problem. Besides the simplicity of the analysis, which mainly relies on decomposing the constraint matrix of the LP into totally balanced matrices, our algorithm, unlike previous work, generalizes to the weighted and partial versions of the basic problem.
Optimizing Autonomous Pipeline Inspection
 IEEE Trans. on Robotics
, 2011
"... Abstract—This paper studies the optimal inspection of autonomous robots in a complex pipeline system. We solve a 3D regionguarding problem to suggest the necessary inspection spots. The proposed hierarchical integer linear programming optimization algorithm seeks the fewest spots necessary to co ..."
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Abstract—This paper studies the optimal inspection of autonomous robots in a complex pipeline system. We solve a 3D regionguarding problem to suggest the necessary inspection spots. The proposed hierarchical integer linear programming optimization algorithm seeks the fewest spots necessary to cover the entire given 3D region. Unlike most existing pipeline inspection systems that focus on designing mobility and control of the explore robots, this paper focuses on global planning of the thorough and automatic inspection of a complex environment. We demonstrate the efficacy of the computation framework using a simulated environment, where scanned pipelines and existing leaks, clogs, and deformation can be thoroughly detected by an autonomous prototype robot. Index Terms—Autonomous pipeline inspection, 3D region guarding. I.
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"... Organized byCOPYRIGHT The material presented at CONTROLO’2006 remains the copyright of the respective authors. By submitting it to the Conference, authors have given APCA permission to be included in the Proceedings in CDROM format. Proceedings published by APCA Associação Portuguesa de Controlo A ..."
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Cited by 3 (0 self)
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Organized byCOPYRIGHT The material presented at CONTROLO’2006 remains the copyright of the respective authors. By submitting it to the Conference, authors have given APCA permission to be included in the Proceedings in CDROM format. Proceedings published by APCA Associação Portuguesa de Controlo Automático (Portuguese