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Sampling Based SensorNetwork Deployment
"... In this paper, we consider the problem of placing networked sensors in a way that guarantees coverage and connectivity. We focus on sampling based deployment and present algorithms that guarantee coverage and connectivity with a small number of sensors. We consider two different scenarios based on t ..."
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In this paper, we consider the problem of placing networked sensors in a way that guarantees coverage and connectivity. We focus on sampling based deployment and present algorithms that guarantee coverage and connectivity with a small number of sensors. We consider two different scenarios based on the flexibility of deployment. If deployment has to be accomplished in one step, like airborne deployment, then the main question becomes how many sensors are needed. If deployment can be implemented in multiple steps, then awareness of coverage and connectivity can be updated. For this case, we present incremental deployment algorithms which consider the current placement to adjust the sampling domain. The algorithms are simple, easy to implement, and require a small number of sensors. We believe the concepts and algorithms presented in this paper will provide a unifying framework for existing and future deployment algorithms which consider many practical issues not considered in the present work.
VCdimension of Exterior Visibility
 IEEE Trans. Pattern Analysis and Machine Intelligence
, 2004
"... In this paper, we study the VapnikChervonenkis (VC)dimension of set systems arising in 2D polygonal and 3D polyhedral configurations where a subset consists of all points visible from one camera. In the past, it has been shown that the VCdimension of planar visibility systems is bounded by 23 if t ..."
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Cited by 13 (1 self)
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In this paper, we study the VapnikChervonenkis (VC)dimension of set systems arising in 2D polygonal and 3D polyhedral configurations where a subset consists of all points visible from one camera. In the past, it has been shown that the VCdimension of planar visibility systems is bounded by 23 if the cameras are allowed to be anywhere inside a polygon without holes [1]. Here, we consider the case of exterior visibility, where the cameras lie on a constrained area outside the polygon and have to observe the entire boundary. We present results for the cases of cameras lying on a circle containing the polygon (VCdimension=2) or lying outside the convex hull of a polygon (VCdimension= 5). The main result of this paper concerns the 3D case: we prove that the VCdimension is unbounded if the cameras lie on a sphere containing the polyhedron, hence the term exterior visibility.
Network abstract linear programming with application to minimumtime formation control
 in IEEE Conference on Decision and Control
, 2007
"... Abstract — We identify a novel class of distributed optimization linear programming. For such problems we propose distributed algorithms for networks with various connectivity and/or memory constraints. Finally, we show how various minimumtime formation control problems can be tackled through appro ..."
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Abstract — We identify a novel class of distributed optimization linear programming. For such problems we propose distributed algorithms for networks with various connectivity and/or memory constraints. Finally, we show how various minimumtime formation control problems can be tackled through appropriate geometric examples of abstract linear programs. I.
VCDimension of Exterior Visibility of Polyhedra
, 2001
"... In this paper, we address the problem of finding the minimal number of viewpoints outside a polyhedron in two or three dimensions such that every point on the exterior of the polyhedron is visible from at least one of the chosen viewpoints. This problem which we call the minimum fortress guard probl ..."
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In this paper, we address the problem of finding the minimal number of viewpoints outside a polyhedron in two or three dimensions such that every point on the exterior of the polyhedron is visible from at least one of the chosen viewpoints. This problem which we call the minimum fortress guard problem (MFGP) is the optimization version of a variant of the artgallery problem (sometimes called the fortress problem with point guards) and has practical importance in surveillance and imagebased rendering. Solutions in the vision and graphics literature are based on image quality constraints and are not concerned with the number of viewpoints needed. The corresponding question for art galleries (minimum number of viewpoints in the interior of a polygon to see the interior of the polygon) which we call the minimum artgallery guard problem (MAGP) has been shown to be NPcomplete. A simple reduction from this problem shows the NPcompleteness of MFGP. Instead of relying on heuristic searches, we address the approximability of the camera placement problem. It is well known (and easy to see) that this problem can be cast as a hitting set problem. While the approximability of generic instances of the hitting set problem is well understood, Bronnimann and Goodrich[3] presented improved approximation algorithms for the problem in the case that the input instances have bounded VapnikChervonenkis (VC) dimension.
Algorithms for Distributed and Mobile Sensing
, 2004
"... Sensing remote, complex and large environments is an important task that arises in diverse applications including planetary exploration, monitoring forest fires and the surveillance of large factories. Currently, automation of such sensing tasks in complex environments is achieved either by deployin ..."
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Cited by 2 (0 self)
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Sensing remote, complex and large environments is an important task that arises in diverse applications including planetary exploration, monitoring forest fires and the surveillance of large factories. Currently, automation of such sensing tasks in complex environments is achieved either by deploying many stationary sensors to the environment, or by mounting a sensor on a mobile device and using the device to sense the environment. The
Convex Approximation by Spherical Patches
 23RD EUROPEAN WORKSHOP ON COMPUTATIONAL GEOMETRY
, 2007
"... Given points in convex position in three dimensions, we want to find an approximating convex surface consisting of spherical patches, such that all points are within some specified tolerance bound ɛ of the approximating surface. We describe a greedy algorithm which constructs an approximating surfac ..."
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Given points in convex position in three dimensions, we want to find an approximating convex surface consisting of spherical patches, such that all points are within some specified tolerance bound ɛ of the approximating surface. We describe a greedy algorithm which constructs an approximating surface whose spherical patches are associated to the faces of an inscribed polytope. We show that deciding whether an approximation with not more than a given number of spherical patches exists is NPhard.
How to cover a point set with a Vshape of minimum width
 Proc. Algorithms Data Stuctures Symp. (WADS’11
, 2011
"... Abstract. A balanced Vshape is a polygonal region in the plane contained in the union of two crossing equalwidth strips. It is delimited by two pairs of parallel rays that emanate from two points x, y, are contained in the strip boundaries, and are mirrorsymmetric with respect to the line xy. Th ..."
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Abstract. A balanced Vshape is a polygonal region in the plane contained in the union of two crossing equalwidth strips. It is delimited by two pairs of parallel rays that emanate from two points x, y, are contained in the strip boundaries, and are mirrorsymmetric with respect to the line xy. The width of a balanced Vshape is the width of the strips. We first present an O(n2 logn) time algorithm to compute, given a set of n points P, a minimumwidth balanced Vshape covering P. We then describe a PTAS for computing a (1+ε)approximation of this Vshape in time O((n/ε) logn+ (n/ε3/2) log2(1/ε)). A much simpler constantfactor approximation algorithm is also described. 1
1 Distributed Abstract Optimization via Constraints Consensus: Theory and Applications
"... Abstract—Distributed abstract programs are a novel class of distributed optimization problems where (i) the number of variables is much smaller than the number of constraints and optimization programs are a generalization of linear programs that captures numerous geometric optimization problems. We ..."
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Abstract—Distributed abstract programs are a novel class of distributed optimization problems where (i) the number of variables is much smaller than the number of constraints and optimization programs are a generalization of linear programs that captures numerous geometric optimization problems. We propose novel constraints consensus algorithms for distributed abstract programs with guaranteed finitetime convergence to a programs and exchanging the solutions among neighboring processors. The proposed algorithms are appropriate for networks with weak timedependent connectivity requirements and tight memory constraints. We show how the constraints consensus algorithms may be applied to suitable target localization and formation control problems. Index Terms—Distributed optimization, linear programs, consensus algorithms, target localization, formation control. I.
Spherical approximation of convex shapes
"... Given points in convex position in three dimensions, we want to find an approximating convex surface consisting of spherical patches, such that all points are within some specified tolerance bound ε of the approximating surface. We describe a greedy algorithm which constructs an approximating surfa ..."
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Given points in convex position in three dimensions, we want to find an approximating convex surface consisting of spherical patches, such that all points are within some specified tolerance bound ε of the approximating surface. We describe a greedy algorithm which constructs an approximating surface whose spherical patches are associated to the faces of an inscribed polytope formed from a subset of the input points. We show that deciding whether an approximation with not more than a given number of spherical patches exists is NPhard by a reduction from planar 3SAT.