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On Finding a Guard that Sees Most and a Shop that Sells Most
 In Proc. 15th ACMSIAM Sympos. Discrete Algorithms
, 2003
"... We present a nearquadratic time algorithm that computes a point inside a simple polygon P having approximately the largest visibility polygon inside P , and nearlinear time algorithm for nding the point that will have approximately the largest Voronoi region when added to an npoint set. We a ..."
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Cited by 28 (1 self)
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We present a nearquadratic time algorithm that computes a point inside a simple polygon P having approximately the largest visibility polygon inside P , and nearlinear time algorithm for nding the point that will have approximately the largest Voronoi region when added to an npoint set. We apply the same technique to nd the translation that approximately maximizes the area of intersection of two polygonal regions in nearquadratic time.
Computing the Maximum Overlap of Two Convex Polygons Under Translations
 THEORY OF COMPUTING SYSTEMS
, 1996
"... Let P be a convex polygon in the plane with n vertices and let Q be a convex polygon with m vertices. We prove that the maximum number of combinatorially distinct place ments of Q with respect to P under translations is O(n 2 + m + rain(rim + nm)), and we give an example showing that this bound ..."
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Cited by 25 (7 self)
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Let P be a convex polygon in the plane with n vertices and let Q be a convex polygon with m vertices. We prove that the maximum number of combinatorially distinct place ments of Q with respect to P under translations is O(n 2 + m + rain(rim + nm)), and we give an example showing that this bound is tight in the worst case. Second, we present an O((n + m) log(n + m)) algorithm for determining a translation of Q that maximizes the area of overlap of P and Q.
Exact and Distributed Algorithms for Collaborative Camera Control
 In The Workshop on Algorithmic Foundations of Robotics
, 2002
"... We propose the ShareCam Problem: controlling a single robotic pan, tilt, zoom camera based on simultaneous frame requests from n online users. To solve it, we propose a new piecewise linear metric, Intersection Over Maximum (IOM), for the degree of satisfaction for each users. To maximize overall sa ..."
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Cited by 12 (11 self)
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We propose the ShareCam Problem: controlling a single robotic pan, tilt, zoom camera based on simultaneous frame requests from n online users. To solve it, we propose a new piecewise linear metric, Intersection Over Maximum (IOM), for the degree of satisfaction for each users. To maximize overall satisfaction, we present several algorithms. For a discrete set of m distinct zoom levels, we give an exact algorithm that runs in O(n m) time. The algorithm can be distributed to run in O(nm) time at each client and in O(n log n + mn) time at the server.
Geometric optimization and sums of algebraic functions
, 2009
"... We present a new optimization technique that yields the first FPTAS for several geometric problems. These problems reduce to optimizing a sum of nonnegative, constant descriptioncomplexity algebraic functions. We first give an FPTAS for optimizing such a sum of algebraic functions, and then we appl ..."
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Cited by 4 (0 self)
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We present a new optimization technique that yields the first FPTAS for several geometric problems. These problems reduce to optimizing a sum of nonnegative, constant descriptioncomplexity algebraic functions. We first give an FPTAS for optimizing such a sum of algebraic functions, and then we apply it to several geometric optimization problems. We obtain the first FPTAS for two fundamental geometric shape matching problems in fixed dimension: maximizing the volume of overlap of two polyhedra under rigid motions, and minimizing their symmetric difference. We obtain the first FPTAS for other problems in fixed dimension, such as computing an optimal ray in a weighted subdivision, finding the largest axially symmetric subset of a polyhedron, and computing minimumarea hulls. 1
Probabilistic Matching of Planar Regions ∗
, 902
"... We analyze a probabilistic algorithm for matching shapes modeled by planar regions under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is close to maximal. I ..."
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Cited by 2 (2 self)
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We analyze a probabilistic algorithm for matching shapes modeled by planar regions under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is close to maximal. In the case of polygons, we give a time bound that does not depend significantly on the number of vertices. 1
Aligning two convex figures to minimize area or perimeter ∗
, 2009
"... Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ϕP of P that minimizes the convex hull of ϕP ∪ Q. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ϕP and Q to interse ..."
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Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ϕP of P that minimizes the convex hull of ϕP ∪ Q. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ϕP and Q to intersect or we restrict their interiors to remain disjoint; and we either allow reorienting P or require its orientation to be fixed. In the case without reorientations, we achieve exact nearlinear time algorithms for all versions of the problem. In the case with reorientations, we compute a (1 + ε)approximation in time O(ε −1/2 log n + ε −3/2 log a (1/ε)) if the two sets are convex polygons with n vertices in total, where a ∈ {0, 1, 2} depending on the version of the problem. 1
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"... was preceded by a oneday workshop entitled “CGAL Innovations and Applications: Robust Geometric ..."
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was preceded by a oneday workshop entitled “CGAL Innovations and Applications: Robust Geometric
Probabilistic Matching of polygons ∗
"... We analyze a probabilistic algorithm for matching plane compact sets with sufficiently nice boundaries under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is ..."
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We analyze a probabilistic algorithm for matching plane compact sets with sufficiently nice boundaries under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is close to maximal. We give a time bound that does not depend on the number of vertices in the case of polygons. 1
Matching solid shapes in arbitrary dimension via random sampling ∗
, 2012
"... We give simple probabilistic algorithms that approximately maximize the volume of overlap of two solid, i.e. fulldimensional, shapes under translations and rigid motions. The shapes are subsets of Rd where d ≥ 2. The algorithms approximate with respect to an prespecified additive error and succeed ..."
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We give simple probabilistic algorithms that approximately maximize the volume of overlap of two solid, i.e. fulldimensional, shapes under translations and rigid motions. The shapes are subsets of Rd where d ≥ 2. The algorithms approximate with respect to an prespecified additive error and succeed with high probability. Apart from measurability assumptions, we only require that points from the shapes can be generated uniformly at random. An important example are shapes given as finite unions of simplices that have pairwise disjoint interiors. 1