Results 1  10
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19
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 29 (3 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 29 (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 14 (12 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.
Maximizing the Overlap of Two Planar Convex Sets under Rigid Motions
, 2006
"... Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any ε> 0, we compute a rigid motion such that the area of overlap is at least 1−ε times the maximum possible overlap. Our algorith ..."
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Cited by 11 (5 self)
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Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any ε> 0, we compute a rigid motion such that the area of overlap is at least 1−ε times the maximum possible overlap. Our algorithm uses O(1/ε) extreme point and line intersection queries on P and Q, plus O((1/ε 2) log(1/ε)) running time. If only translations are allowed, the extra running time reduces to O((1/ε) log(1/ε)). If P and Q are convex polygons with n vertices in total that are given in an array or balanced tree, the total running time is O((1/ε) log n + (1/ε 2) log(1/ε)) for rigid motions and O((1/ε) log n + (1/ε) log(1/ε)) for translations.
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 8 (1 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
Maximizing the Area of Overlap of two Unions Of Disks under rigid motion
, 2004
"... Let A and B be two sets of n resp. m (m n) disjoint unit disks in the plane. We consider the problem of nding a rigid motion of A that maximizes the total area of its overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations, ..."
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Cited by 6 (0 self)
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Let A and B be two sets of n resp. m (m n) disjoint unit disks in the plane. We consider the problem of nding a rigid motion of A that maximizes the total area of its overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations, and hence, we turn our attention to approximation algorithms. First, we give a deterministic (1 )approximation algorithm for the maximum area of overlap under rigid motion that runs in O((n ) log m)) time. If is the diameter of set A, we get an (1 )approximation in O( 3 ) time. Under the condition that the maximum is at least a constant fraction of the area of A, we give a probabilistic (1 ) approximation algorithm that runs in O((m m) time and succeeds with high probability. Our algorithms generalize to the case where A and B consist of possibly intersecting disks of different radii provided that (i) the ratio of the radii of any two disks in A[B is bounded, and (ii) within each set, the maximum number of disks with a nonempty intersection is bounded.
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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Cited by 4 (2 self)
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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.
Probabilistic Matching of Planar Regions
, 2009
"... 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 4 (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.
A geometric method for determining intersection relations between a movable convex object and a set of planar polygons
 In: Robotics, IEEE Transactions on
, 2004
"... Abstract—In this paper, we investigate how to topologically and geometrically characterize the intersection relations between a movable convex polygon and a set of possibly overlapping polygons fixed in the plane. More specifically, a subset is called an intersection relation if there exists a pl ..."
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Cited by 3 (2 self)
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Abstract—In this paper, we investigate how to topologically and geometrically characterize the intersection relations between a movable convex polygon and a set of possibly overlapping polygons fixed in the plane. More specifically, a subset is called an intersection relation if there exists a placement of that intersects, and only intersects, . The objective of this paper is to design an efficient algorithm that finds a finite and discrete representation of all of the intersection relations between and . Past related research only focuses on the complexity of the free space of the configuration space between and and how to move or place an object in this free space. However, there are many applications that require the knowledge of not only the free space, but also the intersection relations. Examples are presented to demonstrate the rich applications of the formulated problem on intersection relations. Index Terms—Configuration space, critical curves and points, geometric and algebraic structure, intersection relation. I.
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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Cited by 2 (0 self)
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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