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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 12 (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 9 (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
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.
Overlap of Convex Polytopes under Rigid Motion
, 2012
"... We present an algorithm to compute an approximate overlap of two convex polytopes P1 and P2 in R³ under rigid motion. Given any ε ∈ (0, 1/2], our algorithm runs in O(ε −3 n log 3.5 n) time with probability 1 − n −O(1) and returns a (1 − ε)approximate maximum overlap, provided that the maximum overl ..."
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Cited by 1 (0 self)
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We present an algorithm to compute an approximate overlap of two convex polytopes P1 and P2 in R³ under rigid motion. Given any ε ∈ (0, 1/2], our algorithm runs in O(ε −3 n log 3.5 n) time with probability 1 − n −O(1) and returns a (1 − ε)approximate maximum overlap, provided that the maximum overlap is at least λ · max{P1, P2} for some given constant λ ∈ (0, 1].
Largest Inscribed Rectangles in Convex Polygons
"... We consider approximation algorithms for the problem of computing an inscribed rectangle having largest area in a convex polygon on n vertices. If the order of the vertices of the polygon is given, we present a randomized algorithm that computes an inscribed rectangle with area at least (1 − ɛ) time ..."
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Cited by 1 (1 self)
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We consider approximation algorithms for the problem of computing an inscribed rectangle having largest area in a convex polygon on n vertices. If the order of the vertices of the polygon is given, we present a randomized algorithm that computes an inscribed rectangle with area at least (1 − ɛ) times the optimum with probability t in time O ( 1/ɛ log n) for any constant t < 1. We further give a deterministic approximation algorithm that computes an inscribed rectangle of area at least (1 − ɛ) times the optimum in running time O (1/ɛ² log n) and show how this running time can be slightly improved.
Finding Largest Rectangles in Convex Polygons
, 2014
"... We consider the following geometric optimization problem: find a maximumarea rectangle and a maximumperimeter rectangle contained in a given convex polygon with n vertices. We give exact algorithms that solve these problems in time O(n3). We also give (1 − ε)approximation algorithms that take tim ..."
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We consider the following geometric optimization problem: find a maximumarea rectangle and a maximumperimeter rectangle contained in a given convex polygon with n vertices. We give exact algorithms that solve these problems in time O(n3). We also give (1 − ε)approximation algorithms that take time O(ε−3/2 +ε−1/2 logn) for maximizing the area and O(ε−3 +ε−1 logn) for maximizing the perimeter.