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Polygon Decomposition for Efficient Construction of Minkowski Sums
, 2000
"... Several algorithms for computing the Minkowski sum of two polygons in the plane begin by decomposing each polygon into convex subpolygons. We examine different methods for decomposing polygons by their suitability for efficient construction of Minkowski sums. We study and experiment with various ..."
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Cited by 42 (8 self)
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Several algorithms for computing the Minkowski sum of two polygons in the plane begin by decomposing each polygon into convex subpolygons. We examine different methods for decomposing polygons by their suitability for efficient construction of Minkowski sums. We study and experiment with various wellknown decompositions as well as with several new decomposition schemes. We report on our experiments with various decompositions and different input polygons. Among our findings are that in general: (i) triangulations are too costly (ii) what constitutes a good decomposition for one of the input polygons depends on the other input polygon  consequently, we develop a procedure for simultaneously decomposing the two polygons such that a "mixed" objective function is minimized, (iii) there are optimal decomposition algorithms that significantly expedite the Minkowskisum computation, but the decomposition itself is expensive to compute  in such cases simple heuristics that approximate the optimal decomposition perform very well.
Approximating minimumweight triangulations in three dimensions.
 Discrete Comput. Geom.,
, 1999
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A QuasiPolynomial Time Approximation Scheme for Minimum Weight Triangulation
 Proceedings of the 38th ACM Symposium on Theory of Computing
, 2006
"... The Minimum Weight Triangulation problem is to find a triangulation T of minimum length for a given set of points P in the Euclidean plane. It was one of the few longstanding open problems from the famous list of twelve problems with unknown complexity status, published by Garey and Johnson [8] in 1 ..."
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Cited by 10 (1 self)
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The Minimum Weight Triangulation problem is to find a triangulation T of minimum length for a given set of points P in the Euclidean plane. It was one of the few longstanding open problems from the famous list of twelve problems with unknown complexity status, published by Garey and Johnson [8] in 1979. Very recently the problem was shown to be NPhard by Mulzer and Rote. In this paper, we present a quasipolynomial time approximation scheme for Minimum Weight Triangulation.
Approximate Minimum Weight Steiner Triangulation in Three Dimensions
 In Proceedings of the Tenth Annual ACMSIAM Symposium on Discrete Algorithms
, 1999
"... Difficulty of minimum weight triangulation of a point set in R 2 is well known. In this paper we study the minimum weight triangulation problem for polyhedra and general obstacle set in three dimensions. The weight of a triangulation in three dimensions is assumed to be the total surface area of a ..."
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Cited by 3 (0 self)
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Difficulty of minimum weight triangulation of a point set in R 2 is well known. In this paper we study the minimum weight triangulation problem for polyhedra and general obstacle set in three dimensions. The weight of a triangulation in three dimensions is assumed to be the total surface area of all triangles involved. It is shown that a polyhedron P of size n can be triangulated with O(n 2 log n) tetrahedra in time O(n 2 log 3 n) approximating the minimum weight triangulation of P within a constant factor. No such prior result is known. The same bounds also hold for a 3D point set triangulation allowing Steiner points. We consider another setting called general obstacle set, where the convex hull of a set of n triangles is triangulated conforming to the input triangles. In this case we show that our method produces a triangulation of size O(n 3 log n) in time O(n 3 log 3 n) approximating the weight of the minimum weight triangulation within a constant factor. This is a ...
Quadtree Decomposition, Steiner Triangulation, and Ray shooting
 IN PROC. 9TH INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION
, 1998
"... We present a new quadtreebased decomposition of a polygon possibly with holes. For a polygon of n vertices, a truncated decomposition can be computed in O(n log n) time which yields a Steiner triangulation of the interior of the polygon that has O(n log n) size and approximates the minimum weig ..."
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Cited by 1 (1 self)
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We present a new quadtreebased decomposition of a polygon possibly with holes. For a polygon of n vertices, a truncated decomposition can be computed in O(n log n) time which yields a Steiner triangulation of the interior of the polygon that has O(n log n) size and approximates the minimum weight Steiner triangulation (MWST) to within a constant factor. An approximate MWST is good for ray shooting in the average case as defined by Aronov and Fortune. The untruncated decomposition also yields an approximate MWST. Moreover, we show that this triangulation supports querysensitive ray shooting as defined by Mitchell, Mount, and Suri. Hence, there exists a Steiner triangulation that is simultaneously good for ray shooting in the querysensitive sense and in the average case.
Minimum weight convex Steiner partitions
, 2009
"... New tight bounds are presented on the minimum length of planar straight line graphs connecting n given points in the plane and having convex faces. Specifically, we show that the minimum length of a convex Steiner partition for n points in the plane is at most O(log n / loglog n) times longer than a ..."
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New tight bounds are presented on the minimum length of planar straight line graphs connecting n given points in the plane and having convex faces. Specifically, we show that the minimum length of a convex Steiner partition for n points in the plane is at most O(log n / loglog n) times longer than a Euclidean minimum spanning tree (EMST), and this bound is the best possible. Without Steiner points, the corresponding bound is known to be Θ(log n), attained for n vertices of a pseudotriangle. We also show that the minimum length convex Steiner partition of n points along a pseudotriangle is at most O(log log n) times longer than an EMST, and this bound is also the best possible. Our methods are constructive and lead to O(n log n) time algorithms for computing convex Steiner partitions having O(n) Steiner points and weight within the above worstcase bounds in both cases.
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
, 2006
"... Abstract. We provide efficient constant factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum o ..."
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Abstract. We provide efficient constant factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves partitioning the plane minus the input points into subsets with exactly three boundary components, with approximately minimum total length. In the Euclidean case, these curves are squares; in the hyperbolic case, they combine our Euclidean square pants decomposition with our tree clustering method for general metric spaces. 1
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
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TOPOLOGICAL EFFECTS ON MINIMUM WEIGHT STEINER TRIANGULATIONS
"... Abstract. Let mwt(X) denote the sum of the Euclidean edge lengths of a minimum weight triangulation of a point set X ∈ R 2. We investigate a curious property of some npoint sets X, which allow for an (n + 1) st point P (called a Steiner point) to give mwt(X ∪ {P}) < mwt(X). We call the regions o ..."
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Abstract. Let mwt(X) denote the sum of the Euclidean edge lengths of a minimum weight triangulation of a point set X ∈ R 2. We investigate a curious property of some npoint sets X, which allow for an (n + 1) st point P (called a Steiner point) to give mwt(X ∪ {P}) < mwt(X). We call the regions of the plane where such a P reduces the length of the minimum weight triangulation Steiner reducing regions. We demonstrate by example that these Steiner reducing regions may have many disconnected components or fail to be simply connected. By examining randomly generated point sets, we show that the surprising topology of these Steiner reducing regions is more common than one might expect. 1.