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Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 88 (20 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Efficient Computation of Location Depth Contours By Methods of Combinatorial Geometry
"... The concept of location depth was introduced as a way to extend the univariate notion of ranking to a bivariate configuration of data points. It has been used successfully for robust estimation, hypothesis testing, and graphical display. The depth contours form a collection of nested polygons, and t ..."
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Cited by 13 (4 self)
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The concept of location depth was introduced as a way to extend the univariate notion of ranking to a bivariate configuration of data points. It has been used successfully for robust estimation, hypothesis testing, and graphical display. The depth contours form a collection of nested polygons, and the center of the deepest contour is called the Tukey median. The only available implemented algorithms for the depth contours and the Tukey median are slow, which limits their usefulness. In this paper we describe an optimal algorithm which computes all bivariate depth contours in O(n 2 ) time and space, using topological sweep of the dual arrangement of lines. Once these contours are known, the location depth of any point is computed in O(log 2 n) time. We provide fast implementations of these algorithms to allow their use in everyday statistical practice.
Finding an optimal path without growing the tree
 6th Annual European Sym. on Algorithms, Springer LNCS
, 1998
"... For problems on computing an optimal path as well as its length in a certain setting, the “standard” approach for finding an actual optimal path is by building (or “growing”) a singlesource optimal path tree. In this paper, we study a class of optimal path problems with the following phenomenon: Th ..."
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Cited by 3 (0 self)
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For problems on computing an optimal path as well as its length in a certain setting, the “standard” approach for finding an actual optimal path is by building (or “growing”) a singlesource optimal path tree. In this paper, we study a class of optimal path problems with the following phenomenon: The space complexity of the algorithms for reporting the lengths of singlesource optimal paths for these problems is asymptotically smaller than the space complexity of the “standard ” treegrowing algorithms for finding actual optimal paths. We present a general and efficient algorithmic paradigm for finding an actual optimal path for such problems without having to grow a singlesource optimal path tree. Our paradigm is based on the “marriagebeforeconquer ” strategy, the pruneandsearch technique, and a new data structure called clipped trees. The paradigm enables us to compute an actual path for a number of optimal path problems and dynamic programming problems in computational geometry, graph theory, and combinatorial optimization. Our algorithmic
The Exact Fitting . . .
 COMPUT. GEOM. THEORY APPL
, 1992
"... Let S be a family of n points in E a. The exact fitting problem asks for finding a hyperplane containing the maximum number of points of $. In this paper, we present an O (rain j' " " }) ,aha log , n d time algorithm where m denoted the number of points the hyperplane. This a ..."
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Let S be a family of n points in E a. The exact fitting problem asks for finding a hyperplane containing the maximum number of points of $. In this paper, we present an O (rain j' " " }) ,aha log , n d time algorithm where m denoted the number of points the hyperplane. This algorithm is based on upper bounds on the maximum number of incidences between families of points and families ofhyperplanes in E d and on an algorithm to compute these incidences. We also show how the upper bound on the maximum number of incidences between families of points and families of hyperplanes can be used to derive new bounds on some wellknown problems in discrete geometry.
On the Volume and Resolution of 3Dimensional Convex Graph Drawing (Extended Abstract)
"... We address the problem of drawing a 3connected planar graph as a convex polyhedron in R³. We give an efficient algorithm for producing such a realization using O(n) volume under the vertexresolution rule. Each vertex in the drawing resulting from this method is guaranteed to need no more than O(n ..."
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We address the problem of drawing a 3connected planar graph as a convex polyhedron in R³. We give an efficient algorithm for producing such a realization using O(n) volume under the vertexresolution rule. Each vertex in the drawing resulting from this method is guaranteed to need no more than O(n log n) bits to represent (as a pair of rational numbers). This solves an open problem of Cohen, Eades, Lin, and Ruskey. We also show that under the angularresolution rule drawing a 3connected planar graph as a convex polyhedron in R³ requires at least exponential volume in the worst case.
On the number of tetrahedra with minimum, unit, and distinct volumes in threespace ∗
, 710
"... We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3space, and in general in d dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by n points in R3 is at most 2 3n3 − O(n2), and there are point sets for ..."
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We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3space, and in general in d dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by n points in R3 is at most 2 3n3 − O(n2), and there are point sets for which this number is 3 16n3 − O(n2). We also present an O(n3) time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O’Rourke, and Seidel. In general, for every k, d ∈ N, 1 ≤ k ≤ d, the maximum number of kdimensional simplices of minimum (nonzero) volume spanned by n points in R d is Θ(n k). (ii) The number of unitvolume tetrahedra determined by n points in R 3 is O(n 7/2), and there are point sets for which this number is Ω(n 3 log log n). (iii) For every d ∈ N, the minimum number of distinct volumes of all fulldimensional simplices determined by n points in R d, not all on a hyperplane, is Θ(n). 1