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On levels in arrangements of surfaces in three dimensions
 Proc. 16th ACMSIAM Sympos. Discrete Algorithms
, 2005
"... A favorite open problem in combinatorial geometry is to determine the worstcase complexity of a level in an arrangement. Up to now, nontrivial upper bounds in three dimensions are known only for the linear cases of planes and triangles. We propose the first technique that can deal with more general ..."
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A favorite open problem in combinatorial geometry is to determine the worstcase complexity of a level in an arrangement. Up to now, nontrivial upper bounds in three dimensions are known only for the linear cases of planes and triangles. We propose the first technique that can deal with more general surfaces in three dimensions. For example, in an arrangement of n “pseudoplanes” or “pseudospherical patches ” (where the main criterion is that each triple of surfaces has at most two common intersections), we prove that there are at most O(n 2.997) vertices at any given level. 1
Tangencies between families of disjoint regions in the plane
, 2010
"... Let C be a family of n convex bodies in the plane, which can be decomposed into k subfamilies of pairwise disjoint sets. It is shown that the number of tangencies between the members of C is at most O(kn), and that this bound cannot be improved. If we only assume that our sets are connected and vert ..."
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Let C be a family of n convex bodies in the plane, which can be decomposed into k subfamilies of pairwise disjoint sets. It is shown that the number of tangencies between the members of C is at most O(kn), and that this bound cannot be improved. If we only assume that our sets are connected and vertically convex, that is, their intersection with any vertical line is either a segment or the empty set, then the number of tangencies can be superlinear in n, but it cannot exceed O(n log 2 n). Our results imply a new upper bound on the number of regular intersection points on the boundary of ∪C. 1
Crossings between curves with many tangencies
"... Abstract. Let A and B be two families of twoway infinite xmonotone curves, no three of which pass through the same point. Assume that every curve in A lies above every curve in B and that there are m pairs of curves, one from A and the other from B, that are tangent to each other. Then the number ..."
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Abstract. Let A and B be two families of twoway infinite xmonotone curves, no three of which pass through the same point. Assume that every curve in A lies above every curve in B and that there are m pairs of curves, one from A and the other from B, that are tangent to each other. Then the number of proper crossings among the members of A∪B is at least (1/2 − o(1))m lnm. This bound is almost tight. 1
Staying in the middle: Exact and approximate medians in R¹ and R² for moving points
 IN PROC. OF THE CANADIAN CONFERENCE ON COMPUTATIONAL GEOMETRY
, 2003
"... Many divideandconquer based geometric algorithms and orderstatistics problems ask for a point that lies “in the middle” of a given point set. We study several fundamental problems of this type for moving points in one and two dimensions. In particular, we show how to kinetically maintain the medi ..."
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Many divideandconquer based geometric algorithms and orderstatistics problems ask for a point that lies “in the middle” of a given point set. We study several fundamental problems of this type for moving points in one and two dimensions. In particular, we show how to kinetically maintain the median of a set of n points moving on the real line, and a center point of a set of n points moving in the plane, that is, a point such that any line through it has at most 2n/3 on either side of it. Since the maintenance of exact medians and center points can be quite expensive, we also show how to maintain εapproximate medians and center points and argue that the latter can be made to be much more stable under motion. These results are based on a new algorithm to maintain an εapproximation of a range space under insertions and deletions, which is of independent interest. All our approximation algorithms run in nearlinear time.
Kinetic and Dynamic Data Structures for Convex Hulls and Upper Envelopes ∗
, 2005
"... Let S be a set of n moving points in the plane. We present a kinetic and dynamic (randomized) data structure for maintaining the convex hull of S. The structure uses O(n) space, and processes an expected number of O(n 2 βs+2(n)log n) critical events, each in O(log 2 n) expected time, including O(n) ..."
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Let S be a set of n moving points in the plane. We present a kinetic and dynamic (randomized) data structure for maintaining the convex hull of S. The structure uses O(n) space, and processes an expected number of O(n 2 βs+2(n)log n) critical events, each in O(log 2 n) expected time, including O(n) insertions, deletions, and changes in the flight plans of the points. Here s is the maximum number of times where any specific triple of points can become collinear, βs(q) = λs(q)/q, and λs(q) is the maximum length of DavenportSchinzel sequences of order s on n symbols. Compared with the previous solution of Basch, Guibas and Hershberger [8], our structure uses simpler certificates, uses roughly the same resources, and is also dynamic.
ACKNOWLEDGMENTS
, 1997
"... The technical appendix preparation was coordinated by Sally C. Curtin in the Division of Vital Statistics under the general direction of Kenneth G. Keppel, Acting Chief of the Reproductive Statistics Branch. The vital statistics computer file on which it is based were prepared by staff from the Divi ..."
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The technical appendix preparation was coordinated by Sally C. Curtin in the Division of Vital Statistics under the general direction of Kenneth G. Keppel, Acting Chief of the Reproductive Statistics Branch. The vital statistics computer file on which it is based were prepared by staff from the Division of Vital Statistics, Division of Data Processing, Division of Data Services, and the Office of Research and Methodology.
Abstract Region Counting Circles
"... The region counting distances, introduced by Demaine, Iacono and Langerman [5], associate to any pair of points p, q the number of items of a dataset S contained in a region R(p, q) surrounding p, q. We define region counting disks and circles, and study the complexity of these objects. In particula ..."
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The region counting distances, introduced by Demaine, Iacono and Langerman [5], associate to any pair of points p, q the number of items of a dataset S contained in a region R(p, q) surrounding p, q. We define region counting disks and circles, and study the complexity of these objects. In particular, we prove that for some wide class of regions R(p, q), the complexity of a region counting circle of radius k is either at least as large as the complexity of the klevel in an arrangement of lines, or is linear in S. We give a complete characterization of regions falling into one of these two cases. Algorithms to compute ɛapproximations of region counting distances and approximations of region counting circles are presented. 1
Abstract On Levels in Arrangements of Surfaces in Three Dimensions
"... A favorite open problem in combinatorial geonmtry is to determine the worstcase complexity of a level in an arrangement. Up to now, nontrivial upper bounds in three dimensions are known only for the linear cases of planes and triangles. We propose the first technique that can deal with more genera ..."
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A favorite open problem in combinatorial geonmtry is to determine the worstcase complexity of a level in an arrangement. Up to now, nontrivial upper bounds in three dimensions are known only for the linear cases of planes and triangles. We propose the first technique that can deal with more general surfaces in three dimensions. For example, in an arrangenmnt of n &quot;pseudoplanes &quot; or &quot;pseudospheres&quot; (where each triple of surfaces has at most two common intersections), we prove that there are at most O(n 299s6) vertices of any given level.