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Staying in the middle: Exact and approximate medians in R 1 and R 2 for moving points, manuscript
 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 med ..."
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Cited by 2 (0 self)
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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. 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
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
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 ..."
Abstract
 Add to MetaCart
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.