TY - JOUR AB - We prove that given n⩾3 convex, compact, and pairwise disjoint sets in the plane, they may be covered with n non-overlapping convex polygons with a total of not more than 6n−9 sides, and with not more than 3n−6 distinct slopes. Furthermore, we construct sets that require 6n−9 sides and 3n−6 slopes for n⩾3. The upper bound on the number of slopes implies a new bound on a recently studied transversal problem. AU - Edelsbrunner, Herbert AU - Robison, Arch AU - Shen, Xiao ID - 4065 IS - 2 JF - Discrete Mathematics SN - 0012-365X TI - Covering convex sets with non-overlapping polygons VL - 81 ER - TY - JOUR AB - We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m 2/3 n 2/3 +n), and that it isO(m 2/3 n 2/3 β(n) +n) forn unit-circles, whereβ(n) (and laterβ(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up toO(m 3/5 n 4/5 β(n) +n). The same bounds (without theβ(n)-terms) hold for the maximum sum of degrees ofm vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees ofm vertices in an arrangement ofn spheres in three dimensions isO(m 4/7 n 9/7 β(m, n) +n 2), in general, andO(m 3/4 n 3/4 β(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances amongm points in three dimensions isO(m 3/2 β(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given. AU - Clarkson, Kenneth AU - Edelsbrunner, Herbert AU - Guibas, Leonidas AU - Sharir, Micha AU - Welzl, Emo ID - 4074 IS - 1 JF - Discrete & Computational Geometry SN - 0179-5376 TI - Combinatorial complexity bounds for arrangements of curves and spheres VL - 5 ER - TY - CONF AB - In this paper we derived combinatorial point selection results for geometric objects defined by pairs of points. In a nutshell, the results say that if many pairs of a set of n points in some fixed dimension each define a geometric object of some type, then there is a point covered by many of these objects. Based on such a result for three-dimensional spheres we show that the combinatorial size of the Delaunay triangulation of a point set in space can be reduced by adding new points. We believe that from a practical point of view this is the most important result of this paper. AU - Chazelle, Bernard AU - Edelsbrunner, Herbert AU - Guibas, Leonidas AU - Hershberger, John AU - Seidel, Raimund AU - Sharir, Micha ID - 4078 SN - 978-0-89791-362-1 T2 - Proceedings of the 6th annual symposium on computational geometry TI - Slimming down by adding; selecting heavily covered points ER - TY - CONF AB - We present an algorithm to compute a Euclidean minimum spanning tree of a given set S of n points in Ed in time O(Td(N, N) logd N), where Td(n, m) is the time required to compute a bichromatic closest pair among n red and m blue points in Ed. If Td(N, N) = Ω(N1+ε), for some fixed ε > 0, then the running time improves to O(Td(N, N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closets pair in expected time O((nm log n log m)2/3+m log2 n + n log2 m) in E3, which yields an O(N4/3log4/3 N) expected time algorithm for computing a Euclidean minimum spanning tree of N points in E3. AU - Agarwal, Pankaj AU - Edelsbrunner, Herbert AU - Schwarzkopf, Otfried AU - Welzl, Emo ID - 4076 SN - 978-0-89791-362-1 T2 - Proceedings of the 6th annual symposium on Computational geometry TI - Euclidean minimum spanning trees and bichromatic closest pairs ER - TY - CONF AB - We prove that for any set S of n points in the plane and n3-α triangles spanned by the points of S there exists a point (not necessarily of S) contained in at least n3-3α/(512 log25 n) of the triangles. This implies that any set of n points in three - dimensional space defines at most 6.4n8/3 log5/3 n halving planes. AU - Aronov, Boris AU - Chazelle, Bernard AU - Edelsbrunner, Herbert AU - Guibas, Leonidas AU - Sharir, Micha AU - Wenger, Rephael ID - 4077 SN - 978-0-89791-362-1 T2 - Proceedings of the 6th annual symposium on Computational geometry TI - Points and triangles in the plane and halving planes in space ER -