@article{4060, abstract = {This paper offers combinatorial results on extremum problems concerning the number of tetrahedra in a tetrahedrization of n points in general position in three dimensions, i.e. such that no four points are co-planar, It also presents an algorithm that in O(n log n) time constructs a tetrahedrization of a set of n points consisting of at most 3n-11 tetrahedra.}, author = {Edelsbrunner, Herbert and Preparata, Franco and West, Douglas}, issn = {1095-855X}, journal = {Journal of Symbolic Computation}, number = {3-4}, pages = {335 -- 347}, publisher = {Elsevier}, title = {{Tetrahedrizing point sets in three dimensions}}, doi = {10.1016/S0747-7171(08)80068-5}, volume = {10}, year = {1990}, } @article{4063, abstract = {This paper describes a general-purpose programming technique, called Simulation of Simplicity, that can be used to cope with degenerate input data for geometric algorithms. It relieves the programmer from the task of providing a consistent treatment for every single special case that can occur. The programs that use the technique tend to be considerably smaller and more robust than those that do not use it. We believe that this technique will become a standard tool in writing geometric software.}, author = {Edelsbrunner, Herbert and Mücke, Ernst}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {1}, pages = {66 -- 104}, publisher = {ACM}, title = {{Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms}}, doi = {10.1145/77635.77639}, volume = {9}, year = {1990}, } @article{4064, abstract = {Given a set of data points pi = (xi, yi ) for 1 ≤ i ≤ n, the least median of squares regression line is a line y = ax + b for which the median of the squared residuals is a minimum over all choices of a and b. An algorithm is described that computes such a line in O(n 2) time and O(n) memory space, thus improving previous upper bounds on the problem. This algorithm is an application of a general method built on top of the topological sweep of line arrangements.}, author = {Edelsbrunner, Herbert and Souvaine, Diane}, issn = {1537-274X}, journal = {Journal of the American Statistical Association}, number = {409}, pages = {115 -- 119}, publisher = {American Statistical Association}, title = {{Computing least median of squares regression lines and guided topological sweep}}, doi = {10.1080/01621459.1990.10475313}, volume = {85}, year = {1990}, } @article{4065, abstract = {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.}, author = {Edelsbrunner, Herbert and Robison, Arch and Shen, Xiao}, issn = {1872-681X}, journal = {Discrete Mathematics}, number = {2}, pages = {153 -- 164}, publisher = {Elsevier}, title = {{Covering convex sets with non-overlapping polygons}}, doi = {10.1016/0012-365X(90)90147-A}, volume = {81}, year = {1990}, } @article{4066, abstract = {We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m 2/3 n logn +n 2); we can calculatem such cells specified by a point in each, in worst-case timeO(m 2/3 n log3 n+n 2 logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m 2/3 n logn+n 2), but this number is onlyO(m 3/5– n 4/5+2 +m+n logm), for any>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m 3/4– n 3/4+3 +m) log2 n+n logn logm) for any>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m 3/4– n 3/4+3 +m] log2 n+n logn logm) for any>0. (v) The maximum number of facets (i.e., (d–1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m 2/3 n d/3 logn+n d–1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.}, author = {Edelsbrunner, Herbert and Guibas, Leonidas and Sharir, Micha}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {197 -- 216}, publisher = {Springer}, title = {{The complexity of many cells in arrangements of planes and related problems}}, doi = {10.1007/BF02187785}, volume = {5}, year = {1990}, } @inproceedings{4067, abstract = {This paper proves an O(m 2/3 n 2/3+m+n) upper bound on the number of incidences between m points and n hyperplanes in four dimensions, assuming all points lie on one side of each hyperplane and the points and hyperplanes satisfy certain natural general position conditions. This result has application to various three-dimensional combinatorial distance problems. For example, it implies the same upper bound for the number of bichromatic minimum distance pairs in a set of m blue and n red points in three-dimensional space. This improves the best previous bound for this problem.}, author = {Edelsbrunner, Herbert and Sharir, Micha}, booktitle = {Proceedings of the International Symposium on Algorithms}, isbn = {978-3-540-52921-7}, location = {Tokyo, Japan}, pages = {419 -- 428}, publisher = {Springer}, title = {{A hyperplane Incidence problem with applications to counting distances}}, doi = {10.1007/3-540-52921-7_91}, volume = {450}, year = {1990}, } @article{4068, abstract = {LetS be a collection ofn convex, closed, and pairwise nonintersecting sets in the Euclidean plane labeled from 1 ton. A pair of permutations (i1i2in−1in)(inin−1i2i1) is called ageometric permutation of S if there is a line that intersects all sets ofS in this order. We prove thatS can realize at most 2n–2 geometric permutations. This upper bound is tight.}, author = {Edelsbrunner, Herbert and Sharir, Micha}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {35 -- 42}, publisher = {Springer}, title = {{The maximum number of ways to stabn convex nonintersecting sets in the plane is 2n−2}}, doi = {10.1007/BF02187778}, volume = {5}, year = {1990}, } @article{4069, abstract = {Let C be a cell complex in d-dimensional Euclidean space whose faces are obtained by orthogonal projection of the faces of a convex polytope in d + 1 dimensions. For example, the Delaunay triangulation of a finite point set is such a cell complex. This paper shows that the in front/behind relation defined for the faces of C with respect to any fixed viewpoint x is acyclic. This result has applications to hidden line/surface removal and other problems in computational geometry.}, author = {Edelsbrunner, Herbert}, issn = {1439-6912}, journal = {Combinatorica}, number = {3}, pages = {251 -- 260}, publisher = {Springer}, title = {{An acyclicity theorem for cell complexes in d dimension}}, doi = {10.1007/BF02122779}, volume = {10}, year = {1990}, } @article{4070, abstract = {Let S be a set of n closed intervals on the x-axis. A ranking assigns to each interval, s, a distinct rank, p(s)∊ [1, 2,…,n]. We say that s can see t if p(s)