Let Ë be a set with Ò Û · points in general position in the plane, Û of them white, and of them black, where Û and are even numbers. We show that there exists a matching of points of the same color with straight line segments and no crossings which matches at least ¿¿¿± of the points. We also derive an Ç´Ò ÐÓÓ Òµ algorithm which achieves this guarantee. On… (More)
Given disjoint sets PI, P2 ..... Pd in R a with n points in total, a ham-sandwich cut is a hyperplane that simultaneously bisects the Pi. We present algorithms for finding ham-sandwich cuts in every dimension d > 1. When d = 2, the algorithm is optimal, having complexity O(n). For dimension d > 2, the bound on the running time is proportional to the… (More)
We consider the problem of element distinctness. Here $n$ synchronized processors, each given an integer input, must decide whether these integers are pairwise distinct, while communicating via an infinitely large shared memory. If simultaneous write access to a memory cell is forbidden, then a lower bound of $\Omega(log n)$ on the number of steps easily… (More)
We consider three problems about the illumination of planar regions with oodlights of prescribed angles. Problem 1 is the decision problem: given a wedge W of angle , n points p 1 ; ; p n in the plane and n angles 1 ; ; n summing up to , decide whether W can be illuminated by oodlights of angles 1 ; ; n placed in some order at the points p 1 ; ; p n and… (More)
Many problems can be formulated as the optimization of functions in R 2 which are implicitly defined by an arrangement of lines, halfplanes, or points, for example linear programming in the plane. We present an efficient general approach to find the optimum exactly, for a wide range of functions that possess certain useful properties. To illustrate the… (More)
We discuss several generalizations of the notion of median to points in $R^d$. They arise in Computational Geometry and in Statistics. The notions are compared with respect to some of their mathematical properties. We also consider computational aspects. The issue of computational complexity raises several intriguing questions.
We show that in the deterministic comparison model for parallel computation, p=n processors can select the kth smallest item from a set of n numbers in O(log log n) parallel time. With this result all comparison tasks (selection, merging, sorting) now have upper and lower bounds of the same order in both random and deterministic models. This optimal time… (More)