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We show that if a sequence s/ of natural numbers has no pair of elements whose difference is a positive square, then the density of J / n{l,...,«} is O(l/log«) c »), c n->-oo. This improves previous results which showed that the density converges to zero, but at a slower rate. We use a technique based on the method of Hardy and Littlewood together with a(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)
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)
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)
We show that in the deter-ministic comparison model for parallel computation, n processors can select the k th smallest item from a set of n numbers in O(loglogn) 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 deter-ministic models.