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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)
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.
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)
In this note we point out that polynomial least squares approximations may be unstable in coefficient space and stable inL 2. That is, an ill-conditioned basis may produce large errors in particular coefficients yet theL 2 error is small. An explanation of this phenomenon is offered. Wir zeigen, daß polynomiale Näherungen zu Kleinsten Fehlerquadraten, die(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)