. Given the n complex coe#cients of a degree n - 1 complex polynomial, we wish to evaluate the polynomial at a large number m # n of points on the complex plane. This problem is required by many algebraic computations and so is considered in most basic algorithm texts (e.g., [A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974]). We assume an arithmetic model of computation, where on each step we can execute an arithmetic operation, which is computed exactly. All previous exact algorithms [C. M. Fiduccia, Proceedings 4th Annual ACM Symposium on Theory of Computing, 1972, pp. 88--93; H. T. Kung, Fast Evaluation and Interpolation, Carnegie-Mellon, 1973; A. B. Borodin and I. Munro, The Computational Complexity of Algebraic and Numerical Problems, American Elsevier, 1975; V. Pan, A. Sadikou, E. Landowne, and O. Tiga, Comput. Math. Appl., 25 (1993), pp. 25--30] cost at least work ## log 2 n) per point, and previously, the...

In this paper, we study data structures for use in N-body simulation. We concentrate on the spatial decomposition tree used in particle-cluster force evaluation algorithms such as the Barnes-Hut algorithm. We prove that a k-d tree is asymptotically inferior to a spatially balanced tree. We show that the worst case complexity of the force evaluation algorithm using a k-d tree is \Theta(n log 3 n log L) compared with \Theta(n log L) for an oct-tree. (L is the separation ratio of the set of points.) We also investigate improving the constant factor of the algorithm, and present several methods which improve over the standard oct-tree decomposition. Finally, we consider whether or not the bounding box of a point set should be "tight", and show that it is only safe to use tight bounding boxes for binary decompositions. The results are all directly applicable to practical implementations of N-body algorithms. 1 Introduction The gravitational force computation problem is: given a set of n ...

In this paper we show that if the input points to the geometric closest pair problem are given with limited precision (each coordinate is specified with O(log n) bits), then we can compute the closest pair in O(n log log n) time. We also apply our spatial decomposition technique to the k-nearest neighbor and n-body problems, achieving similar improvements.