No abstract

No abstract

Abstract. We prove a theorem on partitioning point sets in E d (d fixed) and give an efficient construction of partition trees based on it. This yields a simplex range searching structure with linear space, O(n log n) deterministic preprocessing time, and O(nl-lla(logn)ml)) query time. With O(n 1+6) preproeessing time, where 6 is an arbitrary positive constant, a more complicated data structure yields query time O(n 1-1/d(log log n)°Cl)). This attains the lower bounds due to Chazelle [C1] up to polylogarithmic factors, improving and simplifying previous results of Chazelle et al. [CSW].The partition result implies that, for r ~ _< n 1 -~, a (I/r)-approximation of size O(r ~) with respect to simplices for an n-point set in E a can be computed in O(n log r) deterministic time. A (1/r)-cutting of size O(r ~) for a collection of n hyperplanes in E a can be computed in O(n log r) deterministic time, provided that r <_ n ltc2~-~.

ABSTRACT:The Johnson-Lindenstrauss lemma asserts that an n-point set in any Euclidean space can be mapped to a Euclidean space of dimension k = O(ε −2 log n) so that all distances are preserved up to a multiplicative factor between 1 − ε and 1 + ε. Known proofs obtain such a mapping as a linear map R n → R k with a suitable random matrix. We give a simple and self-contained proof of a version of the Johnson-Lindenstrauss lemma that subsumes a basic versions by Indyk and Motwani and a version more suitable for efficient computations due to Achlioptas. (Another proof of this result, slightly different but in a similar spirit, was given independently by Indyk and Naor.) An even more general result was established by Klartag and Mendelson using considerably heavier machinery.Recently, Ailon and Chazelle showed, roughly speaking, that a good mapping can also be obtained by composing a suitable Fourier transform with a linear mapping that has a sparse random matrix M; a mapping of this form can be evaluated very fast. In their result, the nonzero entries of M are normally distributed. We show that the nonzero entries can be chosen as random ±1, which further speeds up the computation. We also discuss the case of embeddings into R k with the 1 norm.

We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected 0(nde4~) time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input). The expectation is over the internal randomizations performed by the algorithm, and holds for any input. The algorithm is presented in an abstract framework, which facilitates its application to several other related problems. The algolcithm has been presented in a previous work by the authors [ShW], but its analysis and the sub exponential complexity bound are new. KEY WORDS: computational geometry, combinatorial optimization, linear programming, randomized increment al algorithms.

The L 2 -discrepancy for anchored axis-parallel boxes has been used in several recent computational studies, mostly related to numerical integration, as a measure of the quality of uniform distribution of a given point set. We point out that if the number of points is not large enough in terms of the dimension (e.g., fewer than 10 4 points in dimension 30) then nearly the lowest possible L 2 -discrepancy is attained by a pathological point set, and hence the L 2 -discrepancy may not be very relevant for relatively small sets. Recently, Hickernell obtained a formula for the expected L 2 -discrepancy of certain randomized low-discrepancy set constructions introduced by Owen. We note that his formula remains valid also for several modifications of these constructions which admit a very simple and efficient implementation. We also report results of computational experiments with various constructions of lowdiscrepancy sets. Finally, we present a fairly precise formula for the performance of a recent algorithm due to Heinrich for computing the L 2 -discrepancy. 1998Academic Press

Let P be a set of n points in ~d (where d is a small fixed positive integer), and let F be a collection of subsets of ~d, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following F-range searching problem: Given P, build a data structure for efficient answering of queries of the form, "Given a 7 ~ F, count (or report) the points of P lying in 7." Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and with O(n 1-x/b+~) query time, where d < b < 2d-3 and ~ > 0 is an arbitrarily small constant. The acutal value of b is related to the problem of partitioning arrangements of algebraic surfaces into cells with a constant description complexity. We present some of the applications of F-range searching problem, including improved ray shooting among triangles in ~3

For several computational problems in homotopy theory, we obtain algorithms with running time polynomial in the input size. In particular, for every fixed k ≥ 2, there is a polynomial-time algorithm that, for a 1-connected topological space X given as a finite simplicial complex, or more generally, as a simplicial set with polynomial-time homology, computes the kth homotopy group π k (X), as well as the first k stages of a Postnikov system of X. Combined with results of an earlier paper, this yields a polynomial-time computation of [X, Y ], i.e., all homotopy classes of continuous mappings X → Y , under the assumption that Y is (k−1)-connected and dim X ≤ 2k − 2. We also obtain a polynomial-time solution of the extension problem, where the input consists of finite simplicial complexes X, Y , where Y is (k−1)-connected and dim X ≤ 2k − 1, plus a subspace A ⊆ X and a (simplicial) map f : A → Y , and the question is the extendability of f to all of X.The algorithms are based on the notion of a simplicial set with polynomial-time homology, which is an enhancement of the notion of a simplicial set with effective homology developed earlier by Sergeraert and his co-workers. Our polynomial-time algorithms are obtained by showing that simplicial sets with polynomial-time homology are closed under various operations, most notably, Cartesian products, twisted Cartesian products, and classifying space. One of the key components is also polynomial-time homology for the Eilenberg-MacLane space K(Z, 1), provided in another recent paper by Krčál, Matoušek, and Sergeraert.