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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.