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We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model: We present two data structures for 2-d orthogonal range emptiness. The first achieves O(n lg lg n) space and O(lg lg n) query time, assuming that the n given points are in(More)
Postoperative cognitive function (POCD) has been subject to extensive research. In the literature, large differences are apparent in methodology such as the test batteries, the interval between sessions, the endpoints to be analysed, statistical methods, and how neuropsychological deficits are defined. Traditionally, intelligence tests or tests developed(More)
A mode of a multiset S is an element a∈S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1:n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i,j) for which a mode(More)
<i>Range selection</i> is the problem of preprocessing an input array <i>A</i> of <i>n</i> unique integers, such that given a query (<i>i, j, k</i>), one can report the <i>k</i>'th smallest integer in the subarray <i>A</i>[<i>i</i>], <i>A</i>[<i>i</i> + 1],..., <i>A</i>[<i>j</i>]. In this paper we consider static data structures in the word-RAM for range(More)
Given a set of points with uncertain locations, we consider the problem of computing the probability of each point lying on the skyline, that is, the probability that it is not dominated by any other input point. If each point's uncertainty is described as a probability distribution over a discrete set of locations, we improve the best known exact solution.(More)
Range reporting on categorical (or colored) data is a well-studied generalization of the classical range reporting problem in which each of the N input points has an associated color (category). A query then asks to report the set of colors of the points in a given rectangular query range, which may be far smaller than the set of all points in the query(More)
For any n > 1 and 0 < ε < 1/2, we show the existence of an n O(1)-point subset X of R n such that any linear map from (X, 2) to m 2 with distortion at most 1 + ε must have m = Ω(min{n, ε −2 log n}). Our lower bound matches the upper bounds provided by the identity matrix and the Johnson-Lindenstrauss lemma [JL84], improving the previous lower bound of Alon(More)
We consider NCA labeling schemes: given a rooted tree T , label the nodes of T with binary strings such that, given the labels of any two nodes, one can determine, by looking only at the labels, the label of their nearest common ancestor. For trees with n nodes we present upper and lower bounds establishing that labels of size (2 ± ǫ) log n, ǫ < 1 are both(More)
For any integers d, n ≥ 2 and 1/(min{n, d}) 0.4999 < ε < 1, we show the existence of a set of n vectors X ⊂ R d such that any embedding f : X → R m satisfying ∀x, y ∈ X, (1 − ε)x − y 2 2 ≤ f (x) − f (y) 2 2 ≤ (1 + ε)x − y 2 2 must have m = Ω(ε −2 lg n). This lower bound matches the upper bound given by the Johnson-Lindenstrauss lemma [JL84]. Furthermore ,(More)