Kasper Green Larsen

Learn More
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)
<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)
In this paper, we study the cell probe complexity of evaluating an n-degree polynomial P over a finite field F of size at least n<sup>1+&#x03A9;(1)</sup>. More specifically, we show that any static data structure for evaluating P(x), where x &#x2208; F, must use &#x03A9;(lg |F|/ lg(Sw/n lg |F|)) cell probes to answer a query, where S denotes the space of(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)
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)
We study the query complexity of determining a hidden permutation. More specifically, we study the problem of learning a secret (z, π) consisting of a binary string z of length n and a permutation π of [n]. The secret must be unveiled by asking queries x ∈ {0, 1} n , and for each query asked, we are returned the score f z,π (x) defined as f z,π (x) := max{i(More)
Range reporting is a one of the most fundamental topics in spatial databases and computational geometry. In this class of problems, the input consists of a set of geometric objects, such as points, line segments, rectangles etc. The goal is to preprocess the input set into a data structure, such that given a query range, one can efficiently report all input(More)
The mode of a multiset of labels, is a label that occurs at least as often as any other label. The input to the range mode problem is an array A of size n. A range query [i, j] must return the mode of the sub-array A[i], A[i + 1],. .. , A[j]. We prove that any data structure that uses S memory cells of w bits needs Ω(log n log(Sw/n)) time to answer a range(More)
In orthogonal range reporting we are to preprocess N points in d-dimensional space so that the points inside a d-dimensional axis-aligned query box can be reported efficiently. This is a fundamental problem in various fields, including spatial databases and computational geometry. In this paper we provide a number of improvements for three and higher(More)