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- Timothy M. Chan, Kasper Green Larsen, Mihai Patrascu
- Symposium on Computational Geometry
- 2011

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)

- Kasper Green Larsen
- 2012 IEEE 53rd Annual Symposium on Foundations of…
- 2012

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+Ω(1)</sup>. More specifically, we show that any static data structure for evaluating P(x), where x ∈ F, must use Ω(lg |F|/ lg(Sw/n lg |F|)) cell probes to answer a query, where S denotes the space of… (More)

- Kasper Green Larsen
- STOC
- 2012

In this paper we develop a new technique for proving lower bounds on the update time and query time of dynamic data structures in the cell probe model. With this technique, we prove the highest lower bound to date for any explicit problem, namely a lower bound of t<sub>q</sub>=Ω((lg n/lg(wt<sub>u</sub>))<sup>2</sup>). Here n is the number of update… (More)

- Peyman Afshani, Lars Arge, Kasper Green Larsen
- Symposium on Computational Geometry
- 2010

Orthogonal range reporting is the problem of storing a set of <i>n</i> points in <i>d</i>-dimensional space, such that the <i>k</i> points in an axis-orthogonal query box can be reported efficiently. While the 2-d version of the problem was completely characterized in the pointer machine model more than two decades ago, this is not the case in higher… (More)

- Peyman Afshani, Lars Arge, Kasper Green Larsen
- Symposium on Computational Geometry
- 2012

In this paper, we consider two fundamental problems in the pointer machine model of computation, namely orthogonal range reporting and rectangle stabbing. Orthogonal range reporting is the problem of storing a set of <i>n</i> points in d-dimensional space in a data structure, such that the t points in an axis-aligned query rectangle can be reported… (More)

- Kasper Green Larsen, Freek van Walderveen
- SODA
- 2013

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)

- Kasper Green Larsen, Jelani Nelson
- ArXiv
- 2016

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)

- Peyman Afshani, Manindra Agrawal, Benjamin Doerr, Carola Doerr, Kasper Green Larsen, Kurt Mehlhorn
- Space-Efficient Data Structures, Streams, and…
- 2013

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)