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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
- 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)

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
- 2011 IEEE 52nd Annual Symposium on Foundations of…
- 2011

In this paper we establish an intimate connection between dynamic range searching in the group model and combinatorial discrepancy. Our result states that, for a broad class of range searching data structures (including all known upper bounds), it must hold that $t_ut_q = \Omega(\disc^2/\lg n)$ where $t_u$ is the worst case update time, $t_q$ the worst case… (More)

- Peyman Afshani, Lars Arge, Kasper Green Larsen
- 2009 50th Annual IEEE Symposium on Foundations of…
- 2009

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)

- Peyman Afshani, Pankaj K. Agarwal, Lars Arge, Kasper Green Larsen, Jeff M. Phillips
- Theory of Computing Systems
- 2011

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)

- 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)

- 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)

- Kasper Green Larsen, Rasmus Pagh
- SODA
- 2012

Motivated by information retrieval applications, we consider the one-dimensional colored range reporting problem in rank space. The goal is to build a static data structure for sets C1, . . . , Cm ⊆ {1, . . . , σ} that supports queries of the kind: Given indices a, b, report the set ⋃ a≤i≤b Ci. We study the problem in the I/O model, and show that there… (More)