• Corpus ID: 219966630

Duality-based approximation algorithms for depth queries and maximum depth

  title={Duality-based approximation algorithms for depth queries and maximum depth},
  author={Dror Aiger and Haim Kaplan and Micha Sharir},
We design an efficient data structure for computing a suitably defined approximate depth of any query point in the arrangement $\mathcal{A}(S)$ of a collection $S$ of $n$ halfplanes or triangles in the plane or of halfspaces or simplices in higher dimensions. We then use this structure to find a point of an approximate maximum depth in $\mathcal{A}(S)$. Specifically, given an error parameter $\epsilon>0$, we compute, for any query point $q$, an underestimate $d^-(q)$ of the depth of $q$, that… 

Computing Batched Depth Queries and the Depth of a Set of Points

Simplicial depth and Tukey depth are two common measures for expressing the depth of a point q relative to a set P of points in R. We introduce definitions that generalize these notions to express



Range Minima Queries with Respect to a Random Permutation, and Approximate Range Counting

This work proposes an alternative approach to the range-minimum problem, based on cuttings, which achieves better performance and uses, for each permutation, O(n⌊d/2⌋(log’slog n)c/log”n) expected storage and preprocessing time, for some constant c, and answers a range- minimum query in O(log n) expected time.

Approximate range searching

It is shown that if one is willing to allow approximate ranges, then it is possible to do much better than current state-of-the-art results, and empirical evidence is given showing that allowing small relative errors can significantly improve query execution times.

Simplex Range Searching and Its Variants: A Review

A central problem in computational geometry, range searching arises in many applications, and numerous geometric problems can be formulated in terms of range searching. A typical range-searching

Geometric Range Searching and Its Relatives

This volume provides an excellent opportunity to recapitulate the current status of geometric range searching and to summarize the recent progress in this area.

Fixed-dimensional linear programming queries made easy

Two results from Clarkson’s randomized algorithm for linear programming in a fixed dimension d are derived, a simple general method that reduces the problem of answering linear programming queries to theproblem of answering halfspace range queries and a simpler proof of the following.

On Approximate Range Counting and Depth

This work presents a randomized data structure of O(n) expected size which can answer 3D approximate halfspace range counting queries in expected time, and is the first optimal method for the problem in the standard decision tree model.

On approximating the depth and related problems

This paper reduces the problem of finding a disk covering the largest number of red points, while avoiding all the blue points to a near-linear expected-time randomized approximation algorithm and proves that approximate range counting has roughly the same time and space complexity as answering emptiness range queries.

Output Sensitive Algorithms for Approximate Incidences and Their Applications

This work presents efficient output-sensitive approximation algorithms based on simple primal and dual grid decompositions based on duality for approximate incidences between points and lines or circles in the plane, and points and planes, spheres, lines, and circles in three dimensions.

Geometric Approximation Algorithms

This book is the first to cover geometric approximation algorithms in detail, and topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings.