Consider a set of S of n data points in real d-dimensional space, Rd, where distances are measured using any Minkowski metric. In nearest neighbor searching, we preprocess S into a data structure, so that given any query point q∈ Rd, is the closest point of S to q can be reported quickly. Given any positive real ϵ, data point p is a (1 +ϵ)-approximate nearest neighbor of q if its distance from q is within a factor of (1 + ϵ) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of n points in Rd in O(dn log n) time and O(dn) space, so that given a query point q ∈ Rd, and ϵ > 0, a (1 + ϵ)-approximate nearest neighbor of q can be computed in O(cd, ϵ log n) time, where cd,ϵ≤d ⌈1 + 6d/ϵ⌉d is a factor depending only on dimension and ϵ. In general, we show that given an integer k ≥ 1, (1 + ϵ)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.

/pdf/an-optimal-algorithm-for-approximate-nearest-neighbor-29l6kwjnzm.pdf

An optimal algorithm for approximate nearest neighbor searching fixed dimensions

In this paper, we present a formal approach to reciprocal n-body collision avoidance, where multiple mobile robots need to avoid collisions with each other while moving in a common workspace In our formulation, each robot acts fully independently, and does not communicate with other robots Based on the definition of velocity obstacles [5], we derive sufficient conditions for collision-free motion by reducing the problem to solving a low-dimensional linear program We test our approach on several dense and complex simulation scenarios involving thousands of robots and compute collision-free actions for all of them in only a few milliseconds To the best of our knowledge, this method is the first that can guarantee local collision-free motion for a large number of robots in a cluttered workspace

/pdf/reciprocal-n-body-collision-avoidance-3g7xt37qbz.pdf

Reciprocal n-Body Collision Avoidance

Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles Although small angles inherent in the input geometry cannot be removed, one would like to triangulate a domain without creating any new small angles Unfortunately, this problem is not always soluble A compromise is necessary A Delaunay refinement algorithm is presented that can create a mesh in which most angles are 30^o or greater and no angle is smaller than arcsin[(3/2)sin(@f/2)]~(3/4)@f, where @f=<60^ois the smallest angle separating two segments of the input domain New angles smaller than 30^o appear only near input angles smaller than 60^o In practice, the algorithm's performance is better than these bounds suggest Another new result is that Ruppert's analysis technique can be used to reanalyze one of Chew's algorithms Chew proved that his algorithm produces no angle smaller than 30^o (barring small input angles), but without any guarantees on grading or number of triangles He conjectures that his algorithm offers such guarantees His conjecture is conditionally confirmed here: if the angle bound is relaxed to less than 265^o, Chew's algorithm produces meshes (of domains without small input angles) that are nicely graded and size-optimal

Delaunay refinement algorithms for triangular mesh generation

The pivotal problem of comorbidity research lies in the psychometric foundation it rests on, that is, latent variable theory, in which a mental disorder is viewed as a latent variable that causes a constellation of symptoms. From this perspective, comorbidity is a (bi)directional relationship between multiple latent variables. We argue that such a latent variable perspective encounters serious problems in the study of comorbidity, and offer a radically different conceptualization in terms of a network approach, where comorbidity is hypothesized to arise from direct relations between symptoms of multiple disorders. We propose a method to visualize comorbidity networks and, based on an empirical network for major depression and generalized anxiety, we argue that this approach generates realistic hypotheses about pathways to comorbidity, overlapping symptoms, and diagnostic boundaries, that are not naturally accommodated by latent variable models: Some pathways to comorbidity through the symptom space are more likely than others; those pathways generally have the same direction (i.e., from symptoms of one disorder to symptoms of the other); overlapping symptoms play an important role in comorbidity; and boundaries between diagnostic categories are necessarily fuzzy.

/pdf/comorbidity-a-network-perspective-3hmyj5htzt.pdf

Comorbidity: a network perspective.

Full-text indexes provide fast substring search over large text collections. A serious problem of these indexes has traditionally been their space consumption. A recent trend is to develop indexes that exploit the compressibility of the text, so that their size is a function of the compressed text length. This concept has evolved into self-indexes, which in addition contain enough information to reproduce any text portion, so they replace the text. The exciting possibility of an index that takes space close to that of the compressed text, replaces it, and in addition provides fast search over it, has triggered a wealth of activity and produced surprising results in a very short time, which radically changed the status of this area in less than 5 years. The most successful indexes nowadays are able to obtain almost optimal space and search time simultaneously.In this article we present the main concepts underlying (compressed) self-indexes. We explain the relationship between text entropy and regularities that show up in index structures and permit compressing them. Then we cover the most relevant self-indexes, focusing on how they exploit text compressibility to achieve compact structures that can efficiently solve various search problems. Our aim is to give the background to understand and follow the developments in this area.

/pdf/compressed-full-text-indexes-4cxd0n1jff.pdf

Compressed full-text indexes

This introduction to computational geometry focuses on algorithms. Motivation is provided from the application areas as all techniques are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement.

Computational Geometry: Algorithms and Applications

https://www.cse.ust.hk/~golin/pubs/TCS_Vor_game.pdf

Competitive facility location: the Voronoi game

Given a set of compact sites on a sphere, we show that their spherical Voronoi diagram can be computed by computing two planar Voronoi diagrams of suitably transformed sites in the plane. We also show that a planar furthest-site Voronoi diagram can always be obtained as a portion of a nearest-site Voronoi diagram of a set of transformed sites. Two immediate applications are an O(n logn) algorithm for the spherical Voronoi diagram of a set of circular arcs on the sphere, and an O(n logn) algorithm for the furthest-site Voronoi diagram for a set of circular arcs in the plane.

Voronoi diagrams on the sphere

In the one-round Voronoi game, the first player chooses
 an n-point set W in a square Q, and then the
 second player places another n-point set B into
 Q. The payoff for the second player is the fraction of
 the area of Q occupied by the regions of the points of
 B in the Voronoi diagram of W \cup B.
 We give a (randomized) strategy for the second player that always
 guarantees him a payoff of at least ½ + α,
 for a constant α > 0 and every large enough n. This
 contrasts with the one-dimensional situation, with
 Q=[0,1], where the first player can always win more
 than ½.

/pdf/the-one-round-voronoi-game-1mv0nzmf14.pdf

The one-round Voronoi game

(MATH) In the one-round Voronoi game, the first player chooses an n-point set $\PFRST$ in a square $Q$, and then the second player places another n-point set $\PSCND$ into $Q$. The payoff for the second player is the fraction of the area of $Q$ occupied by the regions of the points of $\PSCND$ in the Voronoi diagram of $\PFRST\cup\PSCND$. We give a strategy for the second player that always guarantees him a payoff of at least $\frac12+\alpha$, for a constant $\alpha>0$ independent of n. This contrasts with the one-dimensional situation, with $Q=[0,1]$, where the first player can always win more than 1/2.

/pdf/the-one-round-voronoi-game-230u3pdu7a.pdf

Otfried Cheong

Papers

Computational Geometry: Algorithms and Applications

Competitive facility location: the Voronoi game

Voronoi diagrams on the sphere

The one-round Voronoi game

The one-round Voronoi game