Boris Aronov

On Approximating the Depth and Related Problems

In this paper, we study the problem of finding a disk covering the largest number of red points, while avoiding all the blue points. We reduce it to the question of finding a deepest point in an arrangement of pseudodisks and provide a near-linear expected-time randomized approximation algorithm for this problem. As an application of our techniques, we show how to solve linear programming with violations approximately. We also prove that approximate range counting has roughly the same time and space complexity as answering emptiness range queries.

Minkowski-type theorems and least-squares clustering

Abstract. Dissecting Euclidean d -space with the power diagram of n weighted point sites partitions a given m -point set into clusters, one cluster for each region of the diagram. In this manner, an assignment of points to sites is induced. We show the equivalence of such assignments to constrained Euclidean least-squares assignments. As a corollary, there always exists a power diagram whose regions partition a given d -dimensional m -point set into clusters of prescribed sizes, no matter where the sites are placed. Another consequence is that constrained least-squares assignments can be computed by finding suitable weights for the sites. In the plane, this takes roughly O(n2m) time and optimal space O(m) , which improves on previous methods. We further show that a constrained least-squares assignment can be computed by solving a specially structured linear program in n+1 dimensions. This leads to an algorithm for iteratively improving the weights, based on the gradient-descent method. Besides having the obvious optimization property, least-squares assignments are shown to be useful in solving a certain transportation problem, and in finding a least-squares fitting of two point sets where translation and scaling are allowed. Finally, we extend the concept of a constrained least-squares assignment to continuous distributions of points, thereby obtaining existence results for power diagrams with prescribed region volumes. These results are related to Minkowskis theorem for convex polytopes. The aforementioned iterative method for approximating the desired power diagram applies to continuous distributions as well.

Quasi-planar graphs have a linear number of edges

A graph is calledquasi-planar if it can be drawn in the plane so that no three of its edges are pairwise crossing. It is shown that the maximum number of edges of a quasi-planar graph withn vertices isO(n).

On compatible triangulations of simple polygons

Abstract It is well known that, given two simple n -sided polygons, it may not be possible to triangulate the two polygons in a compatible fashion, if ones choice of triangulation vertices is restricted to polygon corners. Is it always possible to produce compatible triangulations if additional vertices inside the polygon are allowed? We give a positive answer and construct a pair of such triangulations with O( n 2 ) new triangulation vertices. Moreover, we show that there exists a ‘universal’ way of triangulating an n -sided polygon with O( n 2 ) extra triangulation vertices. Finally, we also show that creating compatible triangulations requires a quadratic number of extra vertices in the worst case.

Small-Size

We show the existence of

\eps

\varepsilon

-Nets for Axis-Parallel Rectangles and Boxes

-nets of size

Nearest neighbor searching under uncertainty II

O\left(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon}\right)

Line transversals of balls and smallest enclosing cylinders in three dimensions

for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane and “fat” triangular ranges and for point sets in

Fréchet distance for curves, revisited

\boldsymbol{R}^3