In this report, the implementation of an algorithm for computing the 2D Voronoi diagram on a coarse-grained multicomputer (CGM) is presented. The used metric is Euclidean. In the coarsegrained model, each processor has O  p ≫O 1 local memory, where n is the size of the data and p the number of processors. In one communication round, each processor can exchange data with any other processor,...

This paper presents a family of layered feed-forward networks that is able to uniformly approximate functions on any metric space, and also on a wide variety of non-metric spaces. Non-Euclidean input spaces are frequently encountered in practice, while usual approximation schemes are guaranteed to work only on Euclidean metric spaces. Theoretical foundations are provided, as well as practical a...

ABSTRACT There are various useful metrics for finding the distance between two points in Euclidean space. Metrics for finding the distance between two rigid body locations in Euclidean space depend on both the coordinate frame and units used. A metric independent of these choices is desirable. This paper presents a metric for a finite set of rigid body displacements. The methodology uses the pr...

We present an optimal Θ(n log n) algorithm for determining shortest paths according to the L1 (L∞) metric in the presence of disjoint polygonal obstacles in the plane. Our algorithm requires only linear O(n) space to build a planar subdivision (a Shortest Path Map) with respect to a fixed source point such that the length of a shortest path from the source to any query point can be reported in ...

This report provides a mathematically thorough review and investigation of Metric Multidimensional scaling (MDS) through the analysis of Euclidean distances in input and output spaces. By combining a geometric approach with modern linear algebra and multivariate analysis, Metric MDS is viewed as a Euclidean distance embedding transformation that converts between coordinate and coordinate-free r...

We initiate the study of metric embeddings with outliers. Given some metric space (X, ρ) we wish to find a small set of outlier points K ⊂ X and either an isometric or a low-distortion embedding of (X \K, ρ) into some target metric space. This is a natural problem that captures scenarios where a small fraction of points in the input corresponds to noise. For the case of isometric embeddings we ...

We present a polynomial time approximation scheme (PTAS) for the Steiner tree problem with polygonal obstacles in the plane with running time O(n log n), where n denotes the number of terminals plus obstacle vertices. To this end, we show how a planar spanner of size O(n log n) can be constructed that contains a (1 + ǫ)-approximation of the optimal tree. Then one can find an approximately optim...