Related papers: Realizing Metric Spaces with Convex Obstacles
The definition of weighted distance measure involves weights. The paper proposes a weighted distance measure without the help of weights. Here, weights are intrinsically added to the measure, and for this, the concept of metric space is…
Associated to any finite metric space are a large number of objects and quantities which provide some degree of structural or geometric information about the space. In this paper we show that in the setting of subsets of weighted Hamming…
Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion…
In algorithms for finite metric spaces, it is common to assume that the distance between two points can be computed in constant time, and complexity bounds are expressed only in terms of the number of points of the metric space. We…
We show that in complete metric spaces, $4$-hyperconvexity is equivalent to finite hyperconvexity. Moreover, every complete, almost $n$-hyperconvex metric space is $n$-hyperconvex. This generalizes among others results of Lindenstrauss and…
Trajectory optimization offers mature tools for motion planning in high-dimensional spaces under dynamic constraints. However, when facing complex configuration spaces, cluttered with obstacles, roboticists typically fall back to…
A new method of metric space investigation, based on classification of its finite subspaces, is suggested. It admits to derive information on metric space properties which is encoded in metric. The method describes geometry in terms of only…
We introduce the convex matching distance, a novel metric for comparing functions with values in the real plane. This metric measures the maximal bottleneck distance between the persistence diagrams associated with the convex combinations…
In this paper, we present a new approach of creating PTAS to the TSP problems by defining a bounded-curvature surface embedded spaces. Using this definition we prove: - A bounded-curvature surface embedded spaces TSP admits to a PTAS. -…
We address the problem of merging graph and feature-space information while learning a metric from structured data. Existing algorithms tackle the problem in an asymmetric way, by either extracting vectorized summaries of the graph…
This paper establishes a metric framework for Spencer complexes based on the geometric theory of compatible pairs $(D,\lambda)$ in principal bundle constraint systems, solving fundamental technical problems in computing Spencer cohomology…
Many algorithmic problems can be solved (almost) as efficiently in metric spaces of bounded doubling dimension as in Euclidean space. Unfortunately, the metric space defined by points in a simple polygon equipped with the geodesic distance…
The main contribution of this paper is the proof of the convexity of the omni-directional tethered robot workspace (namely, the set of all tether-length-admissible robot configurations), as well as a set of distance-optimal tethered path…
One of the basic problems in discrete geometry is to determine the most efficient packing of congruent replicas of a given convex set $K$ in the plane or in space. The most commonly used measure of efficiency is density. Several types of…
Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…
We investigate approximation algorithms for several fundamental optimization problems on geometric packing. The geometric objects considered are very generic, namely $d$-dimensional convex fat objects. Our main contribution is a versatile…
Weighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal-Pruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric set-cover).…
Many years ago John Tyrell a lecturer at King's college London challenged his Ph.D. students with the following puzzle: show that there is a unique triangle of minimal perimeter with exactly one vertex to lie on one of three given lines,…
We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a…
In recent years, considerable advances have been made in the study of properties of metric spaces in terms of their doubling dimension. This line of research has not only enhanced our understanding of finite metrics, but has also resulted…