Related papers: Distance geometry with and without the graph
The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the…
Computing the diameter of the intersection graphs of objects is a basic problem in computational geometry. Previous works showed that the complexity of computing the diameter mainly depends on the object types: for unit disks and squares in…
We present an algorithm for computation of cell adjacencies for well-based cylindrical algebraic decomposition. Cell adjacency information can be used to compute topological operations e.g. closure, boundary, connected components, and…
In this note we resolve three conjectures from [M. Dehmer, S. Pickl, Y. Shi, G. Yu, \emph{New inequalities for network distance measures by using graph spectra}, Discrete Appl. Math. 252 (2019), 17--27] on the comparison of distance…
Graph drawing research traditionally focuses on producing geometric embeddings of graphs satisfying various aesthetic constraints. After the geometric embedding is specified, there is an additional step that is often overlooked or ignored:…
Distances are pervasive in machine learning. They serve as similarity measures, loss functions, and learning targets; it is said that a good distance measure solves a task. When defining distances, the triangle inequality has proven to be a…
We consider mappings satisfying an upper bound for the distortion of families of curves. We establish lower bounds for the distortion of distances under such mappings. As applications, we obtain theorems on the discreteness of the limit…
Finding correspondences between 3D shapes is a crucial problem in computer vision and graphics, which is for example relevant for tasks like shape interpolation, pose transfer, or texture transfer. An often neglected but essential property…
The pursue of what are properties that can be identified to permit an automated reasoning program to generate and find new and interesting theorems is an interesting research goal (pun intended). The automatic discovery of new theorems is a…
Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include…
Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. These problems appear in Physics, Probability, Biology, Finance, or Industry, and the study of solutions and free boundaries…
Real networks are finite metric spaces. Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. Other forms of network geometry are the geometry of latent spaces underlying many networks, and…
We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…
The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the…
Let $G$ be a finite, simple connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The remoteness $\rho(G)$ of $G$ is the maximum of the average distances…
The concept of viewing graph solvability has gained significant interest in the context of structure-from-motion. A viewing graph is a mathematical structure where nodes are associated to cameras and edges represent the epipolar geometry…
We consider the problem of embedding the nodes of a hypergraph into Euclidean space under the assumption that the interactions arose through closeness to unknown hyperedge centres. In this way, we tackle the inverse problem associated with…
The minimum distance is one of the most important combinatorial characterizations of a code. The maximum likelihood decoding problem is one of the most important algorithmic problems of a code. While these problems are known to be hard for…
Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…
Computational difficulty of quadratic matching and the Gromov-Wasserstein distance has led to various approximation and relaxation schemes. One of such methods, relying on the notion of distance profiles, has been widely used in practice,…