Related papers: Shortest paths in planar domains with hyperbolic t…
We introduce the heat method for computing the shortest geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of…
We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on…
Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately…
The length of the shortest closed geodesic in a hyperbolic surface $X$ is called the systole of $X.$ When $X$ is an $n$-times punctured sphere $\hat{ \mathbb{C}} \setminus A$ where $A \subset \hat{\mathbb{C}}$ is a finite set of cardinality…
Dijkstra's algorithm is the standard method for computing shortest paths on arbitrary graphs. However, it is slow for large graphs, taking at least linear time. It has been long known that for real world road networks, creating a hierarchy…
Geodesic distance, commonly called shortest path length, has proved useful in a great variety of disciplines. It has been playing a significant role in search engine at present and so attracted considerable attention at the last few…
Geodesic problems involve computing trajectories between prescribed initial and final states to minimize a user-defined measure of distance, cost, or energy. They arise throughout physics and engineering -- for instance, in determining…
We consider the problem of computing shortest paths in a dense motion-planning roadmap $\mathcal{G}$. We assume that~$n$, the number of vertices of $\mathcal{G}$, is very large. Thus, using any path-planning algorithm that directly searches…
A common approach to compute distances on continuous surfaces is by considering a discretized polygonal mesh approximating the surface and estimating distances on the polygon. We show that exact geodesic distances restricted to the polygon…
The shortest path problem is related to many dynamic processes on networks, ranging from routing in communication networks to signaling in molecular interaction networks. When the network is fully known, the shortest path problem can be…
We characterize geodesic paths in the $n$-dimensional unit sphere under sup norm. A geodesic path between two points is a shortest curve joining the two points.
We present the first polynomial time approximation algorithm for computing shortest paths in weighted three-dimensional domains. Given a polyhedral domain $\D$, consisting of $n$ tetrahedra with positive weights, and a real number…
Embedding into hyperbolic space is emerging as an effective representation technique for datasets that exhibit hierarchical structure. This development motivates the need for algorithms that are able to effectively extract knowledge and…
Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we establish an upper bound of the length of an $n^{th}$ shortest…
By "geodesic" we mean any sequence of vertices $(v_1,v_2,...,v_k)$ of a graph $G$ that constitute a shortest path from $v_1$ to $v_k$. We propose a novel, natural algorithm to enumerate all geodesics of $G$, and pit it (using Mathematica)…
We propose a path planning methodology for a mobile robot navigating through an obstacle-filled environment to generate a reference path that is traceable with moderate sensing efforts. The desired reference path is characterized as the…
We study inequalities between the hyperbolic metric and intrinsic metrics in convex polygonal domains in the complex plane. Special attention is paid to the triangular ratio metric in rectangles. A local study leads to an investigation of…
We consider certain inequalities among the Apollonian metric, the Apollonian inner metric, the $j$ metric and the quasihyperbolic metric. We verify that whether these inequalities can occur in simply connected planar domains and in proper…
Probabilistic Latent Variable Models (LVMs) excel at modeling complex, high-dimensional data through lower-dimensional representations. Recent advances show that equipping these latent representations with a Riemannian metric unlocks…
A geodesic is a shortest path which connects a pair of vertices of a graph G. In this paper we define the geodesic subpath number gpn(G) of a graph G as the number of geodesics in G. The number of subtrees and subpaths are already studied…