Related papers: The Discrete Fr\'echet Gap
An important problem in geometric computing is defining and computing similarity between two geometric shapes, e.g. point sets, curves and surfaces, etc. Important geometric and topological information of many shapes can be captured by…
In this note, we present a novel measure of similarity between two functions. It quantifies how the sub-optimality gaps of two functions convert to each other, and unifies several existing notions of functional similarity. We show that it…
Consider the continuum of points along the edges of a network, i.e., a connected, undirected graph with positive edge weights. We measure the distance between these points in terms of the weighted shortest path distance, called the network…
The Hausdorff distance is a relatively new measure of similarity of graphs. The notion of the Hausdorff distance considers a special kind of a common subgraph of the compared graphs and depends on the structural properties outside of the…
In the stochastic exploration of geometrically embedded graphs, intuition suggests that providing a shortcut between a pair of nodes reduces the mean first passage time of the entire graph. Counterintuitively, we find a Braess paradox…
For most optimisation methods an essential assumption is the vector space structure of the feasible set. This condition is not fulfilled if we consider optimisation problems over the sphere. We present an algorithm for solving a special…
The advances of modern localization techniques and the wide spread of mobile devices have provided us great opportunities to collect and mine human mobility trajectories. In this work, we focus on passively collected trajectories, which are…
In this paper we present an interesting gadget based on the chain pair simplification problem under the discrete Fr\'echet distance (CPS-3F), which allows the construction of arbitrarily long paths that must be chosen in the simplification…
The Levenshtein distance is an important tool for the comparison of symbolic sequences, with many appearances in genome research, linguistics and other areas. For efficient applications, an approximation by a distance of smaller…
In this manuscript we consider random objects being measured in multiple metric spaces, which may arise when those objects may be measured in multiple distinct ways. In this new multivariate setting, we define a Fr\'echet covariance and…
Fr\'echet means, conceptually appealing, generalize the Euclidean expectation to general metric spaces. We explore how well Fr\'echet means can be estimated from independent and identically distributed samples and uncover a fundamental…
Similarity measures are used extensively in machine learning and data science algorithms. The newly proposed graph Relative Hausdorff (RH) distance is a lightweight yet nuanced similarity measure for quantifying the closeness of two graphs.…
To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Fr\'echet mean. In this…
Fr\'echet mean and variance provide a way of obtaining mean and variance for general metric space valued random variables and can be used for statistical analysis of data objects that lie in abstract spaces devoid of algebraic structure and…
A novel algorithm is proposed for quantitative comparisons between compact surfaces embedded in the three-dimensional Euclidian space. The key idea is to identify those objects with the associated surface measures and compute a weak…
We examine the possibility of approximating Maximum Vertex-Disjoint Shortest Paths. In this problem, the input is an edge-weighted (directed or undirected) $n$-vertex graph $G$ along with $k$ terminal pairs…
Fr\'echet regression is becoming a mainstay in modern data analysis for analyzing non-traditional data types belonging to general metric spaces. This novel regression method is especially useful in the analysis of complex health data such…
Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of…
We study two popular ways to sketch the shortest path distances of an input graph. The first is distance preservers, which are sparse subgraphs that agree with the distances of the original graph on a given set of demand pairs. Prior work…
We consider the problem of determining the number of distinct distances between two point sets in $\mathbb{R}^2$ where one point set $\mathcal{P}_1$ of size $m$ lies on a real algebraic curve of fixed degree $r$, and the other point set…