Related papers: Efficient Fr\'echet distance queries for segments
In machine learning and particularly in topological data analysis, $\epsilon$-graphs are important tools but are generally hard to compute as the distance calculation between n points takes time O(n^2) classically. Recently, quantum…
We study the problem of computing the diameter and the mean distance of a continuous graph, i.e., a connected graph where all points along the edges, instead of only the vertices, must be taken into account. It is known that for continuous…
We study a map matching problem, the task of finding in an embedded graph a path that has low distance to a given curve in R^2. The Fr\'echet distance is a common measure for this problem. Efficient methods exist to compute the best path…
We consider the following problem: Preprocess a set $\mathcal{S}$ of $n$ axis-parallel boxes in $\mathbb{R}^d$ so that given a query of an axis-parallel box in $\mathbb{R}^d$, the pairs of boxes of $\mathcal{S}$ whose intersection…
The string indexing problem is a fundamental computational problem with numerous applications, including information retrieval and bioinformatics. It aims to efficiently solve the pattern matching problem: given a text T of length n for…
Given a polyline on $n$ vertices, the polyline simplification problem asks for a minimum size subsequence of these vertices defining a new polyline whose distance to the original polyline is at most a given threshold under some distance…
Let $P$ be a set of $n$ points in the plane. In this paper we study a new variant of the circular separability problem in which a point set $P$ is preprocessed so that one can quickly answer queries of the following form: Given a geometric…
In the $(1+\varepsilon,r)$-approximate near-neighbor problem for curves (ANNC) under some distance measure $\delta$, the goal is to construct a data structure for a given set $\mathcal{C}$ of curves that supports approximate near-neighbor…
Clustering is the task of partitioning a given set of geometric objects. This is thoroughly studied when the objects are points in the euclidean space. There are also several approaches for points in general metric spaces. In this thesis we…
For fixed $h \geq 2$, we consider the task of adding to a graph $G$ a set of weighted shortcut edges on the same vertex set, such that the length of a shortest $h$-hop path between any pair of vertices in the augmented graph is exactly the…
We present an efficient dynamic data structure that supports geodesic nearest neighbor queries for a set $S$ of point sites in a static simple polygon $P$. Our data structure allows us to insert a new site in $S$, delete a site from $S$,…
Merge trees are a common topological descriptor for data with a hierarchical component, such as terrains and scalar fields. The interleaving distance, in turn, is a common distance for comparing merge trees. However, the interleaving…
Let $P$ be a convex polygon in the plane, and let $T$ be a triangulation of $P$. An edge $e$ in $T$ is called a diagonal if it is shared by two triangles in $T$. A flip of a diagonal $e$ is the operation of removing $e$ and adding the…
We propose a geometric structure induced by any given convex polygon $P$, called $Nest(P)$, which is an arrangement of $\Theta(n^2)$ line segments, each of which is parallel to an edge of $P$, where $n$ denotes the number of edges of $P$.…
We consider succinct data structures for representing a set of $n$ horizontal line segments in the plane given in rank space to support \emph{segment access}, \emph{segment selection}, and \emph{segment rank} queries. A segment access query…
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 revisit the classic problem of simplex range searching and related problems in computational geometry. We present a collection of new results which improve previous bounds by multiple logarithmic factors that were caused by the use of…
We consider a variant of two-point Euclidean shortest path query problem: given a polygonal domain, build a data structure for two-point shortest path query, provided that query points always lie on the boundary of the domain. As a main…
CAT(0) metric spaces constitute a far-reaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path {\gamma}(x,y). In this…
In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time.…