In 2012 Driemel et al. \cite{DBLP:journals/dcg/DriemelHW12} introduced the concept of $c$-packed curves as a realistic input model. In the case when $c$ is a constant they gave a near linear time $(1+\varepsilon)$-approximation algorithm for computing the Fr\'echet distance between two $c$-packed polygonal curves. Since then a number of papers have used the model. In this paper we consider the problem

Previously we showed the family of 3-periodics in the elliptic billiard (confocal pair) is the image under a variable similarity transform of poristic triangles (those with non-concentric, fixed incircle and circumcircle). Both families conserve the ratio of inradius to circumradius and therefore also the sum of cosines. This is consisten with the fact that a similarity preserves angles. Here we study

In a recent article (Auer et al, Algorithmica 2016) it was claimed that every outer-1-planar graph has a planar visibility representation of area $O(n\log n)$. In this paper, we show that this is wrong: There are outer-1-planar graphs that require $\Omega(n^2)$ area in any planar drawing.

Generalised persistence functions (gp-functions) are defined on $(\mathbb{R}, \le)$-indexed diagrams in a given category. A sufficient condition for stability is also introduced. In the category of graphs, a standard way of producing gp-functions is proposed: steady and ranging sets for a given feature. The example of steady and ranging hubs is studied in depth; their meaning is investigated in three

Let $V$ be a finite set of vertices in the plane and $S$ be a finite set of polygonal obstacles. We show how to construct a plane $2$-spanner of the visibility graph of $V$ with respect to $S$. As this graph can have unbounded degree, we modify it in three easy-to-follow steps, in order to bound the degree to $7$ at the cost of slightly increasing the spanning ratio to 6.

The visual inspection of a hexahedral mesh with respect to element quality is difficult due to clutter and occlusions that are produced when rendering all element faces or their edges simultaneously. Current approaches overcome this problem by using focus on specific elements that are then rendered opaque, and carving away all elements occluding their view. In this work, we make use of advanced GPU

Constrained Optimization solution algorithms are restricted to point based solutions. In practice, single or multiple objectives must be satisfied, wherein both the objective function and constraints can be non-convex resulting in multiple optimal solutions. Real world scenarios include intersecting surfaces as Implicit Functions, Hyperspectral Unmixing and Pareto Optimal fronts. Local or global convexification

We prove a Kunneth theorem for the Vietoris-Rips homology and cohomology of a semi-uniform space. We then interpret this result for graphs, where we show that the Kunneth theorem holds for graphs with respect to the strong graph product. We finish by computing the Vietoris-Rips cohomology of the torus endowed with diferent semi-uniform structures.

We show that, for every set of $n$ points in $d$-dimensional unit cube, there is an empty axis-parallel box of volume at least $\Omega(d/n)$. In the opposite direction, we give a construction without an empty axis-parallel box of volume $O(d^2\log d/n)$. These improve on the previous best bounds of $\Omega(\log d/n)$ and $O(2^{7d}/n)$ respectively.

Fractional cascading is one of the influential techniques in data structures, as it provides a general framework for solving the important iterative search problem. In the problem, the input is a graph $G$ with constant degree and a set of values for every vertex of $G$. The goal is to preprocess $G$ such that when given a query value $q$, and a connected subgraph $\pi$ of $G$, we can find the predecessor

In this paper we introduce and describe an implementation of curved surface geometries within the Dune framework for grid-based discretizations. Therefore, we employ the abstraction of geometries as local-functions bound to a grid element, and the abstraction of a grid as connectivity of elements together with a grid-function that can be localized to the elements to provide element local parametrizations

Triangulation algorithms that conform to a set of non-intersecting input segments typically proceed in an incremental fashion, by inserting points first, and then segments. Inserting a segment amounts to delete all the triangles it intersects, define two polygons that fill the so generated hole and have the segment as shared basis, and then re-triangulate each polygon separately. In this paper we prove

This paper presents a methodology aiming at easing considerably the generation of high-quality meshes for complex 3D domains. We show that the whole mesh generation process can be controlled with only five parameters to generate in one stroke quality meshes for arbitrary geometries. The main idea is to build a meshsize field $h(x)$ taking local features of the geometry, such as curvatures, into account

This paper presents a new method to generate tool paths for machining freeform surfaces represented either as parametric surfaces or as triangular meshes. This method allows for the optimal tradeoff between the preferred feed direction field and the constant scallop height, and yields a minimized overall path length. The optimality is achieved by formulating tool path planning as a Poisson problem

We introduce trimmed NURBS surfaces with accurate boundary control, briefly called ABC-surfaces, as a solution to the notorious problem of constructing watertight or smooth ($G^1$ and $G^2)$ multi-patch surfaces within the function range of standard CAD/CAM systems and the associated file exchange formats. Our construction is based on the appropriate blend of a base surface, which traces out the intended

In 2015, Driemel, Krivo\v{s}ija and Sohler introduced the $(k,\ell)$-median problem for clustering polygonal curves under the Fr\'echet distance. Given a set of input curves, the problem asks to find $k$ median curves of at most $\ell$ vertices each that minimize the sum of Fr\'echet distances over all input curves to their closest median curve. A major shortcoming of their algorithm is that the input

Multidimensional Projection is a fundamental tool for high-dimensional data analytics and visualization. With very few exceptions, projection techniques are designed to map data from a high-dimensional space to a visual space so as to preserve some dissimilarity (similarity) measure, such as the Euclidean distance for example. In fact, although adopting distinct mathematical formulations designed to

If a graph can be drawn on the torus so that every two independent edges cross an even number of times, then the graph can be embedded on the torus.

This paper presents the first purely numerical (i.e., non-algebraic) subdivision algorithm for the isotopic approximation of a simple arrangement of curves. The arrangement is "simple" in the sense that any three curves have no common intersection, any two curves intersect transversally, and each curve is non-singular. A curve is given as the zero set of an analytic function $f:\mathbb{R}^2\rightarrow

Molecular robotics is challenging, so it seems best to keep it simple. We consider an abstract molecular robotics model based on simple folding instructions that execute asynchronously. Turning Machines are a simple 1D to 2D folding model, also easily generalisable to 2D to 3D folding. A Turning Machine starts out as a line of connected monomers in the discrete plane, each with an associated turning

Answering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene triangle. Our construction is based on Kleetopes, and shows that there exists an integer $i$ such that all convex $i$-iterated Kleetopes have a scalene face. However,

We utilize classical facts from topology to show that the classification problem in machine learning is always solvable under very mild conditions. Furthermore, given a training dataset, we show how topological formalism can be used to suggest the appropriate architectural choices for neural networks designed to be trained as classifiers on the data. Finally, we show how the architecture of a neural

Searching topological similarity between a pair of shapes or data is an important problem in data analysis and visualization. The problem of computing similarity measures using scalar topology has been studied extensively and proven useful in shape and data matching. Even though multi-field (or multivariate) topology-based techniques reveal richer topological features, research on computing similarity

Indirect methods for visual SLAM are gaining popularity due to their robustness to varying environments. ORB-SLAM2 is a benchmark method in this domain, however, the computation of descriptors in ORB-SLAM2 is time-consuming and the descriptors cannot be reused unless a frame is selected as a keyframe. To overcome these problems, we present FastORB-SLAM which is light-weight and efficient as it tracks

The problem of finding a diagonal flip path between two triangulations has been studied for nearly a century in the combinatorial (topological) setting and for decades in the geometric setting. In the geometric setting, finding a diagonal flip path between two triangulations that minimizes the number of diagonal flips over all such paths is NP-complete. However, when restricted to lattice triangulations

A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. When allowing empty lenses (a face in the arrangement induced by two edges that is bounded by a 2-cycle), two independent edges may cross arbitrarily many times in a star-simple drawing. We consider star-simple drawings of

Every finite graph admits a \emph{simple (topological) drawing}, that is, a drawing where every pair of edges intersects in at most one point. However, in combination with other restrictions simple drawings do not universally exist. For instance, \emph{$k$-planar graphs} are those graphs that can be drawn so that every edge has at most $k$ crossings (i.e., they admit a \emph{$k$-plane drawing}). It

We introduce a model for random geodesic drawings of the complete bipartite graph $K_{n,n}$ on the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$, where we select the vertices in each bipartite class of $K_{n,n}$ with respect to two non-degenerate probability measures on $\mathbb{S}^2$. It has been proved recently that many such measures give drawings whose crossing number approximates the Zarankiewicz

Stress minimization is among the best studied force-directed graph layout methods because it reliably yields high-quality layouts. It thus comes as a surprise that a novel approach based on stochastic gradient descent (Zheng, Pawar and Goodman, TVCG 2019) is claimed to improve on state-of-the-art approaches based on majorization. We present experimental evidence that the new approach does not actually

An ortho-radial grid is described by concentric circles and straight-line spokes emanating from the circles' center. An ortho-radial drawing is the analog of an orthogonal drawing on an ortho-radial grid. Such a drawing has an unbounded outer face and a central face that contains the origin. Building on the notion of an ortho-radial representation (Barth et al., SoCG, 2017), we describe an integer-linear

Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role for studying rigidity properties of mechanism in material designs. In this paper, we design an algorithm which solves this problem. It is based on the computations

We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. The problem of realizing $A$ by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of

This work uses projective mapping with n=349 consumers to evaluate nine commercial chocolate chips cookies, including a blind duplicate of a multinational best-selling brand and seven private labels. The data obtained are processed using both a statistical technique, Multiple Factor Analysis, and two geometric techniques, the recently appeared method SensoGraph using Gabriel clustering, and a novel

We study the planar orthogonal drawing style within the framework of partial representation extension. Let $(G,H,{\Gamma}_H )$ be a partial orthogonal drawing, i.e., G is a graph, $H\subseteq G$ is a subgraph and ${\Gamma}_H$ is a planar orthogonal drawing of H. We show that the existence of an orthogonal drawing ${\Gamma}_G$ of $G$ that extends ${\Gamma}_H$ can be tested in linear time. If such a

We give the first ANN-data structure for time series under the continuous Fr\'echet distance, where ANN stands for approximate near neighbor. Given a parameter $\epsilon \in (0,1]$, the data structure can be used to preprocess $n$ curves in Euclidean $\mathbb{R}$ (aka time series), each of complexity $m$, to answer queries with a curve of complexity $k$ by either returning a curve that lies within

The $2$-layer drawing model is a well-established paradigm to visualize bipartite graphs. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of $k$-planar graphs has been considered only for $k=1$ in this context. We provide several contributions that address this gap in the literature. First, we show tight density bounds for the classes

An interesting class of orthogonal representations consists of the so-called turn-regular ones, i.e., those that do not contain any pair of reflex corners that "point to each other" inside a face. For such a representation H it is possible to compute in linear time a minimum-area drawing, i.e., a drawing of minimum area over all possible assignments of vertex and bend coordinates of H. In contrast

K\'{a}rolyi, Pach, and T\'{o}th proved that every 2-edge-colored straight-line drawing of the complete graph contains a monochromatic plane spanning tree. It is open if this statement generalizes to other classes of drawings, specifically, to simple drawings of the complete graph. These are drawings where edges are represented by Jordan arcs, any two of which intersect at most once. We present two

Let F:R^n -> R be a feedforward ReLU neural network. It is well-known that for any choice of parameters, F is continuous and piecewise (affine) linear. We lay some foundations for a systematic investigation of how the architecture of F impacts the geometry and topology of its possible decision regions for binary classification tasks. Following the classical progression for smooth functions in differential

The fundamental result of Li, Long, and Srinivasan on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, combinatorics and data analysis (described as `the pinnacle of a long sequence of papers'). The goal of this paper is to give a simpler, self-contained, modular proof of this result for finite set systems. The only ingredient we assume

We proposed, in a recent paper (10.1002/nme.5987), a fast 3D parallel Delaunay kernel for tetrahedral mesh generation. This kernel was however incomplete in the sense that it lacked the necessary mesh improvement tools. The present paper builds on that previous work and proposes a fast parallel mesh improvement stage that delivers high-quality tetrahedral meshes compared to alternative open-source

An $h$-queue layout of a graph $G$ consists of a linear order of its vertices and a partition of its edges into $h$ queues, such that no two independent edges of the same queue nest. The minimum $h$ such that $G$ admits an $h$-queue layout is the queue number of $G$. We present two fixed-parameter tractable algorithms that exploit structural properties of graphs to compute optimal queue layouts. As

In a planar L-drawing of a directed graph (digraph) each edge e is represented as a polyline composed of a vertical segment starting at the tail of e and a horizontal segment ending at the head of e. Distinct edges may overlap, but not cross. Our main focus is on bimodal graphs, i.e., digraphs admitting a planar embedding in which the incoming and outgoing edges around each vertex are contiguous. We

Jigsaw puzzle solving, the problem of constructing a coherent whole from a set of non-overlapping unordered fragments, is fundamental to numerous applications, and yet most of the literature has focused thus far on less realistic puzzles whose pieces are identical squares. Here we formalize a new type of jigsaw puzzle where the pieces are general convex polygons generated by cutting through a global

Motivated by the fact that in a space where shortest paths are unique, no two shortest paths meet twice, we study a question posed by Greg Bodwin: Given a geodetic graph $G$, i.e., an unweighted graph in which the shortest path between any pair of vertices is unique, is there a philogeodetic drawing of $G$, i.e., a drawing of $G$ in which the curves of any two shortest paths meet at most once? We answer

Consider the natural question of how to measure the similarity of curves in the plane by a quantity that is invariant under translations of the curves. Such a measure is justified whenever we aim to quantify the similarity of the curves' shapes rather than their positioning in the plane, e.g., to compare the similarity of handwritten characters. Perhaps the most natural such notion is the (discrete)

Minimal paths are considered as a powerful and efficient tool for boundary detection and image segmentation due to its global optimality and well-established numerical solutions such as fast marching algorithm. In this paper, we introduce a flexible interactive image segmentation model based on the minimal geodesic framework in conjunction with region-based homogeneity enhancement. A key ingredient

A classic problem in unsupervised learning and data analysis is to find simpler and easy-to-visualize representations of the data that preserve its essential properties. A widely-used method to preserve the underlying hierarchical structure of the data while reducing its complexity is to find an embedding of the data into a tree or an ultrametric. The most popular algorithms for this task are the classic

We consider the problem of stretching pseudolines in a planar straight-line drawing to straight lines while preserving the straightness and the combinatorial embedding of the drawing. We answer open questions by Mchedlidze et al. by showing that not all instances with two pseudolines are stretchable. On the positive side, for $k\geq 2$ pseudolines intersecting in a single point, we prove that in case

Given a polygon $P$ in the plane that can be translated, rotated and enlarged arbitrarily inside a unit square, the goal is to find a set of lines such that at least one of them always hits $P$ and the number of lines is minimized. We prove the solution is always a regular grid or a set of equidistant parallel lines, whose distance depends on $P$.

Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data. With few exceptions, graph drawing algorithms typically optimize one such criterion, usually at the expense of others. We propose a layout approach, Graph Drawing via Gradient Descent, $(GD)^2$, that can handle multiple

Battle, Harary, and Kodama (1962) and independently Tutte (1963) proved that the complete graph with nine vertices is not biplanar. Aiming towards simplicity and brevity, in this note we provide a short proof of this claim.

Network equilibrium models represent a versatile tool for the analysis of interconnected objects and their relationships. They have been widely employed in both science and engineering to study the behavior of complex systems under various conditions, including external perturbations and damage. In this paper, network equilibrium models are revisited through graph-theory laws and attributes with special

The problem considered in this paper is the weighted obnoxious facility location in the convex hull of demand points. The objective function is to maximize the smallest weighted distance between a facility and a set of demand points. Three new optimal solution approaches are proposed. Two variants of the "Big Triangle Small Triangle" global optimization method, and a procedure based on intersection

In standard rounding, we want to map each value $X$ in a large continuous space (e.g., $R$) to a nearby point $P$ from a discrete subset (e.g., $Z$). This process seems to be inherently discontinuous in the sense that two consecutive noisy measurements $X_1$ and $X_2$ of the same value may be extremely close to each other and yet they can be rounded to different points $P_1\ne P_2$, which is undesirable

In this paper we propose an algorithm for the approximate k-Nearest-Neighbors problem. According to the existing researches, there are two kinds of approximation criterion. One is the distance criteria, and the other is the recall criteria. All former algorithms suffer the problem that there are no theoretical guarantees for the two approximation criterion. The algorithm proposed in this paper unifies

A low-dimensional version of our main result is the following `converse' of the Conway-Gordon-Sachs Theorem on intrinsic linking of the graph $K_6$ in 3-space: For any integer $z$ there are 6 points $1,2,3,4,5,6$ in 3-space, of which every two $i,j$ are joint by a polygonal line $ij$, the interior of one polygonal line is disjoint with any other polygonal line, the linking coefficient of any pair disjoint

We present an algorithm to find near-optimal weighted Steiner minimal trees in the plane. The algorithm is implemented in Evolver programming language, which already contains many built-in energy minimisation routines. Some are invoked in the program, which enable it to consist of only 183 lines of source code. Our algorithm reproduces the physical experiment of a soap film detaching from connected

Given a planar point set $X$, we study the convex shells and the convex layers. We prove that when $X$ consists of points independently and uniformly sampled inside a convex polygon with $k$ vertices, the expected number of vertices on the first $t$ convex shells is $O(kt\log{n})$ for $t=O(\sqrt{n})$, and the expected number of vertices on the first $t$ convex layers is $O\left(kt^{3}\log{\frac{n}{t^2}}\right)$

Proximity complexes and filtrations are a central construction in topological data analysis. Built using distance functions or more generally metrics, they are often used to infer connectivity information from point clouds. We investigate proximity complexes and filtrations built over the Chebyshev metric, also known as the maximum metric or $\ell_{\infty}$ metric, rather than the classical Euclidean