Spanners for metric spaces have been extensively studied, both in general metrics and in restricted classes, perhaps most notably in low-dimensional Euclidean spaces -- due to their numerous applications. Euclidean spanners can be viewed as means of compressing the $\binom{n}{2}$ pairwise distances of a $d$-dimensional Euclidean space into $O(n) = O_{\epsilon,d}(n)$ spanner edges, so that the spanner

We provide, for any $r\in (0,1)$, lower and upper bounds on the maximal density of a packing in the Euclidean plane of discs of radius $1$ and $r$. The lower bounds are mostly folk, but the upper bounds improve the best previously known ones for any $r\in[0.11,0.74]$. For many values of $r$, this gives a fairly good idea of the exact maximum density. In particular, we get new intervals for $r$ which

We consider a variant of the $k$-center clustering problem in $\Re^d$, where the centers can be divided into two subsets, one, the red centers of size $p$, and the other, the blue centers of size $q$, where $p+q=k$, and such that each red center and each blue center must be apart a distance of at least some given $\alpha \geq 0$, with the aim of minimizing the covering radius. We provide a bi-criteria

This short contribution presents a method for generating $N$-point spherical configurations with low mesh ratios. The method extends Caspar-Klug icosahedral point-grids to non-icosahedral nets through the use of planar barycentric coordinates, which are subsequently interpreted as spherical area coordinates for spherical point sets. The proposed procedure may be applied iteratively and is parameterised

We consider the problem 2-Dim-Bounding-Surface. 2-Dim-Bounded-Surface asks whether or not there is a subcomplex $S$ of a simplicial complex $K$ homeomorphic to a given compact, connected surface bounded by a given subcomplex $B\subset K$. 2-Dim-Bounding-Surface is NP-hard. We show it is fixed-parameter tractable with respect to the treewidth of the 1-skeleton of the simplicial complex $K$. Using some

We implement an algorithm RSHT (Random Simple-Homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines elementary simplicial collapses with pure elementary expansions. For triangulated d-manifolds with d < 7, we show that RSHT reduces to (random) bistellar

In this paper we study the following problem: Given $k$ disjoint sets of points, $P_1, \ldots, P_k$ on the plane, find a minimum cardinality set $\mathcal{T}$ of arbitrary rectangles such that each rectangle contains points of just one set $P_i$ but not the others. We prove the NP-hardness of this problem.

In this paper, we consider the Minimum-Load $k$-Clustering/Facility Location (MLkC) problem where we are given a set $P$ of $n$ points in a metric space that we have to cluster and an integer $k$ that denotes the number of clusters. Additionally, we are given a set $F$ of cluster centers in the same metric space. The goal is to select a set $C\subseteq F$ of $k$ centers and assign each point in $P$

Curves are essential concepts that enable compounded aesthetic curves, e.g., to assemble complex silhouettes, match a specific curvature profile in industrial design, and construct smooth, comfortable, and safe trajectories in vehicle-robot navigation systems. New mechanisms able to encode, generate, evaluate, and deform aesthetic curves are expected to improve the throughput and the quality of industrial

We develop a method for analyzing spatiotemporal anomalies in geospatial data using topological data analysis (TDA). To do this, we use persistent homology (PH), a tool from TDA that allows one to algorithmically detect geometric voids in a data set and quantify the persistence of these voids. We construct an efficient filtered simplicial complex (FSC) such that the voids in our FSC are in one-to-one

The purpose of the current study is to investigate a special case of art gallery problem, namely Sculpture Garden Problem. In the said problem, for a given polygon $P$, the ultimate goal is to place the minimum number of guards to define the interior polygon $P$ by applying a monotone Boolean formula composed of the guards. As the findings indicate, the conjecture about the issue that in the worst

We extend the theory of PAC learning in a way which allows to model a rich variety of learning tasks where the data satisfy special properties that ease the learning process. For example, tasks where the distance of the data from the decision boundary is bounded away from zero. The basic and simple idea is to consider partial concepts: these are functions that can be undefined on certain parts of the

We study a generalization of $k$-center clustering, first introduced by Kavand et. al., where instead of one set of centers, we have two types of centers, $p$ red and $q$ blue, and where each red center is at least $\alpha$ distant from each blue center. The goal is to minimize the covering radius. We provide an approximation algorithm for this problem, and a polynomial time algorithm for the constrained

We study the $c$-approximate near neighbor problem under the continuous Fr\'echet distance: Given a set of $n$ polygonal curves with $m$ vertices, a radius $\delta > 0$, and a parameter $k \leq m$, we want to preprocess the curves into a data structure that, given a query curve $q$ with $k$ vertices, either returns an input curve with Fr\'echet distance at most $c\cdot \delta$ to $q$, or returns that

This paper presents a unified computational framework for the estimation of distances, geodesics and barycenters of merge trees. We extend recent work on the edit distance [106] and introduce a new metric, called the Wasserstein distance between merge trees, which is purposely designed to enable efficient computations of geodesics and barycenters. Specifically, our new distance is strictly equivalent

Parts fabricated by additive manufacturing (AM) are often fabricated first as a near-net shape, a combination of intended nominal geometry and sacrificial support structures, which need to be removed in a subsequent post-processing stage using subtractive manufacturing (SM). In this paper, we present a framework for optimizing the build orientation with respect to removability of support structures

Geometry processing of 3D objects is of primary interest in many areas of computer vision and graphics, including robot navigation, 3D object recognition, classification, feature extraction, etc. The recent introduction of cheap range sensors has created a great interest in many new areas, driving the need for developing efficient algorithms for 3D object processing. Previously, in order to capture

In the Euclidean $k$-Means problem we are given a collection of $n$ points $D$ in an Euclidean space and a positive integer $k$. Our goal is to identify a collection of $k$ points in the same space (centers) so as to minimize the sum of the squared Euclidean distances between each point in $D$ and the closest center. This problem is known to be APX-hard and the current best approximation ratio is a

We consider the following geometric optimization problem: Given $ n $ axis-aligned rectangles in the plane, the goal is to find a set of horizontal segments of minimum total length such that each rectangle is stabbed. A segment stabs a rectangle if it intersects both its left and right edge. As such, this stabbing problem falls into the category of weighted geometric set cover problems for which techniques

The greedy spanner in a low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh showed that the greedy spanner in $\mathbb{R}^2$ admits a sublinear separator in a strong sense: any subgraph

In this paper we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode encodes the rank invariant as a $\mathbb{Z}$-linear combination of rank invariants of indicator modules supported

It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance. The question of whether the JL method is optimal for practical measures of distortion was recently raised in \cite{BFN19} (NeurIPS'19). They provided upper bounds

We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. It is known that the problem admits an algorithm with running time f(c).n^{O(c)}, where n is the size of the graph G and c is the size of the two-dimensional

We introduce a novel learning-based, visibility-aware, surface reconstruction method for large-scale, defect-laden point clouds. Our approach can cope with the scale and variety of point cloud defects encountered in real-life Multi-View Stereo (MVS) acquisitions. Our method relies on a 3D Delaunay tetrahedralization whose cells are classified as inside or outside the surface by a graph neural network

We introduce the axiomatic theory of Spherical Occlusion Diagrams as a tool to study certain combinatorial properties of polyhedra in $\mathbb R^3$, which are of central interest in the context Art Gallery problems for polyhedra and other visibility-related problems in discrete and computational geometry.

We introduce giotto-ph, a high-performance, open-source software package for the computation of Vietoris--Rips barcodes. giotto-ph is based on Morozov and Nigmetov's lockfree (multicore) implementation of Ulrich Bauer's Ripser package. It also contains a re-implementation of Boissonnat and Pritam's "Edge Collapser", implemented so far only in the GUDHI library. Our contribution is twofold: on the one

In the problem Localization and trilateration with minimum number of landmarks, we faced 3-guard and classic Art Gallery Problem. The goal of the art gallery problem is to find the minimum number of guards within a simple polygon to observe and protect the entire of it. It has many applications in Robotics, Telecommunication and so on and there are some approaches to handle the art gallery problem

Floor planning is an important and difficult task in architecture. When planning office buildings, rooms that belong to the same organisational unit should be placed close to each other. This leads to the following NP-hard mathematical optimization problem. Given the outline of each floor, a list of room sizes, and, for each room, the unit to which it belongs, the aim is to compute floor plans such

The following geometric vehicle scheduling problem has been considered: given continuous curves $f_1, \ldots, f_n : \mathbb{R} \rightarrow \mathbb{R}^2$, find non-negative delays $t_1, \ldots, t_n$ minimizing $\max \{ t_1, \ldots, t_n \}$ such that, for every distinct $i$ {and $j$} and every time $t$, $| f_j (t - t_j) - f_i (t - t_i) | > \ell$, where~$\ell$ is a given safety distance. We study a variant

Reeb graphs are widely used in a range of fields for the purposes of analyzing and comparing complex spaces via a simpler combinatorial object. Further, they are closely related to extended persistence diagrams, which largely but not completely encode the information of the Reeb graph. In this paper, we investigate the effect on the persistence diagram of a particular continuous operation on Reeb graphs;

The notion of persistence partial matching, as a generalization of partial matchings between persistence modules, is introduced. We study how to obtain a persistence partial matching $\mathcal{G}_f$, and a partial matching $\mathcal{M}_f$, induced by a morphism $f$ between persistence modules, both being linear with respect to direct sums of morphisms. Some of their properties are also provided, including

Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into 4 triangles by joining the midpoints of its edges. We show the existence of a uniform $\delta>0$ such that, at any step of the subdivision, all the triangle angles lie in the interval $(\delta, \pi -\delta)$. Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision

In this paper, we present an algorithm for computing a feedback vertex set of a unit disk graph of size $k$, if it exists, which runs in time $2^{O(\sqrt{k})}(n+m)$, where $n$ and $m$ denote the numbers of vertices and edges, respectively. This improves the $2^{O(\sqrt{k}\log k)}n^{O(1)}$-time algorithm for this problem on unit disk graphs by Fomin et al. [ICALP 2017]. Moreover, our algorithm is optimal

We characterize the triples of interior angles that are possible in non-self-crossing triangles with circular-arc sides, and we prove that a given cyclic sequence of angles can be realized by a non-self-crossing polygon with circular-arc sides whenever all angles are at most pi. As a consequence of these results, we prove that every cactus has a planar Lombardi drawing (a drawing with edges depicted

We introduce a method for jointly registering ensembles of partitioned datasets in a way which is both geometrically coherent and partition-aware. Once such a registration has been defined, one can group partition blocks across datasets in order to extract summary statistics, generalizing the commonly used order statistics for scalar-valued data. By modeling a partitioned dataset as an unordered $k$-tuple

We study the median slope selection problem in the oblivious RAM model. In this model memory accesses have to be independent of the data processed, i.e., an adversary cannot use observed access patterns to derive additional information about the input. We show how to modify the randomized algorithm of Matou\v{s}ek (1991) to obtain an oblivious version with O(n log^2 n) expected time for n points in

In the minimum constraint removal ($MCR$), there is no feasible path to move from the starting point towards the goal and, the minimum constraints should be removed in order to find a collision-free path. It has been proved that $MCR$ problem is $NP-hard$ when constraints have arbitrary shapes or even they are in shape of convex polygons. However, it has a simple linear solution when constraints are

Given a convex polyhedral surface P, we define a tailoring as excising from P a simple polygonal domain that contains one vertex v, and whose boundary can be sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In particular, a digon-tailoring cuts off from P a digon containing v, a subset of P bounded by two equal-length geodesic segments that share endpoints, and can then zip

In the two-dimensional orthogonal colored range counting problem, we preprocess a set, $P$, of $n$ colored points on the plane, such that given an orthogonal query rectangle, the number of distinct colors of the points contained in this rectangle can be computed efficiently. For this problem, we design three new solutions, and the bounds of each can be expressed in some form of time-space tradeoff

We revisit the fundamental problem of I/O-efficiently computing $r$-way separators on planar graphs. An $r$-way separator divides a planar graph with $N$ vertices into $O(r)$ regions of size $O(N/r)$ and $O(\sqrt {Nr})$ boundary vertices in total, where boundary vertices are vertices that are adjacent to more than one region. Such separators are used in I/O-efficient solutions to many fundamental problems

Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The

We define a persistent cohomology invariant called persistent cup-length which is able to extract non trivial information about the evolution of the cohomology ring structure across a filtration. We also devise algorithms for the computation of this invariant and we furthermore show that the persistent cup-length is 2-Lipschitz continuous with respect to the homotopy interleaving and Gromov-Hausdorff

In this paper, we study the online Euclidean spanners problem for points in $\mathbb{R}^d$. Suppose we are given a sequence of $n$ points $(s_1,s_2,\ldots, s_n)$ in $\mathbb{R}^d$, where point $s_i$ is presented in step~$i$ for $i=1,\ldots, n$. The objective of an online algorithm is to maintain a geometric $t$-spanner on $S_i=\{s_1,\ldots, s_i\}$ for each step~$i$. First, we establish a lower bound

We consider the time-convex hull problem in the presence of two orthogonal highways H. In this problem, the travelling speed on the highway is faster than off the highway, and the time-convex hull of a point set P is the closure of P with respect to the inclusion of shortest time-paths. In this paper, we provide the algorithm for constructing the time-convex hull with two orthogonal highways. We reach

Seminal works on light spanners over the years provide spanners with optimal or near-optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Two shortcomings of all previous work on light spanners are: (1) The techniques are ad hoc per graph class, and thus can't be applied broadly (e.g., some require large stretch and are thus suitable to general

Resolving an open question from 2006, we prove the existence of light-weight bounded-degree spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple $\mathcal{O}(\log^*n)$-round distributed algorithm that given a unit ball graph $G$ with $n$ vertices and a positive constant $\epsilon < 1$ finds a $(1+\epsilon)$-spanner with constant bounds on its maximum degree

Let $S$ be a set of $n$ sites, each associated with a point in $\mathbb{R}^2$ and a radius $r_s$ and let $\mathcal{D}(S)$ be the disk graph on $S$. We consider the problem of designing data structures that maintain the connectivity structure of $\mathcal{D}(S)$ while allowing the insertion and deletion of sites. For unit disk graphs we describe a data structure that has $O(\log^2n)$ amortized update

We prove NP-completeness of Yin-Yang / Shiromaru-Kuromaru pencil-and-paper puzzles. Viewed as a graph partitioning problem, we prove NP-completeness of partitioning a rectangular grid graph into two induced trees (normal Yin-Yang), or into two induced connected subgraphs (Yin-Yang without $2 \times 2$ rule), subject to some vertices being pre-assigned to a specific tree/subgraph.

Many clustering algorithms are guided by certain cost functions such as the widely-used $k$-means cost. These algorithms divide data points into clusters with often complicated boundaries, creating difficulties in explaining the clustering decision. In a recent work, Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML'20) introduced explainable clustering, where the cluster boundaries are axis-parallel

Efficient algorithms for solving optimal transport problems are important for measuring and optimizing distances between functions. In the $L^2$ semi-discrete context, this problem consists of finding a map from a continuous density function to a discrete set of points so as to minimize the transport cost, using the squared Euclidean distance as the cost function. This has important applications in

In this paper, we present our heuristic solutions to the problems of finding the maximum and minimum area polygons with a given set of vertices. Our solutions are based mostly on two simple algorithmic paradigms: greedy method and local search. The greedy heuristic starts with a simple polygon and adds vertices one by one, according to a weight function. A crucial ingredient to obtain good solutions

A realizer, commonly known as Schnyder woods, of a triangulation is a partition of its interior edges into three oriented rooted trees. A flip in a realizer is a local operation that transforms one realizer into another. Two types of flips in a realizer have been introduced: colored flips and cycle flips. A corresponding flip graph is defined for each of these two types of flips. The vertex sets are

We show how to edge-unfold a new class of convex polyhedra, specifically a new class of prismatoids (the convex hull of two parallel convex polygons, called the top and base), by constructing a nonoverlapping "petal unfolding" in two new cases: (1) when the top and base are sufficiently far from each other; and (2) when the base is a rectangle and all other faces are nonobtuse triangles. The latter

Consider two axis-aligned rectilinear simple polygons in the domain consisting of axis-aligned rectilinear obstacles in the plane such that the bounding boxes, one for each obstacle and one for each polygon, are disjoint. We present an algorithm that computes a minimum-link rectilinear shortest path connecting the two polygons in $O((N+n)\log (N+n))$ time using $O(N+n)$ space, where $n$ is the number

In this paper, we present a linear-time approximation scheme for $k$-means clustering of \emph{incomplete} data points in $d$-dimensional Euclidean space. An \emph{incomplete} data point with $\Delta>0$ unspecified entries is represented as an axis-parallel affine subspaces of dimension $\Delta$. The distance between two incomplete data points is defined as the Euclidean distance between two closest

We present simpler algorithms for two closely related morphing problems, both based on the barycentric interpolation paradigm introduced by Floater and Gotsman, which is in turn based on Floater's asymmetric extension of Tutte's classical spring-embedding theorem. First, we give a much simpler algorithm to construct piecewise-linear morphs between planar straight-line graphs. Specifically, given isomorphic

Consider the geometric range space $(X, \mathcal{H}_d)$ where $X \subset \mathbb{R}^d$ and $\mathcal{H}_d$ is the set of ranges defined by $d$-dimensional halfspaces. In this setting we consider that $X$ is the disjoint union of a red and blue set. For each halfspace $h \in \mathcal{H}_d$ define a function $\Phi(h)$ that measures the "difference" between the fraction of red and fraction of blue points

We describe a purely image-based method for finding geometric constructions with a ruler and compass in the Euclidea geometric game. The method is based on adapting the Mask R-CNN state-of-the-art image processing neural architecture and adding a tree-based search procedure to it. In a supervised setting, the method learns to solve all 68 kinds of geometric construction problems from the first six

Let $P$ be a crossing-free polygon and $\mathcal C$ a set of shortcuts, where each shortcut is a directed straight-line segment connecting two vertices of $P$. A shortcut hull of $P$ is another crossing-free polygon that encloses $P$ and whose oriented boundary is composed of elements from $\mathcal C$. Shortcut hulls find their application in geo-related problems such as the simplification of contour

In this paper, we study two extensions of the maximum bichromatic separating rectangle (MBSR) problem introduced in \cite{Armaselu-CCCG, Armaselu-arXiv}. One of the extensions, introduced in \cite{Armaselu-FWCG}, is called \textit{MBSR with outliers} or MBSR-O, and is a more general version of the MBSR problem in which the optimal rectangle is allowed to contain up to $k$ outliers, where $k$ is given