Following the seminal work of Erlebach and van Leeuwen in SODA 2008, we introduce the minimum ply covering problem. Given a set P of points and a set S of geometric objects, both in the plane, our goal is to find a subset S′ of S that covers all points of P while minimizing the maximum number of objects covering any point in the plane (not only points of P). For objects that are unit squares and unit

Given an angle γ>0, a geometric path (v1,…,vk) is called angle-monotone with width γ if, for any two integers 1≤i,j

We study the problem of finding maximum-area and maximum-perimeter rectangles that are inscribed in polygons in the plane. There has been a fair amount of work on this problem when the rectangles have to be axis-aligned or when the polygons are convex. We consider this problem in polygons with n vertices that are not necessarily convex, possibly with holes, and with no restriction on the orientation

A hiker is lost in a forest of unknown shape. What is a good path for the hiker to follow in order to escape from the forest within a reasonable amount of time? The hiker's dilemma clearly is: Should one start exploring the area close-by and expand the search radii gradually? Or should one rather pick some direction and run straight on? We employ a competitive analysis to prove that a certain spiral

When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability

The database skyline query (or non-domination query) has a spatial form: Given a set P with n point sites, and a point set S of m locations of interest, a site p∈P is a skyline point if and only if for each q∈P∖{p}, there exists at least one location s∈S that is closer to p than to q. We reduce the problem of determining skyline points to the problem of finding sites that have non-empty cells in an

Over the past years there has been an enormous increase in the amount of data generated on a daily basis. A critical task in handling the information overload is locating the most interesting objects of a dataset according to a specific configuration or ranking function. Our work is based on the concept of dominance which compares data objects based on maximization preferences on the attribute values

In this paper, we address the minimum-area rectangular and square annulus problem, which asks a rectangular or square annulus of minimum area, either in a fixed orientation or over all orientations, that encloses a set P of n input points in the plane. To our best knowledge, no nontrivial results on the problem have been discussed in the literature, while its minimum-width variants have been intensively

We present the first static and dynamic external memory data structures for variants of 2-d orthogonal range skyline reporting with worst-case logarithmic query and update I/O-complexity. The results are obtained by using persistent data structures and by extending the attrition priorities queues of Sundar (1989) [26] to also support real-time concatenation, a result of independent interest. We show

We say that a triangle T tiles a polygon A, if A can be dissected into finitely many nonoverlapping triangles similar to T. We are concerned with the question whether the triangle of angles π/N,π/N, (N−2)π/N tiles the regular N-gon. It is easy to see that if N=3,4 or 6, then the answer is affirmative. We show the same in the cases N=5,8,10 and 12.

The design and implementation of theoretically-sound robot motion planning algorithms is challenging. Within the framework of resolution-exact algorithms, it is possible to exploit soft predicates for collision detection. The design of soft predicates is a balancing act between their implementability and their accuracy/effectivity. In this paper, we focus on the class of planar polygonal rigid robots

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of n vertices and h holes. We introduce a graph of oriented distances to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and

A universal cover for a family T of triangles is a convex set that contains a congruent copy of each triangle T∈T. We conjecture that for any family T of triangles of bounded diameter there is a triangle that forms a universal cover for T of smallest possible area. We prove this conjecture for all families of two triangles, and for the family of triangles that fit in the unit disk.

Let F be a family of pseudo-disks in the plane, and P be a finite subset of F. Consider the hyper-graph H(P,F) whose vertices are the pseudo-disks in P and the edges are all subsets of P of the form {D∈P|D∩S≠∅}, where S is a pseudo-disk in F. We give an upper bound of O(nk3) for the number of edges in H(P,F) of cardinality at most k. This generalizes a result of Buzaglo et al. [4]. As an application

Let R be a connected and closed region in the plane and let S be a set of n points (representing mobile sensors) in the interior of R. We think of R's boundary as a barrier which needs to be monitored. This gives rise to the barrier coverage problem, where one needs to move the sensors to the boundary of R, so that every point on the boundary is covered by one of the sensors. We focus on the variant

Let D be a set of n pairwise disjoint disks in the plane. Consider the metric space in which the distance between any two disks D and D′ in D is the length of the shortest line segment that connects D and D′. For any real number ε>0, we show how to obtain a (1+ε)-spanner for this metric space that has at most (2π/ε)⋅n edges. The previously best known result is by Bose et al. (2011) [1]. Their (1+ε)-spanner

We improve the theorem of Beck giving a lower bound on the number of k-flats spanned by a set of points in real space, and improve the bound of Elekes and Tóth on the number of incidences between points and k-flats in real space.

We are interested in the following problem of covering the plane by a sequence of congruent circular disks with a constraint on the distance between consecutive disks. Let (Dn)n∈N be a sequence of closed unit circular disks such that ∪n∈NDn=R2 with the condition that for n≥2, the center of the disk Dn lies in Dn−1. What is a “most economical” or an optimal way of placing Dn for all n∈N? We answer this

Range closest-pair (RCP) search is a range-search variant of the classical closest-pair problem, which aims to store a given set S of points into some space-efficient data structure such that when a query range Q is specified, the closest pair in S∩Q can be reported quickly. RCP search has received attention over years, but the primary focus was only on R2. In this paper, we study RCP search in higher

Given a set of points S⊆R2, a subset X⊆S with |X|=k is called k-gon if all points of X lie on the boundary of the convex hull of X, and k-hole if, in addition, no point of S∖X lies in the convex hull of X. We use computer assistance to show that every set of 17 points in general position admits two disjoint 5-holes, that is, holes with disjoint respective convex hulls. This answers a question of Hosono

A beacon b∈Rd is a point-shaped object in d-dimensional space that can exert a magnetic pull on any other point-shaped object p∈Rd. This object p then moves greedily towards b. The motion stops when p gets stuck at an obstacle or when p reaches b. By placing beacons inside a d-dimensional polyhedron P, we can implement a scheme to route point-shaped objects between any two locations in P. We can also

An ε-approximate incidence between a point and some geometric object (line, circle, plane, sphere) occurs when the point and the object lie at distance at most ε from each other. Given a set of points and a set of objects, computing the approximate incidences between them is a major step in many database and web-based applications in computer vision and graphics, including robust model fitting, approximate

In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes as input an upward planar drawing ΓH of a subgraph H of a directed graph G and asks whether ΓH can be extended to an upward planar drawing of G. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing. We

We consider defining the embedding of a triangle mesh into R3, up to translation, rotation, and scale, by its vector of dihedral angles. On the theoretical side, we show that locally the map from realizable vectors of dihedrals to mesh embeddings is one-to-one almost everywhere. On the implementation side, we are interested in using the dihedral parameterization in shape analysis. This demands a way

Let P be a set of n points in the plane in general position. We show that at least ⌊n/3⌋ plane spanning trees can be packed into the complete geometric graph on P. This improves the previous best known lower bound Ω(n). Towards our proof of this lower bound we show that the center of a set of points, in the d-dimensional space in general position, is of dimension either 0 or d.

We consider a new trajectory planning problem involving a simple articulated probe. The probe is modeled as two line segments ab and bc, with a joint at the common point b, where bc is of fixed length r and ab is of arbitrarily large length. Initially, ab and bc are collinear. Given a set of obstacles in the form of n line segments and a target point t, the probe is to first be inserted in straight

The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that well-chosen (finite) sets of PDs can differentiate between geometric simplicial complexes, providing a method for representing complex shapes

A witness set W in a polygon P is a subset of P such that any set G⊂P that guards W is guaranteed to guard P. We study the problem of finding a minimum witness set for an orthogonal polygon under three models of orthogonal visibility. It is known that not all simple polygons admit a finite witness set under the traditional line-segment visibility and, if a polygon admits a finite minimal witness set

The range closest-pair (RCP) problem, as a range-search version of the classical closest-pair problem, aims to store a dataset of points in some data structure such that whenever a query range Q is given, the closest-pair inside Q can be reported efficiently. In this paper, we study the approximate version of the RCP problem with two approximation criteria. The first criterion is in terms of the query

The alpha complex, a subset of the Delaunay triangulation, has been extensively used as the underlying representation for biomolecular structures. We propose a GPU-based parallel algorithm for the computation of the alpha complex, which exploits the knowledge of typical spatial distribution and sizes of atoms in a biomolecule. Unlike existing methods, this algorithm does not require prior construction

In the minimum constraint removal problem, we are given a set of overlapping geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable and no sub-linear approximations are known even for simple obstacles such as rectangles and

Erdős-Beck's theorem states that n points in the plane with at most n−x points collinear define at least cxn lines for some positive constant c. It implies n points in the plane define Θ(n2) lines unless most of the points (i.e. n−o(n) points) are collinear. In this paper, we will present two ways to extend this result to higher dimensions. Given a set S of n points in Rd, we want to estimate a lower

We show that, as a consequence of a new result of Pór on universal Tverberg partitions, any large-enough set P of points in Rd has a (d+2)-sized subset whose Radon point has half-space depth at least cd⋅|P|, where cd∈(0,1) depends only on d. We then give two applications of this result. The first is to computing weak ϵ-nets by random sampling. The second is to show that given any set P of points in

We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Instead of defining these proximity graphs using circles, we use an arbitrary convex shape C. Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k-DGC(S), has vertex set S, and edges defined as follows. Given p,q∈S, pq is an edge of k-DGC(S) provided there exists some homothet of C

We study the complexity of symmetric assembly puzzles: given a collection of simple polygons, can we translate, rotate, and possibly flip them so that their interior-disjoint union is line symmetric? On the negative side, we show that the problem is strongly NP-complete even if the pieces are all polyominos. On the positive side, we show that the problem can be solved in polynomial time if the number

We present two universal hinge patterns that enable a strip of material to fold into any connected surface made up of unit squares on the 3D cube grid—for example, the surface of any polycube. The folding is efficient: for target surfaces topologically equivalent to a sphere, the strip needs to have only twice the target surface area, and the folding stacks at most two layers of material anywhere.

We consider a repulsion actuator located in an n-sided convex environment full of point particles. When the actuator is activated, all the particles move away from the actuator. We study the problem of gathering all the particles to a point. We give an O(n2) time algorithm to compute all the actuator locations that gather the particles to one point with one activation, and an O(n) time algorithm to

We augment a tree T with a shortcut pq to minimize the largest distance between any two points along the resulting augmented tree T+pq. We study this problem in a continuous and geometric setting where T is a geometric tree in the Euclidean plane, a shortcut is a line segment connecting any two points along the edges of T, and we consider all points on T+pq (i.e., vertices and points along edges) when

There are many representations of planar graphs, but few are as elegant as Turán's (1984): it is simple and practical, uses only 4 bits per edge, can handle self-loops and multi-edges, and can store any specified embedding. Its main disadvantage has been that “it does not allow efficient searching” (Jacobson, 1989). In this paper we show how to add a sublinear number of bits to Turán's representation

It is proved that every series-parallel digraph whose maximum vertex degree is Δ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of Δ distinct slopes. The construction is worst-case optimal in terms of the number of slopes, and it gives rise to drawings with optimal angular resolution πΔ. A variant of the drawing algorithm is used to show that (non-directed)

Consider k robots initially located at a point inside a region T. Each robot can move anywhere in T independently of the other robots with maximum speed one. The goal of the robots is to evacuate T through an exit at an unknown location on the boundary of T. The objective is to minimize the evacuation time, which is defined as the time the last robot reaches the exit. We consider the face-to-face communication

Globally non-positively curved, or CAT(0), polyhedral complexes arise in a number of applications, including evolutionary biology and robotics. These spaces have unique shortest paths and are composed of Euclidean polyhedra, yet many algorithms and properties of shortest paths and convex hulls in Euclidean space fail to transfer over. We give an algorithm, using linear programming, to compute the convex

Persistent homology allows for tracking topological features, like loops, holes and their higher-dimensional analogues, along a single-parameter family of nested shapes. Computing descriptors for complex data characterized by multiple parameters is becoming a major challenging task in several applications, including physics, chemistry, medicine, and geography. Multiparameter persistent homology generalizes

The unit disk graph (UDG) is a widely employed model for the study of wireless networks. In this model, wireless nodes are represented by points in the plane and there is an edge between two points if and only if their Euclidean distance is at most one. A hop spanner for the UDG is a spanning subgraph H such that for every edge (p,q) in the UDG the topological shortest path between p and q in H has

Given a set of polygonal curves, we present algorithms for computing a middle curve that serves as a representative for the entire set of curves. We require that the middle curve consists of vertices of the input curves and that it minimizes the maximum discrete Fréchet distance to all input curves. We consider three different variants of a middle curve depending on in which order vertices of the input

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 detail. We present an O(n3m)-time algorithm that takes a polygonal curve of n vertices and produces a set of consistent simplifications for m scales while minimizing

Consider the circle C of length 1 and a circular arc A of length ℓ∈(0,1). It is shown that there exists k=k(ℓ)∈N, and a schedule for k runners along the circle with k constant but distinct positive speeds so that at any time t≥0, at least one of the k runners is not in A. On the other hand, we show the following. Assume that k runners 1,2,…,k, with constant rationally independent (thus distinct) speeds

According to a result of Arkin et al. (2016), given n point pairs in the plane, there exists a simple polygonal cycle that separates the two points in each pair to different sides; moreover, a O(n)-factor approximation with respect to the minimum length can be computed in polynomial time. Here the following results are obtained: (I) We extend the problem to geometric hypergraphs and obtain the following

For a set O of n points in Rd, the skyline consists of the subset of all points of O where no point is dominated by any other point of O. Suppose that each point oi∈O has an associated probability of existence pi∈(0,1]. The problem of computing the skyline with the maximum probability of occurrence is considered. It is shown that in Rd, d≥3, the problem is NP-hard and that the desired skyline cannot

Consider a set P of n points on the boundary of an axis-aligned square Q. We study the boundary-anchored packing problem on P in which the goal is to find a set of interior-disjoint axis-aligned rectangles in Q such that each rectangle is anchored (has a corner at some point in P), each point in P is used to anchor at most one rectangle, and the total area of the rectangles is maximized. Here, a rectangle

Art Gallery Localization (AGL) is the problem of placing a set T of broadcast towers in a simple polygon P in order for a point to locate itself in the interior. For any point p∈P: for each tower t∈T∩V(p) (where V(p) denotes the visibility polygon of p) the point p receives the coordinates of t and the Euclidean distance between t and p. From this information p can determine its coordinates. We study

Given a set of data points sampled from some underlying space, there are two important challenges in geometric and topological data analysis when dealing with sampled data: reconstruction – how to assemble discrete samples into global structures, and inference – how to extract geometric and topological information from data that are high-dimensional, incomplete and noisy. Niyogi et al. (2008) have

We study the problem of k-dominance in a set of d-dimensional vectors, prove bounds on the number of maxima (skyline vectors), under both worst-case and average-case models, perform experimental evaluation using synthetic and real-world data, and explore an application of k-dominant skyline for extracting a small set of top-ranked vectors in high dimensions where the full skylines can be unmanageably

In a visibility representation of a graph G, the vertices are represented by non-overlapping geometric objects, while the edges are represented as segments that only intersect the geometric objects associated with their end-vertices. Given a set P of n points, an Anchored Visibility Representation of a graph G with n vertices is a visibility representation such that for each vertex v of G, the geometric

In the Geometric Conflict Free Covering, we are given a set of points P, a set of closed geometric objects O and a conflict graph CGO with O as vertex set. An edge (Oi,Oj) in CGO denotes conflict between Oi and Oj and at most one among these can be part of any feasible solution. A set of objects is conflict free if they form an independent set in CGO. The objective is to find a conflict free set of

We study the problem of connecting two points in a simple polygon with a self-approaching path. A self-approaching path is a directed curve such that the Euclidean distance between a point moving along the path and any future position does not increase, that is, for all points a, b, and c that appear in that order along the curve, |ac|≥|bc|. We analyze properties of self-approaching paths inside simple

We consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes. We may preprocess P to obtain a label and a routing table for each vertex of P. Then, we must be able to route a data packet between any two vertices p and q of P, where each step must use only the label of the target node q and the routing table of the current node. For

Suppose we are searching for a target point t in a certain restricted search space. To pinpoint the location of t, we issue query points q1,…,qn in the search space. As a response, we obtain an ordering of the query points by distance to t. This restricts possible locations of t. In this paper, we consider the case where the search space is the unit interval [0,1]. We define the accuracy of a query

In this paper we present a package for implementing exact kinetic data structures built on objects which move along polynomial trajectories. We discuss how the package design was influenced by various considerations, including extensibility, support for multiple kinetic data structures, access to existing data structures and algorithms in CGAL, as well as debugging. Due to the similarity between the

For a set S of points in ℝ(d), an s-spanner is a subgraph of the complete graph with node set S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε(d)) edges, where ε is a specified parameter. The key property of this spanner is that it