Let P=(p1,…,pn) be a geometric path in the plane. For a real number 0<γ<180°, P is called an angle-monotone path with width γ if there exists a closed wedge of angle γ such that the vector of every edge (pi,pi+1) of P lies in this wedge. Let G be a geometric graph in the plane. The graph G is called angle-monotone with width γ if there exists an angle-monotone path with width γ between any two vertices

We consider a bichromatic two-center problem for pairs of points. Given a set S of n pairs of points in the plane, for every pair, we want to assign a red color to one point and a blue color to the other, in such a way that the value max⁡{r1,r2} is minimized, where r1 (resp., r2) is the radius of the smallest enclosing disk of all red (resp., blue) points. Previously, an exact algorithm of O(n3log2⁡n)

Given two shapes A and B in the plane with Hausdorff distance 1, is there a shape S with Hausdorff distance 1/2 to and from A and B? The answer is always yes, and depending on convexity of A and/or B, S may be convex, connected, or disconnected. We show that our result can be generalized to give an interpolated shape between A and B for any interpolation variable α between 0 and 1, and prove that the

Let G=(V,E) be an edge weighted geometric graph (not necessarily planar) such that every edge is horizontal or vertical. The weight of an edge uv∈E is the L1-distance between its endpoints. Let WG(u,v) denote the length of a shortest path between a pair of vertices u and v in G. The graph G is said to be a Manhattan network for a given point set P in the plane if P⊆V and ∀p,q∈P, WG(p,q)=‖pq‖1. In addition

A class of counting problems asks for the number of regions of a central hyperplane arrangement. By duality, this is the same as counting the vertices of a zonotope. Efficient algorithms are known that solve this problem by computing the vertices of a zonotope from its set of generators. Here, we give an efficient algorithm, based on a linear optimization oracle, that performs the inverse task and

Let S be a set of n weighted points in the plane and let R be a query range in the plane. In the range closest pair problem, we want to report the closest pair in the set R∩S. In the range minimum weight problem, we want to report the minimum weight of any point in the set R∩S. We show that these two query problems are equivalent for query ranges that are squares, for data structures having Ω(log⁡n)

We study Erdős's distinct distances problem under ℓp metrics with integer p. We prove that, for every ε>0 and n points in R2, there exists a point that spans Ω(n6/7−ε) distinct distances with the other n−1 points. This improves upon the previous best bound of Ω(n4/5). We also characterize the sets that span an asymptotically minimal number of distinct distances under the ℓ1 and ℓ∞ metrics.

A unit disk graph G on a given set P of points in the plane is a geometric graph where an edge exists between two points p,q∈P if and only if |pq|≤1. A spanning subgraph G′ of G is a k-hop spanner if and only if for every edge pq∈G, there is a path between p,q in G′ with at most k edges. We obtain the following results for unit disk graphs in the plane. (i) Every n-vertex unit disk graph has a 5-hop

The partition of a problem into smaller sub-problems satisfying certain properties is often a key ingredient in the design of divide-and-conquer algorithms. For questions related to location, the partition problem can be modeled, in geometric terms, as finding a subdivision of a planar map – which represents, say, a geographical area – into regions subject to certain conditions while optimizing some

Given a set of open axis-aligned disjoint rectangles, each of which plays as both an obstacle and a target, we seek to find shortest obstacle-avoiding rectilinear paths from a query to the nearest target and the farthest target. The distance to a target is determined by the point on the target achieving the minimum or maximum geodesic distance among all points on the boundary of the target. This problem

We consider the problem of approximating a two-dimensional shape contour (or curve segment) using discrete assembly systems, which allow to build geometric structures based on limited sets of node and edge types subject to edge length and orientation restrictions. We show that already deciding feasibility of such approximation problems is NP-hard, and remains intractable even for very simple setups

We consider the problem of determining if a pair of undirected graphs 〈Gv,Gh〉, which share the same vertex set, has a representation using opaque geometric shapes for vertices, and vertical (respectively, horizontal) visibility between shapes to determine the edges of Gv (respectively, Gh). While such a simultaneous visibility representation of two graphs can be determined efficiently if the direction

We study Snipperclips, a computer puzzle game whose objective is to create a target shape with two tools. The tools start as constant-complexity shapes, and each tool can snip (i.e., subtract its current shape from) the other tool. We study the computational problem of, given a target shape represented by a polygonal domain of n vertices, is it possible to create it as one of the tools' shape via a

In this paper, we develop a method for setting lower and upper bounds on growth constants of polyominoes and polycubes whose enumerating sequences are so-called quasi sub- or super-multiplicative. The method is based on concatenation arguments, applied directly or recursively. Inter alia, we demonstrate the method on general polycubes, tree polyominoes and polycubes, and convex polyominoes.

Given a set P of n data points and an integer k, a fundamental computational task is to find a smaller subset Q⊆P of only k points which approximately preserves the geometry of P. Here we consider the problem of finding the subset Q of k points which best captures the convex hull of P, where our error measure is the sum of the distances of the points in P to the convex hull of Q. We generalize the

We prove that any finite polyhedral manifold in 3D can be continuously flattened into 2D while preserving intrinsic distances and avoiding crossings, answering a 19-year-old open problem, if we extend standard folding models to allow for countably infinite creases. The most general cases previously known to be continuously flattenable were convex polyhedra and semi-orthogonal polyhedra. For non-orientable

Slanted and curved nonograms are a new type of picture puzzles introduced by Van de Kerkhof et al. (2019). They consist of an arrangement of lines or curves within a frame B, where some of the cells need to be colored in order to obtain the solution picture. For solving the puzzle, up to two clues need to be attached as numeric labels to each line on either side of B. In this paper we study the algorithmic

We study the problem of finding maximum-area triangles that can be inscribed in a polygon in the plane. We consider eight versions of the problem: we use either convex polygons or simple polygons as the container; we require the triangles to have either one corner with a fixed angle or all three corners with fixed angles; we either allow reorienting the triangle or require its orientation to be fixed

We study a game where two players take turns selecting points of a convex geometry until the convex closure of the jointly selected points contains all the points of a given winning set. The winner of the game is the last player able to move. We develop a structure theory for these games and use it to determine the nim number for several classes of convex geometries, including one-dimensional affine

We study the algorithmic problem of computing drawings of graphs in which (i) each vertex is a disk with fixed radius ρ, (ii) each edge is a straight-line segment connecting the centers of the two disks representing its end-vertices, (iii) no two disks intersect, and (iv) the distance between an edge segment and the center of a non-incident disk, called edge-vertex resolution, is at least ρ. We call

We show that every polycube tree can be unfolded with a 4×4 refinement of the grid faces. This is the first constant refinement unfolding result for polycube trees that are not required to be well-separated.

Given a simple polygon P on n vertices, nr of which are reflex, and a set D of m pairwise intersecting geodesic disks in P, we show that at most 14 points in P suffice to pierce all the disks in D and these points can be computed in O(n+mlog⁡nr) time.

We consider the following problem: Let L be an arrangement of n lines in R3 in general position colored red, green, and blue. Does there exist a vertical plane P such that a line in P simultaneously bisects all three classes of points induced by the intersection of lines in L with P? Recently, Schnider used topological methods to prove that such a cross-section always exists. In this work, we give

The Minimum Area Spanning Tree Problem (MASTP) is defined in terms of a complete undirected graph G, where every vertex represents a point in the two dimensional Euclidean plane. Associated to each edge, there is a disk placed right at its midpoint, with diameter matching the length of the edge. MASTP seeks a spanning tree of G that minimizes the area in the union of the disks associated to its edges

I present a generalization of Chew's first algorithm for Delaunay mesh refinement. I split the line segments of an input planar straight line graph (PSLG) such that the lengths of split segments are asymptotically proportional to the local feature size at their endpoints. By employing prior algorithms, I then refine the truly or constrained Delaunay triangulation of the PSLG by inserting off-center

Donald Knuth introduced abstract CC systems to represent configurations of points in a plane with a given orientation (clockwise or counterclockwise) of all triples of points. We present efficient enumeration of all non-isomorphic CC systems with at most 12 points. Our algorithm is based on Faradžev-Read type enumeration, enhanced with the homomorphism principle and SAT solving, enabling us to enumerate

3D Face recognition is being extensively recognized as a biometric performance refers to its non-intrusive environment. In spite of large research on 2-D face recognition, it suffers from low recognition rate due to illumination variations, pose changes, poor image quality, occlusions and facial expression variations, while 3D face models are insensitive to all these conditions. In this paper, we present

We show that for any compact convex body Q in the plane, the average distance from the Fermat-Weber center of Q to the points in Q is at most 99−50336⋅Δ(Q)<0.3444⋅Δ(Q), where Δ(Q) denotes the diameter of Q. This improves upon the previous bound of 2(4−3)13⋅Δ(Q)<0.3490⋅Δ(Q). The average distance from the Fermat-Weber center of Q is calculated by comparing it with that of a circular sector of radius

We consider the planar Euclidean two-center problem in which given n points in the plane we are to find two congruent disks of the smallest radius covering the points. We present a deterministic O(nlog⁡n)-time algorithm for the case that the centers of the two optimal disks are close to each other, that is, the overlap of the two optimal disks is a constant fraction of the disk area. We also present

An articulated probe is modeled in the plane as two line segments, ab and bc, joined at b, with ab being very long, and bc of some small length r. We investigate a trajectory planning problem involving the articulated two-segment probe where the length r of bc can be customized. Consider a set P of simple polygonal obstacles with a total of n vertices, a target point t located in the free space such

We present Cgal implementations of two algorithms for computing straight skeletons in the plane, based on exact arithmetic. One code, named Surfer2, can handle multiplicatively weighted planar straight-line graphs (PSLGs) while our second code, Monos, is specifically targeted at monotone polygons. Both codes are available on GitHub. We discuss algorithmic as well as implementational and engineering

Let P be a path graph of n vertices embedded in a metric space. We consider the problem of adding a new edge to P to minimize the radius of the resulting graph. Previously, a similar problem for minimizing the diameter of the graph was solved in O(nlog⁡n) time. To the best of our knowledge, the problem of minimizing the radius has not been studied before. In this paper, we present an O(n) time algorithm

Let S be a set of n red and n blue points in general position in R3. Let τ be a tetrahedron with vertices in S. We say that τ is empty if it does not contain any point of S in its interior. We say that τ is balanced if two of its vertices are blue, and two of its vertices are red. In this paper we show that S spans Ω(n5/2) empty balanced tetrahedra.

Beacon attraction, or simply attraction, is a movement system whereby a point moves in a free space so as to always locally minimize its Euclidean distance to an activated beacon (also a point). This results in the point moving directly towards the beacon when it can, and otherwise sliding along the edge of an obstacle or being stuck (unable to move). When the point can reach the activated beacon by

In this paper, we study a generalization of Voronoi diagram, called the Influence-based Voronoi Diagram (IVD). The input consists of a point set P in Rd, a collection C={C1,C2,…,Cn} where each Ci⊆P, i=1,2,…,n, is a cluster of points of P, and an influence function F(C,q) measuring the influence from a set C of points to any point q in Rd, and the goal is to construct an influence-based Voronoi diagram

An annulus is, informally, a ring-shaped region, often described by two concentric circles. The maximum-width empty annulus problem asks to find an annulus of a certain shape with the maximum possible width that avoids a given set of n points in the plane. This problem can also be interpreted as the problem of finding an optimal location of a ring-shaped obnoxious facility among the input points. In

We consider geometric hypergraphs whose vertex set is a finite set of points (e.g., in the plane), and whose hyperedges are the intersections of this set with a family of geometric regions (e.g., axis-parallel rectangles). A typical coloring problem for such geometric hypergraphs asks, given an integer k, for the existence of an integer m=m(k), such that every set of points can be k-colored such that

We address the problems of constructing the Euclidean Minimum Spanning Tree (EMST) of points in the plane with mutually dependent location uncertainties, testing its stability, and computing its total weight. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), an expressive and computationally efficient worst-case, first order linear approximation

We introduce new distance measures for comparing straight-line embedded graphs based on the Fréchet distance and the weak Fréchet distance. These graph distances are defined using continuous mappings and thus take the combinatorial structure as well as the geometric embeddings of the graphs into account. We present a general algorithmic approach for computing these graph distances. Although we show

We give a simple algorithm to obtain the tight span of a finite subset of the Manhattan plane.

Assume we are given a set of parallel line segments in the plane, and we wish to place a point on each line segment such that the resulting point set maximizes or minimizes the area of the largest or smallest triangle in the set. We analyze the complexity of the four resulting computational problems, and we show that three of them admit polynomial-time algorithms, while the fourth is NP-hard. Specifically

The problem of nearest-neighbor condensation aims to reduce the size of a training set of a nearest-neighbor classifier while maintaining its classification accuracy. Although many condensation techniques have been proposed, few bounds have been proved on the amount of reduction achieved. In this paper, we present one of the first theoretical results for practical nearest-neighbor condensation algorithms

Let {p1,…,pn} and {q1,…,qn} be two sets of n labeled points in general position in the plane. We say that these two point sets have the same order type if for every triple of indices (i,j,k), pk is above the directed line from pi to pj if and only if qk is above the directed line from qi to qj. In this paper we give the first non-trivial lower bounds on the number of different order types of n points

Let S be a set of n points in general position in the plane. Suppose that each point of S has been assigned one of k≥3 possible colors and that there is the same number, m, of points of each color class. This means n=km. A polygon with vertices on S is empty if it does not contain points of S in its interior; and it is rainbow if all its vertices have different colors. Let f(k,m) be the minimum number

In 2006, Alberto Bressan [1] suggested the following problem. Suppose a circular fire spreads in the Euclidean plane at unit speed. The task is to build, in real time, barrier curves to contain the fire. At each time t the total length of all barriers built so far must not exceed t⋅v, where v is a speed constant. How large a speed v is needed? He proved that speed v>2 is sufficient, and that v>1 is

We study the number of edge flips required to transform any triangulation of an n-vertex spiral polygon into any other. We improve the upper bound from 4n−6 to 3n−9 flips and show a lower bound of 2n−8 flips. Instead of using a single canonical triangulation as the intermediate point in the transformation between two triangulations, we use a family of closely-connected triangulations.

Let D be a set of n disks in the plane. The disk graph GD for D is the undirected graph with vertex set D in which two disks are joined by an edge if and only if they intersect. The directed transmission graph GD→ for D is the directed graph with vertex set D in which there is an edge from a disk D1∈D to a disk D2∈D if and only if D1 contains the center of D2. Given D and two non-intersecting disks

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.