Computer Science - Computational Geometry Publications (50)


Computer Science - Computational Geometry Publications

The approaches taken to describe and develop spatial discretisations of the domains required for geophysical simulation models are commonly ad hoc, model or application specific and under-documented. This is particularly acute for simulation models that are flexible in their use of multi-scale, anisotropic, fully unstructured meshes where a relatively large number of heterogeneous parameters are required to constrain their full description. As a consequence, it can be difficult to reproduce simulations, ensure a provenance in model data handling and initialisation, and a challenge to conduct model intercomparisons rigorously. Read More

Geophysical model domains typically contain irregular, complex fractal-like boundaries and physical processes that act over a wide range of scales. Constructing geographically constrained boundary-conforming spatial discretizations of these domains with flexible use of anisotropically, fully unstructured meshes is a challenge. The problem contains a wide range of scales and a relatively large, heterogeneous constraint parameter space. Read More

Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called \emph{nerve} of a cover of the domain. Read More

In this paper we extend the works of Tancer and of Malgouyres and Franc\'es, showing that $(d,k)$-collapsibility is NP-complete for $d\geq k+2$ except $(2,0)$. By $(d,k)$-collapsibility we mean the following problem: determine whether a given $d$-dimensional simplicial complex can be collapsed to some $k$-dimensional subcomplex. The question of establishing the complexity status of $(d,k)$-collapsibility was asked by Tancer, who proved NP-completeness of $(d,0)$ and $(d,1)$-collapsibility (for $d\geq 3$). Read More

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e. Read More

The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in $\mathbb{R}^2$ and on surfaces embedded in $\mathbb{R}^3$ as detailed in our experimental companion paper. Read More

The concept of derivative coordinate functions proved useful in the formulation of analytic fractal functions to represent smooth symmetric binary fractal trees [1]. In this paper we introduce a new geometry that defines the fractal space around these fractal trees. We present the canonical and degenerate form of this fractal space and extend the fractal geometrical space to R3 specifically and Rn by a recurrence relation. Read More

A map $f\colon K\to \mathbb R^d$ of a simplicial complex is an almost embedding if $f(\sigma)\cap f(\tau)=\emptyset$ whenever $\sigma,\tau$ are disjoint simplices of $K$. Theorem. Fix integers $d,k\ge2$ such that $d=\frac{3k}2+1$. Read More

We study self-approaching paths that are contained in a simple polygon. A self-approaching path is a directed curve connecting two points 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, $|ac| \ge |bc|$. We analyze the properties, and present a characterization of shortest self-approaching paths. Read More

Given a set of points in the plane, we want to establish a connection network between these points that consists of several disjoint layers. Motivated by sensor networks, we want that each layer is spanning and plane, and that no edge is very long (when compared to the minimum length needed to obtain a spanning graph). We consider two different approaches: first we show an almost optimal centralized approach to extract two graphs. Read More

Let $P$ be a set of $n$ points in the plane. We consider the problem of partitioning $P$ into two subsets $P_1$ and $P_2$ such that the sum of the perimeters of $\text{CH}(P_1)$ and $\text{CH}(P_2)$ is minimized, where $\text{CH}(P_i)$ denotes the convex hull of $P_i$. The problem was first studied by Mitchell and Wynters in 1991 who gave an $O(n^2)$ time algorithm. Read More

Graphs and network data are ubiquitous across a wide spectrum of scientific and application domains. Often in practice, an input graph can be considered as an observed snapshot of a (potentially continuous) hidden domain or process. Subsequent analysis, processing, and inferences are then performed on this observed graph. Read More

Non-rigid registration is challenging because it is ill-posed with high degrees of freedom and is thus sensitive to noise and outliers. We propose a robust non-rigid registration method using reweighted sparsities on position and transformation to estimate the deformations between 3-D shapes. We formulate the energy function with dual sparsities on both the data term and the smoothness term, and define the smoothness constraint using local rigidity. Read More

In this paper, we consider the strict self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-handed Tile Assembly Model (2HAM). We are particularly interested in a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpinski's carpet). We present a 2HAM system that strictly self-assembles the discrete Sierpinski's carpet with scale factor 1. Read More

We present an $(1+\varepsilon)$-approximation algorithm with quasi-polynomial running time for computing the maximum weight independent set of polygons out of a given set of polygons in the plane (specifically, the running time is $n^{O( \mathrm{poly}( \log n, 1/\varepsilon))}$). Contrasting this, the best known polynomial time algorithm for the problem has an approximation ratio of~$n^{\varepsilon}$. Surprisingly, we can extend the algorithm to the problem of computing the maximum weight subset of the given set of polygons whose intersection graph fulfills some sparsity condition. Read More

Given a rectilinear domain $\mathcal{P}$ of $h$ pairwise-disjoint rectilinear obstacles with a total of $n$ vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points $s$ and $t$ in $\mathcal{P}$. Three types of bicriteria rectilinear paths are considered: minimum-link shortest paths, shortest minimum-link paths, and minimum-cost paths where the cost of a path is a non-decreasing function of both the number of edges and the length of the path. The one-point and two-point path queries are also considered. Read More

Given a set $S$ of $n$ line segments in the plane, we say that a region $\mathcal{R}\subseteq \mathbb{R}^2$ is a {\em stabber} for $S$ if $\mathcal{R}$ contains exactly one endpoint of each segment of $S$. In this paper we provide optimal or near-optimal algorithms for reporting all combinatorially different stabbers for several shapes of stabbers. Specifically, we consider the case in which the stabber can be described as the intersection of axis-parallel halfplanes (thus the stabbers are halfplanes, strips, quadrants, $3$-sided rectangles, or rectangles). Read More

We describe a set of $\Delta -1$ slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree $\Delta \geq 4$; this establishes a new upper bound of $\Delta-1$ on the 1-bend planar slope number. By universal we mean that every planar graph of degree $\Delta$ has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was $\frac{3}{2} (\Delta -1)$ (the known lower bound being $\frac{3}{4} (\Delta -1)$); secondly, all the known algorithms to construct 1-bend planar drawings with $O(\Delta)$ slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is $\frac{\pi}{(\Delta-1)}$. Read More

3D shape models are naturally parameterized using vertices and faces, \ie, composed of polygons forming a surface. However, current 3D learning paradigms for predictive and generative tasks using convolutional neural networks focus on a voxelized representation of the object. Lifting convolution operators from the traditional 2D to 3D results in high computational overhead with little additional benefit as most of the geometry information is contained on the surface boundary. Read More

We study data structures for storing a set of polygonal curves in ${\rm R}^d$ such that, given a query curve, we can efficiently retrieve similar curves from the set, where similarity is measured using the discrete Fr\'echet distance or the dynamic time warping distance. To this end we devise the first locality-sensitive hashing schemes for these distance measures. A major challenge is posed by the fact that these distance measures internally optimize the alignment between the curves. Read More

We answer the following question dating back to J.E. Littlewood (1885 - 1977): Can two lions catch a man in a bounded area with rectifiable lakes? The lions and the man are all assumed to be points moving with at most unit speed. Read More

Let $s$ be a point in a polygonal domain $\mathcal{P}$ of $h-1$ holes and $n$ vertices. We consider a quickest visibility query problem. Given a query point $q$ in $\mathcal{P}$, the goal is to find a shortest path in $\mathcal{P}$ to move from $s$ to see $q$ as quickly as possible. Read More

As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as {\em functional distortion} or {\em interleaving} that are provably more discriminative than the so-called {\em bottleneck distance}, being true metrics whereas the latter is only a pseudo-metric. Read More

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 of pieces is a fixed constant. Read More

A tensor $T$, in a given tensor space, is said to be $h$-identifiable if it admits a unique decomposition as a sum of $h$ rank one tensors. A criterion for $h$-identifiability is called effective if it is satisfied in a dense, open subset of the set of rank $h$ tensors. In this paper we give effective $h$-identifiability criteria for a large class of tensors. Read More

The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back to more than fifty years ago and, recently, it has driven increasing attention in the areas of graph theory, graph algorithms, graph drawing, and computational geometry. Read More

A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of $2$-level polytopes of a given dimension $d$, and provide complete experimental results for $d \leqslant 7$. Read More

We define the visual complexity of a plane graph drawing to be the number of geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g. Read More

The Euclidean TSP with neighborhoods (TSPN) problem seeks a shortest tour that visits a given collection of $n$ regions ({\em neighborhoods}). We present the first polynomial-time approximation scheme for TSPN for a set of regions given by arbitrary disjoint fat regions in the plane. This improves substantially upon the known approximation algorithms, and is the first PTAS for TSPN on regions of non-comparable sizes. Read More

In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NP-hard. In this paper, we present new approximation results for the TSPN, including (1) a constant-factor approximation algorithm for the case of arbitrary connected neighborhoods having comparable diameters; and (2) a PTAS for the important special case of disjoint unit disk neighborhoods (or nearly disjoint, nearly-unit disks). Read More

In an $\mathsf{L}$-embedding of a graph, each vertex is represented by an $\mathsf{L}$-segment, and two segments intersect each other if and only if the corresponding vertices are adjacent in the graph. If the corner of each $\mathsf{L}$-segment in an $\mathsf{L}$-embedding lies on a straight line, we call it a monotone $\mathsf{L}$-embedding. In this paper we give a full characterization of monotone $\mathsf{L}$-embeddings by introducing a new class of graphs which we call "non-jumping" graphs. Read More

If $\varphi$ and $\psi$ are two continuous real-valued functions defined on a compact topological space $X$ and $G$ is a subgroup of the group of all homeomorphisms of $X$ onto itself, the natural pseudo-distance $d_G(\varphi,\psi)$ is defined as the infimum of $\mathcal{L}(g)=\|\varphi-\psi \circ g \|_\infty$, as $g$ varies in $G$. In this paper, we make a first step towards extending the study of this concept to the case of Lie groups, by assuming $X=G=S^1$. In particular, we study the set of the optimal homeomorphisms for $d_G$, i. Read More

Given a set $S$ of $n$ static points and a free point $p$ in the Euclidean plane, we study a new variation of the minimum enclosing circle problem, in which a dynamic weight that equals to the reciprocal of the distance from the free point $p$ to the undetermined circle center is included. In this work, we prove the optimal solution of the new problem is unique and lies on the boundary of the farthest-point Voronoi diagram of $S$, once $p$ does not coincide with any vertex of the convex hull of $S$. We propose a tree structure constructed from the boundary of the farthest-point Voronoi diagram and use the hierarchical relationship between edges to locate the optimal solution. Read More

A graph is $k$-planar $(k \geq 1)$ if it can be drawn in the plane such that no edge is crossed more than $k$ times. A graph is $k$-quasi planar $(k \geq 2)$ if it can be drawn in the plane with no $k$ pairwise crossing edges. The families of $k$-planar and $k$-quasi planar graphs have been widely studied in the literature, and several bounds have been proven on their edge density. Read More

In the General Position Subset Selection (GPSS) problem, the goal is to find the largest possible subset of a set of points, such that no three of its members are collinear. If $s_{\textrm{GPSS}}$ is the size the optimal solution, $\sqrt{s_{\textrm{GPSS}}}$ is the current best guarantee for the size of the solution obtained using a polynomial time algorithm. In this paper we present an algorithm for GPSS to improve this bound based on the number of collinear pairs of points. Read More

We present a new algorithm for the widely used density-based clustering method DBscan. Our algorithm computes the DBscan-clustering in $O(n\log n)$ time in $\mathbb{R}^2$, irrespective of the scale parameter $\varepsilon$ (and assuming the second parameter MinPts is set to a fixed constant, as is the case in practice). Experiments show that the new algorithm is not only fast in theory, but that a slightly simplified version is competitive in practice and much less sensitive to the choice of $\varepsilon$ than the original DBscan algorithm. Read More

The Gapminder project set out to use statistics to dispel simplistic notions about global development. In the same spirit, we use persistent homology, a technique from computational algebraic topology, to explore the relationship between country development and geography. For each country, two statistics, gross domestic product per capita and average life expectancy, were used to quantify the development. Read More

We investigate the PPI algorithm as a means for computing ap- proximate convex hull. We explain how the algorithm computes the curvature of points and prove consistency and convergence. We also extend the algorithm to compute approximate convex hulls described in terms of hyperplanes. Read More

A straight-line drawing $\Gamma$ of a graph $G=(V,E)$ is a drawing of $G$ in the Euclidean plane, where every vertex in $G$ is mapped to a distinct point, and every edge in $G$ is mapped to a straight line segment between their endpoints. A path $P$ in $\Gamma$ is called increasing-chord if for every four points (not necessarily vertices) $a,b,c,d$ on $P$ in this order, the Euclidean distance between $b,c$ is at most the Euclidean distance between $a,d$. A spanning tree $T$ rooted at some vertex $r$ in $\Gamma$ is called increasing-chord if $T$ contains an increasing-chord path from $r$ to every vertex in $T$. Read More

We introduce the persistent homotopy type distance dHT to compare real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of dHT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopically equivalent. This distance is interesting in connection with persistent homology. Read More

Schnyder woods are particularly elegant combinatorial structures with numerous applications concerning planar triangulations and more generally 3-connected planar maps. We propose a simple generalization of Schnyder woods from the plane to maps on orientable surfaces of any genus with a special emphasis on the toroidal case. We provide a natural partition of the set of Schnyder woods of a given map into distributive lattices depending on the surface homology. Read More

For a set of points in the plane, a \emph{crossing family} is a set of line segments, each joining two of the points, such that any two line segments cross. We investigate the following generalization of crossing families: a \emph{spoke set} is a set of lines drawn through a point set such that each unbounded region of the induced line arrangement contains at least one point of the point set. We show that every point set has a spoke set of size $\sqrt{\frac{n}{8}}$. Read More

For a distribution function $F$ on $\mathbb{R}^d$ and a point $q\in \mathbb{R}^d$, the \emph{spherical depth} $\SphD(q;F)$ is defined to be the probability that a point $q$ is contained inside a random closed hyper-ball obtained from a pair of points from $F$. The spherical depth $\SphD(q;S)$ is also defined for an arbitrary data set $S\subseteq \mathbb{R}^d$ and $q\in \mathbb{R}^d$. This definition is based on counting all of the closed hyper-balls, obtained from pairs of points in $S$, that contain $q$. Read More

Given a set of $n$ points $P$ in the plane, the first layer $L_1$ of $P$ is formed by the points that appear on $P$'s convex hull. In general, a point belongs to layer $L_i$, if it lies on the convex hull of the set $P \setminus \bigcup_{jRead More

Edge bundling is an important concept, heavily used for graph visualization purposes. To enable the comparison with other established nearly-planarity models in graph drawing, we formulate a new edge-bundling model which is inspired by the recently introduced fan-planar graphs. In particular, we restrict the bundling to the endsegments of the edges. Read More

We introduce a new geometric spanner, $\delta$-Greedy, whose construction is based on a generalization of the known Path-Greedy and Gap-Greedy spanners. The $\delta$-Greedy spanner combines the most desirable properties of geometric spanners both in theory and in practice. More specifically, it has the same theoretical and practical properties as the Path-Greedy spanner: a natural definition, small degree, linear number of edges, low weight, and strong $(1+\varepsilon)$-spanner for every $\varepsilon>0$. Read More

A celebrated technique for finding near neighbors for the angular distance involves using a set of \textit{random} hyperplanes to partition the space into hash regions [Charikar, STOC 2002]. Experiments later showed that using a set of \textit{orthogonal} hyperplanes, thereby partitioning the space into the Voronoi regions induced by a hypercube, leads to even better results [Terasawa and Tanaka, WADS 2007]. However, no theoretical explanation for this improvement was ever given, and it remained unclear how the resulting hypercube hash method scales in high dimensions. Read More

A graph $G$ is called B$_k$-VPG (resp., B$_k$-EPG), for some constant $k\geq 0$, if it has a string representation on a grid such that each vertex is an orthogonal path with at most $k$ bends and two vertices are adjacent in $G$ if and only if the corresponding strings intersect (resp., the corresponding strings share at least one grid edge). Read More

Airborne laser scanning (lidar) point clouds can be process to extract tree-level information over large forested landscapes. Existing procedures typically detect more than 90% of overstory trees, yet they barely detect 60% of understory trees because of reduced number of lidar points penetrating the top canopy layer. Although understory trees provide limited financial value, they offer habitat for numerous wildlife species and are important for stand development. Read More