# Computer Science - Computational Geometry Publications (50) Home Article Categories

## Computer Science - Computational Geometry Publications

2017Apr

We devise dynamic algorithms for the following (weak) visibility polygon computation problems: * Maintaining visibility polygon of a fixed point located interior to simple polygon amid vertex insertions and deletions to simple polygon. * Answering visibility polygon query corresponding to any point located exterior to simple polygon amid vertex insertions and deletions to simple polygon. * Maintaining weak visibility polygon of a fixed line segment located interior to simple polygon amid vertex insertions to simple polygon. Read More

We focus on the analysis of planar shapes and solid objects having thin features and propose a new mathematical model to characterize them. Based on our model, that we call an epsilon-shape, we show how thin parts can be effectively and efficiently detected by an algorithm, and propose a novel approach to thicken these features while leaving all the other parts of the shape unchanged. When compared with state-of-the-art solutions, our proposal proves to be particularly flexible, efficient and stable, and does not require any unintuitive parameter to fine-tune the process. Read More

We say that an algorithm is stable if small changes in the input result in small changes in the output. Algorithm stability plays an important role when analyzing and visualizing time-varying data. However, so far, there are only few theoretical results on the stability of algorithms, possibly due to a lack of theoretical analysis tools. Read More

A pseudocircle is a simple closed curve on some surface. Arrangements of pseudocircles were introduced by Gr\"unbaum, who defined them as collections of pseudocircles that pairwise intersect in exactly two points, at which they cross. There are several variations on this notion in the literature, one of which requires that no three pseudocircles have a point in common. Read More

Motivated by map labeling, we study the problem in which we are given a collection of $n$ disks $D_1, \dots, D_n$ in the plane that grow at possibly different speeds. Whenever two disks meet, the one with the lower index disappears. This problem was introduced by Funke, Krumpe, and Storandt [IWOCA 2016]. Read More

In this paper, we consider a coverage problem for uncertain points in a tree. Let T be a tree containing a set P of n (weighted) demand points, and the location of each demand point P_i\in P is uncertain but is known to appear in one of m_i points on T each associated with a probability. Given a covering range \lambda, the problem is to find a minimum number of points (called centers) on T to build facilities for serving (or covering) these demand points in the sense that for each uncertain point P_i\in P, the expected distance from P_i to at least one center is no more than $\lambda$. Read More

We give algorithms with running time $2^{O({\sqrt{k}\log{k}})} \cdot n^{O(1)}$ for the following problems. Given an $n$-vertex unit disk graph $G$ and an integer $k$, decide whether $G$ contains (1) a path on exactly/at least $k$ vertices, (2) a cycle on exactly $k$ vertices, (3) a cycle on at least $k$ vertices, (4) a feedback vertex set of size at most $k$, and (5) a set of $k$ pairwise vertex-disjoint cycles. For the first three problems, no subexponential time parameterized algorithms were previously known. Read More

We investigate several computational problems related to stochastic convex hull (SCH). Given a stochastic dataset consisting of $n$ points in $\mathbb{R}^d$ each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e. Read More

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational geometry. Given a simple polygon $P$ and an integer $k$, the goal is to decide if there exists a set $G$ of $k$ guards within $P$ such that every point $p\in P$ is seen by at least one guard $g\in G$. Read More

In this paper, we consider the problems for covering multiple intervals on a line. Given a set B of m line segments (called "barriers") on a horizontal line L and another set S of n horizontal line segments of the same length in the plane, we want to move all segments of S to L so that their union covers all barriers and the maximum movement of all segments of S is minimized. Previously, an O(n^3 log n)-time algorithm was given for the problem but only for the special case m = 1. Read More

We present an algorithm for computation of cell adjacencies for well-based cylindrical algebraic decomposition. Cell adjacency information can be used to compute topological operations e.g. Read More

The dispersion problem has been widely studied in computational geometry and facility location, and is closely related to the packing problem. The goal is to locate n points (e.g. Read More

We consider the path planning problem for a 2-link robot amidst polygonal obstacles. Our robot is parametrizable by the lengths $\ell_1, \ell_2>0$ of its two links, the thickness $\tau \ge 0$ of the links, and an angle $\kappa$ that constrains the angle between the 2 links to be strictly greater than $\kappa$. The case $\tau>0$ and $\kappa \ge 0$ corresponds to "thick non-crossing" robots. Read More

We present a simplified treatment of stability of filtrations on finite spaces. Interestingly, we can lift the stability result for combinatorial filtrations from [CSEM06] to the case when two filtrations live on different spaces without directly invoking the concept of interleaving. Read More

Let $P$ be a finite set of points in the plane and $S$ a set of non-crossing line segments with endpoints in $P$. The visibility graph of $P$ with respect to $S$, denoted $Vis(P,S)$, has vertex set $P$ and an edge for each pair of vertices $u,v$ in $P$ for which no line segment of $S$ properly intersects $uv$. We show that the constrained half-$\theta_6$-graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) is a plane 2-spanner of $Vis(P,S)$. Read More

We study an extension of active learning in which the learning algorithm may ask the annotator to compare the distances of two examples from the boundary of their label-class. For example, in a recommendation system application (say for restaurants), the annotator may be asked whether she liked or disliked a specific restaurant (a label query); or which one of two restaurants did she like more (a comparison query). We focus on the class of half spaces, and show that under natural assumptions, such as large margin or bounded bit-description of the input examples, it is possible to reveal all the labels of a sample of size $n$ using approximately $O(\log n)$ queries. Read More

We address the problem of restoring a high-quality image from an observed image sequence strongly distorted by atmospheric turbulence. A novel algorithm is proposed in this paper to reduce geometric distortion as well as space-and-time-varying blur due to strong turbulence. By considering a suitable energy functional, our algorithm first obtains a sharp reference image and a subsampled image sequence containing sharp and mildly distorted image frames with respect to the reference image. Read More

2017Apr

It is a long standing open problem whether Yao-Yao graphs $\mathsf{YY}_{k}$ are all spanners. Bauer and Damian \cite{bauer2013infinite} showed that all $\mathsf{YY}_{6k}$ for $k \geq 6$ are spanners. Li and Zhan \cite{li2016almost} generalized their result and proved that all even Yao-Yao graphs $\mathsf{YY}_{2k}$ are spanners (for $k\geq 42$). Read More

2017Apr
Affiliations: 1DGA-TN, 2Ensta Bretagne, Lab-Sticc

This papers shows that using separators, which is a pair of two complementary contractors, we can easily and efficiently solve the localization problem of a robot with sonar measurements in an unstructured environment. We introduce separators associated with the Minkowski sum and the Minkowski difference in order to facilitate the resolution. A test-case is given in order to illustrate the principle of the approach. Read More

We present a deterministic algorithm that computes the diameter of a directed planar graph with real arc lengths in $\tilde{O}(n^{5/3})$ time. This improves the recent breakthrough result of Cabello (SODA'17), both by improving the running time (from $\tilde{O}(n^{11/6})$), and by using a deterministic algorithm. It is in fact the first truly subquadratic deterministic algorithm for this problem. Read More

2017Apr

LSH (locality sensitive hashing) had emerged as a powerful technique in nearest-neighbor search in high dimensions [IM98, HIM12]. Given a point set $P$ in a metric space, and given parameters $r$ and $\varepsilon > 0$, the task is to preprocess the point set, such that given a query point $q$, one can quickly decide if $q$ is in distance at most $\leq r$ or $\geq (1+\varepsilon)r$ from the point set $P$. Once such a near-neighbor data-structure is available, one can reduce the general nearest-neighbor search to logarithmic number of queries in such structures [IM98, Har01, HIM12]. Read More

In this paper, we are concerned with the problem of creating flattening maps of simply-connected open surfaces in $\mathbb{R}^3$. Using a natural principle of density diffusion in physics, we propose an effective algorithm for computing density-equalizing flattening maps with any prescribed density distribution. By varying the initial density distribution, a large variety of mappings with different properties can be achieved. Read More

2017Apr

The problem of constrained $k$-center clustering has attracted significant attention in the past decades. In this paper, we study balanced $k$-center cluster where the size of each cluster is constrained by the given lower and upper bounds. The problem is motivated by the applications in processing and analyzing large-scale data in high dimension. Read More

A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational framework to determine {\delta}(d, k) for small instances. Read More

We present the first algorithm for finding holes in high dimensional data that runs in polynomial time with respect to the number of dimensions. Previous algorithms are exponential. Finding large empty rectangles or boxes in a set of points in 2D and 3D space has been well studied. Read More

We present an algorithm to support the dynamic embedding in the plane of a dynamic graph. An edge can be inserted across a face between two vertices on the face boundary (we call such a vertex pair linkable), and edges can be deleted. The planar embedding can also be changed locally by flipping components that are connected to the rest of the graph by at most two vertices. Read More

We exhibit relations between van Kampen-Flores, Conway-Gordon-Sachs and Radon theorems, by presenting direct proofs of some implications between them. The key idea is an interesting relation between the van Kampen and the Conway-Gordon-Sachs numbers for restrictions of a map of $(d+2)$-simplex to $\mathbb R^d$ to the $(d+1)$-face and to the $[d/2]$-skeleton. Read More

We construct a dense point set of $n$ points in the plane with $ne^{\Omega\left({\sqrt{\log n}}\right)}$ halving lines. This improves the bound $O(n\log n)$ of Edelsbrunner, Valtr and Welzl from 1997. We also observe that the upper bound on the maximum number of halving lines of dense point set can be improved to $O(n^{7/6})$. Read More

The $r$-fold analogues of Whitney trick were `in the air' since 1960s. However, only in this century they were stated, proved and applied to obtain interesting results, most notably by Mabillard and Wagner. Here we prove and apply a version of the $r$-fold Whitney trick when general position $r$-tuple intersections have positive dimension. Read More

2017Apr

In this paper we introduce a novel algorithm to combine two or more cellular complexes, providing a minimal fragmentation of the cells of the resulting complex. This algorithm has several important applications, including Boolean operations over big geometric data, like the detailed geometric modeling of buildings, and the smooth combination of 3D meshes. The algorithm operates over the LAR representation of argument complexes, based on sparse matrices, so being well-suited for implementation on last generation accelerators and GPGPU applications. Read More

2017Mar

We study the minimum diameter problem for a set of inexact points. By inexact, we mean that the precise location of the points is not known. Instead, the location of each point is restricted to a contineus region ($\impre$ model) or a finite set of points ($\indec$ model). Read More

The computation of (i) $\varepsilon$-kernels, (ii) approximate diameter, and (iii) approximate bichromatic closest pair are fundamental problems in geometric approximation. In this paper, we describe new algorithms that offer significant improvements to their running times. In each case the input is a set of $n$ points in $R^d$ for a constant dimension $d \geq 3$ and an approximation parameter $\varepsilon > 0$. Read More

Sending a message through an unknown network is a difficult problem. In this paper, we consider the case in which, during a preprocessing phase, we assign a label and a routing table to each node. The routing strategy must then decide where to send the package using only the label of the target node and the routing table of the node the message is currently at. Read More

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

This article introduces a theory of proximal nerve complexes and nerve spokes, restricted to the triangulation of finite regions in the Euclidean plane. A nerve complex is a collection of filled triangles with a common vertex, covering a finite region of the plane. Structures called $k$-spokes, $k\geq 1$, are a natural extension of nerve complexes. 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