# Computer Science - Computational Geometry Publications (50)

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## Computer Science - Computational Geometry Publications

We study the problem of computing the \textsc{Maxima} of a set of $n$ $d$-dimensional points. For dimensions 2 and 3, there are algorithms to solve the problem with order-oblivious instance-optimal running time. However, in higher dimensions there is still room for improvements. Read More

We introduce the dynamic conflict-free coloring problem for a set $S$ of intervals in $\mathbb{R}^1$ with respect to points, where the goal is to maintain a conflict-free coloring for $S$ under insertions and deletions. We investigate trade-offs between the number of colors used and the number of intervals that are recolored upon insertion or deletion of an interval. Our results include: - a lower bound on the number of recolorings as a function of the number of colors, which implies that with $O(1)$ recolorings per update the worst-case number of colors is $\Omega(\log n/\log\log n)$, and that any strategy using $O(1/\varepsilon)$ colors needs $\Omega(\varepsilon n^{\varepsilon})$ recolorings; - a coloring strategy that uses $O(\log n)$ colors at the cost of $O(\log n)$ recolorings, and another strategy that uses $O(1/\varepsilon)$ colors at the cost of $O(n^{\varepsilon}/\varepsilon)$ recolorings; - stronger upper and lower bounds for special cases. Read More

In this paper, we propose a framework to reconstruct 3D models from raw scanned points by learning the prior knowledge of a specific class of objects. Unlike previous work that heuristically specifies particular regularities and defines parametric models, our shape priors are learned directly from existing 3D models under a framework based on affinity propagation. Given a database of 3D models within the same class of objects, we build a comprehensive library of 3D local shape priors. Read More

Solutions of partial differential equations (PDEs) on manifolds have provided important applications in different fields in science and engineering. Existing methods are majorly based on discretization of manifolds as implicit functions, triangle meshes, or point clouds, where the manifold structure is approximated by either zero level set of an implicit function or a set of points. In many applications, manifolds might be only provided as an inter-point distance matrix with possible missing values. Read More

Consider a pair of plane straight-line graphs, whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. Read More

A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. Read More

We study the following local-to-global phenomenon: Let $B$ and $R$ be two finite sets of (blue and red) points in the Euclidean plane $\mathbb{R}^2$. Suppose that in each "neighborhood" of a red point, the number of blue points is at least as large as the number of red points. We show that in this case the total number of blue points is at least one fifth of the total number of red points. Read More

Readability criteria have been addressed as a measurement of the quality of graph visualizations. In this paper, we argue that readability criteria are necessary but not sufficient. We propose a new kind of criteria, namely faithfulness, to evaluate the quality of graph layouts. Read More

We present an algorithm that cuts any collection of n disjoint triangles in R^3 into O(n^{7/4} polylog n) triangular fragments such that all cycles in the depth-order relation are eliminated. The running time of our algorithm is O(n^{3.69}). Read More

This paper investigates efficient techniques to collect and concentrate an under-actuated particle swarm despite obstacles. Concentrating a swarm of particles is of critical importance in health-care for targeted drug delivery, where micro-scale particles must be steered to a goal location. Individual particles must be small in order to navigate through micro-vasculature, but decreasing size brings new challenges. Read More

This paper presents a non-parametric approach for segmenting trees from airborne LiDAR data in deciduous forests. Based on the LiDAR point cloud, the approach collects crown information such as steepness and height on-the-fly to delineate crown boundaries, and most importantly, does not require a priori assumptions of crown shape and size. The approach segments trees iteratively starting from the tallest within a given area to the smallest until all trees have been segmented. Read More

Airborne LiDAR point cloud of a forest contains three dimensional data, from which vertical stand structure (including information about under-story trees) can be derived. This paper presents a segmentation approach for multi-story stands that strips the point cloud to its canopy layers, identifies individual tree segments within each layer using a DSM-based tree identification method as a building block, and combines the segments of immediate layers in order to fix potential over-segmentation of tree crowns across the layers. We introduce local layering that analyzes the vertical distributions of LiDAR points within their local neighborhoods in order to locally determine the height thresholds for layering the canopy. Read More

We prove the computational intractability of rotating and placing $n$ square tiles into a $1 \times n$ array such that adjacent tiles are compatible--either equal edge colors, as in edge-matching puzzles, or matching tab/pocket shapes, as in jigsaw puzzles. Beyond basic NP-hardness, we prove that it is NP-hard even to approximately maximize the number of placed tiles (allowing blanks), while satisfying the compatibility constraint between nonblank tiles, within a factor of 0.9999999851. Read More

**Affiliations:**

^{1}DATASHAPE,

^{2}DATASHAPE,

^{3}LSTA

Approximations of Laplace-Beltrami operators on manifolds through graph Lapla-cians have become popular tools in data analysis and machine learning. These discretized operators usually depend on bandwidth parameters whose tuning remains a theoretical and practical problem. In this paper, we address this problem for the unnormalized graph Laplacian by establishing an oracle inequality that opens the door to a well-founded data-driven procedure for the bandwidth selection. Read More

A graph drawing is $\textit{greedy}$ if, for every ordered pair of vertices $(x,y)$, there is a path from $x$ to $y$ such that the Euclidean distance to $y$ decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination "greedily" forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed. Read More

Human identification remains to be one of the challenging tasks in computer vision community due to drastic changes in visual features across different viewpoints, lighting conditions, occlusion, etc. Most of the literature has been focused on exploring human re-identification across viewpoints that are not too drastically different in nature. Cameras usually capture oblique or side views of humans, leaving room for a lot of geometric and visual reasoning. Read More

The $c$-approximate Near Neighbor problem in high dimensional spaces has been mainly addressed by Locality Sensitive Hashing (LSH), which offers polynomial dependence on the dimension, query time sublinear in the size of the dataset, and subquadratic space requirement. For practical applications, linear space is typically imperative. Most previous work in the linear space regime focuses on the case that $c$ exceeds $1$ by a constant term. Read More

In this paper, we show that any $B_2$-VPG graph (i.e., an intersection graph of orthogonal curves with at most 2 bends) can be decomposed into $O(\log n)$ outerstring graphs or $O(\log^3 n)$ permutation graphs. Read More

We address the NP-hard problem of finding a non-overlapping dense packing pattern for n Unequal Circle items in a two-dimensional Square Container (PUC-SC) such that the size of the container is minimized. Based on our previous work on an Action Space based Global Optimization (ASGO) that approximates each circle item as a square item to efficiently find the large unoccupied spaces, we propose an optimization algorithm based on the Partitioned Action Space and Partitioned Circle Items (PAS-PCI). The PAS is to partition the narrow action space on the long side to find two equal action spaces to fully utilize the unoccupied spaces. Read More

In this paper, we study the dominance relation under a stochastic setting. Let $\mathcal{S}$ be a set of $n$ colored stochastic points in $\mathbb{R}^d$, each of which is associated with an existence probability. We investigate the problem of computing the probability that a realization of $\mathcal{S}$ contains inter-color dominances, which we call the \textit{colored stochastic dominance} (CSD) problem. Read More

We consider the problem of planning a collision-free path of a robot in the presence of risk zones. The robot is allowed to travel in these zones but is penalized in a super-linear fashion for consecutive accumulative time spent there. We recently suggested a natural cost function that balances path length and risk-exposure time. Read More

Let $T$ be a tree space (or tree network) represented by a weighted tree with $t$ vertices, and $S$ be a set of $n$ stochastic points in $T$, each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the $\ell$-threshold probability and the expectation of the closest-pair distance of a realization of $S$. Read More

We consider edge insertion and deletion operations that increase the connectivity of a given planar straight-line graph (PSLG), while minimizing the total edge length of the output. We show that every connected PSLG $G=(V,E)$ in general position can be augmented to a 2-connected PSLG $(V,E\cup E^+)$ by adding new edges of total Euclidean length $\|E^+\|\leq 2\|E\|$, and this bound is the best possible. An optimal edge set $E^+$ can be computed in $O(|V|^4)$ time; however the problem becomes NP-hard when $G$ is disconnected. Read More

This paper addresses 3D shape recognition. Recent work typically represents a 3D shape as a set of binary variables corresponding to 3D voxels of a uniform 3D grid centered on the shape, and resorts to deep convolutional neural networks(CNNs) for modeling these binary variables. Robust learning of such CNNs is currently limited by the small datasets of 3D shapes available, an order of magnitude smaller than other common datasets in computer vision. Read More

Near neighbor problems are fundamental in algorithms for high-dimensional Euclidean spaces. While classical approaches suffer from the curse of dimensionality, locality sensitive hashing (LSH) can effectively solve a-approximate r-near neighbor problem, and has been proven to be optimal in the worst case. However, for real-world data sets, LSH can naturally benefit from well-dispersed data and low doubling dimension, leading to significantly improved performance. Read More

An important theorem of Banaszczyk (Random Structures & Algorithms `98) states that for any sequence of vectors of $\ell_2$ norm at most $1/5$ and any convex body $K$ of Gaussian measure $1/2$ in $\mathbb{R}^n$, there exists a signed combination of these vectors which lands inside $K$. A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. Read More

The crossing number is the smallest number of pairwise edge-crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions, like treewidth or pathwidth. Read More

We consider several classes of intersection graphs of line segments in the plane and prove new equality and separation results between those classes. In particular, we show that: (1) intersection graphs of grounded segments and intersection graphs of downward rays form the same graph class, (2) not every intersection graph of rays is an intersection graph of downward rays, and (3) not every intersection graph of rays is an outer segment graph. The first result answers an open problem posed by Cabello and Jej\v{c}i\v{c}. Read More

**Authors:**Cody Barnson

^{1}, Dawn Chandler

^{2}, Qiao Chen

^{3}, Christina Chung

^{4}, Andrew Coccimiglio

^{5}, Sean La

^{6}, Lily Li

^{7}, Aïna Linn

^{8}, Anna Lubiw

^{9}, Clare Lyle

^{10}, Shikha Mahajan

^{11}, Gregory Mierzwinski

^{12}, Simon Pratt

^{13}, Yoon Su

^{14}, Yoo, Hongbo Zhang, Kevin Zhang

**Affiliations:**

^{1}Matthias,

^{2}Matthias,

^{3}Matthias,

^{4}Matthias,

^{5}Matthias,

^{6}Matthias,

^{7}Matthias,

^{8}Matthias,

^{9}Matthias,

^{10}Matthias,

^{11}Matthias,

^{12}Matthias,

^{13}Matthias,

^{14}Matthias

We consider the conjecture by Aichholzer, Aurenhammer, Hurtado, and Krasser that any two points sets with the same cardinality and the same size convex hull can be triangulated in the "same" way, more precisely via \emph{compatible triangulations}. We show counterexamples to various strengthened versions of this conjecture. Read More

We introduce the dune-curvilineargrid module. The module provides the self-contained, parallel grid manager, as well as the underlying elementary curvilinear geometry module dune-curvilineargeometry. This work is motivated by the need for reliable and scalable electromagnetic design of nanooptical devices. Read More

Delaunay has shown that the Delaunay complex of a finite set of points $P$ of Euclidean space $\mathbb{R}^m$ triangulates the convex hull of $P$, provided that $P$ satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay's genericity assumption no longer guarantees that the Delaunay complex will yield a triangulation; stronger assumptions on $P$ are required. Read More

We present in this paper a framework for mapping data onto real and complex projective spaces. The resulting projective coordinates provide a multi-scale representation of the data, and capture low dimensional underlying topological features. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Read More

In this paper, we present a new method for computing approximate geodesic distances. We introduce the wave method for approximating geodesic distances from a point on a manifold mesh. Our method involves the solution of two linear systems of equations. Read More

We consider the problem of digitalizing Euclidean line segments from $\mathbb{R}^d$ to $\mathbb{Z}^d$. Christ {\em et al.} (DCG, 2012) showed how to construct a set of {\em consistent digital segment} (CDS) for $d=2$: a collection of segments connecting any two points in $\mathbb{Z}^2$ that satisfies the natural extension of the Euclidean axioms to $\mathbb{Z}^d$. Read More

Let $C$ be the unit circle in $\mathbb{R}^2$. We can view $C$ as a plane graph whose vertices are all the points on $C$, and the distance between any two points on $C$ is the length of the smaller arc between them. We consider a graph augmentation problem on $C$, where we want to place $k\geq 1$ \emph{shortcuts} on $C$ such that the diameter of the resulting graph is minimized. Read More

Shape analysis is very often performed by segmenting the shape into smooth surface parts that can be further classified using a set of predefined primitives such as planes, cylinders or spheres. Hence the shape is generally assumed to be manifold and smooth or to be an assembly of primitive parts. In this paper we propose an approach which does not make any assumption on the shape properties but rather learns its characteristics through a statistical analysis of local shape variations. Read More

We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m=3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. Read More

Let $P$ be a set of $n$ points in the plane. We show how to find, for a given integer $k>0$, the smallest-area axis-parallel rectangle that covers $k$ points of $P$ in $O(nk^2 \log n+ n\log^2 n)$ time. We also consider the problem of, given a value $\alpha>0$, covering as many points of $P$ as possible with an axis-parallel rectangle of area at most $\alpha$. Read More

Development of additive manufacturing in last decade greatly improves tissue engineering. During the manufacturing of porous scaffold, simplified but functionally equivalent models are getting focused for practically reasons. Scaffolds can be classified into regular porous scaffolds and irregular porous scaffolds. Read More

In the polytope membership problem, a convex polytope $K$ in $R^d$ is given, and the objective is to preprocess $K$ into a data structure so that, given a query point $q \in R^d$, it is possible to determine efficiently whether $q \in K$. We consider this problem in an approximate setting and assume that $d$ is a constant. Given an approximation parameter $\varepsilon > 0$, the query can be answered either way if the distance from $q$ to $K$'s boundary is at most $\varepsilon$ times $K$'s diameter. Read More

We show that the KLS constant for $n$-dimensional isotropic logconcave measures is $O(n^{1/4})$, improving on the current best bound of $O(n^{1/3}\sqrt{\log n})$$.$ As corollaries we obtain the same improved bound on the thin-shell estimate, Poincare constant and exponential concentration constant and an alternative proof of this bound for the isotropic constant; it also follows that the ball walk for sampling from an isotropic logconcave density in ${\bf R}^{n}$ converges in $O^{*}(n^{2.5})$ steps from a warm start. Read More

Cellular operators are increasingly turning towards renewable energy (RE) as an alternative to using traditional electricity in order to reduce operational expenditure and carbon footprint. Due to the randomness in both RE generation and mobile traffic at each base station (BS), a surplus or shortfall of energy may occur at any given time. To increase energy self-reliance and minimize the network's energy cost, the operator needs to efficiently exploit the RE generated across all BSs. Read More

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, where 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. Read More

This paper applies the randomized incremental construction (RIC) framework to computing the Hausdorff Voronoi diagram of a family of k clusters of points in the plane. The total number of points is n. The diagram is a generalization of Voronoi diagrams based on the Hausdorff distance function. Read More

We present a simple sublinear time algorithm for testing the following geometric property. Let $P_1, .. Read More

We establish a cutting lemma for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definable families of sets on the plane in $o$-minimal expansions of fields. Using it, we generalize the results in [J. Fox, J. Read More

In this paper, we propose a novel curvature-penalized minimal path model via an orientation-lifted Finsler metric and the Euler elastica curve. The original minimal path model computes the globally minimal geodesic by solving an Eikonal partial differential equation (PDE). Essentially, this first-order model is unable to penalize curvature which is related to the path rigidity property in the classical active contour models. Read More

The discovery of physical laws consistent with empirical observations lies at the heart of (applied) science and engineering. These laws typically take the form of nonlinear differential equations depending on parameters, dynamical systems theory provides, through the appropriate normal forms, an "intrinsic", prototypical characterization of the types of dynamical regimes accessible to a given model. Using an implementation of data-informed geometry learning we directly reconstruct the relevant "normal forms": a quantitative mapping from empirical observations to prototypical realizations of the underlying dynamics. Read More

An efficient algorithm to enumerate the vertices of a two-dimensional (2D) projection of a polytope, is presented in this paper. The proposed algorithm uses the support function of the polytope to be projected and enumerated for vertices. The complexity of our algorithm is linear in the number of vertices of the projected polytope and we show empirically that the performance is significantly better in comparison to some known efficient algorithms of projection and enumeration. Read More

We consider a problem of dispersing points on disjoint intervals on a line. Given n pairwise disjoint intervals sorted on a line, we want to find a point in each interval such that the minimum pairwise distance of these points is maximized. Based on a greedy strategy, we present a linear time algorithm for the problem. Read More