Computational Geometry-Theory and Applications最新文献

A geometric graph is a graph whose vertex set is a set of points in general position in the plane, and its edges are straight line segments joining these points. We show that for every integer $k\ge 2$, there exists a constant $c>0$ such that the following holds. The edges of every dense geometric graph, with sufficiently many vertices, can be colored with k colors, such that the number of pairs of edges of the same color that cross is at most $(1/k-c)$ times the total number of pairs of edges that cross. The case when $k=2$ and G is a complete geometric graph, was proved by Aichholzer et al. (2019) [2].

In this article, we present a construction of a spanner on a set of n points in $R^d$ that we call a heavy path WSPD spanner. The construction is parameterized by a constant $s>2$ called the separation ratio. The size of the graph is $O\left(s^{d}n\right)$ and the spanning ratio is at most $1+2/s+2/(s-1)$. We also show that this graph has a hop spanning ratio of at most $2\lg n+1$.

We present a memoryless local routing algorithm for heavy path WSPD spanners. The routing algorithm requires a vertex v of the graph to store $O(\mathrm{deg}(v)\log n)$ bits of information, where $\mathrm{deg}\left(v\right)$ is the degree of v. The routing ratio is at most $1+4/s+1/(s-1)$ and at least $1+4/s$ in the worst case. The number of edges on the routing path is bounded by $2\lg n+1$.

We then show that the heavy path WSPD spanner can be constructed in metric spaces of bounded doubling dimension. These metric spaces have been studied in computational geometry as a generalization of Euclidean space. We show that, in a metric space with doubling dimension λ, the heavy path WSPD spanner has size $O\left(s^{\lambda }n\right)$ where s is the separation ratio. The spanning ratio and hop spanning ratio are the same as in the Euclidean case.

Finally, we show that the local routing algorithm works in the bounded doubling dimension case. The vertices require the same amount of storage, but the routing ratio becomes at most $1+(2+\frac{\tau }{\tau -1})/s+1/(s-1)$ in the worst case, where $\tau \ge 11$ is a constant related to the doubling dimension.

In this paper, we study the online class cover problem where a (finite or infinite) family $F$ of geometric objects and a set $P_r$ of red points in $R^d$ are given a prior, and blue points from $R^d$ arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from $F$ that do not cover any points of $P_r$. The objective of the problem is to place a minimum number of objects. When $F$ consists of axis-parallel unit squares in $R^2$, we prove that the competitive ratio of any deterministic online algorithm is $\Omega (\log |P_r\left|\right)$, and also propose an $O(\log |P_r\left|\right)$-competitive deterministic algorithm for the problem.

Quad-surfaces are polyhedral surfaces with quadrilateral faces and the combinatorics of a square grid. Isometric deformation of the quad-surfaces can be thought of as transformations that keep all the involved quadrilaterals rigid. Among quad-surfaces, those capable of non-trivial isometric deformations are identified as flexible, marking flexibility as a core topic in discrete differential geometry. The study of quad-surfaces and their flexibility is not only theoretically intriguing but also finds practical applications in fields like membrane theory, origami, architecture and robotics.

A generic quad-surface is rigid, however, certain subclasses exhibit a 1-parameter family of flexibility. One of such subclasses is the T-hedra which are originally introduced by Graf and Sauer in 1931.

This article provides a synthetic and an analytic description of T-hedra and their smooth counterparts namely, the T-surfaces. In the next step the parametrization of their isometric deformation is obtained and their deformability range is discussed. The given parametrizations and isometric deformations are provided for general T-hedra and T-surfaces. However, specific subclasses are extensively examined and explored, particularly those that encompass notable and well-known structures, including the Miura fold, surfaces of revolution and molding surfaces.

We design the first dynamic distance oracles for interval graphs, which are intersection graphs of a set of intervals on the real line, and for proper interval graphs, which are intersection graphs of a set of intervals in which no interval is properly contained in another.

For proper interval graphs, we design a linear space data structure which supports distance queries (computing the distance between two query vertices) and vertex insertion or deletion in $O(\lg n)$ worst-case time, where n is the number of vertices currently in G. Under incremental (insertion only) or decremental (deletion only) settings in general interval graphs, we design linear space data structures that support distance queries in $O(\lg n)$ worst-case time and vertex insertion or deletion in $O(\lg n)$ amortized time, where n is the maximum number of vertices in the graph. Under fully dynamic settings in general interval graphs, we design a data structure that represents an interval graph G in $O\left(n\right)$ words of space to support distance queries in $O(n\lg n/S(n\left)\right)$ worst-case time and vertex insertion or deletion in $O\left(S\right(n)+\lg n)$ worst-case time, where n is the number of vertices currently in G and $S\left(n\right)$ is an arbitrary function that satisfies $S\left(n\right)=\Omega \left(1\right)$ and $S\left(n\right)=O\left(n\right)$. This implies an $O\left(n\right)$-word solution with $O\left(\sqrt{n\lg n}\right)$-time support for both distance queries and updates. All four data structures can answer shortest path queries by reporting the vertices in the shortest path between two query vertices in $O(\lg n)$ worst-case time per vertex.

We also study the hardness of supporting distance queries under updates over an intersection graph of 3D axis-aligned line segments, which generalizes our problem to 3D. Finally, we solve the problem of computing the diameter of a dynamic connected interval graph.

Finding the smallest d-chain with a specific $(d-1)$-boundary in a simplicial complex is known as the Minimum Bounded Chain problem (MBC_d). MBC_d is NP-hard for all $d\ge 2$. In this paper, we prove that it is also W[1]-hard for all $d\ge 2$, if we parameterize the problem by solution size. We also give an algorithm solving MBC₁ in polynomial time and introduce and implement two fixed parameter tractable (FPT) algorithms solving MBC_d for all d. The first algorithm uses a shortest path approach and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.

Tessellations are an important tool to model the microstructure of cellular and polycrystalline materials. Classical tessellation models include the Voronoi diagram and the Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram.

The Euclidean k-Steiner tree problem asks for a minimum-cost network connecting n given points in the plane, allowing at most k additional nodes referred to as Steiner points. In the classical Steiner tree problem in which there is no restriction on the number of nodes, every Steiner point must be of degree 3. The k-Steiner problem differs in that Steiner points of degree 4 may be included in an optimal solution. This simple change leads to a number of complexities when attempting to create a generation algorithm for optimal k-Steiner trees, which has proven to be a powerful component of the flagship algorithm, namely GeoSteiner, for solving the classical Steiner tree problem. In the present paper we firstly extend the basic framework of GeoSteiner's generation algorithm to include degree-4 Steiner points. We then introduce a number of novel results restricting the structural and geometric properties of optimal k-Steiner trees, and then show how these properties may be used as topological pruning methods underpinning our generation algorithm. Finally, we present experimental data to show the effectiveness of our pruning methods in reducing the number of sub-optimal solution topologies.

In this study, we lay the groundwork for a systematic investigation of the rigidity and flexibility of rigid origami by using the mathematical model referred to as the panel-point model. Rigid origami is commonly known as a type of panel-hinge structure where rigid polygonal panels are connected by rotational hinges, and its motion and stability are often investigated from the perspective of its consistency constraints representing the rigidity and connection conditions of panels. In the proposed methodology, vertex coordinates are directly treated as the variables to represent the rigid origami in the panel-point model, and these variables are constrained by the conditions for the out-of-plane and in-plane rigidity of panels. This model offers several advantages including: 1) the simplicity of polynomial consistency constraints; 2) the ease of incorporating displacement boundary conditions; and 3) the straightforwardness of numerical simulation and visualization. It is anticipated that the presented theories in this article are valuable to a broad audience, including mathematicians, engineers, and architects.

Given a set of n colored points with k colors in the plane, we study the problem of computing a maximum-width rainbow-bisecting empty annulus (of objects specifically axis-parallel square, axis-parallel rectangle and circle) problem. We call a region rainbow if it contains at least one point of each color. The maximum-width rainbow-bisecting empty annulus problem asks to find an annulus A of a particular shape with maximum possible width such that A does not contain any input points and it bisects the input point set into two parts, each of which is a rainbow. We compute a maximum-width rainbow-bisecting empty axis-parallel square, axis-parallel rectangular and circular annulus in $O\left(n^3\right)$ time using $O\left(n\right)$ space, in $O(k^2{n}^2\log n)$ time using $O(n\log n)$ space and in $O\left(n^3\right)$ time using $O\left(n^2\right)$ space respectively.