Computational Geometry-Theory and Applications最新文献

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.

In colored range counting (CRC), the input is a set of points where each point is assigned a “color” (or a “category”) and the goal is to store them in a data structure such that the number of distinct categories inside a given query range can be counted efficiently. CRC has strong motivations as it allows data structure to deal with categorical data.

However, colors (i.e., the categories) in the CRC problem do not have any internal structure, whereas this is not the case for many datasets in practice where hierarchical categories exist or where a single input belongs to multiple categories. Motivated by these, we consider variants of the problem where such structures can be represented. We define two variants of the problem called hierarchical range counting (HCC) and sub-category colored range counting (SCRC) and consider hierarchical structures that can either be a DAG or a tree. We show that the two problems on some special trees are in fact equivalent to other well-known problems in the literature. Based on these, we also give efficient data structures when the underlying hierarchy can be represented as a tree. We show a conditional lower bound for the general case when the existing hierarchy can be any DAG, through a reduction from the orthogonal vectors problem.

Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations by additionally handling addition and deletion of cells in a filtered topological space and by processing changes in a single batch. We show that the complexity of our scheme scales with the number of elementary changes to the filtration which as a result is often less expensive than the full persistent homology computation. Finally, we perform computational experiments demonstrating practical speedups in several situations including feature generation and optimization guided by persistent homology.

Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have emerged as a way to provide a more precise and expressive control over the solution, relying on persistent homology to quantify and reduce its roughness. All such existing techniques back-propagate gradients through the persistence diagram, which is a summary of the topological features of a function. Their downside is that they provide information only at the critical points of the function. We propose a method that instead builds on persistence-sensitive simplification and translates the required changes to the persistence diagram into changes on large subsets of the domain, including both critical and regular points. This approach enables a faster and more precise topological regularization, the benefits of which we illustrate with experimental evidence.

We define a $C\left(k\right)$ to be a family of k sets $F_1,\dots ,F_k$ such that $\text{conv}(F_i\cup F_{i+1})\cap \text{conv}(F_j\cup F_{j+1})=\varnothing$ when $\{i,i+1\}\cap \{j,j+1\}=\varnothing$ (indices are taken modulo k). We show that if $F$ is a family of compact, convex sets that does not contain a $C\left(k\right)$, then there are $k-2$ lines that pierce $F$. Additionally, we give an example of a family of compact, convex sets that contains no $C\left(k\right)$ and cannot be pierced by $\lceil \frac{k}{2}\rceil -1$ lines.