Computational Geometry-Theory and Applications最新文献

A t-spanner is a subgraph of a graph G in which the length of the shortest path between two vertices never exceeds t times the length of the shortest path between them in G. A geometric graph is one whose vertices are points and whose edges are line segments between the corresponding points. Geometric t-spanners are t-spanners of the complete geometric graph on a given point set. Besides approximating the distance between points, we may ask a geometric t-spanner to be planar, have low degree, or low total edge length.

One famous algorithm used to generate spanners is path-greedy, which scans pairs of vertices in non-decreasing order of edge length and adds the edge between them unless the current set of added edges already connects them with a path that t-approximates the edge length. Graphs from this algorithm are called path-greedy spanners. This work analyzes properties of path-greedy geometric spanners under different conditions. Specifically, we answer an open problem regarding the planarity and degree of path-greedy t-spanners for points in convex position in 2D. Further, we show a simple and efficient way to reduce the degree of a geometric spanner by adding extra points.

An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in the graph. Recently, Biedl and Mehrabi (Graph Drawing 2017) studied non-blocking grid obstacle representations of graphs in which the vertices of the graph are mapped onto points in the plane while the straight-line segments representing the adjacency between the vertices are replaced by the $L_1$ (Manhattan) shortest paths in the plane that avoid obstacles.

In this paper, we introduce the notion of geodesic obstacle representations of graphs with the main goal of providing a generalized model, which comes naturally when viewing line segments as shortest paths in the Euclidean plane. To this end, we extend the definition of obstacle representation by allowing some obstacles-avoiding shortest path between the corresponding points in the underlying metric space whenever the vertices are adjacent in the graph. We consider both general and plane variants of geodesic obstacle representations (in a similar sense to obstacle representations) under any polyhedral distance function in $R^d$ as well as shortest path distances in graphs. Our results generalize and unify the notions of obstacle representations, plane obstacle representations and grid obstacle representations, leading to a number of questions on such representations.

Persistent homology, an algebraic method for discerning structure in abstract data, relies on the construction of a sequence of nested topological spaces known as a filtration. Two-parameter persistent homology allows the analysis of data simultaneously filtered by two parameters, but requires a bifiltration—a sequence of topological spaces simultaneously indexed by two parameters. To apply two-parameter persistence to digital images, we first must consider bifiltrations constructed from digital images, which have scarcely been studied. We introduce the value-offset bifiltration for grayscale digital image data. We present efficient algorithms for computing this bifiltration with respect to the taxicab distance and for approximating it with respect to the Euclidean distance. We analyze the runtime complexity of our algorithms, demonstrate the results on sample images, and contrast the bifiltrations obtained from real images with those obtained from random noise.

Given a set of n points in the plane, the Unit Disk Cover (UDC) problem asks to compute the minimum number of unit disks required to cover the points, along with a placement of the disks. The problem is NP-hard and several approximation algorithms have been designed over the last three decades. In this paper, we have engineered and experimentally compared practical performances of some of these algorithms on massive pointsets. The goal is to investigate which algorithms run fast and give good approximation in practice.

We present a simple 7-approximation algorithm for UDC that runs in $O\left(n\right)$ expected time and uses $O\left(s\right)$ extra space, where s denotes the size of the generated cover. In our experiments, it turned out to be the speediest of all. We also present two heuristics to reduce the sizes of covers generated by it without slowing it down by much.

To our knowledge, this is the first work that experimentally compares algorithms for the UDC problem. Experiments with them using massive pointsets (in the order of millions) throw light on their practical uses. We share the engineered algorithms via GitHub¹ for broader uses and future research in the domain of geometric optimization.

We consider the planar two-center problem for a convex polygon: given a convex polygon in the plane, find two congruent disks of minimum radius whose union contains the polygon. We present an $O(n\log n)$-time algorithm for the two-center problem for a convex polygon, where n is the number of vertices of the polygon. This improves upon the previous best algorithm for the problem.

Compression aims to reduce the size of an input, while maintaining its relevant properties. For multi-parameter persistent homology, compression is a necessary step in any computational pipeline, since standard constructions lead to large inputs, and computational tasks in this area tend to be expensive. We propose two compression methods for chain complexes of free 2-parameter persistence modules. The first method extends the multi-chunk algorithm for one-parameter persistent homology, returning the smallest chain complex among all the ones quasi-isomorphic to the input. The second method produces minimal presentations of the homology of the input; it is based on an algorithm of Lesnick and Wright, but incorporates several improvements that lead to substantial performance gains. The two methods are complementary, and can be combined to compute minimal presentations for complexes with millions of generators in a few seconds. The methods have been implemented, and the software is publicly available. We report on experimental evaluations, which demonstrate substantial improvements in performance compared to previously available compression strategies.

A weighted bisector graph is a geometric graph whose faces are bounded by edges that are portions of multiplicatively weighted bisectors of pairs of (point) sites such that each of its faces is defined by exactly one site. A prominent example of a bisector graph is the multiplicatively weighted Voronoi diagram of a finite set of points which induces a tessellation of the plane into Voronoi faces bounded by circular arcs and straight-line segments. Several algorithms for computing various types of bisector graphs are known. In this paper we reverse the problem: Given a partition $G$ of the plane into faces, find a set of points and suitable weights such that $G$ is a bisector graph of the weighted points, if a solution exists. If $G$ is a graph that is regular of degree three then we can decide in $O\left(m\right)$ time whether it is a bisector graph, where m denotes the combinatorial complexity of $G$. In the same time we can identify up to two candidate solutions such that $G$ could be their multiplicatively weighted Voronoi diagram. Additionally, we show that it is possible to recognize $G$ as a multiplicatively weighted Voronoi diagram and find all possible solutions in $O(m\log m)$ time if $G$ is given by a set of disconnected lines and circles.

We present subquadratic algorithms in the algebraic decision-tree model for several 3Sum-hard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle $\Delta \in C$, the number of intersection points between the segments of A and those of B that lie in Δ. We present solutions in the algebraic decision-tree model whose cost is $O\left(n^{60/31+\epsilon }\right)$, for any $\epsilon >0$.

Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal et al. (2021) [3]. A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the order type of the lines, a “handicap” that turns out to be beneficial for speeding up our algorithm.

Two sets of labeled points $\{p_1,\dots ,p_n\}$ and $\{q_1,\dots ,q_n\}$ are of the same labeled order type if, for every $i,j,k$, the triples $(p_i,p_j,p_k)$ and $(q_i,q_j,q_k)$ have the same orientation. In the 1980's, Goodman, Pollack, and Sturmfels showed that (i) the number of labeled order types on n points is of order $n^{4n+o\left(n\right)}$, (ii) all order types can be realized with double-exponential integer coordinates, and that (iii) certain order types indeed require double-exponential integer coordinates. In 2018, Caraballo, Díaz-Báñez, Fabila-Monroy, Hidalgo-Toscano, Leaños, and Montejano showed that at least $n^{3n+o\left(n\right)}$ labeled n-point order types can be realized on an integer grid of polynomial size. In this article, we improve their result by showing that at least $n^{4n+o\left(n\right)}$ labeled n-point order types can be realized on an integer grid of polynomial size, which is asymptotically tight in the exponent. Finally we conclude that there are $n^{3n+o\left(n\right)}$ order types in the unlabeled setting and that $n^{3n+o\left(n\right)}$ of them can be realized on an integer grid of polynomial size.

Computational topology research of the past two decades has emphasized combinatorial techniques while numerical methods such as numerical linear algebra remain underutilized. While the combinatorial techniques have been very successful in diverse areas, for some applications, it is worth considering the numerical counterparts. We discuss one such application. Harmonic forms are elements of the kernel of the Hodge Laplacian operator and contain information about the topology of the manifold. If a particular cohomology class is chosen, the closed differential form with the smallest norm in that class is a harmonic form. We use these well-known facts to give an algorithm for solving the following problem: given a piecewise flat manifold simplicial complex (with or without boundary) and a closed piecewise polynomial differential form representing a cohomology class, find the discrete harmonic form in that cohomology class. We give a least squares based algorithm to solve this problem and show that the computed form satisfies the finite element exterior calculus (FEEC) equations for being a harmonic form. The piecewise polynomial spaces used are the spaces of trimmed polynomial forms, that is, arbitrary degree polynomial generalizations of Whitney forms which are used in FEEC. We also survey other methods for finding harmonic forms.