Computational Geometry-Theory and Applications最新文献

In this work, we study a least-squares formulation of the source localization problem given time-of-arrival measurements. We show that the formulation, albeit non-convex in general, is globally strongly convex under certain condition on the geometric configuration of the anchors and the source and on the measurement noise. Next, we derive a characterization of the critical points of the least-squares formulation, leading to a bound on the maximum number of critical points under a very mild assumption on the measurement noise. In particular, the result provides a sufficient condition for the critical points of the least-squares formulation to be isolated. Prior to our work, the isolation of the critical points is treated as an assumption without any justification in the localization literature. The said characterization also leads to an algorithm that can find a global optimum of the least-squares formulation by searching through all critical points. We then establish an upper bound of the estimation error of the least-squares estimator. Finally, our numerical results corroborate the theoretical findings and show that our proposed algorithm can obtain a global solution regardless of the geometric configuration of the anchors and the source.

A classical tensegrity model consists of an embedded graph in a vector space with rigid bars representing edges, and an assignment of a stress to every edge such that at every vertex of the graph the stresses sum up to zero. The tensegrity frameworks have been recently extended from the two dimensional graph case to the multidimensional setting. We study the multidimensional tensegrities using tools from toric geometry. We introduce a link between self-stresses and Chow rings on toric varieties. More precisely, for a given rational tensegrity framework $F$, we construct a glued toric surface $X_F$. We show that the abelian group of tensegrities on $F$ is isomorphic to a subgroup of the Chow group $A^1(X_F;Q)$. In the case of planar frameworks, we show how to explicitly carry out the computation of tensegrities via classical tools in toric geometry.

We initiate the study of a generalization of the class cover problem [Cannon and Cowen [1], Bereg et al. [2]] the generalized class cover problem, where we are allowed to misclassify some points provided we pay an associated positive penalty for every misclassified point. Two versions: single coverage and multiple coverage, of the generalized class cover problem are investigated. We study five different variants of both versions of the generalized class cover problem with axis-parallel strips and axis-parallel half-strips extending to different directions in the plane, thus extending similar work by Bereg et al. (2012) [2] on the class cover problem. We prove that the multiple coverage version of the generalize class cover problem with axis-parallel strips are in $P$, whereas the single coverage version is $\mathrm{NP}$-hard. A factor 2 approximation algorithm is provided for the later problem. The $\mathrm{APX}$-hardness result is also shown for the single coverage version. For half-strips extending to exactly one direction, both the single and multiple coverage versions can be solved in polynomial time using dynamic programming. In the case of half-strips extending to two orthogonal directions, we prove the class cover problem is $\mathrm{NP}$-hard followed by $\mathrm{APX}$-hard. This gives improve hardness results compare to Bereg et al. (2012) [2], where they proved the class cover problem with half-strips oriented in four different directions is $\mathrm{NP}$-hard. These $\mathrm{NP}$- and $\mathrm{APX}$-hardness results can directly apply to both single and multiple versions. Finally, constant factor approximation algorithms are provided for half-strips extending to more than one direction.

A 2D rigidity circuit is a minimal graph $G=(V,E)$ supporting a non-trivial stress in any generic placement of its vertices in the Euclidean plane. All 2D rigidity circuits can be constructed from $K_4$ graphs using combinatorial resultant (CR) operations. A combinatorial resultant tree (CR-tree) is a rooted binary tree capturing the structure of such a construction.

The CR operation has a specific algebraic interpretation, where an essentially unique circuit polynomial is associated to each circuit graph. Performing Sylvester resultant operations on these polynomials is in one-to-one correspondence with CR operations on circuit graphs. This mixed combinatorial/algebraic approach led recently to an effective algorithm for computing circuit polynomials. Its complexity analysis remains an open problem, but it is known to be influenced by the depth and shape of CR-trees in ways that have only partially been investigated.

In this paper, we present an effective algorithm for enumerating all the CR-trees of a given circuit with n vertices. Our algorithm has been fully implemented in Mathematica and allows for computational experimentation with various optimality criteria in the resulting, potentially exponentially large collections of CR-trees.

This paper uses the technology of weighted triangulations to study discrete versions of the Laplacian on piecewise Euclidean manifolds. Given a collection of Euclidean simplices glued together along their boundary, a geometric structure on the Poincaré dual may be constructed by considering weights at the vertices. We show that this is equivalent to specifying sphere radii at vertices and generalized intersection angles at edges, or by specifying a certain way of dividing the edges. This geometric structure gives rise to a discrete Laplacian operator acting on functions on the vertices. We study these geometric structures in some detail, considering when dual volumes are nondegenerate, which corresponds to weighted Delaunay triangulations in dimension 2, and how one might find such nondegenerate weighted triangulations. Finally, we talk briefly about the possibilities of discrete Riemannian manifolds.

A geometric graph is a combinatorial graph, endowed with a geometry that is inherited from its embedding in a Euclidean space. Formulation of a meaningful measure of (dis-)similarity in both the combinatorial and geometric structures of two such geometric graphs is a challenging problem in pattern recognition. We study two notions of distance measures for geometric graphs, called the geometric edit distance (GED) and geometric graph distance (GGD). While the former is based on the idea of editing one graph to transform it into the other graph, the latter is inspired by inexact matching of the graphs. For decades, both notions have been lending themselves well as measures of similarity between attributed graphs. If used without any modification, however, they fail to provide a meaningful distance measure for geometric graphs—even cease to be a metric. We have curated their associated cost functions for the context of geometric graphs. Alongside studying the metric properties of GED and GGD, we investigate how the two notions compare. We further our understanding of the computational aspects of GGD by showing that the distance is $\mathrm{NP}$-hard to compute, even if the graphs are planar and arbitrary cost coefficients are allowed.

As a computationally tractable alternative, we propose in this paper the Graph Mover's Distance (GMD), which has been formulated as an instance of the earth mover's distance. The computation of the GMD between two geometric graphs with at most n vertices takes only $O\left(n^3\right)$-time. The GMD demonstrates extremely promising empirical evidence at recognizing letter drawings.

A framework, which is a (possibly infinite) graph with a realization of its vertices in the plane, is called flexible if it can be continuously deformed while preserving the edge lengths. We focus on flexibility of frameworks in which 4-cycles form parallelograms. For the class of frameworks considered in this paper (allowing triangles), we prove that the following are equivalent: flexibility, infinitesimal flexibility, the existence of at least two classes of an equivalence relation based on 3- and 4-cycles and being a non-trivial subgraph of the Cartesian product of graphs. We study the algorithmic aspects and the rotationally symmetric version of the problem. The results are illustrated on frameworks obtained from tessellations by regular polygons.

We introduce a new variant of the art gallery problem that comes from safety issues. In this variant we are not interested in guard sets of smallest cardinality, but in guard sets with largest possible distances between these guards. To the best of our knowledge, this variant has not been considered before. We call it the Dispersive Art Gallery Problem. In particular, in the dispersive art gallery problem we are given a polygon $P$ and a real number ℓ, and want to decide whether $P$ has a guard set such that every pair of guards in this set is at least a distance of ℓ apart.

In this paper, we study the vertex guard variant of this problem for the class of polyominoes. We consider rectangular visibility and distances as geodesics in the $L_1$-metric. Our results are as follows. We give a (simple) thin polyomino such that every guard set has minimum pairwise distances of at most 3. On the positive side, we describe an algorithm that computes guard sets for simple polyominoes that match this upper bound, i.e., the algorithm constructs worst-case optimal solutions. We also study the computational complexity of computing guard sets that maximize the smallest distance between all pairs of guards within the guard sets. We prove that deciding whether there exists a guard set realizing a minimum pairwise distance for all pairs of guards of at least 5 in a given polyomino is NP-complete.

We also present an optimal dynamic programming approach that computes a guard set that maximizes the minimum pairwise distance between guards in tree-shaped polyominoes, i.e., computes optimal solutions. Because the shapes constructed in the NP-hardness reduction are thin as well (but have holes), this result completes the case for thin polyominoes.

Let T be a tree with n vertices in a metric space. We consider the problem of adding one shortcut edge to T to minimize the radius of the resulting graph.

For the continuous version of the problem where a center may be a point in the interior of an edge of the graph we give a linear time algorithm. In the case when the center is restricted to lie on a vertex, the discrete version, we give an $O(n\log n)$ expected time algorithm.

Previously linear-time algorithms were known for the special case when the input graph is a path.

We study a lower bound for the constant of the Szemerédi–Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies $I(P,L)\ge (c+o(1\left)\right)\left|P{|}^{2/3}\right|L{|}^{2/3}$, with $c\approx 1.27$. Our technique is based on studying a variety of properties of Euler's totient function. We also improve the current best constant for Elekes's construction from 1 to about 1.27. From an expository perspective, this is the first full analysis of the constant of Erdős's construction.