Computational Geometry-Theory and Applications最新文献

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.

We examine rectangle packing problems where only the areas $a_1,\dots ,a_n$ of the rectangles to be packed are given while their aspect ratios may be chosen from a given interval $[\frac{1}{\gamma },\gamma ]$. In particular, we ask for the smallest possible size of a rectangle R such that, under these constraints, any collection $a_1,\dots ,a_n$ of rectangle areas of total size 1 can be packed into R. As for standard square packing problems, which are contained as a special case for $\gamma =1$, this question leads us to three different answers, depending on whether the aspect ratio of R is given or whether we may choose it either with or without knowing the areas $a_1,\dots ,a_n$. Generalizing known results for square packing problems, we provide upper and lower bounds for the size of R with respect to all three variants of the problem, which are tight at least for larger values of γ. Moreover, we show how to improve these bounds on the size of R if we restrict ourselves to instances where the largest element in $a_1,\dots ,a_n$ is bounded.

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.