Computational Geometry-Theory and Applications最新文献

A polycube graph is a polyhedron composed of cubes glued together along whole faces, whose surface is a 2-manifold. A polycube graph is 3-separated if no two boxes of degree 3 or higher are adjacent, and no grid edge is entirely surrounded by boxes (i.e., there is no cycle of length 4). We show that every 3-separated polycube graph can be unfolded with a $7\times 7$ refinement of the grid faces. This result extends the class of well-separated polycube graphs known to have an unfolding by allowing boxes of degree 2 to be adjacent to each other and to higher degree boxes.

Given a simple polygon P on n vertices and a set $D$ of m pairwise intersecting geodesic disks in P, we show that five points in P are always sufficient to pierce all the disks in $D$. The points can be computed in $O\left(\right(n+m)\log n_r)$ time, where $n_r$ is the number of the reflex vertices of P. This improves the previous bound of 14, obtained by Bose, Carmi, and Shermer [1].

In Topological Data Analysis, filtered chain complexes enter the persistence pipeline between the initial filtering of data and the final persistence invariants extraction. It is known that they admit a tame class of indecomposables, called interval spheres. In this paper, we provide an algorithm to decompose filtered chain complexes into such interval spheres. This algorithm provides geometric insights into various aspects of the standard persistence algorithm and two of its runtime optimizations. Moreover, since it works for any filtered chain complexes, our algorithm can be applied in more general cases. As an application, we show how to decompose filtered kernels with it.

Operations on the cohomology of spaces are important tools enhancing the descriptive power of this computable invariant. For cohomology with mod 2 coefficients, Steenrod squares are the most significant of these operations. Their effective computation relies on formulas defining a cup-i construction, a structure on (co)chains which is important in its own right, having connections to lattice field theory, convex geometry and higher category theory among others. In this article we present new formulas defining a cup-i construction, and use them to introduce a fast algorithm for the computation of Steenrod squares on the cohomology of finite simplicial complexes. In forthcoming work we use these formulas to axiomatically characterize the cup-i construction they define, showing additionally that all other formulas in the literature define the same cup-i construction up to isomorphism.

We consider the problem of computing the Fréchet distance between two curves for which the exact locations of the vertices are unknown. Each vertex may be placed in a given uncertainty region for that vertex, and the objective is to place vertices so as to minimise the Fréchet distance. This problem was recently shown to be NP-hard in 2D, and it is unclear how to compute an optimal vertex placement at all.

We present the first general algorithmic framework for this problem. We prove that it results in a polynomial-time algorithm for curves in 1D with intervals as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D in the case that vertices are placed to maximise the Fréchet distance.

We also study the weak Fréchet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than the regular Fréchet distance—and indeed we can easily prove that the problem is NP-hard in 2D—the optimal placement of vertices in 1D can be computed in polynomial time. Finally, we investigate the discrete weak Fréchet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D.

Higher Dimensional Automata (HDA) are higher dimensional relatives to transition systems in concurrency theory taking into account to which degree various actions commute. Mathematically, they take the form of labelled cubical complexes. It is important to know, and challenging from a geometric/topological perspective, whether the space of directed paths (executions in the model) between two vertices (states) is connected; more generally, to estimate higher connectivity of these path spaces.

This paper presents an approach for such an estimation for particularly simple HDA arising from PV programs and modelling the access of a number of processors to a number of resources with given limited capacity each. It defines the spare capacity of a concurrent program with prescribed periods of access of the processors to the resources using only the syntax of individual programs and the capacities of shared resources. It shows that the connectivity of spaces of directed paths can be estimated (from above) by spare capacities. Moreover, spare capacities can also be used to detect deadlocks and critical states in such a simple HDA.

The key theoretical ingredient is a transition from the calculation of local connectivity bounds (of the upper links of vertices of an HDA) to global ones by applying a version of the nerve lemma due to Anders Björner.

We examine volume computation of general-dimensional polytopes and more general convex bodies, defined by the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest for the public in general and for policy makers in particular. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be estimated by computing volumes of convex bodies.

We design and implement practical algorithms in the exact and approximate setting, and experimentally juxtapose them in order to study the trade-off of exactness and accuracy for speed. We also experimentally find an efficient parameter-tuning to achieve a sufficiently good estimation of the probability density of each copula. Our C++ software, based on Eigen and available on github, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises.

In the acrophobic guard watchtower problem for a polyhedral terrain, a square axis-aligned platform is placed on the top of a tower whose bottom end-point lies on the surface of the terrain. As in the standard watchtower problem, the objective is to minimize the height (i.e., the length) of the watchtower such that every point on the surface of the terrain is weakly visible from the platform placed on the top of the tower. In this paper, we show that in $R^2$ the problem can be solved in $O\left(n\right)$ time, and in $R^3$ it takes $O(n\log n)$ time, where n is the total number of vertices of the terrain.

We show that any embedding of a planar graph can be encoded succinctly while efficiently answering a number of topological queries near-optimally. More precisely, we build on a succinct representation that encodes an embedding of m edges within 4m bits, which is close to the information-theoretic lower bound of about 3.58m. With $4m+o\left(m\right)$ bits of space, we show how to answer a number of topological queries relating nodes, edges, and faces, most of them in any time in $\omega \left(1\right)$. Indeed, $3.58m+o\left(m\right)$ bits suffice if the graph has no self-loops and no nodes of degree one. Further, we show that with $O\left(m\right)$ bits of space we can solve all those operations in $O\left(1\right)$ time.

Recent work in mathematical neuroscience has calculated the directed graph homology of the directed simplicial complex given by the brain's sparse adjacency graph, the so called connectome. These biological connectomes show an abundance of both high-dimensional directed simplices and Betti-numbers in all viable dimensions – in contrast to Erdős–Rényi-graphs of comparable size and density. An analysis of synthetically trained connectomes reveals similar findings, raising questions about the graphs comparability and the nature of origin of the simplices.

We present a new method capable of delivering insight into the emergence of simplices and thus simplicial abundance. Our approach allows to easily distinguish simplex-rich connectomes of different origin. The method relies on the novel concept of an almost-d-simplex, that is, a simplex missing exactly one edge, and consequently the almost-d-simplex closing probability by dimension. We also describe a fast algorithm to identify almost-d-simplices in a given graph. Applying this method to biological and artificial data allows us to identify a mechanism responsible for simplex emergence, and suggests this mechanism is responsible for the simplex signature of the excitatory subnetwork of a statistical reconstruction of the mouse primary visual cortex. Our highly optimized code for this new method is publicly available.