Discrete & Computational Geometry最新文献

Given a lattice $L\subseteq Z^m$ and a subset $A\subseteq R^m$ , we say that a point in A is lonely if it is not equivalent modulo $L$ to another point of A. We are interested in identifying lonely points for specific choices of $L$ when A is a dilated standard simplex, and in conditions on $L$ which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.

For three distinct infinite families $\left(R_m\right)$ , $\left(S_m\right)$ , and $\left(T_m\right)$ of non-arithmetic 1-cusped hyperbolic Coxeter 3-orbifolds, we prove incommensurability for a pair of elements $X_k$ and $Y_l$ belonging to the same sequence and for most pairs belonging two different ones. We investigate this problem first by means of the Vinberg space and the Vinberg form, a quadratic space associated to each of the corresponding fundamental Coxeter prism groups, which allows us to deduce some partial results. The complete proof is based on the analytic behavior of another commensurability invariant. It is given by the cusp density, and we prove and exploit its strict monotonicity.

Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class $\exists R$ plays a crucial role in the study of geometric problems. Sometimes $\exists R$ is referred to as the 'real analog' of NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, $\exists R$ deals with existentially quantified real variables. In analogy to ${\Pi }_2^p$ and ${\Sigma }_2^p$ in the famous polynomial hierarchy, we study the complexity classes $\forall \exists R$ and $\exists \forall R$ with real variables. Our main interest is the Area Universality problem, where we are given a plane graph G, and ask if for each assignment of areas to the inner faces of G, there exists a straight-line drawing of G realizing the assigned areas. We conjecture that Area Universality is $\forall \exists R$-complete and support this conjecture by proving $\exists R$- and $\forall \exists R$-completeness of two variants of Area Universality. To this end, we introduce tools to prove $\forall \exists R$-hardness and membership. Finally, we present geometric problems as candidates for $\forall \exists R$-complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.

Updating an abstract Voronoi diagram in linear time, after deletion of one site, has been an open problem in a long time; similarly, for any concrete Voronoi diagram of generalized (non-point) sites. In this paper we present a simple, expected linear-time algorithm to update an abstract Voronoi diagram after deletion of one site. To achieve this result, we use the concept of a Voronoi-like diagram, a relaxed Voronoi structure of independent interest. Voronoi-like diagrams serve as intermediate structures, which are considerably simpler to compute, thus, making an expected linear-time construction possible. We formalize the concept and prove that it is robust under insertion, therefore, enabling its use in incremental constructions. The time-complexity analysis introduces a variant to backwards analysis, which is applicable to order-dependent structures. We further extend the technique to compute in expected linear time: the order-$(k+1)$ subdivision within an order-k Voronoi region, and the farthest abstract Voronoi diagram, after the order of its regions at infinity is known.

A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of $G+e$ extending D(G). As a result of Levi's Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles $A$ and a pseudosegment  $\sigma$ , it can be decided in polynomial time whether there exists a pseudocircle ${\Phi }_{\sigma }$ extending $\sigma$ for which $A\cup \left\{{\Phi }_{\sigma }\right\}$ is again an arrangement of pseudocircles.

We construct an example of a group $G=Z^2\times G_0$ for a finite abelian group $G_0$, a subset E of $G_0$, and two finite subsets $F_1,F_2$ of G, such that it is undecidable in ZFC whether $Z^2\times E$ can be tiled by translations of $F_1,F_2$. In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of E by the tiles $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $Z^2$). A similar construction also applies for $G=Z^d$ for sufficiently large d. If one allows the group $G_0$ to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.

We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique.

Given a finite set $A\subset R^d$, let ${\text{Cov}}_{r,k}$ denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.