Discrete & Computational Geometry最新文献

The order type of a point set in (mathbb {R}^d) maps each ((d{+}1))-tuple of points to its orientation (e.g., clockwise or counterclockwise in (mathbb {R}^2)). Two point sets X and Y have the same order type if there exists a bijection f from X to Y for which every ((d{+}1))-tuple ((a_1,a_2,ldots ,a_{d+1})) of X and the corresponding tuple ((f(a_1),f(a_2),ldots ,f(a_{d+1}))) in Y have the same orientation. In this paper we investigate the complexity of determining whether two point sets have the same order type. We provide an (O(n^d)) algorithm for this task, thereby improving upon the (O(n^{lfloor {3d/2}rfloor })) algorithm of Goodman and Pollack (SIAM J. Comput. 12(3):484–507, 1983). The algorithm uses only order type queries and also works for abstract order types (or acyclic oriented matroids). Our algorithm is optimal, both in the abstract setting and for realizable points sets if the algorithm only uses order type queries.

The volume of a Meissner polyhedron is computed in terms of the lengths of its dual edges. This allows to reformulate the Meissner conjecture regarding constant width bodies with minimal volume as a series of explicit finite dimensional problems. A direct consequence is the minimality of the volume of Meissner tetrahedras among Meissner pyramids.

We consider point sets in the real projective plane ({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}}) and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erdős–Szekeres-type problems. We provide asymptotically tight bounds for a variant of the Erdős–Szekeres theorem about point sets in convex position in ({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}}), which was initiated by Harborth and Möller in 1994. The notion of convex position in ({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}}) agrees with the definition of convex sets introduced by Steinitz in 1913. For (k ge 3), an (affine) k-hole in a finite set (S subseteq {mathbb {R}}^2) is a set of k points from S in convex position with no point of S in the interior of their convex hull. After introducing a new notion of k-holes for points sets from ({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}}), called projective k-holes, we find arbitrarily large finite sets of points from ({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}}) with no projective 8-holes, providing an analogue of a classical planar construction by Horton from 1983. We also prove that they contain only quadratically many projective k-holes for (k le 7). On the other hand, we show that the number of k-holes can be substantially larger in ({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}}) than in ({mathbb {R}}^2) by constructing, for every (k in {3,dots ,6}), sets of n points from ({mathbb {R}}^2 subset {{,mathrm{{mathbb {R}}{mathcal {P}}^2},}}) with (Omega (n^{3-3/5k})) projective k-holes and only (O(n^2)) affine k-holes. Last but not least, we prove several other results, for example about projective holes in random point sets in ({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}}) and about some algorithmic aspects. The study of extremal problems about point sets in ({{,mathrm{{mathbb {R}}{mathcal {P}}^2},}}) opens a new area of research, which we support by posing several open problems.

A consistent path system in a graph G is an intersection-closed collection of paths, with exactly one path between any two vertices in G. We call G metrizable if every consistent path system in it is the system of geodesic paths defined by assigning some positive lengths to its edges. We show that metrizable graphs are, in essence, subdivisions of a small family of basic graphs with additional compliant edges. In particular, we show that every metrizable graph with 11 vertices or more is outerplanar plus one vertex.

We study the problem of estimating the convex hull of the image (f(X)subset {mathbb {R}}^n) of a compact set (Xsubset {mathbb {R}}^m) with smooth boundary through a smooth function (f:{mathbb {R}}^mrightarrow {mathbb {R}}^n). Assuming that f is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of f(X) and the convex hull of the images (f(x_i)) of M sampled inputs (x_i) on the boundary of X. When applied to the problem of geometric inference from a random sample, our results give error bounds that are tighter and more general than in previous work. We present applications to the problems of robust optimization, of reachability analysis of dynamical systems, and of robust trajectory optimization under bounded uncertainty.

We study some discrete invariants of Newton non-degenerate polynomial maps (f: {mathbb {K}}^n rightarrow {mathbb {K}}^n) defined over an algebraically closed field of Puiseux series ({mathbb {K}}), equipped with a non-trivial valuation. It is known that the set ({mathcal {S}}(f)) of points at which f is not finite forms an algebraic hypersurface in ({mathbb {K}}^n). The coordinate-wise valuation of ({mathcal {S}}(f)cap ({mathbb {K}}^*)^n) is a piecewise-linear object in ({mathbb {R}}^n), which we call the tropical non-properness set of f. We show that the tropical polynomial map corresponding to f has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of f. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of ({mathcal {S}}(f)) in terms of multivariate resultants.

We approximate boundaries of convex polytopes (Xsubset {mathbb {R}}^n) by smooth hypersurfaces (Y=Y_varepsilon ) with positive mean curvatures and, by using basic geometric relations between the scalar curvatures of Riemannian manifolds and the mean curvatures of their boundaries, establish lower bound on the dihedral angles of X.

The lattice of noncrossing partitions is well-known for its wide variety of combinatorial appearances and properties. For example, the lattice is rank-symmetric and enumerated by the Catalan numbers. In this article, we introduce a large family of new noncrossing partition lattices with both of these properties, each parametrized by a configuration of n points in the plane.

We study the problem of finding a minimum homology basis, that is, a lightest set of cycles that generates the 1-dimensional homology classes with (mathbb {Z}_2) coefficients in a given simplicial complex K. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al. (LATIN 2018: Theoretical Informatics, Springer International Publishing, Cham, 2018), runs in (O(N m^{omega -1} + n m g)) time, where N denotes the total number of simplices in K, m denotes the number of edges in K, n denotes the number of vertices in K, g denotes the rank of the 1-homology group of K, and (omega ) denotes the exponent of matrix multiplication. In this paper, we present three conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex K. The first algorithm runs in (tilde{O}(m^omega )) time, the second algorithm runs in (O(N m^{omega -1})) time and the third algorithm runs in (tilde{O}(N^2,g + N m g{^2} + m g{^3})) time which is nearly quadratic time when (g=O(1)). We also study the problem of finding a minimum cycle basis in an undirected graph G with n vertices and m edges. The best known algorithm for this problem runs in (O(m^omega )) time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in (tilde{O}(m^omega )) time. We also provide a practical implementation of computing the minimum homology basis for general weighted complexes. The implementation is broadly based on the algorithmic ideas described in this paper, differing in its use of practical subroutines. Of these subroutines, the more costly step makes use of a parallel implementation, thus potentially addressing the issue of scale. We compare results against the currently known state of the art implementation (ShortLoop).

A 3-dimensional polytope P is k-equiprojective when the projection of P along any line that is not parallel to a facet of P is a polygon with k vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective polytopes. It has been shown recently that the number of combinatorial types of k-equiprojective polytopes is at least linear as a function of k. Here, it is shown that there are at least (k^{3k/2+o(k)}) such combinatorial types as k goes to infinity. This relies on the Goodman–Pollack lower bound on the number of order types of point configurations and on new constructions of equiprojective polytopes via Minkowski sums.