Discrete & Computational Geometry最新文献

For (din {mathbb {N}}), a compact sphere packing of Euclidean space ({mathbb {R}}^{d}) is a set of spheres in ({mathbb {R}}^{d}) with disjoint interiors so that the contact hypergraph of the packing is the vertex scheme of a homogeneous simplicial d-complex that covers all of ({mathbb {R}}^{d}). We are motivated by the question: For (d,nin {mathbb {N}}) with (d,nge 2), how many configurations of numbers (0<r_{0}<r_{1}<cdots <r_{n-1}=1) can occur as the radii of spheres in a compact sphere packing of ({mathbb {R}}^{d}) wherein there occur exactly n sizes of sphere? We introduce what we call ‘heteroperturbative sets’ of labeled triangulations of unit spheres and we discuss the existence of non-trivial examples of heteroperturbative sets. For a fixed heteroperturbative set, we discuss how a compact sphere packing may be associated to the heteroperturbative set or not. We proceed to show, for (d,nin {mathbb {N}}) with (d,nge 2) and for a fixed heteroperturbative set, that the collection of all configurations of n distinct positive numbers that can occur as the radii of spheres in a compact packing is finite, when taken over all compact sphere packings of ({mathbb {R}}^{d}) which have exactly n sizes of sphere and which are associated to the fixed heteroperturbative set.

Let (textrm{R}) be a real closed field and (textrm{C}) the algebraic closure of (textrm{R}). We give an algorithm for computing a semi-algebraic basis for the first homology group, (textrm{H}_1(S,{mathbb {F}})), with coefficients in a field ({mathbb {F}}), of any given semi-algebraic set (S subset textrm{R}^k) defined by a closed formula. The complexity of the algorithm is bounded singly exponentially. More precisely, if the given quantifier-free formula involves s polynomials whose degrees are bounded by d, the complexity of the algorithm is bounded by ((s d)^{k^{O(1)}}). This algorithm generalizes well known algorithms having singly exponential complexity for computing a semi-algebraic basis of the zeroth homology group of semi-algebraic sets, which is equivalent to the problem of computing a set of points meeting every semi-algebraically connected component of the given semi-algebraic set at a unique point. It is not known how to compute such a basis for the higher homology groups with singly exponential complexity. As an intermediate step in our algorithm we construct a semi-algebraic subset (Gamma ) of the given semi-algebraic set S, such that (textrm{H}_q(S,Gamma ) = 0) for (q=0,1). We relate this construction to a basic theorem in complex algebraic geometry stating that for any affine variety X of dimension n, there exists Zariski closed subsets

with (dim _textrm{C}Z^{(i)} le i), and (textrm{H}_q(X,Z^{(i)}) = 0) for (0 le q le i). We conjecture a quantitative version of this result in the semi-algebraic category, with X and (Z^{(i)}) replaced by closed semi-algebraic sets. We make initial progress on this conjecture by proving the existence of (Z^{(0)}) and (Z^{(1)}) with complexity bounded singly exponentially (previously, such an algorithm was known only for constructing (Z^{(0)})).

Developing an algorithm for computing the Betti numbers of semi-algebraic sets with singly exponential complexity has been a holy grail in algorithmic semi-algebraic geometry and only partial results are known. In this paper we consider the more general problem of computing the image under the homology functor of a continuous semi-algebraic map (f:X rightarrow Y) between closed and bounded semi-algebraic sets. For every fixed (ell ge 0) we give an algorithm with singly exponential complexity that computes bases of the homology groups (text{ H}_i(X), text{ H}_i(Y)) (with rational coefficients) and a matrix with respect to these bases of the induced linear maps (text{ H}_i(f):text{ H}_i(X) rightarrow text{ H}_i(Y), 0 le i le ell ). We generalize this algorithm to more general (zigzag) diagrams of continuous semi-algebraic maps between closed and bounded semi-algebraic sets and give a singly exponential algorithm for computing the homology functors on such diagrams. This allows us to give an algorithm with singly exponential complexity for computing barcodes of semi-algebraic zigzag persistent homology in small dimensions.

This paper considers an extremal version of the Erdős distinct distances problem. For a point set (P subset {mathbb {R}}^d), let (Delta (P)) denote the set of all Euclidean distances determined by P. Our main result is the following: if (Delta (A^d) ll |A|^2) and (d ge 5), then there exists (A' subset A) with (|A'| ge |A|/2) such that (|A'-A'| ll |A| log |A|). This is one part of a more general result, which says that, if the growth of (|Delta (A^d)|) is restricted, it must be the case that A has some additive structure. More specifically, for any two integers k, n, we have the following information: if

then there exists (A' subset A) with (|A'| ge |A|/2) and

These results are higher dimensional analogues of a result of Hanson [4], who considered the two-dimensional case.

We develop a method for subdividing polyhedral complexes in a way that restricts the possible recession cones and allows one to work with a fixed class of polyhedron. We use these results to construct locally finite completions of rational polyhedral complexes whose recession cones lie in a fixed fan, locally finite polytopal completions of polytopal complexes, and locally finite zonotopal completions of zonotopal complexes.

Given a set of n labeled points in general position in the plane, we remove all of its points one by one. At each step, one point from the convex hull of the remaining set is erased. In how many ways can the process be carried out? The answer obviously depends on the point set. If the points are in convex position, there are exactly n! ways, which is the maximum number of ways for n points. But what is the minimum number? It is shown that this number is (roughly) at least (3^n) and at most (12.29^n).

The convexity of a set can be generalized to the two weaker notions of positive reach and r-convexity; both describe the regularity of a set’s boundary. For any compact subset of ({{mathbb {R}}}^d), we provide methods for computing upper bounds on these quantities from point cloud data. The bounds converge to the respective quantities as the sampling scale of the point cloud decreases, and the rate of convergence for the bound on the reach is given under a weak regularity condition. We also introduce the (beta )-reach, a generalization of the reach that excludes small-scale features of size less than a parameter (beta in [0,infty )). Numerical studies suggest how the (beta )-reach can be used in high-dimension to infer the reach and other geometric properties of smooth submanifolds.

We study the cone of completely positive (cp) matrices for the first interesting case (n = 5). This is a semialgebraic set for which the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem.

Abstract

Almost 50 years ago Erdős and Purdy asked the following question: Given n points in the plane, how many triangles can be approximate congruent to equilateral triangles? They pointed out that by dividing the points evenly into three small clusters built around the three vertices of a fixed equilateral triangle, one gets at least (leftlfloor frac{n}{3} rightrfloor cdot leftlfloor frac{n+1}{3} rightrfloor cdot leftlfloor frac{n+2}{3} rightrfloor ) such approximate copies. In this paper we provide a matching upper bound and thereby answer their question. More generally, for every triangle T we determine the maximum number of approximate congruent triangles to T in a point set of size n. Parts of our proof are based on hypergraph Turán theory: for each point set in the plane and a triangle T, we construct a 3-uniform hypergraph (mathcal {H}=mathcal {H}(T)) , which contains no hypergraph as a subgraph from a family of forbidden hypergraphs (mathcal {F}=mathcal {F}(T)) . Our upper bound on the number of edges of (mathcal {H}) will determine the maximum number of triangles that are approximate congruent to T.

We investigate how the complexity of Euclidean TSP for point sets P inside the strip ((-infty ,+infty )times [0,delta ]) depends on the strip width (delta ). We obtain two main results.

• For the case where the points have distinct integer x-coordinates, we prove that a shortest bitonic tour (which can be computed in (O(nlog ^2 n)) time using an existing algorithm) is guaranteed to be a shortest tour overall when (delta leqslant 2sqrt{2}), a bound which is best possible.

• We present an algorithm that is fixed-parameter tractable with respect to (delta ). Our algorithm has running time (2^{O(sqrt{delta })} n + O(delta ^2 n^2)) for sparse point sets, where each (1times delta ) rectangle inside the strip contains O(1) points. For random point sets, where the points are chosen uniformly at random from the rectangle ([0,n]times [0,delta ]), it has an expected running time of (2^{O(sqrt{delta })} n). These results generalise to point sets P inside a hypercylinder of width (delta ). In this case, the factors (2^{O(sqrt{delta })}) become (2^{O(delta ^{1-1/d})}).