Showing papers in "Discrete and Computational Geometry in 2008"

Finding the Homology of Submanifolds with High Confidence from Random Samples

Abstract: Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high-dimensional spaces. We consider the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data are “noisy” and lie near rather than on the submanifold in question.

Generating All Vertices of a Polyhedron Is Hard

Abstract: We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P=NP. As a corollary, we solve in the negative two well-known generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating maximal feasible subsystems the complexity remains open. (ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard. Yet, in the case of bounded polyhedra the complexity remains open. Equiva lently, the complexity of generating vertices and extreme rays of polyhedra remains open.

Empty Convex Hexagons in Planar Point Sets

Abstract: Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmative. We show that every set that contains the vertex set of a convex 9-gon also contains an empty convex hexagon.

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Abstract: Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have “nicely shaped” triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions. The main contributions are rigorous, theory-tested definitions of CDTs and piecewise linear complexes (geometric domains that incorporate nonconvex faces with “internal” boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.

On the Hardness of Computing Intersection, Union and Minkowski Sum of Polytopes

Abstract: For polytopes P 1,P 2⊂ℝd , we consider the intersection P 1∩P 2, the convex hull of the union CH(P 1∪P 2), and the Minkowski sum P 1+P 2. For the Minkowski sum, we prove that enumerating the facets of P 1+P 2 is NP-hard if P 1 and P 2 are specified by facets, or if P 1 is specified by vertices and P 2 is a polyhedral cone specified by facets. For the intersection, we prove that computing the facets or the vertices of the intersection of two polytopes is NP-hard if one of them is given by vertices and the other by facets. Also, computing the vertices of the intersection of two polytopes given by vertices is shown to be NP-hard. Analogous results for computing the convex hull of the union of two polytopes follow from polar duality. All of the hardness results are established by showing that the appropriate decision version, for each of these problems, is NP-complete.

Reconstruction Using Witness Complexes

Abstract: We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions.

Ehrhart Series and Lattice Triangulations

Abstract: We express the generating function for lattice points in a rational polyhedral cone with a simplicial subdivision in terms of multivariate analogues of the h-polynomials of the subdivision and “local contributions” of the links of its nonunimodular faces. We also compute new examples of nonunimodal h *-vectors of reflexive polytopes.

An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions

Abstract: We present an optimal-time algorithm for computing (an implicit representation of) the shortest-path map from a fixed source s on the surface of a convex polytope P in three dimensions. Our algorithm runs in O(nlog n) time and requires O(nlog n) space, where n is the number of edges of P. The algorithm is based on the O(nlog n) algorithm of Hershberger and Suri for shortest paths in the plane (Hershberger, J., Suri, S. in SIAM J. Comput. 28(6):2215–2256, 1999), and similarly follows the continuous Dijkstra paradigm, which propagates a “wavefront” from s along ∂P. This is effected by generalizing the concept of conforming subdivision of the free space introduced by Hershberger and Suri and by adapting it for the case of a convex polytope in ℝ3, allowing the algorithm to accomplish the propagation in discrete steps, between the “transparent” edges of the subdivision. The algorithm constructs a dynamic version of Mount’s data structure (Mount, D.M. in Discrete Comput. Geom. 2:153–174, 1987) that implicitly encodes the shortest paths from s to all other points of the surface. This structure allows us to answer single-source shortest-path queries, where the length of the path, as well as its combinatorial type, can be reported in O(log n) time; the actual path can be reported in additional O(k) time, where k is the number of polytope edges crossed by the path. The algorithm generalizes to the case of m source points to yield an implicit representation of the geodesic Voronoi diagram of m sites on the surface of P, in time O((n+m)log (n+m)), so that the site closest to a query point can be reported in time O(log (n+m)).

Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

Abstract: The detour and spanning ratio of a graph G embedded in $\mathbb{E}^{d}$ measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(nlog n) time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we obtain O(nlog 2 n)-time algorithms for computing the detour or spanning ratio of planar trees and cycles. Finally, we develop subquadratic algorithms for computing the detour and spanning ratio for paths, cycles, and trees embedded in $\mathbb{E}^{3}$, and show that computing the detour in $\mathbb{E}^{3}$ is at least as hard as Hopcroft’s problem.

Schnyder Woods and Orthogonal Surfaces

Abstract: In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated notions. We use these connections for an intuitive proof of the Brightwell-Trotter Theorem which says, that the face lattice of a 3-polytope minus one face has order dimension three. Our proof yields a linear time algorithm for the construction of the three linear orders that realize the face lattice. Coplanar orthogonal surfaces are in correspondence with a large class of convex straight line drawings of 3-connected planar graphs. We show that Schnyder’s face counting approach with weighted faces can be used to construct all coplanar orthogonal surfaces and hence the corresponding drawings. Appropriate weights are computable in linear time.

Improved Bounds on the Union Complexity of Fat Objects

Abstract: We introduce a new class of fat, not necessarily convex or polygonal, objects in the plane, namely locally γ-fat objects. We prove that the union complexity of any set of n such objects is O(λ s+2(n)log 2 n). This improves the best known bound, and extends it to a more general class of objects.

Robust Shape Fitting via Peeling and Grating Coresets

Abstract: Let P be a set of n points in ℝ d . A subset $\mathcal {S}$ of P is called a (k,e)-kernel if for every direction, the directional width of $\mathcal {S}$ e-approximates that of P, when k “outliers” can be ignored in that direction. We show that a (k,e)-kernel of P of size O(k/e(d−1)/2) can be computed in time O(n+k2/ed−1). The new algorithm works by repeatedly “peeling” away (0,e)-kernels from the point set. We also present a simple e-approximation algorithm for fitting various shapes through a set of points with at most k outliers. The algorithm is incremental and works by repeatedly “grating” critical points into a working set, till the working set provides the required approximation. We prove that the size of the working set is independent of n, and thus results in a simple and practical, near-linear e-approximation algorithm for shape fitting with outliers in low dimensions. We demonstrate the practicality of our algorithms by showing their empirical performance on various inputs and problems.

There Are Integral Heptagons, no Three Points on a Line, no Four on a Circle

Abstract: We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erdős.

Slicing Convex Sets and Measures by a Hyperplane

Abstract: Given convex bodies K1,…,Kd in ℝd and numbers α1,…,αd∈[0,1], we give a sufficient condition for existence and uniqueness of an (oriented) halfspace H with Vol (H∩Ki)=αi⋅Vol Ki for every i. The result is extended from convex bodies to measures.

A Centrally Symmetric Version of the Cyclic Polytope

Abstract: We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and an integer 1≤j 0 and at most $(1-2^{-d}+o(1)){n\choose j+1}$ as n grows. We show that c1(d)≥1−(d−1)−1 and conjecture that the bound is best possible.

Pure Point Diffractive Substitution Delone Sets Have the Meyer Property

Abstract: We prove that a primitive substitution Delone set, which is pure point diffractive, is a Meyer set. This answers a question of J.C. Lagarias. We also show that for primitive substitution Delone sets, being a Meyer set is equivalent to having a relatively dense set of Bragg peaks. The proof is based on tiling dynamical systems and the connection between the diffraction and dynamical spectra.

Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings

Abstract: Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into ${\Bbb{R}}^{d}$, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in ${\Bbb{R}}^{d}$ by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v∈S, which is the exponential map to S from the tangent space at v. We characterize the cut locus (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of three-polytopes into ${\Bbb{R}}^{2}$. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic nonpolynomial complexity of nonconvex manifolds.

Norm Bounds for Ehrhart Polynomial Roots

Abstract: M. Beck et al. found that the roots of the Ehrhart polynomial of a d-dimensional lattice polytope are bounded above in norm by 1+(d+1)!. We provide an improved bound which is quadratic in d and applies to a larger family of polynomials.

Odd Crossing Number and Crossing Number Are Not the Same

Abstract: The crossing number of a graph is the minimum number of edge intersections in a plane drawing of a graph, where each intersection is counted separately. If instead we count the number of pairs of edges that intersect an odd number of times, we obtain the odd crossing number. We show that there is a graph for which these two concepts differ, answering a well-known open question on crossing numbers. To derive the result we study drawings of maps (graphs with rotation systems).

On Average Distortion of Embedding Metrics into the Line

Abstract: We introduce and study the notion of the average distortion of a nonexpanding embedding of one metric space into another. Less sensitive than the multiplicative metric distortion, the average distortion captures well the global picture and, overall, is a quite interesting new measure of metric proximity, related to the concentration of measure phenomenon. The paper mostly deals with embeddings into the real line with a low (as much as it is possible) average distortion. Our main technical contribution is that the shortest-path metrics of special (e.g., planar, bounded treewidth, etc.) undirected weighted graphs can be embedded into the line with constant average distortion. This has implications, e.g., on the value of the MinCut–MaxFlow gap in uniform-demand multicommodity flows on such graphs.

Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

Abstract: We prove Helly-type theorems for line transversals to disjoint unit balls in ℝd. In particular, we show that a family of n≥2d disjoint unit balls in ℝd has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n≥4d−1 disjoint unit balls in ℝd admits a line transversal if any subfamily of size 4d−1 admits a transversal.

Pin-Collinear Body-and-Pin Frameworks and the Molecular Conjecture

Abstract: T.-S. Tay and W. Whiteley independently characterized the multigraphs which can be realized as an infinitesimally rigid d-dimensional body-and-hinge framework. In 1984 they jointly conjectured that each graph in this family can be realized as an infinitesimally rigid framework with the additional property that the hinges incident to each body lie in a common hyperplane. This conjecture has become known as the Molecular Conjecture because of its implication for the rigidity of molecules in 3-dimensional space. Whiteley gave a partial solution for the 2-dimensional form of the conjecture in 1989 by showing that it holds for multigraphs G=(V,E) in the family which have the minimum number of edges, i.e. satisfy 2|E|=3|V|−3. In this paper, we give a complete solution for the 2-dimensional version of the Molecular Conjecture. Our proof relies on a new formula for the maximum rank of a pin-collinear body-and-pin realization of a multigraph as a 2-dimensional bar-and-joint framework.

Certificates of Positivity in the Bernstein Basis

Abstract: Let $P\in\mathbb{Z[X]}$ be a polynomial of degree p with coefficients in the monomial basis of bit-size bounded by τ. If P is positive on [−1,1], we obtain a certificate of positivity (i.e., a description of P making obvious that it is positive) of bit-size O(p 4(τ+log 2 p)). Previous comparable results had a bit-size complexity exponential in p and τ (Powers and Reznick in Trans. Am. Math. Soc. 352(10):4677–4692, 2000; Powers and Reznick in J. Pure Appl. Algebra 164:221–229, 2001).

A Variational Proof of Alexandrov’s Convex Cap Theorem

Abstract: We give a variational proof of the existence and uniqueness of a convex cap with the given metric on the boundary. The proof uses the concavity of the total scalar curvature functional (also called Hilbert-Einstein functional) on the space of generalized convex caps. As a by-product, we prove that generalized convex caps with the fixed metric on the boundary are globally rigid, that is uniquely determined by their curvatures.

An Inscribing Model for Random Polytopes

Abstract: For convex bodies K with $\mathcal {C}^{2}$ boundary in ℝ d , we explore random polytopes with vertices chosen along the boundary of K. In particular, we determine asymptotic properties of the volume of these random polytopes. We provide results concerning the variance and higher moments of this functional, as well as an analogous central limit theorem.

Cores of Simplicial Complexes

Abstract: We introduce a method to reduce the study of the topology of a simplicial complex to that of a simpler one. Applying this method to complexes arising from graphs, we give topological meaning to classical graph invariants. As a consequence, we answer some questions raised in (Ehrenborg and Hetyei in Eur. J. Comb. 27(6):906---923, 2006) on the independence complex and the dominance complex of a forest and obtain improved algorithms to compute their homotopy types.

Siegel’s Lemma and Sum-Distinct Sets

Abstract: Let L(x)=a1x1+a2x2+⋅⋅⋅+anxn, n≥2, be a linear form with integer coefficients a1,a2,…,an which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a1,a2,…,an. The main result of this paper asserts that there exist linearly independent vectors x1,…,x n−1∈ℤn such that L(xi)=0, i=1,…,n−1, and $$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$ where a=(a1,a2,…,an) and $$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$ This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdos–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry.

Line Transversals to Disjoint Balls

Abstract: We prove that the set of directions of lines intersecting three disjoint balls in ℝ3 in a given order is a strictly convex subset of $\mathbb {S}^{2}$. We then generalize this result to n disjoint balls in ℝd. As a consequence, we can improve upon several old and new results on line transversals to disjoint balls in arbitrary dimension, such as bounds on the number of connected components and Helly-type theorems.

Visibility Graphs of Point Sets in the Plane

Abstract: The visibility graph $\mathcal {V}(X)$ of a discrete point set X⊂ℝ2 has vertex set X and an edge xy for every two points x,y∈X whenever there is no other point in X on the line segment between x and y. We show that for every graph G, there is a point set X∈ℝ2, such that the subgraph of $\mathcal {V}(X\cup \mathbb {Z}^{2})$ induced by X is isomorphic to G. As a consequence, we show that there are visibility graphs of arbitrary high chromatic number with clique number 6 settling a question by Kara, Por and Wood.