Showing papers in "Discrete and Computational Geometry in 2002"

Topological Persistence and Simplification

Abstract: We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise depending on its lifetime or persistence within the filtration. We give fast algorithms for computing persistence and experimental evidence for their speed and utility.

A Polytope Related to Empirical Distributions, Plane Trees, Parking Functions, and the Associahedron

Abstract: The volume of the n -dimensional polytope ? n (x):= {y ? R n : y i ? 0 and y 1 + · · · + y i ≤ x 1 + · · ·+ x i for all 1 ≤ i ≤ n } for arbitrary x:=(x 1 , . . ., x n ) with x i >0 for all i defines a polynomial in variables x i which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as the volumes of chambers in two different polytopal subdivisions of ? n (x) . The first of these subdivisions generalizes to a class of polytopes called sections of order cones. In the second subdivision the chambers are indexed in a natural way by rooted binary trees with n+1 vertices, and the configuration of these chambers provides a representation of another polytope with many applications, the associahedron .

New Similarity Measures between Polylines with Applications to Morphing and Polygon Sweeping

Abstract: We introduce two new related metrics, the geodesic width and the link width, for measuring the "distance" between two nonintersecting polylines in the plane. If the two polylines have n vertices in total, we present algorithms to compute the geodesic width of the two polylines in O(n 2 log n) time using O(n 2) space and the link width in O(n 3 log n) time using O(n 2) working space where n is the total number of edges of the polylines. Our computation of these metrics relies on two closely related combinatorial strutures: the shortest-path diagram and the link diagram of a simple polygon. The shortest-path (resp., link) diagram encodes the Euclidean (resp., link) shortest path distance between all pairs of points on the boundary of the polygon. We use these algorithms to solve two problems:

A Deterministic Algorithm for Approximating the Mixed Discriminant and Mixed Volume, and a Combinatorial Corollary

Abstract: We present a deterministic polynomial-time algorithm that computes the mixed discriminant of an n -tuple of positive semidefinite matrices to within an exponential multiplicative factor. To this end we extend the notion of doubly stochastic matrix scaling to a larger class of n -tuples of positive semidefinite matrices, and provide a polynomial-time algorithm for this scaling. As a corollary, we obtain a deterministic polynomial algorithm that computes the mixed volume of n convex bodies in R n to within an error which depends only on the dimension. This answers a question of Dyer, Gritzmann and Hufnagel. A ``side benefit'' is a generalization of Rado's theorem on the existence of a linearly independent transversal.

Cutting Circles into Pseudo-Segments and Improved Bounds for Incidences% and Complexity of Many Faces

Abstract: We show that n arbitrary circles in the plane can be cut into O(n 3/2+? ) arcs, for any ?>0 , such that any pair of arcs intersects at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree.

Degrees of Real Wronski Maps

Abstract: We study the map which sends vectors of polynomials into their Wronski determinants. This defines a projection map of a Grassmann variety which we call a Wronski map. Our main result is computation of degrees of the real Wronski maps. Connections with real algebraic geometry and control theory are described.

Visibility Queries and Maintenance in Simple Polygons

Abstract: In this paper we explore some novel aspects of visibility for stationary and moving points inside a simple polygon P . We provide a mechanism for expressing the visibility polygon from a point as the disjoint union of logarithmically many canonical pieces using a quadratic-space data structure. This allows us to report visibility polygons in time proportional to their size, but without the cubic space overhead of earlier methods. The same canonical decomposition can be used to determine visibility within a frustum, or to compute various attributes of the visibility polygon efficiently. By exploring the connection between visibility polygons and shortest-path trees, we obtain a kinetic algorithm that can track the visibility polygon as the viewpoint moves along polygonal paths inside P , at a polylogarithmic cost per combinatorial change in the visibility or in the flight plan of the point. The combination of the static and kinetic algorithms leads to a new static algorithm in which we can trade off space for increased overhead in the query time. As another application, we obtain an algorithm which computes the weak visibility polygon from a query segment inside P in output-sensitive time.

A Lower Bound on the Distortion of Embedding Planar Metrics into Euclidean Space

Abstract: We exhibit a simple infinite family of series-parallel graphs that cannot be metrically embedded into Euclidean space with distortion smaller than $\Omega(\sqrt{\log n})$ . This matches Rao's [14] general upper bound for metric embedding of planar graphs into Euclidean space, thus resolving the question how well do planar metrics embed in Euclidean spaces?

Finding Sets of Points without Empty Convex 6-Gons

Abstract: . Erdos asked whether every large enough set of points in general position in the plane contains six points that form a convex 6-gon without any points from the set in its interior. In this note we show how a set of 29 points was found that contains no empty convex 6-gon. To this end a fast incremental algorithm for finding such 6-gons was designed and implemented and a heuristic search approach was used to find promising sets. The experiments also led to two observations that might be useful in proving that large sets always contain an empty convex 6-gon.

Recognizing String Graphs Is Decidable

Abstract: A graph is called a string graph if its vertices can be represented by continuous curves ("strings") in the plane so that two of them cross each other if and only if the corresponding vertices are adjacent. It is shown that there exists a recursive function f(n) with the property that every string graph of n vertices has a representation in which any two curves cross at most f(n) times. We obtain as a corollary that there is an algorithm for deciding whether a given graph is a string graph. This solves an old problem of Benzer (1959), Sinden (1966), and Graham (1971).

Box-Trees and R-Trees with Near-Optimal Query Time

Abstract: A box-tree is a bounding-volume hierarchy that uses axis-aligned boxes as bounding volumes. The query complexity of a box-tree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing box-trees with small worst-case query complexity with respect to queries with axis-parallel boxes and with points. We also prove lower bounds on the worst-case query complexity for box-trees, which show that our results are optimal or close to optimal. Finally, we present algorithms to convert box-trees to R-trees, resulting in R-trees with (almost) optimal query complexity.

Hyperplane projections of the unit ball of ℓpn

Abstract: Let B p n ={x?\R n ;\; \sum i=1 n |x i | p ≤ 1} , 1≤ p\le+?fty . We study the extreme values of the volume of the orthogonal projection of B p n onto hyperplanes H\subset \R n . For a fixed H , we prove that the ratio vol(P H B p n )/ vol(B p n-1 ) is non-decreasing in p?[1,+?fty] .

A Cube Tiling of Dimension Eight with No Facesharing

Abstract: A cube tiling of eight-dimensional space in which no pair of cubes share a complete common seven-dimensional face is constructed. Together with a result of Perron, this shows that the first dimension in which such a tiling can exist is seven or eight.

Densest Packing of Equal Spheres in Hyperbolic Space

Abstract: . We propose a method to analyze the density of packings of spheres of fixed radius in the hyperbolic space of any dimension m≥ 2 , and prove that for all but countably many radii, optimally dense packings must have low symmetry.

Generation of Oriented Matroids--A Graph Theoretical Approach

Abstract: We discuss methods for the generation of oriented matroids and of isomorphism classes of oriented matroids. Our methods are based on single element extensions and graph theoretical representations of oriented matroids, and all these methods work in general rank and for non-uniform and uniform oriented matroids as well. We consider two types of graphs, cocircuit graphs and tope graphs, and discuss the single element extensions in terms of localizations which can be viewed as partitions of the vertex sets of the graphs. Whereas localizations of the cocircuit graph are well characterized, there is no graph theoretical characterization known for localizations of the tope graph. In this paper we prove a connectedness property for tope graph localizations and use this for the design of algorithms for the generation of single element extensions by use of tope graphs. Furthermore, we discuss similar algorithms which use the cocircuit graph. The characterization of localizations of cocircuit graphs finally leads to a backtracking algorithm which is a simple and efficient method for the generation of single element extensions. We compare this method with a recent algorithm of Bokowski and Guedes de Oliveira for uniform oriented matroids.

NP -Hardness of Largest Contained and Smallest Containing Simplices for V- and H-Polytopes

Abstract: The problem of finding a d -simplex of maximum volume in an arbitrary d -dimensional V -polytope, for arbitrary d , was shown by Gritzmann et al. [GKL] in 1995 to be NP-hard. They conjectured that the corresponding problem for H -polytopes is also NP-hard. This paper presents a unified way of proving the NP-hardness of both these problems. The approach also yields NP-hardness proofs for the problems of finding d -simplices of minimum volume containing d -dimensional V - or H -polytopes. The polytopes that play the key role in the hardness proofs are truncations of simplices. A construction is presented which associates a truncated simplex to a given directed graph, and the hardness results follow from the hardness of detecting whether a directed graph has a partition into directed triangles.

Untangling a Polygon

Abstract: The following problem was raised by M. Watanabe. Let P be a self-intersecting closed polygon with n vertices in general position. How manys steps does it take to untangle P , i.e., to turn it into a simple polygon, if in each step we can arbitrarily relocate one of its vertices. It is shown that in some cases one has to move all but at most O((n log n) 2/3 ) vertices. On the other hand, every polygon P can be untangled in at most $n-\Omega(\sqrt{n})$ steps. Some related questions are also considered.

A Space of Cyclohedra

Abstract: . The real points of the Deligne—Knudsen—Mumford moduli space \overline \cal M n 0 of marked points on the sphere have a natural tiling by associahedra. We extend this idea to construct an aspherical space tiled by cyclohedra . We explore the structure of this space, coming from blow-ups of hyperplane arrangements, as well as discuss possibilities of its role in knot theory and mathematical physics.

Fractional Power Series Solutions for Systems of Equations

Abstract: In this paper we extend the explorations in [8] to include the fractional power series expansions of k equations in d variables, where d>k . An analog of Newton's polygon construction which uses the Minkowski sum P of the Newton polytopes P 1 ,...,P k of the k equations is given for computing such series expansions. If the Newton polytopes of these equations are the same, then the common domains of convergence for the solutions correspond to the vertices of a certain fiber polytope Σ(P) . In general, our results suggest the existence of a ``mixed fiber polytope'' of k polytopes. It is also indicated that there may be a relationship between these mixed fiber polytopes and a generalization of the discriminant, which we call the mixed discriminant.

Equipartition of Two Measures by a 4-Fan

Abstract: We show that, given two probability measures in the plane, there exists a 4-fan that simultaneously equipartitions them. In other words, there is a point and four half-lines emanating from it such that each of the four sectors have measure \frac 1 4 in both measures.

In between k-Sets, j-Facets, and i-Faces: (i, j)-Partitions ∗

Abstract: . Let S be a finite set of points in general position in R d . We call a pair (A,B) of subsets of S an (i,j) -partition of S if |A|=i , |B|=j and there is an oriented hyperplane h with S $\cap$ h=A and with B the set of points from S on the positive side of h . (i,j) -Partitions generalize the notions of k -sets (these are (0,k) -partitions) and j -facets ((d,j) -partitions) of point sets as well as the notion of i -faces of the convex hull of S ((i+1,0) -partitions). In oriented matroid terminology, (i,j) -partitions are covectors where the number of 0 's is i and the numbers of + 's is j . We obtain linear relations among the numbers of (i,j) -partitions, mainly by means of a correspondence between (i-1) -faces of so-called k -set polytopes on the one side and (i,j) -partitions for certain j 's on the other side. We also describe the changes of the numbers of (i,j) -partitions during continuous motion of the underlying point set. This allows us to demonstrate that in dimensions exceeding 3 , the vector of the numbers of k -sets does not determine the vector of the numbers of j -facets—nor vice versa. Finally, we provide formulas for the numbers of (i,j) -partitions of points on the moment curve in R d .

Infinitesimal Hopf Algebras and the cd-Index of Polytopes

Abstract: Infinitesimal bialgebras were introduced by Joni and Rota [JR]. The basic theory of these objects was developed in [Aff1] and [Aff2]. In this paper we present a simple proof of the existence of the cd -index of polytopes, based on the theory of infinitesimal Hopf algebras. For the purpose of this work, the main examples of infinitesimal Hopf algebras are provided by the algebra \ppp of all posets and the algebra k ab > of noncommutative polynomials. We show that k ab > satisfies the following universal property: given a graded infinitesimal bialgebra A and a morphism of algebras ? A \colon A? k , there exists a unique morphism of graded infinitesimal bialgebras ?\colon A ? k ab > such that ?_{1,0}?=?_A, where ?_{1,0} is evaluation at (1,0). When the universal property is applied to the algebra of posets and the usual zeta function ?_{\ppp}(P)=1, one obtains the \abindex of posets ?\colon \ppp?k ab >. The notion of antipode is used to define an analog of the Mobius function of posets for more general infinitesimal Hopf algebras than \ppp , and this in turn is used to define a canonical infinitesimal Hopf subalgebra, called the eulerian subalgebra. All eulerian posets belong to the eulerian subalgebra of \ppp . The eulerian subalgebra of k ab > is precisely the algebra spanned by c=a+b and d=ab+ba . The existence of the cd -index of eulerian posets is then an immediate consequence of the simple fact that eulerian subalgebras are preserved under morphisms of infinitesimal Hopf algebras. The theory also provides a version of the generalized Dehn--Sommerville equations for more general infinitesimal Hopf algebras than k ab >.

Lattice Points, Dedekind Sums, and Ehrhart Polynomials of Lattice Polyhedra

Abstract: Let ? be a simplex of R N with vertices in the integral lattice Z N . The number of lattice points of m? (={m? : ? ? ?}) is a polynomial function L(?,m) of m ? 0 . In this paper we present: (i) a formula for the coefficients of the polynomial L(?,t) in terms of the elementary symmetric functions; (ii) a hyperbolic cotangent expression for the generating functions of the sequence L(?,m) , m ? 0 ; (iii) an explicit formula for the coefficients of the polynomial L(?,t) in terms of torsion. As an application of (i), the coefficient for the lattice n -simplex of R n with the vertices (0,. . ., 0, a j , 0,. . . ,0) (1≤ j≤ n) plus the origin is explicitly expressed in terms of Dedekind sums; and when n=2 , it reduces to the reciprocity law about Dedekind sums. The whole exposition is elementary and self-contained.

A Short Simplicial h -Vector and the Upper Bound Theorem

Abstract: The Upper Bound Conjecture is verified for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes--a short simplicial h -vector.

Combinatorics of the Toric Hilbert Scheme

Abstract: The toric Hilbert scheme is a parameter space for all ideals with the same multigraded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in P 4 whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of the point configuration defining the toric ideal. Inspired by the recent discovery of a disconnected Baues graph, we close with results that suggest the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme.

The Combinatorial and Topological Complexity of a Single Cell

Abstract: . The problem of bounding the combinatorial complexity of a single connected component (a single cell) of the complement of a set of n geometric objects in R k , each object of constant description complexity, is an important problem in computational geometry which has attracted much attention over the past decade. It has been conjectured that the combinatorial complexity of a single cell is bounded by a function much closer to O(n k-1 ) rather than O(n k ) which is the bound for the combinatorial complexity of the whole arrangement. Until now, this was known to be true only for k ≤ 3 and only for some special cases in higher dimensions. A classic result in real algebraic geometry due to Oleinik and Petrovsky [15], Thom [18], and Milnor [14], bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. However, until now no better bounds were known if we restricted attention to a single connected component of a basic semi-algebraic set. In this paper we show how these two problems are related. We prove a new bound on the sum of the Betti numbers of one connected component of a basic semi-algebraic set which is an improvement over the Oleinik—Petrovsky—Thom—Milnor bound. This also implies that the topological complexity of a single cell, measured by the sum of the Betti numbers, is bounded by O(n k-1 ) . The techniques used for proving this topological result combined with those developed by Halperin and Sharir for the single cell problem in three dimensions allow us to prove a bound of O(n k-1+e ) on the combinatorial complexity of a single cell. Finally, we show that under a certain natural geometric assumption on the objects (namely, that whenever they intersect, the intersection is robustly transversal) it is possible to prove a bound of O(n k-1 ) on the combinatorial complexity of a single cell.

Combinatorial Geometry Problems in Pattern Recognition

Abstract: Combinatorial geometry problems motivated by point pattern matching algorithms are considered, and the classical exact matching situation and several variants are discussed.

The Number of Congruent Simplices in a Point Set

Abstract: For 1 ≤ k≤ d-1 , let f k (d) (n) be the maximum possible number of k-simplices spanned by a set of n points in ? d that are congruent to a given k-simplex. We prove that $$f_2^{(3)} (n) = O(n^{5/3} 2^{O(\alpha ^2 (n))} )$$ , f 2 (4) (n) = O(n 2+?), for any ? > 0, f 2 (5) (n) = ?(n 7/3), and f 3 (4) (n) = O(n 20/9+?), for any ? > 0. We also derive a recurrence to bound f k (d) (n) for arbitrary values of k and d, and use it to derive the bound f k (d) (n) = O(n d/2+ ?), for any ? > 0, for d ≤ 7 and k ≤ d ? 2. Following Erd?s and Purdy, we conjecture that this bound holds for larger values of d as well, and for k ≤ d ? 2.

A Commutative Algebra for Oriented Matroids

Abstract: Let V be a vector space of dimension d over a field K and let A be a central arrangement of hyperplanes in V. To answer a question posed by K. Aomoto, P. Orlik and H. Terao construct a commutative K -algebra U(A) in terms of the equations for the hyperplanes of A. In the course of their work the following question naturally occurred: \circ Is U(A) determined by the intersection lattice L(A) of the hyperplanes of A? We give a negative answer to this question. The theory of oriented matroids gives rise to a combinatorial analogue of the algebra of Orlik--Terao, which is the main tool of our proofs.

Triangulated Spheres and Colored Cliques

Abstract: In [1] a generalization of Hall's theorem was proved for families of hypergraphs. The proof used Sperner's lemma. In [5] Meshulam proved an extension of this result, using homology and the nerve theorem. In this paper we show how the triangulations method can be used to derive Meshulam's results. As in [1], the proof is based on results on extensions of triangulations from the sphere to the full ball. A typical result of this type is that any triangulation of the (d-1) -dimensional sphere S d-1 can be extended to a triangulation of the ball B d , by adding one point at a time, having degree at most 2d to its predecessors.