Showing papers in "Discrete and Computational Geometry in 2006"
TL;DR: If a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5( v-2); and the crossing number of any graph is at least $\frac73e-\frac{25}3(v-2).$ Both bounds are tight up to an additive constant.
Abstract: Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e > 4v edges is at least ce3/v2, where c > 0 is an absolute constant. This result, known as the "Crossing Lemma," has found many important applications in discrete and computational geometry. It is tight up to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with c > 1024/31827 > 0.032. The proof has two new ingredients, interesting in their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5(v-2); and (2) the crossing number of any graph is at least $\frac73e-\frac{25}3(v-2).$ Both bounds are tight up to an additive constant (the latter one in the range $4v\le e\le 5v$).
128 citations
TL;DR: An algorithm for finding points of locally maximum elevation, which is invariant under rigid motions and can be used to define features such as lines of discontinuous or continuous but non-smooth elevation.
Abstract: Given a smoothly embedded 2-manifold in ${\Bbb R}^3,$ we define the elevation of a point as the height difference to a canonically defined second point on the same manifold. Our definition is invariant under rigid motions and can be used to define features such as lines of discontinuous or continuous but non-smooth elevation. We give an algorithm for finding points of locally maximum elevation, which we suggest mark cavities and protrusions and are useful in matching shapes as for example in protein docking.
119 citations
TL;DR: This paper is the first in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing.
Abstract: This paper is the first in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. After some preliminary comments about the face-centered cubic and hexagonal close packings, the history of the Kepler problem is described, including a discussion of various published bounds on the density of sphere packings. There is also a general historical discussion of various proof strategies that have been tried with this problem.
105 citations
TL;DR: In this article, the second in a series of six papers devoted to the proof of the Kepler conjecture is presented, which states that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing.
Abstract: This paper is the second in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. The top level structure of the proof is described. A compact topological space is described. Each point of this space can be described as a finite cluster of balls with additional combinatorial markings. A continuous function on this compact space is defined. It is proved that the Kepler conjecture will follow if the value of this function is never greater than a given explicit constant.
73 citations
TL;DR: The characterization of globally rigid graphs is extended by characterizing globally linked pairs in M-connected graphs, an important family of rigid graphs by determining the number of distinct realizations of an M- connected graph, each of which is equivalent to a given generic realization.
Abstract: A two-dimensional framework (G,p) is a graph G = (V,E) together with a map p: V ź ź2. We view (G,p) as a straight line realization of G in ź2. Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u,v} is globally linked in G if %and for all equivalent frameworks (G,q), the distance between the points corresponding to u and v is the same in all pairs of equivalent generic realizations of G. The graph G is globally rigid if all of its pairs of vertices are globally linked. We extend the characterization of globally rigid graphs given by the first two authors [13] by characterizing globally linked pairs in M-connected graphs, an important family of rigid graphs. As a byproduct we simplify the proof of a result of Connelly [6] which is a key step in the characterization of globally rigid graphs. We also determine the number of distinct realizations of an M-connected graph, each of which is equivalent to a given generic realization. Bounds on this number for minimally rigid graphs were obtained by Borcea and Streinu in [3].
69 citations
TL;DR: A multivariable (infinite) Taylor series expansion is derived to measure a simplicial solid angle in terms of the inner products of its spanning vectors and it is shown that it converges within the natural boundary for solid angles.
Abstract: The dot product formula allows one to measure an angle determined by two vectors, and a formula known to Euler and Lagrange outputs the measure of a solid angle in ${\Bbb R}^3$ given its three spanning vectors. However, there appears to be no closed form expression for the measure of an n-dimensional solid angle for n > 3. We derive a multivariable (infinite) Taylor series expansion to measure a simplicial solid angle in terms of the inner products of its spanning vectors. We then analyze the domain of convergence of this hypergeometric series and show that it converges within the natural boundary for solid angles.
63 citations
TL;DR: The super-Apollonian group has been studied in this paper for strongly integral Apollonian circle packings, where the curvatures of all circles are integral and the curvature x centers of all centers are integral.
Abstract: Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain, where a Descartes configuration is a set of four mutually tangent circles in the Riemann sphere, having disjoint interiors. Part I showed there exists a discrete group, the Apollonian group, acting on a parameter space of (ordered, oriented) Descartes configurations, such that the Descartes configurations in a packing formed an orbit under the action of this group. It is observed there exist infinitely many types of integral Apollonian packings in which all circles had integer curvatures, with the integral structure being related to the integral nature of the Apollonian group. Here we consider the action of a larger discrete group, the super-Apollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric object we call a super-packing. The circles in a super-packing never cross each other but are nested to an arbitrary depth. Certain Apollonian packings and super-packings are strongly integral in the sense that the curvatures of all circles are integral and the curvature x centers of all circles are integral. We show that (up to scale) there are exactly eight different (geometric) strongly integral super-packings, and that each contains a copy of every integral Apollonian circle packing (also up to scale). We show that the super-Apollonian group has finite volume in the group of all automorphisms of the parameter space of Descartes configurations, which is isomorphic to the Lorentz group O(3, 1).
60 citations
TL;DR: In this paper, a packing is compact if every disc D is tangent to a sequence of discs D1, D2, D3, D4, D5, D6, D7, D8, D9, D10, D11, D12, D13, D14, D15, D16, D17, D18, D19, D20, D21, D22, D23, D24, D26, D27, D28, D29, D30, D31, D32, D34
Abstract: We consider packings of the plane using discs of radius 1 and r. A packing is compact if every disc D is tangent to a sequence of discs D1, D2, ..., Dn such that Di is tangent to Di+1. We prove that there are only nine values of r with r < 1 for which such packings are possible. For each of the nine values we describe the possible compact packings.
58 citations
TL;DR: In this article, it was shown that there exist k-neighborly centrally symmetric d-dimensional polytopes with 2(n + d) vertices, where k(d,n)=\Theta\left(\frac{d}{1+\log ((d+n)/d)}\right).
Abstract: We show that there exist k-neighborly centrally symmetric d-dimensional polytopes with 2(n + d) vertices, where $k(d,n)=\Theta\left(\frac{d}{1+\log ((d+n)/d)}\right).$ We also show that this bound is tight.
57 citations
TL;DR: In this paper, the authors considered the problem of constructing rational Apollonian circle packings with integrality properties in terms of the curvatures and centers of the circles, and showed that the dual and super-Apollonian groups are Coxeter groups.
Abstract: This paper gives $n$-dimensional analogues of the Apollonian circle packings in Parts I and II. Those papers considered circle packings described in terms of their Descartes configurations, which are sets of four mutually touching circles. They studied packings that had integrality properties in terms of the curvatures and centers of the circles. Here we consider collections of $n$-dimensional Descartes configurations, which consist of $n+2$ mutually touching spheres. We work in the space $M_D^n$ of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, augmented curvature-center coordinates, as those $(n+2) \times (n+2)$ real matrices $W$ with $W^T Q_{D,n} W = Q_{W,n}$ where $Q_{D,n} = x_1^2 + \cdots + x_{n+2}^2 - ({1}/{n})(x_1 +\cdots + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + \cdots + 2x_{n+2}^2$, and $\bQ_{D,n}$ and $\bQ_{W,n}$ are their corresponding symmetric matrices. On the parameter space $M_D^n$ of augmented curvature-center matrices, the group ${\it Aut}(Q_{D,n})$ acts on the left and ${\it Aut}(Q_{W,n})$ acts on the right. Both these groups are isomorphic to the $(n+2)$-dimensional Lorentz group $O(n+1,1)$, and give two different "geometric" actions. The right action of ${\it Aut}(Q_{W,n})$ (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space $\rr^n$ while the left action of ${\it Aut}(Q_{D,n})$ is defined only on the parameter space $M_D^n$. We introduce $n$-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups in ${\it Aut}(Q_{D,n})$, with the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set $S$ depending on the dimension. We show that the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group, with similar notions for the other two groups. We determine in which dimensions there exist rational Apollonian cluster ensembles (all curvatures are rational) and strongly rational Apollonian sphere ensembles (all augmented curvature-center coordinates are rational).
56 citations
TL;DR: In this paper, the inverse Laplace transforms of rational functions with poles on arrangements of hyperplanes are computed for root systems, and an algorithm to compute the value of inverse Laplacian transforms with poles is presented.
Abstract: This paper presents an algorithm to compute the value of the inverse Laplace transforms of rational functions with poles on arrangements of hyperplanes. As an application, we present an efficient computation of the partition function for classical root systems.
TL;DR: Two algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem are described and rigorous error bounds are provided and some of the new lattices are locally optimal.
Abstract: We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the sense that they approximate optimal covering lattices and optimal packing-covering lattices within any desired accuracy. Both algorithms involve semidefinite programming and are based on Voronoi's reduction theory for positive definite quadratic forms, which describes all possible Delone triangulations of źd. In practice, our implementations reproduce known results in dimensions d ≤ 5 and in particular solve the two problems in these dimensions. For d = 6 our computations produce new best known covering as well as packing-covering lattices, which are closely related to the lattice E*6. For d = 7,8 our approach leads to new best known covering lattices. Although we use numerical methods, we made some effort to transform numerical evidences into rigorous proofs. We provide rigorous error bounds and prove that some of the new lattices are locally optimal.
TL;DR: The first proof that k(3) = 12 was given by Schutte and van der Waerden only in 1953 as mentioned in this paper, which relies on basic calculus and simple spherical geometry.
Abstract: The kissing number k(3) is the maximal number of equal size nonoverlapping spheres in three dimensions that can touch another sphere of the same size. This number was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. The first proof that k(3) = 12 was given by Schutte and van der Waerden only in 1953. In this paper we present a new solution of the Newton--Gregory problem that uses our extension of the Delsarte method. This proof relies on basic calculus and simple spherical geometry.
TL;DR: An elementary proof of the Sylvester-Gallai theorem is given, and it is extended to show that an SG configuration in projective space over the quaternions must be contained in a three-dimensional flat.
Abstract: A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a deep inequality of Hirzebruch. We give an elementary proof of this result, and then extend it to show that an SG configuration in projective space over the quaternions must be contained in a three-dimensional flat.
TL;DR: This work considers the problem of approximate range counting over a stream of d-dimensional points, in the data stream model, and computes a compact summary data structure that can be used to count the number of points inside a query range within additive error ε.
Abstract: We consider the problem of approximate range counting over a stream of d-dimensional points. In the data stream model the algorithm makes a single scan of the data, which is presented in an arbitrary order, and computes a compact summary data structure. The summary, whose size depends on the approximation parameter e, can be used to count the number of points inside a query range within additive error en, where n is the size of the stream seen so far. We present several results, deterministic and randomized, for both rectangle and halfspace ranges.
TL;DR: The proof is based on weighted incidence theory and an inductive procedure which allows us to deal with higher-dimensional interactions effectively and is borrowed from [CES+] where much of the higher- dimensional incidence theoretic motivation comes from.
Abstract: We show that the equation \[ s_{i_1}+s_{i_2}+\cdots+s_{i_d}=s_{i_{d+1}}+\cdots+s_{i_{2d}} \] has $O(N^{2d-2+2^{-d+1}})$ solutions for any strictly convex sequence $\{s_i\}_{i=1}^N$ without any additional arithmetic assumptions. The proof is based on weighted incidence theory and an inductive procedure which allows us to deal with higher-dimensional interactions effectively. The terminology "combinatorial complexity" is borrowed from [CES+] where much of our higher-dimensional incidence theoretic motivation comes from.
TL;DR: This work is the first that provides quality guarantees for Delaunay meshes in the presence of small input angles, and encloses the input edges with a small buffer zone, a union of balls whose sizes are proportional to the local feature sizes at their centers.
Abstract: We propose an algorithm to compute a conforming Delaunay mesh of a bounded domain in ${\Bbb R}^3$ specified by a piecewise linear complex. Arbitrarily small input angles are allowed, and the input complex is not required to be a manifold. Our algorithm encloses the input edges with a small buffer zone, a union of balls whose sizes are proportional to the local feature sizes at their centers. In the output mesh, the radius-edge ratio of the tetrahedra outside the buffer zone is bounded by a constant independent of the domain, while that of the tetrahedra inside the buffer zone is bounded by a constant depending on the smallest input angle. Furthermore, the output mesh is graded. Our work is the first that provides quality guarantees for Delaunay meshes in the presence of small input angles.
TL;DR: It is shown that the values obtained cannot be improved for large n by more than c1/n3 in the area problem and c2/n5 in the perimeter problem, for certain constants c1 and c1.
Abstract: The maximal area of a polygon with n = 2m edges and unit diameter is not known when m ≥ 5, nor is the maximal perimeter of a convex polygon with n = 2m edges and unit diameter known when m ≥ 4. We construct improved polygons in both problems, and show that the values we obtain cannot be improved for large n by more than c1/n3 in the area problem and c2/n5 in the perimeter problem, for certain constants c1 and c2.
TL;DR: This work considers the exact and approximate computational complexity of the multivariate least median-of-squares (LMS) linear regression estimator, which is among the most widely used robust linear statistical estimators.
Abstract: We consider the exact and approximate computational complexity of the multivariate least median-of-squares (LMS) linear regression estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in ${\Bbb R}^d$ and a parameter k, the problem is equivalent to computing the narrowest slab bounded by two parallel hyperplanes that contains k of the points. We present algorithms for the exact and approximate versions of the multivariate LMS problem. We also provide nearly matching lower bounds for these problems. These lower bounds hold under the assumptions that k is Ω(n) and that deciding whether n given points in ${\Bbb R}^d$ are affinely non-degenerate requires Ω(nd) time.
TL;DR: It is shown that the combinatorics of splits discussed in connection with the split decomposition corresponds to the geometric properties of a hyperplane arrangement and a point configuration.
Abstract: This paper sheds new light on split decomposition theory and T-theory from the viewpoint of convex analysis and polyhedral geometry. By regarding finite metrics as discrete concave functions, Bandelt-Dress' split decomposition can be derived as a special case of more general decomposition of polyhedral/discrete concave functions introduced in this paper. It is shown that the combinatorics of splits discussed in connection with the split decomposition corresponds to the geometric properties of a hyperplane arrangement and a point configuration. Using our approach, the split decomposition of metrics can be naturally extended to distance functions, which may violate the triangle inequality, using partial split distances.
TL;DR: The first pseudo-quasipolynomial-time algorithm for no extra information is obtained, which can be viewed as a polynomial- time algorithm given an "extremum oracle" as extra information.
Abstract: A frequently arising problem in computational geometry is when a physical structure, such as an ad-hoc wireless sensor network or a protein backbone, can measure local information about its geometry (e.g., distances, angles, and/or orientations), and the goal is to reconstruct the global geometry from this partial information. More precisely, we are given a graph, the approximate lengths of the edges, and possibly extra information, and our goal is to assign two-dimensional coordinates to the vertices such that the (multiplicative or additive) error on the resulting distances and other information is within a constant factor of the best possible. We obtain the first pseudo-quasipolynomial-time algorithm for this problem given a complete graph of Euclidean distances with additive error and no extra information. For general graphs, the analogous problem is NP-hard even with exact distances. Thus, for general graphs, we consider natural types of extra information that make the problem more tractable, including approximate angles between edges, the order type of vertices, a model of coordinate noise, or knowledge about the range of distance measurements. Our pseudo-quasipolynomial-time algorithm for no extra information can also be viewed as a polynomial-time algorithm given an "extremum oracle" as extra information. We give several approximation algorithms and contrasting hardness results for these scenarios.
TL;DR: A colourful generalization of Liu's simplicial depth is introduced and a parity property and conjecture are proved that the minimum colourful simplicial Depth of any core point in any d-dimensional configuration is d2 + 1 and that the maximum is dd+1 + 1.
Abstract: Inspired by Barany's Colourful Caratheodory Theorem, we introduce a colourful generalization of Liu's simplicial depth. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d2 + 1 and that the maximum is dd+1 + 1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.
TL;DR: It is shown that all combinatorial triangle-free configurations for v ≤ 18 are geometrically realizable and there is a unique smallest astral triangle- free configuration and its Levi graph is the generalized Petersen graph G(18,5).
Abstract: In the paper we show that all combinatorial triangle-free configurations for v ≤ 18 are geometrically realizable. We also show that there is a unique smallest astral (183) triangle-free configuration and its Levi graph is the generalized Petersen graph G(18,5). In addition, we present geometric realizations of the unique flag transitive triangle-free configuration (203) and the unique point transitive triangle-free configuration (213).
TL;DR: It is shown that the combinatorial complexity of the Minkowski sum P ⊕ Q is O(kℓmnα(min{m,n})), where α(·) is the inverse Ackermann function.
Abstract: Let P be a simple polygon with m edges, which is the disjoint union of k simple polygons, all monotone in a common direction u, and let Q be another simple polygon with n edges, which is the disjoint union of l simple polygons, all monotone in a common direction v. We show that the combinatorial complexity of the Minkowski sum P ź Q is O(klmnź(min{m,n})), where ź(·) is the inverse Ackermann function. Some structural properties of the case k = l = 1 have been (implicitly) studied in [17]. We rederive these properties using a different proof, apply them to obtain the above complexity bound for k = l = 1, obtain several additional properties of the sum for this special case, and then use them to derive the general bound.
TL;DR: It is proved that a point in the Euclidean space R5 cannot be surrounded by a finite number of acute simplices, which implies that there does not exist a face-to-face partition of R5 into acute Simplices.
Abstract: We prove that a point in the Euclidean space R5 cannot be surrounded by a finite number of acute simplices. This fact implies that there does not exist a face-to-face partition of R5 into acute simplices. The existence of an acute simplicial partition of Rd for d > 5 is excluded by induction, but for d = 4 this is an open problem.
TL;DR: This conjecture is proved that every finite set S of points in two-dimensional Euclidean space includes two points, a and b, such that either there is no other point in S on the line Ab, or the line ab contains all the points in S.
Abstract: The Sylvester-Gallai theorem asserts that every finite set S of points in two-dimensional Euclidean space includes two points, a and b, such that either there is no other point in S on the line ab, or the line ab contains all the points in S. Chvatal extended the notion of lines to arbitrary metric spaces and made a conjecture that generalizes the Sylvester-Gallai theorem. In the present article we prove this conjecture.
TL;DR: Improved bounds refine results recently obtained by Abrego and Fernandez-Merchant and by Lovasz, Vesztergombi, Wagner, and Welzl on the minimum number of (≤ k)-sets in a set of points in general position.
Abstract: We use circular sequences to give an improved lower bound on the minimum number of (≤ k)-sets in a set of points in general position. We then use this to show that if S is a set of n points in general position, then the number □(S) of convex quadrilaterals determined by the points in S is at least 0.37553(n4) + O(n3). This in turn implies that the rectilinear crossing number cr(Kn) of the complete graph Knis at least 0.37553(n4) + O(n3), and that Sylvester's Four Point Problem Constant is at least 0.37553. These improved bounds refine results recently obtained by Abrego and Fernandez-Merchant and by Lovasz, Vesztergombi, Wagner, and Welzl.
TL;DR: It is shown that tropical secant varieties of ordinary linear spaces correspond to the log-limit sets of ordinary toric varieties, and this characterization is used to reformulate the question of determining Barvinok rank into a question regarding regular subdivisions of products of simplices.
Abstract: In this paper we investigate tropical secant varieties of ordinary linear spaces. These correspond to the log-limit sets of ordinary toric varieties; we show that their interesting parts are combinatorially isomorphic to a certain natural subcomplex of the complex of regular subdivisions of a corresponding point set, and we display the range of behavior of this object. We also use this characterization to reformulate the question of determining Barvinok rank into a question regarding regular subdivisions of products of simplices.
TL;DR: It is proved that each such graph must be isomorphic to a tame graph, of which there are only finitely many up to isomorphism.
Abstract: This paper is the sixth and final part in a series of papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. In a previous paper in this series, a continuous function f on a compact space is defined, certain points in the domain are conjectured to give the global maxima, and the relation between this conjecture and the Kepler conjecture is established. In this paper we consider the set of all points in the domain for which the value of f is at least the conjectured maximum. To each such point, we attach a planar graph. It is proved that each such graph must be isomorphic to a tame graph, of which there are only finitely many up to isomorphism. Linear programming methods are then used to eliminate all possibilities, except for three special cases treated in earlier papers: pentahedral prisms, the face-centered cubic packing, and the hexagonal-close packing. The results of this paper rely on long computer calculations.
TL;DR: It is shown, with an elementary proof, that the number of halving simplices in a set of n points in ℝ4 in general position is O(n4-2/45), which improves the previous bound of O( n4-1/13^{4}).
Abstract: We show, with an elementary proof, that the number of halving simplices in a set of n points in ź4 in general position is O(n4-2/45). This improves the previous bound of O(n4-1/13^{4}). Our main new ingredient is a bound on the maximum number of halving simplices intersecting a fixed 2-plane.