Showing papers in "Discrete and Computational Geometry in 1996"

Optimal output-sensitive convex hull algorithms in two and three dimensions

Abstract: We present simple output-sensitive algorithms that construct the convex hull of a set ofn points in two or three dimensions in worst-case optimalO (n logh) time andO (n) space, whereh denotes the number of vertices of the convex hull.

McLaren's improved snub cube and other new spherical designs in three dimensions

Abstract: Evidence is presented to suggest that, in three dimensions, spherical 6-designs withN points exist forN=24, 26,ź28; 7-designs forN=24, 30, 32, 34,ź36; 8-designs forN=36, 40, 42,ź44; 9-designs forN=48, 50, 52,ź54; 10-designs forN=60, 62, ź64; 11-designs forN=70, 72,ź74; and 12-designs forN=84,ź86. The existence of some of these designs is established analytically, while others are given by very accurate numerical coordinates. The 24-point 7-design was first found by McLaren in 1963, and--although not identified as such by McLaren--consists of the vertices of an "improved" snub cube, obtained from Archimedes' regular snub cube (which is only a 3-design) by slightly shrinking each square face and expanding each triangular face. 5-designs with 23 and 25 points are presented which, taken together with earlier work of Reznick, show that 5 designs exist forN=12, 16, 18, 20,ź22. It is conjectured, albeit with decreasing confidence fortź9, that these lists oft-designs are complete and that no other exist. One of the constructions gives a sequence of putative sphericalt-designs withN=12m points (mź2) whereN=1/2t2(1+o(1)) astźź.

Output-sensitive results on convex hulls, extreme points, and related problems

Abstract: We use known data structures for ray-shooting and linear-programming queries to derive new output-sensitive results on convex hulls, extreme points, and related problems. We show that thef-face convex hull of ann-point setP in a fixed dimensiondź2 can be constructed in $$0\left( {n log f + \left( {nf} \right)^{1 - 1/\left( {\left[ {d/2} \right] + 1} \right)} \log ^{0\left( 1 \right)} n} \right)$$ time; this is optimal if $$f = 0\left( {n^{1/\left[ {d/2} \right]} /\log ^K n} \right)$$ for some sufficiently large constantK. We also show that theh extreme points ofP can be computed in $$0\left( {n log^{0\left( 1 \right)} h + \left( {nh} \right)^{1 - 1/\left( {\left[ {d/2} \right] + 1} \right)} \log ^{0\left( 1 \right)} n} \right)$$ time. These results are then applied to produce an algorithm that computes the vertices of all the convex layers ofP inO(n2źź) time for any constant $$\gamma< 2/\left( {\left[ {d/2} \right]^2 + 1} \right)$$ . Finally, we obtain improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints. In all of our algorithms the input is assumed to be in general position.

Efficient randomized algorithms for some geometric optimization problems

Abstract: In this paper we first prove the following combinatorial bound, concerning the complexity of the vertical decomposition of the minimization diagram of trivariate functions: Let $$\mathcal{F}$$ be a collection ofn totally or partially defined algebraic trivariate functions of constant maximum degree, with the additional property that, for a given pair of functionsf, fźź $$\mathcal{F}$$ , the surfacef(x, y, z)=fź(x, y, z) isxy-monotone (actually, we need a somewhat weaker property). We show that the vertical decomposition of the minimization diagram of $$\mathcal{F}$$ consists ofO(n3+ź) cells (each of constant description complexity), for any ź>0. In the second part of the paper, we present a general technique that yields faster randomized algorithms for solving a number of geometric optimization problems, including (i) computing the width of a point set in 3-space, (ii) computing the minimum-width annulus enclosing a set ofn points in the plane, and (iii) computing the "biggest stick" inside a simple polygon in the plane. Using the above result on vertical decompositions, we show that the expected running time of all three algorithms isO(n3/2+ź), for any ź>0. Our algorithm improves and simplifies previous solutions of all three problems.

Free arrangements and rhombic tilings

Abstract: Let Z be a centrally symmetric polygon with integer side lengths. We answer the following two questions:(1)When is the associated discriminantal hyperplane arrangementfree in the sense of Saito and Terao?(2)When areall of the tilings of Z by unit rhombicoherent in the sense of Billera and Sturmfels? Surprisingly, the answers to these two questions are very similar. Furthermore, by means of an old result of MacMahon on plane partitions and some new results of Elnitsky on rhombic tilings, the answer to the first question helps to answer the second. These results then also give rise to some interesting geometric corollaries. Consideration of the discriminantal arrangements for some particular octagons leads to a previously announced counterexample to the conjecture by Saito [ER2] that the complexified complement of a real free arrangement is aK (�, 1) space.

New lower bounds for Hopcroft's problem

Abstract: We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set ofn points andm hyperplanes in $$\mathbb{R}^d $$ , is any point contained in any hyperplane? We define a general class ofpartitioning algorithms, and show that in the worst case, for allm andn, any such algorithm requires time Ω(n logm + n2/3m2/3 + m logn) in two dimensions, or Ω(n logm + n5/6m1/2 + n1/2m5/6 + m logn) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2O(log*(n+m)) of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was Ω(n logm + m logn). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called amonochromatic cover, and derive lower bounds on its size in the worst case. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover. As a related result, using a straightforward adversary argument, we derive aquadratic lower bound on the complexity of Hopcroft's problem in a surprisingly powerful decision tree model of computation.

Equipartition of mass distributions by hyperplanes

Abstract: We consider the problem of determining the smallest dimensiond=Δ(j, k) such that, for anyj mass distributions inRd, there arek hyperplanes so that each orthant contains a fraction 1/2k of each of the masses. The case Δ(1,2)=2 is very well known. The casek=1 is answered by the ham-sandwich theorem with Δ(j, 1)=j. By using mass distributions on the moment curve the lower bound Δ(j, k)źj(2kź1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Δ(j, k)≤j2kź1. We are able to prove that Δ(j, k)=źj(2kź1/kź for a few pairs (j, k) ((j, 2) forj=3 andj=2n withnź0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Δ(1, 4) (the only case forj=1 in which it is not known whether Δ(1,k)=k); unfortunately the approach fails to give an answer in this case (but we can show Δ(1, 4)≤5).

Inequalities for convex bodies and polar reciprocal lattices inRn II: Application ofK-convexity

Abstract: The paper is a supplement to [2]. LetL be a lattice andU ano-symmetric convex body inRn. The Minkowski functionalźn ofU, the polar bodyU0, the dual latticeL*, the covering radius μ(L, U), and the successive minima źi,i=1, ź,n, are defined in the usual way. Let $$\mathcal{L}_n $$ be the family of all lattices inRn. Given a convex bodyU, we define $$\begin{gathered} mh(U){\text{ }} = {\text{ }}\sup {\text{ }}\max \lambda _i (L,U)\lambda _{n - i + 1} (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n 1 \leqslant i \leqslant n \hfill \\ lh(U){\text{ }} = {\text{ }}\sup {\text{ }}\lambda _1 (L,U) \cdot \mu (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n \hfill \\ \end{gathered} $$ and kh(U) is defined as the smallest positive numbers for which, given arbitrary $$L \in \mathcal{L}_n $$ andxźRn/(L+U), someyźL* with źyźU0źsd(xy,Z) can be found. It is proved $$C_1 n \leqslant jh(U) \leqslant C_2 nK(R_U^n ) \leqslant C_3 n(1 + \log n),$$ , for j=k, l, m, whereC1,C2,C3 are some numerical constants andK(RUn) is theK-convexity constant of the normed space (Rn, źźU). This is an essential strengthening of the bounds obtained in [2]. The bounds for lh(U) are then applied to improve the results of Kannan and Lovasz [5] estimating the lattice width of a convex bodyU by the number of lattice points inU.

Weights on polytopes

Abstract: Theg-theorem describes the possible face-vectors of a simple polytopeP. Much of the author's proof of the necessity of its conditions, while working within the polytope algebra π, in fact only used the spaces of weights onP. Even though this proof was conceptually easier than the original, which employed techniques from algebraic geometry, nevertheless the properties of π which are needed still require some effort to establish, despite a recent simpler approach to π itself. In the earlier paper, doubt was expressed about whether two basic results could be proved directly for weights; later, it appeared that there might also be a possible problem concerning an alternative definition of the product of certain weights. In this paper these questions are settled, in the context of developing an independent theory of an algebra Ω of weights on polytopes. Since the construction of Ω is more approachable than that of π, a yet easier proof of theg-theorem is now available.

A compact piecewise-linear voronoi diagram for convex sites in the plane

Abstract: In the plane the post-office problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a one-dimensional retract of the free space of a robot, are both classically solved by computing a Voronoi diagram. When the sites arek disjoint convex sets we give a compact representation of the Voronoi diagram, usingO (k) line segments, that is sufficient for logarithmic time post-office location queries and motion planning. If these sets are polygons withn total vertices given in standard representations, we compute this diagram optimally in ź (k logn) deterministic time for the Euclidean metric and inO (k logn logm) deterministic time for the convex distance function defined by a convexm-gon.

Mixed volume computation by dynamic lifting applied to polynomial system solving

Abstract: The aim of this paper is to present a flexible approach for the efficient computation of the mixed volume of a tuple of polytopes. In order to compute the mixed volume, a mixed subdivision of the tuple of polytopes is needed, which can be obtained by embedding the polytopes in a higher-dimensional space, i.e., by lifting them. Dynamic lifting is opposed to the static approach. This means that one considers one point at a time and only fixes the value of the lifting function when the point really influences the mixed volume. Conservative lifting functions have been developed for this purpose. This provides us with a deterministic manipulation of the lifting for computing mixed volumes, which rules out randomness conditions. Cost estimates for the algorithm are given. The implications of dynamic lifting on polyhedral homotopy methods for the solution of polynomial systems are investigated and applications are presented.

A construction of inflation rules based on n -fold symmetry

Abstract: In analogy to the well-known tilings of the euclidean plane $$\mathbb{E}^2 $$ by Penrose rhombs (or, to be more precise, to the equivalent tilings by Robinson triangles) we give a construction of an inflation rule based on then-fold symmetryD nfor everyn greater than 3 and not divisible by 3. For givenn the inflation factor η can be any quotient $$\mu _{n,k} : = \sin \left( {k\pi /n} \right)/\sin \left( {\pi /n} \right)$$ as well as any product $$\prod {_{k = 2}^{n/2} \mu _{n,k}^{ak} ,} $$ where $$\alpha _2 ,\alpha _3 ,..., \in \mathbb{N} \cup \left\{ 0 \right\}$$ . The construction is based on the system ofn tangents of the well-known deltoidD, which form angles with the ζ-axis of typevπ/n. None of these tilings permits two linearly independent translations. We conjecture that they have no period at all. For some of them the Fourier transform contains a ℤ-module of Dirac deltas.

Vertical decompositions for triangles in 3-space

Abstract: We prove that, for any constant ɛ>0, the complexity of the vertical decomposition of a set ofn triangles in three-dimensional space isO(n 2+ɛ +K), whereK is the complexity of the arrangement of the triangles. For a single cell the complexity of the vertical decomposition is shown to beO(n 2+ɛ ). These bounds are almost tight in the worst case. We also give a deterministic output-sensitive algorithm for computing the vertical decomposition that runs inO(n 2 logn+V logn) time, whereV is the complexity of the decomposition. The algorithm is reasonably simple (in particular, it tries to perform as much of the computation in two-dimensional spaces as possible) and thus is a good candidate for efficient implementations. The algorithm is extended to compute the vertical decomposition of arrangements ofn algebraic surface patches of constant maximum degree in three-dimensional space in timeO(nλ q (n) logn +V logn), whereV is the combinatorial complexity of the vertical decomposition, λ q (n) is a near-linear function related to Davenport-Schinzel sequences, andq is a constant that depends on the degree of the surface patches and their boundaries. We also present an algorithm with improved running time for the case of triangles which is, however, more complicated than the first algorithm.

Projections of polytopes and the generalized baues conjecture

Abstract: Associated with every projection ź:Pźź(P) of a polytopeP is a partially ordered set of all "locally coherent strings": the families of proper faces ofP that project to valid subdivisions of ź(P), partially ordered by the natural inclusion relation. The "Generalized Baues Conjecture" posed by Billeraet al. [4] asked whether this partially ordered set always has the homotopy type of a sphere of dimension dim(P--dim(ź(P))ź1. We show that this is true in the cases when dim(ź(P))=1 (see[4]) and when dim(P)--dim(ź(P))≤2, but fails in general. For an explicit counterexample we produce a nondegenerate projection of a five-dimensional, simplicial, 2-neighborly polytopeP with 10 vertices and 42 facets to a hexagon ź(P)⊆ź2. The construction of the counterexample is motivated by a geometric analysis of the relation between the fibers in an arbitrary projection of polytopes.

P.L.-Spheres, convex polytopes, and stress

Abstract: We describe here the notion of generalized stress on simplicial complexes, which serves several purposes: it establishes a link between two proofs of the Lower Bound Theorem for simplicial convex polytopes; elucidates some connections between the algebraic tools and the geometric properties of polytopes; leads to an associated natural generalization of infinitesimal motions; behaves well with respect to bistellar operations in the same way that the face ring of a simplicial complex coordinates well with shelling operations, giving rise to a new proof that p.l.-spheres are Cohen-Macaulay; and is dual to the notion of McMullen's weights on simple polytopes which he used to give a simpler, more geometric proof of theg-theorem.

A new technique for analyzing substructures in arrangements of piecewise linear surfaces

Abstract: We present a simple but powerful new probabilistic technique for analyzing the combinatorial complexity of various substructures in arrangements of piecewise-linear surfaces in higher dimensions. We apply the technique (a) to derive new and simpler proofs of the known bounds on the complexity of the lower envelope, of a single cell, or of a zone in arrangements of simplices in higher dimensions, and (b) to obtain improved bounds on the complexity of the vertical decomposition of a single cell in an arrangement of triangles in 3-space, and of several other substructures in such an arrangement (the entire arrangement, all nonconvex cells, and any collection of cells). The latter results also lead to improved algorithms for computing substructures in arrangements of triangles and for translational motion planning in three dimensions.

Accounting for boundary effects in nearest neighbor searching

Abstract: Givenn data points ind-dimensional space, nearest-neighbor searching involves determining the nearest of these data points to a given query point. Most averagecase analyses of nearest-neighbor searching algorithms are made under the simplifying assumption thatd is fixed and thatn is so large relative tod thatboundary effects can be ignored. This means that for any query point the statistical distribution of the data points surrounding it is independent of the location of the query point. However, in many applications of nearest-neighbor searching (such as data compression by vector quantization) this assumption is not met, since the number of data pointsn grows roughly as 2 d .Largely for this reason, the actual performances of many nearest-neighbor algorithms tend to be much better than their theoretical analyses would suggest. We present evidence of why this is the case. We provide an accurate analysis of the number of cells visited in nearest-neighbor searching by the bucketing andk-d tree algorithms. We assumem dpoints uniformly distributed in dimensiond, wherem is a fixed integer ≥2. Further, we assume that distances are measured in theL ∞ metric. Our analysis is tight in the limit asd approaches infinity. Empirical evidence is presented showing that the analysis applies even in low dimensions.

A near-quadratic algorithm for planning the motion of a polygon in a polygonal environment

Abstract: We consider the problem of planning the motion of an arbitraryk-sided polygonal robotB, free to translate and rotate in a polygonal environmentV bounded byn edges. We present an algorithm that constructs a single component of the free configuration space ofB in timeO((kn)2+ź), for any ź>0. This algorithm, combined with some standard techniques in motion planning, yields a solution to the underlying motion-planning problem, within the same running time.

On thecd-variation polynomials of André and simsun permutations

Abstract: We prove a conjecture of Stanley on thecd-index of the semisuspension of the face poset of a simplicial shelling component We give a new signed generalization of Andre permutations, together with a new notion ofcd-variation for signed permutations This generalization not only allows us to compute thecd-index of the face poset of a cube, but also occurs as a natural set of orbit representatives for a signed generalization of the Foata-Strehl commutative group action on the symmetric group From the induction techniques used, it becomes clear that there is more than one way to define classes of permutations andcd-variation such that they allow us to compute thecd-index of the same poset

Triangulations intersect nicely

Abstract: We show that there is a matching between the edges of any two triangulations of a planar point set such that an edge of one triangulation is matched either to the identical edge in the other triangulation or to an edge that crosses it. This theorem also holds for the triangles of the triangulations and in general independence systems. As an application, we give some lower bounds for the minimum-weight triangulation which can be computed in polynomial time by matching and network-flow techniques. We exhibit an easy-to-recognize class of point sets for which the minimum-weight triangulation coincides with the greedy triangulation.

A Probabilistic Analysis of the Power of Arithmetic Filters

Abstract: The assumption of real-number arithmetic, which is at the basis of conventionalgeometric algorithms, has been seriously challenged in recent years,since digital computers do not exhibit such capability. A geometric predicate usually consists of evaluating the sign of some algebraic expression. In most cases, rounded computations yield a reliable result, but sometimes rounded arithmetic introduces errors which may invalidate the algorithms. The rounded arithmetic may produce an incorrect result only if the exact absolute value of the algebraic expression is smaller than some (small) $\varepsilon$, which represents the largest error that may arise in the evaluation of the expression. The threshold $\varepsilon$ depends on the structure of the expression and on the adopted computer arithmetic, assuming that the input operands are error-free. A pair (arithmetic engine,threshold) is an {\em arithmetic filter}. In this paper we develop a general technique for assessing the efficacy of an arithmetic filter. The analysis consists of evaluating both the threshold and the probability of failure of the filter. To exemplify the approach, under the assumption that the input points be chosen randomly in a unit ball or unit cube with uniform density, we analyze the two important predicates "which-side'''' and ''insphere''''. We show that the probability that the absolute values of the corresponding determinants be no larger than some positive value $V$, with emphasis on small $V$, is $\Theta(V)$ for the which-side predicate, while for the insphere predicate it is $\Theta(V^{\frac{2}{3}})$ in dimension 1, $O(V^{\frac{1}{2}})$ in dimension 2, and $O(V^{\frac{1}{2}}\ln \frac{1}{V})$ in higher dimensions. Constants are small, and are given in the paper.

A near-linear algorithm for the planar segment-center problem

Abstract: LetP be a set ofn points in the plane and lete be a segment of fixed length. The segment-center problem is to find a placement ofe (allowing translation and rotation) which minimizes the maximum euclidean distance frome to the points ofP. We present an algorithm that solves the problem in timeO(n 1+e), for any e>0, improving the previous solution of Agarwalet al. [3] by nearly a factor ofO(n).

Nonregular triangulations of products of simplices

Abstract: We exhibit a nonregular triangulation for the product of two tetrahedra. This answers a question by Gel'fand, Kapranov, and Zelevinsky. We also give a complete classification of the symmetry classes of regular triangulations of ź2ן3. Our nonregular triangulation of ź3ן3 can be extended to a nonregular triangulation of the six-dimensional cube. The four-dimensional cube is the smallest cube with a nonregular triangulation.

A Short Proof of an Interesting Helly-Type Theorem*

Abstract: We give a short proof of the theorem that any family of subsets ofR d , with the property that the intersection of any nonempty finite subfamily can be represented as the disjoint union of at mostk closed convex sets, has Helly number at mostk(d+1).

Signable posets and partitionable simplicial complexes

Abstract: The notion of apartitionable simplicial complex is extended to that of asignable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone fans, and oriented matroid polytopes, are all signable. Each of these classes, which are believed to be mutually incomparable, strictly contains the class of convex polytopes. A general sufficient condition, termedtotal signability, for a simplicial complex to satisfy McMullen's Upper Bound Theorem on the numbers of faces, is provided. The simplicial members of each of the three classes above are concluded to be partitionable and to satisfy the upper bound theorem. The computational complexity of face enumeration and of deciding partitionability is discussed. It is shown that under a suitable presentation, the face numbers of a signable simplicial complex can be efficiently computed. In particular, the face numbers of simplicial fans can be computed in polynomial time, extending the analogous statement for convex polytopes.

Computation of betti numbers of monomial ideals associated with cyclic polytopes

Abstract: We give a combinatorial formula for the Betti numbers which appear in a minimal free resolution of the Stanley-Reisner ringk[Δ(P)]=A/IΔ(P) of the boundary complex Δ(P) of an odd-dimensional cyclic polytopePover a fieldk. A corollary to the formula is that the Betti number sequence ofk[Δ(P)] is unimodal and does not depend on the base fieldk.

Extremal theory for convex matchings in convex geometric graphs

Abstract: A convex geometric graphG of ordern consists of the set of vertices of a plane convexn-gonP together with some edges, and/or diagonals ofP as edges. CallG 1-free ifG does not havel disjoint edges in convex position.

The maximum number of odd integral distances between points in the plane

Abstract: Four points in the plane with pairwise odd integral distances do not exist. The maximum number of odd distances betweenn points in the plane is proved to ben2/3+r(r-3)/6 for alln, wherer=1,2,3 andnźr (mod 3). This solves a recently stated problem of Erdos.

Lower bounds for the complexity of the graph of the Hausdorff distance as a function of transformation

Abstract: The Hausdorff distance is a measure defined between two sets in some metric space. This paper investigates how the Hausdorff distance changes as one set is transformed by some transformation group. Algorithms to find the minimum distance as one set is transformed have been described, but few lower bounds are known. We consider the complexity of the graph of the Hausdorff distance as a function of transformation, and exhibit some constructions that give lower bounds for this complexity. We exhibit lower-bound constructions for both sets of points in the plane, and sets of points and line segments; we consider the graph of the directed Hausdorff distance under translation, rigid motion, translation and scaling, and affine transformation. Many of the results can also be extended to the undirected Hausdorff distance. These lower bounds are for the complexity of the graph of the Hausdorff distance, and thus do not necessarily bound algorithms that search this graph; however, they do give an indication of how complex the search may be.

On delaunay oriented matroids for convex distance functions

Abstract: For any finite point setS inEd, an oriented matroid DOM (S) can be defined in terms of howS is partitioned by Euclidean hyperspheres. This oriented matroid is related to the Delaunay triangulation ofS and is realizable, because of thelifting property of Delaunay triangulations. We prove that the same construction of aDelaunay oriented matroid can be performed with respect to any smooth, strictly convex distance function in the planeE2 (Theorem 3.5). For these distances, the existence of a Delaunay oriented matroid cannot follow from a lifting property, because Delaunay triangulations might be nonregular (Theorem 4.2(i). This is related to the fact that the Delaunay oriented matroid can be nonrealizable (Theorem 4.2(ii).