Showing papers in "Discrete and Computational Geometry in 1998"

Improved Bounds for Planar k -Sets and Related Problems

Abstract: We prove an O(n(k+1)1/3) upper bound for planar k -sets. This is the first considerable improvement on this bound after its early solution approximately 27 years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of k -levels in the arrangement of line segments, k convex polygons in the union of n lines, parametric minimum spanning trees, and parametric matroids in general. 26 June, 1998 Editors-in-Chief: la href=../edboard.html#chiefslJacob E. Goodman, Richard Pollackl/al 19n3p373.pdf yes no no yes

Nonperiodicity implies unique composition for self-similar translationally finite Tilings

Abstract: Let T be a translationally finite self-similar tiling of Rd We prove that if T is nonperiodic, then it has the unique composition property More generally, T has the unique composition property modulo the group of its translation symmetries

Primal—Dual Methods for Vertex and Facet Enumeration

Abstract: Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (resp. vertex) to the vertex (resp. halfspace) representation is called vertex enumeration (resp. facet enumeration ). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (resp. facet) enumeration problem is the facet (resp. vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primal—dual algorithms use a solution to the easy direction as an oracle to help solve the seemingly hard direction.

Voronoi Diagrams in Higher Dimensions under Certain Polyhedral Distance Functions

Abstract: The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Specifically, if S is a set of n points in general position in R d , the maximum complexity of its Voronoi diagram under the L ∞ metric, and also under a simplicial distance function, are both shown to be $\Theta(n^{\lceil d/2 \rceil})$ . The upper bound for the case of the L ∞ metric follows from a new upper bound, also proved in this paper, on the maximum complexity of the union of n axis-parallel hypercubes in R d . This complexity is $\Theta(n^{\left\lceil d/2 \right\rceil})$ , for d ≥ 1 , and it improves to $\Theta(n^{\left\lfloor d/2 \right\rfloor})$ , for d ≥ 2 , if all the hypercubes have the same size. Under the L 1 metric, the maximum complexity of the Voronoi diagram of a set of n points in general position in R 3 is shown to be $\Theta(n^2)$ . We also show that the general position assumption is essential, and give examples where the complexity of the diagram increases significantly when the points are in degenerate configurations. (This increase does not occur with an appropriate modification of the diagram definition.) Finally, on-line algorithms are proposed for computing the Voronoi diagram of n points in R d under a simplicial or L ∞ distance function. Their expected randomized complexities are $O(n \log n + n ^{\left\lceil d/2 \right\rceil})$ for simplicial diagrams and $O(n ^{\left\lceil d/2 \right\rceil} \log ^{d-1} n)$ for L ∞ -diagrams.

Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs

Abstract: The main question discussed in this paper is how well a finite metric space of size n can be embedded into a graph with certain topological restrictions

Approximate nearest neighbor queries revisited

Abstract: This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in d -dimensional Euclidean space. For any fixed constant d , a data structure with O( $\varepsilon$ (1-d)/2 n log n) preprocessing time and O( $\varepsilon$ (1-d)/2 log n) query time achieves an approximation factor 1+ $\varepsilon$ for any given 0 < $\varepsilon$ < 1; a variant reduces the $\varepsilon$ -dependence by a factor of $\varepsilon$ -1/2 . For any arbitrary d , a data structure with O(d 2 n log n) preprocessing time and O(d 2 log n) query time achieves an approximation factor O(d 3/2 ) . Applications to various proximity problems are discussed.

The Nature and Meaning of Perturbations in Geometric Computing

Abstract: This paper addresses some fundamental questions concerning perturbations as they are used in computational geometry. How does one define them? What does it mean to compute with them? How can one compute with them? Is it sensible to use them? We define perturbations to be curves, point out that computing with them amounts to computing with limits, and (re)derive some methods of computing with such limits automatically. In principle, a line can always be used as a perturbation curve. We discuss a generic method for choosing such a line that is applicable in many situations.

The Volume as a Metric Invariant of Polyhedra

Abstract: It is proved that in R3 the volume of any polyhedron is a root of some polynomial with coefficients depending only on the combinatorial structure and the metric of the polyhedron. As a consequence, we have a proof of the ``bellows conjecture'' affirming the invariance of the volume of a flexible polyhedron in the process of its flexion.

Distribution of Distances and Triangles in a Point Set and Algorithms for Computing the Largest Common Point Sets

Abstract: This paper considers the following problem, which we call the largest common point set problem (LCP): given two point sets P and Q in the Euclidean plane, find a subset of P with the maximum cardinality that is congruent to some subset of Q . We introduce a combinatorial-geometric quantity λ(P, Q) , which we call the inner product of the distance-multiplicity vectors of P and Q , show its relevance to the complexity of various algorithms for LCP, and give a nontrivial upper bound on λ(P, Q) . We generalize this notion to higher dimensions, give some upper bounds on the quantity, and apply them to algorithms for LCP in higher dimensions. Along the way, we prove a new upper bound on the number of congruent triangles in a point set in four-dimensional space.

Shelling Polyhedral 3-Balls and 4-Polytopes

Abstract: There is a long history of constructions of nonshellable triangulations of three-dimensional (topological) balls. This paper gives a survey of these constructions, including Furch's 1924 construction using knotted curves, which also appears in Bing's 1962 survey of combinatorial approaches to the Poincare conjecture, Newman's 1926 explicit example, and M. E. Rudin's 1958 nonshellable triangulation of a tetrahedron with only 14 vertices (all on the boundary) and 41 facets. Here an (extremely simple) new example is presented: a nonshellable simplicial three-dimensional ball with only 10 vertices and 21 facets. It is further shown that shellings of simplicial 3 -balls and 4 -polytopes can ``get stuck'': simplicial 4 -polytopes are not in general ``extendably shellable.'' Our constructions imply that a Delaunay triangulation algorithm of Beichl and Sullivan, which proceeds along a shelling of a Delaunay triangulation, can get stuck in the three-dimensional version: for example, this may happen if the shelling follows a knotted curve.

On Levels in Arrangements of Lines, Segments, Planes, and Triangles%

Abstract: We consider the problem of bounding the complexity of the k th level in an arrangement of n curves or surfaces, a problem dual to, and an extension of, the well-known k-set problem Among other results, we prove a new bound, O(nk 5/3 ) , on the complexity of the k th level in an arrangement of n planes in R 3 , or on the number of k -sets in a set of n points in three dimensions, and we show that the complexity of the k th level in an arrangement of n line segments in the plane is $O(n\sqrt{k}\alpha(n/k))$ , and that the complexity of the k th level in an arrangement of n triangles in 3-space is O(n 2 k 5/6 α(n/k)) 26 June, 1998 Editors-in-Chief: la href=/edboardhtml#chiefslJacob E Goodman, Richard Pollackl/al 19n3p315pdf yes no no yes

On Functional Separately Convex Hulls

Abstract: Let D be a set of vectors in R d . A function f: R d → R is called D-convex if its restriction to each line parallel to a nonzero vector of D is a convex function. For a set A⊆ R d , the functional D-convex hull of A, denoted by co D (A) , is the intersection of the zero sets of all nonnegative D -convex functions that are 0 on A . We prove some results concerning the structure of functional D -convex hulls, e.g., a Krein—Milman-type theorem and a result on separation of connected components. We give a polynomial-time algorithm for computing co D (A) for a finite point set A (in any fixed dimension) in the case of D being a basis of R d (the case of separate convexity). This research is primarily motivated by questions concerning the so-called rank-one convexity, which is a particular case of D -convexity and is important in the theory of systems of nonlinear partial differential equations and in mathematical modeling of microstructures in solids. As a direct contribution to the study of rank-one convexity, we construct a configuration of 20 symmetric 2 x 2 matrices in a general (stable) position with a nontrivial functionally rank-one convex hull (answering a question of K. Zhang on the existence of higher-dimensional nontrivial configurations of points and matrices).

A Positive Fraction Erdos - Szekeres Theorem

Abstract: We prove a fractional version of the Erdős—Szekeres theorem: for any k there is a constant c k > 0 such that any sufficiently large finite set X⊂R 2 contains k subsets Y 1 , ... ,Y k , each of size ≥ c k |X| , such that every set {y 1 ,...,y k } with y i e Y i is in convex position. The main tool is a lemma stating that any finite set X⊂R d contains ``large'' subsets Y 1 ,...,Y k such that all sets {y 1 ,...,y k } with y i e Y i have the same geometric (order) type. We also prove several related results (e.g., the positive fraction Radon theorem, the positive fraction Tverberg theorem). 26 June, 1998 Editors-in-Chief: la href=../edboard.html#chiefslJacob E. Goodman, Richard Pollackl/al 19n3p335.pdf yes no no yes

An optimal algorithm for closest-pair maintenance

Abstract: Given a set S of n points in {k} -dimensional space, and an Lt metric, the dynamic closest-pair problem is defined as follows: find a closest pair of S after each update of S (the insertion or the deletion of a point). For fixed dimension {k} and fixed metric Lt , we give a data structure of size O(n) that maintains a closest pair of S in O(log n) time per insertion and deletion. The running time of the algorithm is optimal up to a constant factor because Ω (log n) is a lower bound, in an algebraic decision-tree model of computation, on the time complexity of any algorithm that maintains the closest pair (for k=1 ). The algorithm is based on the fair-split tree. The constant factor in the update time is exponential in the dimension. We modify the fair-split tree to reduce it.

How to Cut Pseudoparabolas into Segments

Abstract: Let Γ be a collection of unbounded x -monotone Jordan arcs intersecting at most twice each other, which we call pseudoparabolas, since two axis parallel parabolas intersect at most twice. We investigate how to cut pseudoparabolas into the minimum number of curve segments so that each pair of segments intersect at most once. We give an Ω( n 4/3 ) lower bound and O(n 5/3 ) upper bound on the number of cuts. We give the same bounds for an arrangement of circles. Applying the upper bound, we give an O(n 23/12 ) bound on the complexity of a level in an arrangement of pseudoparabolas, and an O(n 11/6 ) bound on the complexity of a combinatorially concave chain of pseudoparabolas. We also give some upper bounds on the number of transitions of the minimum weight matroid base when the weight of each element changes as a quadratic function of a single parameter.

Note on the Erdos - Szekeres Theorem

Abstract: Let g(n) denote the least integer such that among any g(n) points in general position in the plane there are always n in convex position. In 1935, P. Erdős and G. Szekeres showed that g(n) exists and $2^{n-2}+1\le g(n)\le {2n-4\choose n-2}+1$ . Recently, the upper bound has been slightly improved by Chung and Graham and by Kleitman and Pachter. In this paper we further improve the upper bound to $$g(n)\le {2n-5\choose n-2}+2.$$ 26 June, 1998 Editors-in-Chief: la href=../edboard.html#chiefslJacob E. Goodman, Richard Pollackl/al 19n3p457.pdf yes no no yes

Motion Planning in Environments with Low Obstacle Density

Abstract: We present a simple and efficient paradigm for computing the exact solution of the motion planning problem in environments with a low obstacle density. Such environments frequently occur in practical instances of the motion planning problem. The complexity of the free space for such environments is known to be linear in the number of obstacles. Our paradigm is a new cell decomposition approach to motion planning and exploits properties that follow from the low density of the obstacles in the robot's workspace. These properties allow us to decompose the workspace, subject to some constraints, rather than to decompose the higher-dimensional free configuration space directly. A sequence of uniform steps transforms the workspace decomposition into a free space decomposition of asymptotically the same size. The approach leads to nearly optimal O(n \log n) motion planning algorithms for free-flying robots with any fixed number of degrees of freedom in workspaces with low obstacle density.

Extremal problems for geometric hypergraphs

Abstract: Keywords: geometric hypergraph ; crossing bounds Note: Professor Pach's number: [107]. Also in: Lecture Notes in Computer Science 1178, Springer-Verlag, 1996, 105-114. Reference DCG-ARTICLE-1998-001doi:10.1007/PL00009365 Record created on 2008-11-14, modified on 2017-05-12

Multiregular Point Systems

Abstract: This paper gives several conditions in geometric crystallography which force a structure X in R n to be an ideal crystal An ideal crystal in R n is a finite union of translates of a full-dimensional lattice An (r,R) -set is a discrete set X in R n such that each open ball of radius r contains at most one point of X and each closed ball of radius R contains at least one point of X A multiregular point system X is an (r,R) -set whose points are partitioned into finitely many orbits under the symmetry group Sym(X) of isometries of R n that leave X invariant Every multiregular point system is an ideal crystal and vice versa We present two different types of geometric conditions on a set X that imply that it is a multiregular point system The first is that if X ``looks the same'' when viewed from n+2 points { y i : 1 \leq i \leq n + 2 } , such that one of these points is in the interior of the convex hull of all the others, then X is a multiregular point system The second is a ``local rules'' condition, which asserts that if X is an (r,R) -set and all neighborhoods of X within distance ρ of each x∈X are isometric to one of k given point configurations, and ρ exceeds CRk for C = 2(n 2 +1) log 2 (2R/r+2) , then X is a multiregular point system that has at most k orbits under the action of Sym(X) on R n In particular, ideal crystals have perfect local rules under isometries

Rectification Principles in Additive Number Theory

Abstract: We consider two general principles which allow us to reduce certain additive problems for residue classes modulo a prime to the corresponding problems for integers. 26 June, 1998 Editors-in-Chief: la href=../edboard.html#chiefslJacob E. Goodman, Richard Pollackl/al 19n3p343.pdf yes no no yes

Fitting a set of points by a circle

Abstract: Given a set of points S={p 1 ,. . ., p n } in Euclidean d -dimensional space, we address the problem of computing the d -dimensional annulus of smallest width containing the set. We give a complete characterization of the centers of annuli which are locally minimal in arbitrary dimension and we show that, for d=2 , a locally minimal annulus has two points on the inner circle and two points on the outer circle that interlace anglewise as seen from the center of the annulus. Using this characterization, we show that, given a circular order of the points, there is at most one locally minimal annulus consistent with that order and it can be computed in time O(n log n) using a simple algorithm. Furthermore, when points are in convex position, the problem can be solved in optimal Θ(n) time.

Many polytopes meeting the conjectured Hirsch bound

Abstract: The still open Hirsch conjecture asserts that Δ(d,n) ≤n-d for all n > d ≥ 2 , where Δ (d,n) denotes the maximum edge-diameter of (convex) d -polytopes with n facets. This paper adds to the list of pairs (d,n) that are known to be H -sharp in the sense that Δ (d,n) ≥ n-d . In particular, it is proved that Δ(d,n)≥ n-d for all n > d ≥ 14 .

Largest Placement of One Convex Polygon Inside Another

Abstract: We show that the largest similar copy of a convex polygon P with m edges inside a convex polygon Q with n edges can be computed in O(mn 2 log n) time. We also show that the combinatorial complexity of the space of all similar copies of P inside Q is O(mn 2 ) , and that it can also be computed in O(mn 2 log n) time.

The Discrete 2-Center Problem

Abstract: We present an algorithm for computing the discrete 2-center of a set P of n points in the plane; that is, computing two congruent disks of smallest possible radius, centered at two points of P , whose union covers P . Our algorithm runs in time O(n 4/3 log 5 n) .

Forced Convex n -Gons in the Plane

Abstract: In a seminal paper from 1935, Erdős and Szekeres showed that for each n there exists a least value g(n) such that any subset of g(n) points in the plane in general position must always contain the vertices of a convex n -gon. In particular, they obtained the bounds $$2^{n-2} + 1 \le g(n) \le {{2n-4}\choose{n-2}} +1,$$ which have stood unchanged since then. In this paper we remove the +1 from the upper bound for n ≥ 4 . 26 June, 1998 Editors-in-Chief: la href=../edboard.html#chiefslJacob E. Goodman, Richard Pollackl/al 19n3p367.pdf yes no no yes

On the Volume of the Union of Balls

Abstract: We prove that if some balls in the Euclidean space move continuously in such a way that the distances between their centers decrease, then the volume of their union cannot increase. The proof is based on a formula expressing the derivative of the volume of the union as a linear combination of the derivatives of the distances between the centers with nonnegative coefficients.

Finding Convex Sets Among Points in the Plane

Abstract: Let g(n) denote the least value such that any g(n) points in the plane in general position contain the vertices of a convex n-gon. In 1935, Erdős and Szekeres showed that g(n) exists, and they obtained the bounds 2^(n−2) + 1 ≤ g(n) ≤ (^(2n−4)_(n−2)) + 1. Chung and Graham have recently improved the upper bound by 1; the first improvement since the original Erdős—Szekeres paper. We show that g(n) ≤ (^(2n−4)_(n−2)) + 7 − 2n.

Finite and Uniform Stability of Sphere Packings

Abstract: The main purpose of this paper is to discuss how firm or steady certain known ball packing are, thinking of them as structures. This is closely related to the property of being locally maximally dense. Among other things we show that many of the usual best-known candidates, for the most dense packings with congruent spherical balls, have the property of being uniformly stable, i.e., for a sufficiently small e > 0 every finite rearrangement of the balls of this packing, where no ball is moved more than e , is the identity rearrangement. For example, the lattice packings D d and A d for d ≥ 3 in E d are all uniformly stable. The methods developed here can work for many other packings as well. We also give a construction to show that the densest cubic lattice ball packing in E d for d ≥ 2 is not uniformly stable. A packing of balls is called finitely stable if any finite subfamily of the packing is fixed by its neighbors. If a packing is uniformly stable, then it is finitely stable. On the other hand, the cubic lattice packings mentioned above, which are not uniformly stable, are nevertheless finitely stable.

Geometric Lower Bounds for Parametric Matroid Optimization

Abstract: We relate the sequence of minimum bases of a matroid with linearly varying weights to three problems from combinatorial geometry: k -sets, lower envelopes of line segments, and convex polygons in line arrangements. Using these relations we show new lower bounds on the number of base changes in such sequences: Ω(nr 1/3 ) for a general n -element matroid with rank r , and Ω(mα(n)) for the special case of parametric graph minimum spanning trees. The only previous lower bound was Ω(n log r) for uniform matroids; upper bounds of O(mn 1/2 ) for arbitrary matroids and O(mn 1/2 / log * n) for uniform matroids were also known.

Piercing d-Intervals

Abstract: A (homogeneous) d-interval is a union of d closed intervals in the line Using topological methods, Tardos and Kaiser proved that for any finite collection of d-intervals that contains no k + 1 pairwise disjoint members, there is a set of O(dk) points that intersects each member of the collection Here we give a short, elementary proof of this result A (homogeneous) d-interval is a union of d closed intervals in the line Let H be a finite collection of d-intervals The transversal number τ(H) of H is the minimum number of points that intersect every member of H The matching number ν(H) of H is the maximum number of pairwise disjoint members of H Gyarfas and Lehel [3] proved that τ ≤ O(νd!) and Kaiser [4] proved that τ ≤ O(d2ν) His proof is topological, applies the Borsuk-Ulam theorem and extends and simplifies a result of Tardos [5] Here we give a very short, elementary proof of a similar estimate, using the method of [2] Theorem 1 Let H be a finite family of d-intervals containing no k + 1 pairwise disjoint members Then τ(H) ≤ 2d2k Proof Let H′ be any family of d-intervals obtained from H by possibly duplicating some of its members, and let n denote the cardinality of H′ Note that H′ contains no k + 1 pairwise disjoint members Therefore, by Turan’s Theorem, there are at least n(n − k)/(2k) unordered intersecting pairs of members of H′ Each such intersecting pair supplies at least 2 ordered pairs (p, I), where p is an end point of one of the intervals in a member of H′, I is a different member of H′, and p lies in I Since there are altogether at most 2dn possible choices for p, there is such a point that lies in at least n(n−k) k2dn members of H ′ besides the one in which it is an endpoint of an interval, showing that there is a point that lies in at least n 2dk of the members of H ′ This implies that for any rational weights on the members of H there is a point that lies in at least a fraction 1 2dk of the total weight By the min-max theorem it follows that there is a collection of m points so that each member of H contains at least m/(2dk) of them, and thus contains an interval that contains at least m/(2d2k) of the points Order the points from left to right, and take the set of all points whose rank in this ordering is divisible by dm/(2d2k)e This is a set of at most 2d2k points that intersects each member of H, completing the proof 2 Remarks It may be possible to improve the constant factor in the above proof Kaiser’s estimate is indeed better by roughly a factor of 2; τ(H) ≤ (d2 − d+ 1)ν(H) It will be interesting to decide if the quadratic dependence on d is indeed best possible Higher dimensional extensions are possible, using the techniques in [2], [1] ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel Research supported in part by a USA-Israeli BSF grant and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University Email: noga@mathtauacil