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Showing papers in "Discrete and Computational Geometry in 1997"


Journal ArticleDOI
TL;DR: This article offers fast software-level algorithms for exact addition and multiplication of arbitrary precision floating-point values and proposes a technique for adaptive precision arithmetic that can often speed these algorithms when they are used to perform multiprecision calculations that do not always require exact arithmetic, but must satisfy some error bound.
Abstract: Exact computer arithmetic has a variety of uses, including the robust implementation of geometric algorithms. This article has three purposes. The first is to offer fast software-level algorithms for exact addition and multiplication of arbitrary precision floating-point values. The second is to propose a technique for adaptive precision arithmetic that can often speed these algorithms when they are used to perform multiprecision calculations that do not always require exact arithmetic, but must satisfy some error bound. The third is to use these techniques to develop implementations of several common geometric calculations whose required degree of accuracy depends on their inputs. These robust geometric predicates are adaptive; their running time depends on the degree of uncertainty of the result, and is usually small.

588 citations


Journal ArticleDOI
TL;DR: Borders are derived on the relationship between size and depth for the components of a nearest-neighbor graph and some probabilistic properties of the k-nearest-neIGHbors graph for a random set of points are proved.
Abstract: The "nearest-neighbor" relation, or more generally the "k-nearest-neighbors" relation, defined for a set of points in a metric space, has found many uses in computational geometry and clustering analysis, yet surprisingly little is known about some of its basic properties. In this paper we consider some natural questions that are motivated by geometric embedding problems. We derive bounds on the relationship between size and depth for the components of a nearest-neighbor graph and prove some probabilistic properties of the k-nearest-neighbors graph for a random set of points.

211 citations


Journal ArticleDOI
TL;DR: In this article, the first step of a program to prove the Kepler conjecture on sphere packings is described, and it is shown that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.
Abstract: We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay star. We conjecture that the score of every Delaunay star is at most the score of the stars in the face-centered cubic and hexagonal close packings. This conjecture implies the Kepler conjecture. To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.

140 citations


Journal ArticleDOI
TL;DR: An algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices using O(n) space and requires O (n+h^2\log n) time is given.
Abstract: We give an algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices. The algorithm uses O(n) space and requires $O(n+h^2\log n)$ time.

123 citations


Journal ArticleDOI
TL;DR: An algorithm for computing the 2-center of a set S of n points in the plane (that is, a pair of congruent disks of smallest radius whose union covers S), improving the previous O(n^2\log n) -time algorithm.
Abstract: We present an $O(n\log^{9}n)$ -time algorithm for computing the 2-center of a set S of n points in the plane (that is, a pair of congruent disks of smallest radius whose union covers S), improving the previous $O(n^2\log n)$ -time algorithm of [10].

119 citations


Journal ArticleDOI
TL;DR: Computational methods to find good packings of more than 20 circles are discussed and a new packing of 49 circles settles the proof that when n is a square number, the best packing is the square lattice exactly when n≤ 36.
Abstract: The problem of maximizing the radius of n equal circles that can be packed into a given square is a well-known geometrical problem. An equivalent problem is to find the largest distance d, such that n points can be placed into the square with all mutual distances at least d. Recently, all optimal packings of at most 20 circles in a square were exactly determined. In this paper, computational methods to find good packings of more than 20 circles are discussed. The best packings found with up to 50 circles are displayed. A new packing of 49 circles settles the proof that when n is a square number, the best packing is the square lattice exactly when n≤ 36.

101 citations


Journal ArticleDOI
TL;DR: In this article, the authors introduced m.k/D maxf.kC 1/C 1, m.ki 1/!=2; 1g packings of h.k /D 3k.
Abstract: For each k‚ 1 and corresponding hexagonal number h.k/D 3k.kC 1/C 1, we introduce m.k/D maxf.ki 1/!=2; 1g packings of h.k/ equal disks inside a circle which we call the curved hexagonalpackings. The curved hexagonal packing of 7 disks (kD 1, m.1/D 1) is well known and one of the 19 disks (kD 2, m.2/D 1) has been previously conjectured to be optimal. New curved hexagonal packings of 37, 61, and 91 disks (kD 3, 4, and 5, m.3/D 1, m.4/D 3, and m.5/D 12) were the densest we obtained on a computer using a so-called "billiards" simulation algorithm. A curved hexagonal packing pattern is invariant under a 60 - rotation. For k !1, the density (covering fraction) of curved hexagonal packings tends to … 2 =12. The limit is smaller than the density of the known optimum disk packing in the infinite plane. We found disk configurations that are denser than curved hexagonal packings for 127, 169, and 217 disks (kD 6, 7, and 8). In addition to new packings for h.k/ disks, we present the new packings we found for h.k/C 1 and h.k/i 1 disks for k up to 5, i.e., for 36, 38, 60, 62, 90, and 92 disks. The additional packings show the "tightness" of the curved hexagonal pattern for k• 5: deleting a disk does not change the optimum packing and its quality significantly, but adding a disk causes a substantial rearrangement in the optimum packing and substantially decreases the quality.

90 citations


Journal ArticleDOI
TL;DR: For any 2-coloring of the segments determined by n points in general position in the plane, at least one of the color classes contains a non-self-intersecting spanning tree.
Abstract: For any 2-coloring of the ${n \choose 2}$ segments determined by n points in general position in the plane, at least one of the color classes contains a non-self-intersecting spanning tree. Under the same assumptions, we also prove that there exist $\lfloor (n+1)/3 \rfloor$ pairwise disjoint segments of the same color, and this bound is tight. The above theorems were conjectured by Bialostocki and Dierker. Furthermore, improving an earlier result of Larman et al., we construct a family of m segments in the plane, which has no more than $m^{\log 4/\log 27}$ members that are either pairwise disjoint or pairwise crossing. Finally, we discuss some related problems and generalizations.

77 citations


Journal ArticleDOI
Pavel Valtr1
TL;DR: It is shown that, for any fixed k ≥ 3, any geometric graph on n vertices with no k pairwise parallel edges contains at most O(n) edges, and any geometric graphs on n n verticeswith no k -1 pairwise crossing edges containing at mostO(n log n) edges.
Abstract: A {\em geometric graph\/} is a graph $G=(V,E)$ drawn in the plane so that the vertex set $V$ consists of points in general position and the edge set $E$ consists of straight line segments between points of $V$. Two edges of a geometric graph are said to be {\em parallel\/}, if they are opposite sides of a convex quadrilateral. In this paper we show that, for any fixed $k\ge3$, any geometric graph on $n$ vertices with no $k$ pairwise parallel edges contains at most $O(n)$ edges, and any geometric graph on $n$ vertices with no $k$ pairwise crossing edges contains at most $O(n\log n)$ edges. We also prove a conjecture of Kupitz that any geometric graph on $n$ vertices with no pair of parallel edges contains at most $2n-2$ edges.

73 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that a thrackle has at most twice as many edges as vertices, and that any two distinct arcs either meet at a common vertex or cross at exactly one point interior to both arcs.
Abstract: A thrackle is a graph drawn in the plane so that its edges are represented by Jordan arcs and any two distinct arcs either meet at exactly one common vertex or cross at exactly one point interior to both arcs. About 40 years ago, J. H. Conway conjectured that the number of edges of a thrackle cannot exceed the number of its vertices. We show that a thrackle has at most twice as many edges as vertices. Some related problems and generalizations are also considered.

70 citations


Journal ArticleDOI
TL;DR: The stable mixed volume of the Newton polytopes of a polynomial system is defined and shown to equal (genetically) the number of zeros in affine space Cn.
Abstract: The stable mixed volume of the Newton polytopes of a polynomial system is defined and shown to equal (genetically) the number of zeros in affine space Cn. This result refines earlier bounds by Rojas, Li, and Wang [5], [7], [8]. The homotopies in [4], [9], and [10] extend naturally to a computation of all isolated zeros in Cn

Journal ArticleDOI
TL;DR: A method is presented which reduces a family of problems in combinatorial geometry to purely combinatorsial questions about hypergraphs and proves the tight bound on homogeneous d-intervals, which is linear in k.
Abstract: We present a method which reduces a family of problems in combinatorial geometry (concerning multiple intervals) to purely combinatorial questions about hypergraphs. The main tool is the Borsuk—Ulam theorem together with one of its extensions.

Book ChapterDOI
TL;DR: The second step of a program to prove the Kepler conjecture on sphere packings leads to a decomposition of R3 into polyhedra, which has density at most that of a regular tetrahedron.
Abstract: An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition of ℝ3 into polyhedra. The polyhedra are divided into two classes. The first class of polyhedra, called quasi-regular tetrahedra, have density at most that of a regular tetrahedron. The polyhedra in the remaining class have density at most that of a regular octahedron (about 0.7209).

Journal ArticleDOI
TL;DR: This work proves bounds on the number of halving hyperplanes under the condition that the ratio of largest over smallest distance between any two points is at most δn(1/d), δ some constant.
Abstract: A halving hyperplane of a set S of n points in źd contains d affinely independent points of S so that equally many of the points off the hyperplane lie in each of the two halfspaces. We prove bounds on the number of halving hyperplanes under the condition that the ratio of largest over smallest distance between any two points is at most źn(1/d), ź some constant. Such a set S is called dense. In d = 2 dimensions the number of halving lines for a dense set can be as much as Ω (n log n), and it cannot exceed O (n5/4/log*n). The upper bound improves over the current best bound of O(n3/2/log*n) which holds more generally without any density assumption. In d = 3 dimensions we show that O(n7/3) is an upper bound on the number of halving planes for a dense set. The proof is based on a metric argument that can be extended to d ź 4 dimensions, where it leads to O(nd-2/d) as an upper bound for the number of halving hyperplanes.

Journal ArticleDOI
TL;DR: This is the first algorithm within a polylogarithmic factor of optimal $O(n \log f + f)$ time over the whole range of f to be given for output-sensitive construction of an f-face convex hull of a set of n points in general position in E4.
Abstract: In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E 4 . Our algorithm runs in \(O((n+f)\log^2 f)\) time and uses O(n+f) space. This is the first algorithm within a polylogarithmic factor of optimal \(O(n \log f + f)\) time over the whole range of f. By a standard lifting map, we obtain output-sensitive algorithms for the Voronoi diagram or Delaunay triangulation in E 3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the ``ultimate convex hull algorithm'' of Kirkpatrick and Seidel in E 2 and also leads to improved output-sensitive results on constructing convex hulls in E d for any even constant d > 4.

Journal ArticleDOI
TL;DR: This paper establishes four necessary conditions for recognizing visibility graphs of simple polygons and conjecture that these conditions are sufficient and presents an 0(n2)-time algorithm for testing them.
Abstract: In this paper we establish four necessary conditions for recognizing visibility graphs of simple polygons and conjecture that these conditions are sufficient. We present an 0(n2)-time algorithm for testing the first and second necessary conditions and leave it open whether the third and fourth necessary conditions can be tested in polynomial time. We also show that visibility graphs of simple polygons do not possess the characteristics of a few special classes of graphs

Journal ArticleDOI
TL;DR: It is shown that the permanent of an $n \times n$ nonnegative matrix and the mixed volume of n ellipsoids in Rn can be computed within a 2O(n) factor by probabilistic polynomial time algorithms.
Abstract: We construct a probabilistic polynomial time algorithm that computes the mixed discriminant of given n positive definite $n \times n$ matrices within a 2 O(n) factor. As a corollary, we show that the permanent of an $n \times n$ nonnegative matrix and the mixed volume of n ellipsoids in R n can be computed within a 2 O(n) factor by probabilistic polynomial time algorithms. Since every convex body can be approximated by an ellipsoid, the last algorithm can be used for approximating in polynomial time the mixed volume of n convex bodies in R n within a factor n O(n) .

Journal ArticleDOI
TL;DR: The paper gives a complete classification of the discrete regular polytopes in ordinary space.
Abstract: The three aims of this paper are to obtain the proof by Dress of the completeness of the enumeration of the Grunbaum—Dress polyhedra (that is, the regular apeirohedra, or apeirotopes of rank 3) in ordinary space E 3 in a quicker and more perspicuous way, to give presentations of those of their symmetry groups which are affinely irreducible, and to describe all the discrete regular apeirotopes of rank 4 in E 3 . The paper gives a complete classification of the discrete regular polytopes in ordinary space.

Journal ArticleDOI
Xin He1
TL;DR: If a 4-connected plane graph G has at least four vertices on its external face, then G can be embedded on a grid of size W × H such that W + H ≤n, W ≤ (n + 3)/2 and H ≤ 2(n - 1)/3, where n is the number of vertices of G.
Abstract: A straight line grid embedding of a plane graph G is a drawing of G such that the vertices are drawn at grid points and the edges are drawn as nonintersecting straight line segments. In this paper we show that if a 4-connected plane graph G has at least four vertices on its external face, then G can be embedded on a grid of size W × H such that W + H ≤n, W ≤ (n + 3)/2 and H ≤ 2(n - 1)/3, where n is the number of vertices of G. Such an embedding can be computed in linear time.

Journal ArticleDOI
TL;DR: This encoding is used to improve the upper bound on the number of arrangements of n pseudolines to 2^{0.6974\cdot n^2} and it is shown that the maximal number of halving lines of 10 point in the plane is 13.
Abstract: Given a simple arrangement of n pseudolines in the Euclidean plane, associate with line i the list σ i of the lines crossing i in the order of the crossings on line i. $\sigma_i=(\sigma^i_1,\sigma^i_2,\ldots,\sigma^i_{n-1})$ is a permutation of $\{1,\ldots,n\} - \{i\}$ . The vector (σ 1 ,σ 2 , ...,σ_n) is an encoding for the arrangement. Define $\tau^i_j = 1$ if $\sigma^i_j > i$ and $\tau^i_j = 0$ , otherwise. Let $\tau_i=(\tau^i_1,\tau^i_2,\ldots,\tau^i_{n-1})$ , we show that the vector (τ 1 , τ 2 , ... , τ_n) is already an encoding. We use this encoding to improve the upper bound on the number of arrangements of n pseudolines to $2^{0.6974\cdot n^2}$ . Moreover, we have enumerated arrangements with 10 pseudolines. As a byproduct we determine their exact number and we can show that the maximal number of halving lines of 10 point in the plane is 13.

Journal Article
TL;DR: It is shown that in the worst case, /spl Omega/(n/Sup d/) sidedness queries are required to determine whether a set of n points in R/sup d/ is affinely degenerate, i.e., whether it contains d+1 points on a common hyperplane.
Abstract: We show that in the worst case, /spl Omega/(n/sup d/) sidedness queries are required to determine whether a set of n points in R/sup d/ is affinely degenerate, i.e., whether it contains d+1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on the explicit construction of a point set containing /spl Omega/(n/sup d/) \"collapsible\" simplices, any one of which can be made degenerate without changing the orientation of any other simplex. As an immediate corollary, we have an /spl Omega/(n/sup d/) lower bound on the number of sidedness queries required to determine the order type of a set of n points in R/sup d/. Using similar techniques, we also show that /spl Omega/(n/sup d+1/) in-sphere queries are required to decide the existence of spherical degeneracies in a set of n points in R/sup d/. >

Journal ArticleDOI
TL;DR: This paper proves three lower bounds for variants of the following rangesearching problem: Given n weighted points inRd andn axis-parallel boxes, the sum of the weights within each box is computed.
Abstract: This paper proves three lower bounds for variants of the following rangesearching problem: Given n weighted points inR d andn axis-parallel boxes, compute the sum of the weights within each box: (1) if both additions and subtractions are allowed, we prove that Ω(n log logn) is a lower bound on the number of arithmetic operations; (2) if subtractions are disallowed the lower bound becomes Ω(n(logn/loglogn)d-1), which is nearly optimal; (3) finally, for the case where boxes are replaced by simplices, we establish a quasi-optimal lower bound of Ω(n2-2/(d+1))/polylog(n).

Journal ArticleDOI
TL;DR: The triply exponential upper bound of T. Bisztriczky and G. Fejes Tóth is improved by showing that P3(n) < 16n, the maximum size of a family F with the property that any k members of F are in convex position, but no n are.
Abstract: Let ${\cal F}$ denote a family of pairwise disjoint convex sets in the plane. ${\cal F}$ is said to be in {\em convex position}, if none of its members is contained in the convex hull of the union of the others. For any fixed $k\geq 3$, we estimate $P_k(n)$, the maximum size of a family ${\cal F}$ with the property that any $k$ members of ${\cal F}$ are in convex position, but no $n$ are. In particular, for $k=3$, we improve the triply exponential upper bound of T. Bisztriczky and G. Fejes T\''oth by showing that $P_3(n)>16^n$.

Journal ArticleDOI
TL;DR: Lower and upper bounds for the maximal number of facets of a d-dimensional 0/1-polytope and for the maximum number of vertices that can appear in a two-dimensional projection ( ``shadow'') of such a polytope are provided.
Abstract: We provide lower and upper bounds for the maximal number of facets of a d-dimensional 0/1-polytope, and for the maximal number of vertices that can appear in a two-dimensional projection (``shadow'') of such a polytope.

Journal ArticleDOI
TL;DR: The δ-relativeε-approximation method, developed for the CRCW variant of the PRAM parallel computation model, can be easily implemented to run in $O(\log n(\log\log n)^{d-1})$ time using linear work on an EREW PRAM.
Abstract: We give fast and efficient methods for constructing e-nets and e-approximations for range spaces with bounded VC-exponent. These combinatorial structures have wide applicability to geometric partitioning problems, which are often used in divide-and-conquer constructions in computational geometry algorithms. In addition, we introduce a new deterministic set approximation for range spaces with bounded VC-exponent, which we call the δ-relativee-approximation, and we show how such approximations can be efficiently constructed in parallel. To demonstrate the utility of these constructions we show how they can be used to solve the linear programming problem in \({\Bbb R}^d\) deterministically in \(O((\log\log n)^d)\) time using linear work in the PRAM model of computation, for any fixed constant d. Our method is developed for the CRCW variant of the PRAM parallel computation model, and can be easily implemented to run in \(O(\log n(\log\log n)^{d-1})\) time using linear work on an EREW PRAM.

Journal ArticleDOI
TL;DR: An O(n4)-time and O( n2)-space algorithm that computes a subgraph of the minimum weight triangulation (MWT) of a general point set called the LMT-skeleton, which is able to compute the MWT of substantially larger random point sets than have previously been computed.
Abstract: We present an O(n 4 )-time and O(n 2 )-space algorithm that computes a subgraph of the minimum weight triangulation (MWT) of a general point set. The algorithm works by finding a collection of edges guaranteed to be in any locally minimal triangulation. We call this subgraph the LMT-skeleton. We also give a variant called the modified LMT-skeleton that is both a more complete subgraph of the MWT and is faster to compute requiring only O(n 2 ) time and O(n) space in the expected case for uniform distributions. Several experimental implementations of both approaches have shown that for moderate-sized point sets (up to 350 points^1) the skeletons are connected, enabling an efficient completion of the exact MWT. We are thus able to compute the MWT of substantially larger random point sets than have previously been computed. ^1Though in this paper we summarize some empirical findings for input sets of up to 350 points, a variant of the algorithm has been implemented and tested on up to 40,000 points producing connected subgraphs [2].

Journal ArticleDOI
TL;DR: A new type of Voronoi diagram, called the “closest covered set diagram” based on a convex distance function, is presented, which proves that any collection Pn of points on the plane contains two such that any circle containing them contains n/c elements of Pn, c a constant.
Abstract: A known result in combinatorial geometry states that any collection Pn of points on the plane contains two such that any circle containing them contains n/c elements of Pn, c a constant. We prove: Let ź be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements Si, Sj of ź such that any set Sź homothetic to S that contains them contains n/c elements of ź, c a constant (S is homothetic to S if 5' = źS + v, where ź is a real number greater than 0 and v is a vector of ź2). Our proof method is based on a new type of Voronoi diagram, called the "closest covered set diagram" based on a convex distance function. We also prove that our result does not generalize to higher dimensions; we construct a set ź of n disjoint convex sets in ź3)3 such that for any nonempty subset źHh of ź there is a sphere SH containing all the elements of źH, and no other element of ź.

Journal ArticleDOI
TL;DR: It is shown that every simplicial d-polytope with d+4 vertices is a quotient of a neighborly (2d+4)-polytopes with 2d+8 vertices, using the technique of affine Gale diagrams.
Abstract: We show that every simplicial d-polytope with d+4 vertices is a quotient of a neighborly (2d+4)-polytope with 2d+8 vertices, using the technique of affine Gale diagrams. The result is extended to matroid polytopes.

Journal ArticleDOI
TL;DR: It is proved that for any d, k ≥ 1 there are numbers q = q(d,k) and h = h(d-dimensional Euclidean space) such that the Helly number of ${\cal K}$ is at most h.
Abstract: We prove that for any d, k ≥ 1 there are numbers q = q(d,k) and h = h(d,k) such that the following holds: Let ${\cal K}$ be a family of subsets of the d-dimensional Euclidean space, such that the intersection of any subfamily of ${\cal K}$ consisting of at most q sets can be expressed as a union of at most k convex sets. Then the Helly number of ${\cal K}$ is at most h. We also obtain topological generalizations of some cases of this result. The main result was independently obtained by Alon and Kalai, by a different method.

Journal ArticleDOI
TL;DR: Efficient parametrized algorithms are presented which reach a performance ratio of 11/8 + ɚ for any ɛ > 0 in time and a ratio of $11/8 plus \log\log n / \log n in time O(n \cdot \log^3 n)$ .
Abstract: The classical Steiner tree problem requires a shortest tree spanning a given vertex subset within a graph G=(V,E). An important variant is the Steiner tree problem in rectilinear metric. Only recently two algorithms were found which achieve better approximations than the ``traditional'' one with a factor of 3/2. These algorithms with an approximation ratio of 11/8 are quite slow and run in time $O(n^3)$ and $O(n^{5/2})$ . A new simple implementation reduces the time to $O(n^{3/2})$ . As our main result we present efficient parametrized algorithms which reach a performance ratio of 11/8 + ɛ for any ɛ > 0 in time $O(n \cdot \log^2 n)$ , and a ratio of $11/8 + \log\log n /\log n$ in time $O(n \cdot \log^3 n)$ .