Showing papers in "Discrete and Computational Geometry in 2001"
TL;DR: This article gives a proof of the classical honeycomb conjecture: any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling.
Abstract: This article gives a proof of the classical honeycomb conjecture: any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling.
385 citations
TL;DR: This work shows that Monotone 1-in-3 Satisfiability is NP-complete for planar cubic graphs, and shows NP-completeness for the domino and straight tromino for general areas on the cubic lattice, and for simply connected regions on the four-dimensional hypercubic lattice.
Abstract: It is well known that the question of whether a given finite region can be tiled with a given set of tiles is NP -complete. We show that the same is true for the right tromino and square tetromino on the square lattice, or for the right tromino alone. In the process we show that Monotone 1-in-3 Satisfiability is NP-complete for planar cubic graphs. In higher dimensions we show NP-completeness for the domino and straight tromino for general regions on the cubic lattice, and for simply connected regions on the four-dimensional hypercubic lattice.
153 citations
TL;DR: It is shown that every set of n points in the plane has an element from which there are at least cn6/7 other elements at distinct distances, where c>0 is a constant.
Abstract: It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c>0 is a constant. This improves earlier results of Erd?s, Moser, Beck, Chung, Szemeredi, Trotter, and Szekely.
102 citations
TL;DR: This paper improves the bounds of Erdős, Lovász, et al. on the number of halving hyperplanes in higher dimensions by constructing a set of n points in the plane with ne-Omega k -sets.
Abstract: For any n , k , n\geq 2k>0 , we construct a set of n points in the plane with $ne^{\Omega({\sqrt{\log k}})}$ k -sets. This improves the bounds of Erd?s, Lovasz, et al. As a consequence, we also improve the lower bound for the number of halving hyperplanes in higher dimensions.
95 citations
TL;DR: It is given simple necessary and sufficient conditions for self-affine tiles in R2 to be homeomorphic to a disk.
Abstract: We give simple necessary and sufficient conditions for self-affine tiles in R 2 to be homeomorphic to a disk.
82 citations
TL;DR: A short geometric proof of this result is given, which is used to improve a bound on realizability of weighted graphs as graphs of distances between points in Euclidean space, and its relation to theorems of Bohnenblust, Friedland and Loewy, and Au-Yeung and Poon.
Abstract: Let K n be the cone of positive semidefinite n X n matrices and let A be an affine subspace of the space of symmetric matrices such that the intersection K n ?A is nonempty and bounded. Suppose that n ? 3 and that \codim A = r+2 \choose 2 for some 1 ≤ r ≤ n-2 . Then there is a matrix X ? K n ?A such that rank X ≤ r . We give a short geometric proof of this result, use it to improve a bound on realizability of weighted graphs as graphs of distances between points in Euclidean space, and describe its relation to theorems of Bohnenblust, Friedland and Loewy, and Au-Yeung and Poon.
71 citations
TL;DR: This work studies equipartitions by k-fans of two or more probability measures in the plane, as well as partitions in other prescribed ratios, finding a 4-fan such that one of its sectors contains two-fifths of both measures.
Abstract: A k-fan is a point in the plane and k semilines emanating from it. Motivated by a neat question of Kaneko and Kano, we study equipartitions by k-fans of two or more probability measures in the plane, as well as partitions in other prescribed ratios. One of our results is: for any two measures there is a 4-fan such that one of its sectors contains two-fifths of both measures, and each of the the remaining three sectors contains one-fifth of both measures.
67 citations
TL;DR: In this article, the authors studied integral convex polytopes and their integral decompositions in the sense of the Minkowski sum, and showed that deciding decomposability of integral polygons is NP-complete.
Abstract: Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NP-complete then present a pseudo-polynomial-time algorithm for decomposing polygons. For higher-dimensional polytopes, we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithms include absolute irreducibility testing and factorization of polynomials via their Newton polytopes.
65 citations
TL;DR: It is shown that the growth of plane tessellations and their edge graphs may be controlled from below by upper bounds for the combinatorial curvature.
Abstract: We show that the growth of plane tessellations and their edge graphs may be controlled from below by upper bounds for the combinatorial curvature. Under the assumption that every geodesic path may be extended to infinity we provide explicit estimates of the growth rate and isoperimetric constant of distance balls in negatively curved tessellations. We show that the assumption about geodesics holds for all tessellations with at least p faces meeting in each vertex and at least q edges bounding each face, where (p,q) ? { (3,6), (4,4), (6,3) } .
64 citations
TL;DR: In this article, an algorithm for maintaining an approximating triangulation of a deforming surface in R 3 is described, where the surface is the envelope of an infinite family of spheres defined and controlled by a finite collection of weighted points.
Abstract: This paper describes an algorithm for maintaining an approximating triangulation of a deforming surface in R 3 . The surface is the envelope of an infinite family of spheres defined and controlled by a finite collection of weighted points. The triangulation adapts dynamically to changing shape, curvature, and topology of the surface.
62 citations
TL;DR: It is proved that the maximum number of k -sets in a set S of n points in \Bbb R 3 is O(nk3/2), which improves substantially the previous best known upper bound of O( nk5/3).
Abstract: We prove that the maximum number of k -sets in a set S of n points in \Bbb R 3 is O(nk 3/2 ) . This improves substantially the previous best known upper bound of O(nk 5/3 ) (see [7] and [1]).
TL;DR: The methods developed here are used to show that the n-dimensional chair tiling and the sphinx tiling are pure point diffractive sets.
Abstract: This paper studies ways in which the sets of a partition of a lattice in ?n become regular model sets. The main theorem gives equivalent conditions which assure that a matrix substitution system on a lattice in ?n gives rise to regular model sets (based on p-adic-like internal spaces), and hence to pure point diffractive sets. The methods developed here are used to show that the n-dimensional chair tiling and the sphinx tiling are pure point diffractive.
TL;DR: Any four unit balls in three-dimensional space, whose centers are not collinear, have at most twelve common tangent lines, and this bound is tight.
Abstract: We answer a question of David Larman, by proving the following result. Any four unit balls in three-dimensional space, whose centers are not collinear, have at most twelve common tangent lines. This bound is tight.
TL;DR: This paper presents a lower bound on the area of drawings in which edges are drawn using exactly one circular arc, and gives an algorithm for drawing n-vertex planar graphs such that the edges are sequences of two continuous circular arcs.
Abstract: In this paper we address the problem of drawing planar graphs with circular arcs while maintaining good angular resolution and small drawing area. We present a lower bound on the area of drawings in which edges are drawn using exactly one circular arc. We also give an algorithm for drawing n-vertex planar graphs such that the edges are sequences of two continuous circular arcs. The algorithm runs in O(n) time and embeds the graph on the O(n) × O(n) grid, while maintaining ?(1/d(v)) angular resolution, where d(v) is the degree of vertex v. Since in this case we use circular arcs of infinite radius, this is also the first algorithm that simultaneously achieves good angular resolution, small area, and at most one bend per edge using straight-line segments. Finally, we show how to create drawings in which edges are smooth C1-continuous curves, represented by a sequence of at most three circular arcs.
Journal Article•
TL;DR: An application of Ramsey's theorem then yields the classical Erdos-Szekeres theorem as discussed by the authors, which states that for every integer n ≥ 3 there is an n 0 such that, among any set of n ≥ n 0 points in general position in the plane, there is the vertex set of a convex n-gon.
Abstract: Eszter Klein’s theorem claims that among any 5 points in the plane, no three collinear, there is the vertex set of a convex quadrilateral.An application of Ramsey’s theorem then yields the classical Erdos-Szekeres theorem [19]: For every integer n ≥ 3 there is an N0 such that, among any set of N ≥ N 0 points in general position in the plane, there is the vertex set of a convex n-gon. Let f(n) denote the smallest such number.
TL;DR: This paper gives a very simple algorithm to find a grid drawing of any given 4-connected plane graph G with four or more vertices on the outer face and yields a drawing in a rectangular grid of width \lceil n/2 \rceil - 1 and height \lfloor n/ 2\rfloor if G has n vertices.
Abstract: A grid drawing of a plane graph G is a drawing of G on the plane so that all vertices of G are put on plane grid points and all edges are drawn as straight line segments between their endpoints without any edge-intersection. In this paper we give a very simple algorithm to find a grid drawing of any given 4-connected plane graph G with four or more vertices on the outer face. The algorithm takes time O(n) and yields a drawing in a rectangular grid of width \lceil n/2 \rceil - 1 and height \lfloor n/2\rfloor if G has n vertices. The algorithm is best possible in the sense that there are an infinite number of 4-connected plane graphs, any grid drawings of which need rectangular grids of width \lceil n/2 \rceil - 1 and height \lfloor n/2\rfloor .
TL;DR: A kinetic data structure that maintains the connected components of the union of a set of unit-radius disks moving in the plane and must deal with O(n2 + ε) updates in the worst case, each requiring O(log 2 n) amortized time, for any ε>0 .
Abstract: We describe a kinetic data structure (KDS) that maintains the connected components of the union of a set of unit-radius disks moving in the plane. We assume that the motion of each disk can be specified by a low-degree algebraic trajectory; this trajectory, however, can be modified in an on-line fashion. While the disks move continuously, their connectivity changes at discrete times. Our main result is an O(n) space data structure that takes O(log n\slash \kern -1pt log log n) time per connectivity query of the form ``are disks A and B in the same connected component?'' A straightforward approach based on dynamically maintaining the overlap graph requires Ω (n 2 ) space. Our data structure requires only linear space and must deal with O(n 2 + ? ) updates in the worst case, each requiring O(log 2 n) amortized time, for any ?>0 . This number of updates is close to optimal, since a set of n moving unit disks can undergo Ω (n 2 ) connectivity changes.
TL;DR: In the more restricted but still general setting of LP-type problems, this work proves tail estimates for the sampling lemma, giving Chernoff-type bounds for the number of constraints violated by the solution of a random subset.
Abstract: Random sampling is an efficient method to deal with constrained optimization problems in computational geometry. In a first step, one finds the optimal solution subject to a random subset of the constraints; in many cases, the expected number of constraints still violated by that solution is then significantly smaller than the overall number of constraints that remain. This phenomenon can be exploited in several ways, and typically results in simple and asymptotically fast algorithms.
Very often the analysis of random sampling in this context boils down to a simple identity (the sampling lemma ) which holds in a general framework, yet has not been stated explicitly in the literature.
In the more restricted but still general setting of LP-type problems , we prove tail estimates for the sampling lemma, giving Chernoff-type bounds for the number of constraints violated by the solution of a random subset. As an application, we provide the first theoretical analysis of multiple pricing , a heuristic used in the simplex method for linear programming in order to reduce a large problem to few small ones. This follows from our analysis of a reduction scheme for general LP-type problems, which can be considered as a simplification of an algorithm due to Clarkson. The simplified version needs less random resources and allows a Chernoff-type tail estimate.
TL;DR: New efficient and compact kinetic data structures for maintaining the diameter, width, and smallest area or perimeter bounding rectangle of S are given, showing that Ω(n2) combinatorial changes are possible for these extent functions even if each point is moving with constant velocity.
Abstract: Let S be a set of n moving points in the plane. We give new efficient and compact kinetic data structures for maintaining the diameter, width, and smallest area or perimeter bounding rectangle of S . If the points in S move with algebraic motions, these structures process O(n 2+\eps ) events. We also give constructions showing that Ω(n 2 ) combinatorial changes are possible for these extent functions even if each point is moving with constant velocity. We give a similar construction and upper bound for the convex hull, improving known results.
TL;DR: A vector for a point relative to a point set is introduced, which expresses “how interior” the point is relative to the point set, and a tight upper bound is reviewed on the maximum number of j-facets entered by a directed line.
Abstract: Let S be a set of n points in d-space, no i + 1 points on a common (i?l)-flat for 1 ≤ i ≤ d An oriented (d ? 1)-simplex spanned by d points in S is called a j-facet of S if there are exactly j points from S on the positive side of its affine hull We show: (*) For j ≤ n/2 ? 2, the total number of (≤ j)- facets (ie the number of i- facets with 0 ≤ i ≤ j) in 3 -space is maximized in convex position (where these numbers are known) A large part of this presentation is a preparatory review of some basic properties of the collection of j-facets--some with their proofs--and of relations to well-established concepts and results from the theory of convex polytopes (h-vector, Dehn-Sommerville relations, Upper Bound Theorem, Generalized Lower Bound Theorem) The relations are established via a duality closely related to the Gale transform--similar to previous works by Lee, by Clarkson, and by Mulmuley
A central definition is as follows Given a directed line l and a j-facet F of S, we say that l enters F if l intersects the relative interior of F in a single point, and if l is directed from the positive to the negative side of F One of the results reviewed is a tight upper bound of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaqaabe% qaaiaadQgacqGHRaWkcaWGKbGaeyOeI0IaaGymaaqaaiacaci5aaaa% dsgacWaGasoaaaGHsislcGaGasoaaaaIXaaaaiaawIcacaGLPaaaaa% a!4434! $$\left( \begin{gathered} j + d - 1 \hfill \\ d - 1 \hfill \\ \end{gathered} \right)$$ on the maximum number of j-facets entered by a directed line
Based on these considerations, we also introduce a vector for a point relative to a point set, which--intuitively speaking--expresses "how interior" the point is relative to the point set This concept allows us to show that statement (*) above is equivalent to the Generalized Lower Bound Theorem for d-polytopes with at most d + 4 vertices
TL;DR: This result can be considered a continuous analogue of the Upper Bound Theorem for the maximal number of faces of convex polytopes with a given number of vertices, and so-called h -functions are introduced, continuous counterparts of h -vectors of simplicial convexpolytopes.
Abstract: For an absolutely continuous probability measure μ on R d and a nonnegative integer k , let \tilde s k (μ ,\origin ) denote the probability that the convex hull of k+d+1 random points which are i.i.d. according to μ contains the origin \bf 0 . For d and k given, we determine a tight upper bound on \tilde s k (μ ,\origin ) , and we characterize the measures in R d which attain this bound. As we will see, this result can be considered a continuous analogue of the Upper Bound Theorem for the maximal number of faces of convex polytopes with a given number of vertices. For our proof we introduce so-called h -functions, continuous counterparts of h -vectors of simplicial convex polytopes.
TL;DR: It is proved that a nonperiodic repetitive tiling of the plane is pseudo-self-similar if and only if it has a finite number of derived Vorono\"{\i} tilings up to similarity.
Abstract: A pseudo-self-similar tiling is a hierarchical tiling of Euclidean space which obeys a nonexact substitution rule: the substitution for a tile is not geometrically similar to itself. An example is the Penrose tiling drawn with rhombi. We prove that a nonperiodic repetitive tiling of the plane is pseudo-self-similar if and only if it has a finite number of derived Vorono\"{\i} tilings up to similarity. To establish this characterization, we settle (in the planar case) a conjecture of E. A. Robinson by providing an algorithm which converts any pseudo-self-similar tiling of R 2 into a self-similar tiling of R 2 in such a way that the translation dynamics associated to the two tilings are topologically conjugate.
TL;DR: In this paper, it was shown that a simple closed polygonal chain can be made convex in 3D by O(n) basic O(m) moves, and that any simple closed chain that initially takes the form of a planar polygon can be constructed in three dimensions.
Abstract: This paper studies movements of polygonal chains in three dimensions whose links are not allowed to cross or change length. Our main result is an algorithmic proof that any simple closed chain that initially takes the form of a planar polygon can be made convex in three dimensions. Other results include an algorithm for straightening open chains having a simple orthogonal projection onto some plane, and an algorithm for making convex any open chain initially configured on the surface of a polytope. All our algorithms require only O(n) basic ``moves.''
TL;DR: Using offset as a notion of distance, it is shown how to compute the corresponding nearest- and furthest-site Voronoi diagrams of point sites in the plane using near-optimal deterministic O(n(logn + log2m) +m)-time algorithms.
Abstract: In this paper we develop the concept of a convexpolygon-offset distance function. Using offset as a notion of distance, we show how to compute the corresponding nearest- and furthest-site Voronoi diagrams of point sites in the plane. We provide near-optimal deterministicO(n(logn + log2m) +m)-time algorithms, wheren is the number of points andm is the complexity of the underlying polygon, for computing compact representations of both diagrams.
TL;DR: This work constructs arbitrarily large finite sets A with coD(A) ≠ A whose proper subsets are all equal to their D-convex hull, which implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 matrices.
Abstract: Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: R d ? R is called D-convex, where D is a set of vectors in R d, if its restriction to each line parallel to a nonzero v ? D is convex. The D-convex hull of a compact set A ? R d, denoted by coD(A), is the intersection of the zero sets of all nonnegative D-convex functions that are zero on A. It also equals the zero set of the D-convex envelope of the distance function of A. We give an example of an n-point set A ? R 2 where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from co D(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R 3 (where D is the orthonormal basis of R 3), we construct arbitrarily large finite sets A with co D(A) ? A whose proper subsets are all equal to their D-convex hull. This implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 (or larger) matrices.
TL;DR: A randomized algorithm for computing the trapezoidal decomposition of a simple polygon, which can be viewed as a combination of Chazelle’s algorithm and of simple nonoptimal randomized algorithms due to Clarkson et al.
Abstract: We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's [3] celebrated optimal deterministic algorithm. The new algorithm can be viewed as a combination of Chazelle's algorithm and of simple nonoptimal randomized algorithms due to Clarkson et al. [6], [7], [9] and to Seidel [20]. As in Chazelle's algorithm, it is indispensable to include a bottom-up preprocessing phase, in addition to the actual top-down construction. An essential new idea is the use of random sampling on subchains of the initial polygonal chain, rather than on individual edges as is normally done.
TL;DR: Given a set of n disjoint line segments in the plane, it is shown that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of theTree is 3.
Abstract: Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree is 3. Furthermore, there exist configurations of line segments where any such tree requires degree 3. We provide an O(nlog n) time algorithm for constructing such a tree, and show that this is optimal.
TL;DR: The diameter algorithm appears to be the last one in Clarkson and Shor’s paper that up to now had no deterministic counterpart with a matching running time, and improves previous deterministic algorithms by Ramos and Bespamyatnikh.
Abstract: We describe a deterministic algorithm for computing the diameter of a finite set of points in R 3 , that is, the maximum distance between any pair of points in the set. The algorithm runs in optimal time O(nlog n) for a set of n points. The first optimal, but randomized, algorithm for this problem was proposed more than 10 years ago by Clarkson and Shor [11] in their ground-breaking paper on geometric applications of random sampling. Our algorithm is relatively simple except for a procedure by Matousek [25] for the efficient deterministic construction of epsilon-nets. This work improves previous deterministic algorithms by Ramos [31] and Bespamyatnikh [7], both with running time O(nlog 2 n) . The diameter algorithm appears to be the last one in Clarkson and Shor's paper that up to now had no deterministic counterpart with a matching running time.
TL;DR: This article has classified completely all the simplicial equivelar polyhedra on ≤ 11 vertices such that there exists an equivel polyhedron of type {p,q} and of Euler characteristic n .
Abstract: We know that the polyhedra corresponding to the Platonic solids are equivelar. In this article we have classified completely all the simplicial equivelar polyhedra on ≤ 11 vertices. There are exactly 27 such polyhedra. For each n\geq -4 , we have classified all the (p,q) such that there exists an equivelar polyhedron of type {p,q} and of Euler characteristic n . We have also constructed five types of equivelar polyhedra of Euler characteristic -2m , for each m\geq 2.
TL;DR: An O(n2+δ) -time algorithm is presented, for any δ>0 , that computes a cylindrical shell of width at most 56\opt containing S.
Abstract: Let S be a set of n points in \reals 3 . Let \opt be the width (i.e., thickness) of a minimum-width infinite cylindrical shell (the region between two co-axial cylinders) containing S . We first present an O(n 5 ) -time algorithm for computing \opt , which as far as we know is the first nontrivial algorithm for this problem. We then present an O(n 2+? ) -time algorithm, for any ?>0 , that computes a cylindrical shell of width at most 56\opt containing S .