# Discrete and Computational Geometry (DISCRETE COMPUT GEOM)

Publisher Springer Verlag

## Description

Discrete & Computational Geometry is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It accepts research articles of high quality in discrete geometry and on the design and analysis of geometric algorithms; more specifically, DCG publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometry of numbers, and motion planning, as well as papers with a distinct geometric flavor in such areas as graph theory, mathematical programming, real algebraic geometry, matroids, solid modeling, computer graphics, combinatorial optimization, image processing, pattern recognition, crystallography, VLSI design, and robotics.

• Impact factor
0.65
Show impact factor history

Impact factor
.
Year
• Website
Discrete & Computational Geometry website
• Other titles
Discrete & computational geometry (Online), Discrete and computational geometry
• ISSN
0179-5376
• OCLC
39976194
• Material type
Document, Periodical, Internet resource
• Document type
Internet Resource, Computer File, Journal / Magazine / Newspaper

## Publisher details

• Pre-print
• Author can archive a pre-print version
• Post-print
• Author can archive a post-print version
• Conditions
• Authors own final version only can be archived
• Publisher's version/PDF cannot be used
• On author's website or institutional repository
• On funders designated website/repository after 12 months at the funders request or as a result of legal obligation
• Published source must be acknowledged
• Must link to publisher version
• Set phrase to accompany link to published version (The original publication is available at www.springerlink.com)
• Articles in some journals can be made Open Access on payment of additional charge
• Classification
​ green

## Publications in this journal

• ##### Article: Guest Editor’s Foreword
Discrete and Computational Geometry 08/2013; 47(4).
• ##### Article: Geometric Characterization of Weyl’s Discrepancy Norm in Terms of Its n-Dimensional Unit Balls
[hide abstract]
ABSTRACT: Weyl’s discrepancy measure induces a norm on ℝn which shows a monotonicity and a Lipschitz property when applied to differences of index-shifted sequences. It turns out that its n-dimensional unit ball is a zonotope that results from a multiple sheared projection from the (n+1)-dimensional hypercube which can be interpreted as a discrete differentiation. This characterization reveals that this norm is the canonical metric between sequences of differences of values from the unit interval in the sense that the n-dimensional unit ball of the discrepancy norm equals the space of such sequences.
Discrete and Computational Geometry 08/2013; 48(4).
• ##### Article: Guest Editor’s Foreword
Discrete and Computational Geometry 08/2013; 42(1).
• Source
##### Article: Guest Editors’ Foreword
Discrete and Computational Geometry 08/2013; 42(2).
• Source
##### Article: Fast Enumeration Algorithms for Non-crossing Geometric Graphs
[hide abstract]
ABSTRACT: A non-crossing geometric graph is a graph embedded on a set of points in the plane with non-crossing straight line segments. In this paper we present a general framework for enumerating non-crossing geometric graphs on a given point set. Applying our idea to specific enumeration problems, we obtain faster algorithms for enumerating plane straight-line graphs, non-crossing spanning connected graphs, non-crossing spanning trees, and non-crossing minimally rigid graphs. Our idea also produces efficient enumeration algorithms for other graph classes, for which no algorithm has been reported so far, such as non-crossing matchings, non-crossing red-and-blue matchings, non-crossing k-vertex or k-edge connected graphs, or non-crossing directed spanning trees. The proposed idea is relatively simple and potentially applies to various other problems of non-crossing geometric graphs.
Discrete and Computational Geometry 08/2013;
• Source
##### Article: Guest Editor’s Foreword
Discrete and Computational Geometry 08/2013; 45(4).
• ##### Article: Mandelbrot Set + Symmetry Groups ∗ Higher Dimensions = ?
[hide abstract]
ABSTRACT: We construct the MaṇḍalaBeth (higher dimensional analogs of the Mandelbrot set) with various symmetry groups. The generating function of the MaṇḍalaBeth is the sum (over the symmetry group) of conjugates (by isometries) of 2D projections (generating functions of the Mandelbrot set).
Discrete and Computational Geometry 08/2013; 48(4).
• ##### Article: Allowable Interval Sequences and Line Transversals in the Plane
[hide abstract]
ABSTRACT: Given a family F of n pairwise disjoint compact convex sets in the plane with non-empty interiors, let T(k) denote the property that every subfamily of F of size k has a line transversal, and T the property that the entire family has a line transversal. We illustrate the applicability of allowable interval sequences to problems involving line transversals in the plane by proving two new results and generalizing three old ones. Two of the old results are Klee’s assertion that if F is totally separated then T(3) implies T, and the following variation of Hadwiger’s Transversal Theorem proved by Wenger and (independently) Tverberg: If F is ordered and each four sets of F have some transversal which respects the order on F, then there is a transversal for all of F which respects this order. The third old result (a consequence of an observation made by Kramer) and the first of the new results (which partially settles a 2008 conjecture of Eckhoff) deal with fractional transversals and share the following general form: If F has property T(k) and meets certain other conditions, then there exists a transversal of some m sets in F, with k<m<n. The second new result establishes a link between transversal properties and separation properties of certain families of convex sets.
Discrete and Computational Geometry 08/2013; 48(4).
• Source
##### Article: Ehrhart h ∗-Vectors of Hypersimplices
[hide abstract]
ABSTRACT: We consider the Ehrhart h ∗-vector for the hypersimplex. It is well-known that the sum of the $h_{i}^{*}$ is the normalized volume which equals the Eulerian numbers. The main result is a proof of a conjecture by R. Stanley which gives an interpretation of the $h^{*}_{i}$ coefficients in terms of descents and exceedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes.
Discrete and Computational Geometry 08/2013; 48(4).
• Source
##### Article: Guest Editors’ Foreword
Discrete and Computational Geometry 08/2013; 49(1).
• ##### Article: Some geometric applications of Dilworth’s theorem
[hide abstract]
ABSTRACT: A geometric graph is a graph drawn in the plane such that its edges are closed line segments and no three vertices are collinear. We settle an old question of Avital, Hanani, Erdős, Kupitz, and Perles by showing that every geometric graph withn vertices andm>k 4n edges containsk+1 pairwise disjoint edges. We also prove that, given a set of pointsV and a set of axis-parallel rectangles in the plane, then either there arek+1 rectangles such that no point ofV belongs to more than one of them, or we can find an at most 2·105k 8 element subset ofV meeting all rectangles. This improves a result of Ding, Seymour, and Winkler. Both proofs are based on Dilworth’s theorem on partially ordered sets.
Discrete and Computational Geometry 08/2013; 12(1).
• ##### Article: Topologically Sweeping Visibility Complexes via Pseudotriangulations
[hide abstract]
ABSTRACT: This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using red-black trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of the visibility complex.
Discrete and Computational Geometry 08/2013;
• Source
##### Article: The “Point” Goalie Problem
[hide abstract]
ABSTRACT: Suppose you are given a collection of discs of radius ε and you are asked to place them in the plane so that any straight line that crosses the unit disc will hit one or more of them. Furthermore, you are asked to do this with as few discs as possible. How many do you need? This problem can be cast as an (infinite) integer programming (IP) problem. Often, for such problems, one can obtain a lower bound by considering the linear programming (LP) relaxation. For the above problem the relaxed problem allows discs with positive weight, requires that each line crossing the unit disc accumulates weight 1 from discs it hits, and asks that the total weight be minimized. This LP problem has a simple asymptotically optimal solution with total weight 1/ε. Clearly, this is the best possible since blocking all the lines in a single direction requires total weight 1/ε. Although it is quite intuitive that the IP version cannot have such a “perfect” solution, achieving the lower bound, it is surprisingly difficult to obtain a better lower bound. Here we improve the lower bound to 1.001/ε. The true answer is probably considerably larger. We believe this is the first example of a problem in this class where it has been proven that the LP and IP versions have different answers.
Discrete and Computational Geometry 08/2013; 30(4).
• ##### Article: Properties of ℛ-Sausages
[hide abstract]
ABSTRACT: This paper is concerned with the Steiner ratio. A number of properties about the structure of the flat sausage and ℛ-Sausage convex polytopes yielding the best Steiner ratio in two- and three-dimensional Euclidean space, and the topology of the Steiner Minimal Tree for the corresponding vertex sets, are presented.
Discrete and Computational Geometry 08/2013; 31(4).
• ##### Article: Weighted Derivations and the cd-Index
[hide abstract]
ABSTRACT: Weighted derivations W 1 and W 2 allowed R. Ehrenborg and M. Readdy (Discrete Comput. Geom. 21:389–403, [1999]) to give a recursive description of the cd-indices of the lattices of the regions of the arrangements $\mathcal{A}_{n}$ and ℬn . In part motivated by this, we describe a new basis for the subspace spanned by ab-indices of all simplicial graded posets and determine the action of certain linear maps (associated with weighted derivations W k ) on this basis. Extending the “pyramid” and “prism” operations, we define operations Σk on graded posets and show that relations between the ab-indices (or cd-indices, for an Eulerian poset P) of barycentric subdivisions of Σk (P) and P can be described by using the linear maps associated by weighted derivations W k . Finally, in response to a question from Ehrenborg and Readdy (Discrete Comput. Geom. 21:389–403, [1999]), we obtain the formulae that express the cd-index of the lattice $R_{\mathcal{D}_{n}}$ of regions of the arrangement $\mathcal{D}_{n}$ in terms of the cd-indices of $R_{\mathcal{B}_{n}}$ and $R_{\mathcal{A}_{n-2}}$ .
Discrete and Computational Geometry 08/2013; 39(4).

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