Discrete & Computational Geometry

http://link.springer.com/journal/454

List of Papers (Total 90)

Unknot Diagrams Requiring a Quadratic Number of Reidemeister Moves to Untangle

Given any knot diagram E, we present a sequence of knot diagrams of the same knot type for which the minimum number of Reidemeister moves required to pass to E is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in ℝ2.

Uniform Convergence of Discrete Curvatures from Nets of Curvature Lines

We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.

Streaming Algorithms for Line Simplification

We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p 0,p 1,p 2,… in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we ...

Untangling a Planar Graph

A straight-line drawing δ of a planar graph G need not be plane but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G,δ) denote the minimum number of vertices that need to be moved to untangle δ. We show that shift(G,δ) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a ...

Irreducible Apollonian Configurations and Packings

An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. One well-studied method of forming an Apollonian configuration is to start with three mutually tangent circles and fill a curvilinear triangle with a new circle, then repeat with each newly ...

Polychromatic Colorings of Plane Graphs

We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by ⌊(3g−5)/4⌋ colors so that every color appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and that admit no vertex coloring of this type with more than ⌊(3g+1)/4⌋ colors. We further show that the ...

Refolding Planar Polygons

This paper describes an algorithm for generating a guaranteed intersection-free interpolation sequence between any pair of compatible polygons. Our algorithm builds on prior results from linkage unfolding, and if desired it can ensure that every edge length changes monotonically over the course of the interpolation sequence. The computational machinery that ensures against ...

An Optimization Problem Related to Minkowski’s Successive Minima

The purpose of this paper is to establish an inequality connecting the lattice point enumerator of a 0-symmetric convex body with its successive minima. To this end, we introduce an optimization problem whose solution refines former methods, thus producing a better upper bound. In particular, we show that an analogue of Minkowski’s second theorem on successive minima with the ...

Affine Configurations and Pure Braids

We show that the fundamental group of ordered affine-equivalent configurations with at least five points in the real plane is isomorphic to the pure braid group in as many strands, modulo its centre. In the case of four points, this fundamental group is free with 11 generators.

Small Hop-diameter Sparse Spanners for Doubling Metrics

Given a metric M=(V,d), a graph G=(V,E) is a t-spanner for M if every pair of nodes in V has a “short” path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every pair of nodes has such a short path that also uses at most D edges. We consider the problem of constructing sparse ...

Generating All Vertices of a Polyhedron Is Hard

We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles ...

Efficient Algorithms for Maximum Regression Depth

We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n d ) time and O(n d−1) space algorithm computes undirected depth of all points in d ...

Polyboxes, Cube Tilings and Rigidity

A non-empty subset A of X=X 1×⋅⋅⋅×X d is a (proper) box if A=A 1×⋅⋅⋅×A d and A i ⊂X i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A i =B i , A i =X i ∖B i , A i ∉{B i ,X i ∖B i }. Let F and G be two systems of disjoint boxes. Can one decide whether ∪F=∪G? In general, the answer is ‘no’, but as is shown in ...

Slicing Convex Sets and Measures by a Hyperplane

Given convex bodies K 1,…,K d in ℝ d and numbers α 1,…,α d ∈[0,1], we give a sufficient condition for existence and uniqueness of an (oriented) halfspace H with Vol (H∩K i )=α i ⋅Vol K i for every i. The result is extended from convex bodies to measures.

Improved Bounds on the Union Complexity of Fat Objects

We introduce a new class of fat, not necessarily convex or polygonal, objects in the plane, namely locally γ-fat objects. We prove that the union complexity of any set of n such objects is O(λ s+2(n)log 2 n). This improves the best known bound, and extends it to a more general class of objects.

Lattice Points, Dedekind Sums, and Ehrhart Polynomials of Lattice Polyhedra

Let σ be a simplex of R N with vertices in the integral lattice Z N . The number of lattice points of mσ (={mα : α ∈ σ}) is a polynomial function L(σ,m) of m ≥ 0 . In this paper we present: (i) a formula for the coefficients of the polynomial L(σ,t) in terms of the elementary symmetric functions; (ii) a hyperbolic cotangent expression for the generating functions of the sequence ...

Simultaneous partitions of measures by <Emphasis Type=

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 ...