184 research outputs found

    Upper and Lower Bounds on Long Dual-Paths in Line Arrangements

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    Given a line arrangement A\cal A with nn lines, we show that there exists a path of length n2/3O(n)n^2/3 - O(n) in the dual graph of A\cal A formed by its faces. This bound is tight up to lower order terms. For the bicolored version, we describe an example of a line arrangement with 3k3k blue and 2k2k red lines with no alternating path longer than 14k14k. Further, we show that any line arrangement with nn lines has a coloring such that it has an alternating path of length Ω(n2/logn)\Omega (n^2/ \log n). Our results also hold for pseudoline arrangements.Comment: 19 page

    A History of Flips in Combinatorial Triangulations

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    Given two combinatorial triangulations, how many edge flips are necessary and sufficient to convert one into the other? This question has occupied researchers for over 75 years. We provide a comprehensive survey, including full proofs, of the various attempts to answer it.Comment: Added a paragraph referencing earlier work in the vertex-labelled setting that has implications for the unlabeled settin

    Quasi-infra-red fixed points and renormalisation group invariant trajectories for non-holomorphic soft supersymmetry breaking

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    In the MSSM the quasi-infra-red fixed point for the top-quark Yukawa coupling gives rise to specific predictions for the soft-breaking parameters. We discuss the extent to which these predictions are modified by the introduction of additional ``non-holomorphic'' soft-breaking terms. We also show that in a specific class of theories there exists an RG-invariant trajectory for the ``non-holomorphic'' terms, which can be understood using a holomorphic spurion term.Comment: 24 pages, TeX, two figures. Uses Harvmac (big) and epsf. Minor errors corrected, and the RG trajectory explained in terms of a holomorphic spurion ter

    Maximizing Maximal Angles for Plane Straight-Line Graphs

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    Let G=(S,E)G=(S, E) be a plane straight-line graph on a finite point set SR2S\subset\R^2 in general position. The incident angles of a vertex pSp \in S of GG are the angles between any two edges of GG that appear consecutively in the circular order of the edges incident to pp. A plane straight-line graph is called ϕ\phi-open if each vertex has an incident angle of size at least ϕ\phi. In this paper we study the following type of question: What is the maximum angle ϕ\phi such that for any finite set SR2S\subset\R^2 of points in general position we can find a graph from a certain class of graphs on SS that is ϕ\phi-open? In particular, we consider the classes of triangulations, spanning trees, and paths on SS and give tight bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that were omitted in the previous version are now include

    Embedding Four-directional Paths on Convex Point Sets

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    A directed path whose edges are assigned labels "up", "down", "right", or "left" is called \emph{four-directional}, and \emph{three-directional} if at most three out of the four labels are used. A \emph{direction-consistent embedding} of an \mbox{nn-vertex} four-directional path PP on a set SS of nn points in the plane is a straight-line drawing of PP where each vertex of PP is mapped to a distinct point of SS and every edge points to the direction specified by its label. We study planar direction-consistent embeddings of three- and four-directional paths and provide a complete picture of the problem for convex point sets.Comment: 11 pages, full conference version including all proof

    Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

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    We introduce a new structure for a set of points in the plane and an angle α\alpha, which is similar in flavor to a bounded-degree MST. We name this structure α\alpha-MST. Let PP be a set of points in the plane and let 0<α2π0 < \alpha \le 2\pi be an angle. An α\alpha-ST of PP is a spanning tree of the complete Euclidean graph induced by PP, with the additional property that for each point pPp \in P, the smallest angle around pp containing all the edges adjacent to pp is at most α\alpha. An α\alpha-MST of PP is then an α\alpha-ST of PP of minimum weight. For α<π/3\alpha < \pi/3, an α\alpha-ST does not always exist, and, for απ/3\alpha \ge \pi/3, it always exists. In this paper, we study the problem of computing an α\alpha-MST for several common values of α\alpha. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point pPp \in P, we associate a wedge WpW_p of angle α\alpha and apex pp. The goal is to assign an orientation and a radius rpr_p to each wedge WpW_p, such that the resulting graph is connected and its MST is an α\alpha-MST. (We draw an edge between pp and qq if pWqp \in W_q, qWpq \in W_p, and pqrp,rq|pq| \le r_p, r_q.) Unsurprisingly, the problem of computing an α\alpha-MST is NP-hard, at least for α=π\alpha=\pi and α=2π/3\alpha=2\pi/3. We present constant-factor approximation algorithms for α=π/2,2π/3,π\alpha = \pi/2, 2\pi/3, \pi. One of our major results is a surprising theorem for α=2π/3\alpha = 2\pi/3, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set PP of 3n3n points in the plane and any partitioning of the points into nn triplets, one can orient the wedges of each triplet {\em independently}, such that the graph induced by PP is connected. We apply the theorem to the {\em antenna conversion} problem

    On the number of simple arrangements of five double pseudolines

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    We describe an incremental algorithm to enumerate the isomorphism classes of double pseudoline arrangements. The correction of our algorithm is based on the connectedness under mutations of the spaces of one-extensions of double pseudoline arrangements, proved in this paper. Counting results derived from an implementation of our algorithm are also reported.Comment: 24 pages, 16 figures, 6 table

    Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition

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    A geometric graph is angle-monotone if every pair of vertices has a path between them that---after some rotation---is xx- and yy-monotone. Angle-monotone graphs are 2\sqrt 2-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone---specifically, we prove that the half-θ6\theta_6-graph is generalized angle-monotone. We give a local routing algorithm for Gabriel triangulations that finds a path from any vertex ss to any vertex tt whose length is within 1+21 + \sqrt 2 times the Euclidean distance from ss to tt. Finally, we prove some lower bounds and limits on local routing algorithms on Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Interactive architectural modeling with procedural extrusions

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    We present an interactive procedural modeling system for the exterior of architectural models. Our modeling system is based on procedural extrusions of building footprints. The main novelty of our work is that we can model difficult architectural surfaces in a procedural framework, e.g. curved roofs, overhanging roofs, dormer windows, interior dormer windows, roof constructions with vertical walls, buttresses, chimneys, bay windows, columns, pilasters, and alcoves. We present a user interface to interactively specify procedural extrusions, a sweep plane algorithm to compute a two-manifold architectural surface, and applications to architectural modeling
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