Introduction. A pseudoline is formed from a line by stretching the plane without tearing: it is the image of a line under a homeomorphism of the plane [13]. In arrangements of pseudolines, pairs of pseudolines intersect at most once and cross at their intersections. Pseudoline arrangements can be used to model sorting networks [1], tilings of convex polygons by rhombi [4], and graphs that have distance-preserving embeddings into hypercubes [6]. They are also closely related to oriented matroids [11]. We consider here the visualization of arrangements using well-shaped curves. Primarily, we study weak outerplanar pseudoline arrangements. An arrangement is weak if it does not necessarily have a crossing for every pair of pseudolines [12], and outerplanar if every crossing is part of an unbounded face of the arrangement. We show that these arrangements can be drawn with all curves convex, either as polygonal chains with at most two bends per pseudoline or as semicircles above a line. Arbitrary pseudolines can also be drawn as convex curves, but may require linearly many bends. Related Work. Several results related to the visualization of pseudoline arrangements are known. In wiring diagrams, pseudolines are drawn on parallel horizontal lines, with crossings on short line segments that connect pairs of horizontal lines [10]. The graphs of arrangements have drawings in small grids [8] and the dual graphs of weak arrangements have drawings in which each bounded face is centrally symmetric [5]. The pseudoline arrangements in which each pseudoline is a translated quadrant can be used to visualize learning spaces representing the states of a human learner [7]. Researchers in graph drawing have also studied force-directed methods for schematizing systems of curves representing metro maps by replacing each curve by a spline; these curves are not necessarily pseudolines, but they typically have few crossings [9]. Results. Below we state our results for outerplanar and arbitrary arrangements. Theorem 1. Every weak outerplanar pseudoline arrangement may be represented by
Several results related to the visualization of pseudoline arrangements are known. In , pseudolines are drawn on parallel horizontal lines, with crossings on short line segments that connect pairs of horizontal lines @cite_14 . Using a method based on compaction of wiring diagrams, the graphs of pseudoline arrangements may be given straight line drawings in small grids @cite_0 . The planar dual graph of a weak pseudoline arrangement may be characterized as having drawings in which each bounded face is a centrally symmetric polygon @cite_9 . The pseudoline arrangements in which each pseudoline is a translated quadrant can be used to visualize , representing the possible states of knowledge of a human learner @cite_8 . Researchers in graph drawing have also studied force-directed methods for schematizing systems of curves representing metro maps by replacing each curve by a spline; these curves are not necessarily pseudolines, but they typically have few crossings @cite_3 .
@cite_14 @cite_8 @cite_9 @cite_3 @cite_0
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We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area On 7 6. No known input causes our algorithm to use area Ωn 1+e for any e>0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle. Given a planar triangulated graph (PTG) G, the problem of constructing a floor-plan F such that G is the dual of F and the boundary of F is rectangular is studied. It is shown that if only zero-concave rectilinear modules (CRM) (or rectangular modules) and 1-CRM (i.e., L-shaped) are allowed, there are PTGs that do not admit any floor-plan. However, if 2-bend modules (e.g., T-shaped and Z-shaped) are also allowed, then every biconnected PTG admits a floor-plan. Thus, the employment of 2-bend modules is necessary and sufficient for graph dualization floor-planning. A linear-time algorithm for constructing a 2-CRM floor-plan of an arbitrary PTG is proposed. This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two bipolar orientations that are transversal. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For a random irreducible triangulation with n vertices, the grid size of the drawing is asymptotically with high probability 11n 27x11n 27 up to an additive error of O(n). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size (@?n 2@?-1)x@?n 2@?. Right Angle Crossing (RAC) drawings are polyline drawings where each crossing forms four right angles. RAC drawings have been introduced because cognitive experiments provided evidence that increasing the number of crossings does not decrease the readability of the drawing if the edges cross at right angles. We investigate to what extent RAC drawings can help in overcoming the limitations of widely adopted planar graph drawing conventions, providing both positive and negative results. First, we prove that there exist acyclic planar digraphs not admitting any straight-line upward RAC drawing and that the corresponding decision problem is NP-hard. Also, we show digraphs whose straight-line upward RAC drawings require exponential area. Second, we study if RAC drawings allow us to draw bounded-degree graphs with lower curve complexity than the one required by more constrained drawing conventions. We prove that every graph with vertex-degree at most 6 (at most 3) admits a RAC drawing with curve complexity 2 (resp. 1) and with quadratic area. Third, we consider a natural non-planar generalization of planar embedded graphs. Here we give bounds for curve complexity and area different from the ones known for planar embeddings. Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization; most previous work focuses on straight lines. We present a new algorithm for area-preserving curved schematization of territorial outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs. We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of “curviness”: we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. To evaluate the effectiveness of our design choices, we present a geometric evaluation of the resulting schematizations. Besides the geometric qualities of our algorithm, we also investigate the potential of curved schematization as a concept. We conducted an online user study investigating the effectiveness of curved schematizations compared to straight-line schematizations. While the visual complexity of curved shapes was judged higher than that of straight-line shapes, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents. We establish a duality principle for arrangements of pseudolines in the projective plane, and thereby prove the conjecture of Burr, Grunbaum, and Sloane that the solution of the ''orchard problem'' for pseudoline arrangements and the solution [email protected]?(p) of the dual problem are equal. In this article, we study automated simplification and schematization of territorial outlines. We present a quadratic-time simplification algorithm based on an operation called edge-move. We prove that the number of edges of any nonconvex simple polygon can be reduced with this operation. Moreover, edge-moves preserve area and topology and do not introduce new orientations. The latter property in particular makes the algorithm highly suitable for schematization in which all resulting lines are required to be parallel to one of a given set of lines (orientations). To obtain such a result, we need only to preprocess the input to use only lines that are parallel to one of the given set. We present an algorithm to enforce such orientation restrictions, again without changing area or topology. Experiments show that our algorithms obtain results of high visual quality. Visualization methods that arrange data objects in 2D or 3D layouts have followed two main schools, methods oriented for graph layout and methods oriented for vectorial embedding. We show the two previously separate approaches are tied by an optimization equivalence, making it possible to relate methods from the two approaches and to build new methods that take the best of both worlds. In detail, we prove a theorem of optimization equivalences between β- and γ-, as well as α- and Renyi-divergences through a connection scalar. Through the equivalences we represent several nonlinear dimensionality reduction and graph drawing methods in a generalized stochastic neighbor embedding setting, where information divergences are minimized between similarities in input and output spaces, and the optimal connection scalar provides a natural choice for the tradeoff between attractive and repulsive forces. We give two examples of developing new visualization methods through the equivalences: 1) We develop weighted symmetric stochastic neighbor embedding (ws-SNE) from Elastic Embedding and analyze its benefits, good performance for both vectorial and network data; in experiments ws-SNE has good performance across data sets of different types, whereas comparison methods fail for some of the data sets; 2) we develop a γ-divergence version of a PolyLog layout method; the new method is scale invariant in the output space and makes it possible to efficiently use large-scale smoothed neighborhoods. The topics discussed are minimization of the area occupied by a layout and related results concerning networks flow and rectilinear representation of planar graphs, based on a graph model of floorplans and layouts. Arbitrary floorplans are allowed. Given an arbitrary floorplan and the areas of the embedded building blocks, the existence and uniqueness of a zero wasted area layout are proved, and characterized by a necessary and sufficient condition. Based on this condition, a scheme is described to generate zero-wasted-area layouts. Given a family of dual network pairs for which the product of dual arc lengths are invariant, it is proved that the minimal product of their longest paths is not smaller than the maximal product of their shortest paths. It is also shown that the maximal product of the flows in such a family of dual network pairs is given by the total sum of the arc length product of each individual pair of dual arcs. An efficient procedure to derive a rectilinear representation for any planar graph is presented. > Multiple objects in the plane are called pseudodisks if the boundaries of any two of them intersect transversely at most twice. Given a set of @math (possibly intersecting) convex pseudodisks of @math complexity each and two points @math and @math in the plane, we present an efficient algorithm for computing a shortest @math -to- @math path avoiding the pseudodisks. After the union of the pseudodisks is computed, which can be done in @math randomized time or @math deterministic time, our algorithm runs in @math deterministic time, where @math is the size of the extended visibility graph of the union of the pseudodisks. Note that @math in the worst case. In over two decades, the previously best algorithms for this problem have not improved on the bound of @math time, even when all the pseudodisks are pairwise disjoint disks. Our technique is also applicable to a motion planning problem of finding a shortest path to translate a convex object in the plane from one location to anot... We study a family of closely-related distributed graph problems, which we call degree splitting, where roughly speaking the objective is to partition (or orient) the edges such that each node's degree is split almost uniformly. Our findings lead to answers for a number of problems, a sampling of which includes: • We present a poly log n round deterministic algorithm for (2Δ−1)·(1+o(1))-edge-coloring, where Δ denotes the maximum degree. Modulo the 1 + o(1) factor, this settles one of the long-standing open problems of the area from the 1990's (see e.g. Panconesi and Srinivasan [PODC'92]). Indeed, a weaker requirement of (2Δ − 1) · poly log Δ-edge-coloring in poly log n rounds was asked for in the 4th open question in the Distributed Graph Coloring book by Barenboim and Elkin. • We show that sinkless orientation---i.e., orienting edges such that each node has at least one out-going edge---on Δ-regular graphs can be solved in O(logΔ log n) rounds randomized and in O(logΔ n) rounds deterministically. These prove the corresponding lower bounds by [STOC'16] and Chang, Kopelowitz, and Pettie [FOCS'16] to be tight. Moreover, these show that sinkless orientation exhibits an exponential separation between its randomized and deterministic complexities, akin to the results of for Δ-coloring Δ-regular trees. • We present a randomized O(log4 n) round algorithm for orienting a-arboricity graphs with maximum out-degree a(1 + e). This can be also turned into a decomposition into a(1 + e) forests when a = Ω(log n) and into a(1 + e) pseduo-forests when a = o(log n). Obtaining an efficient distributed decomposition into less than 2a forests was stated as the 10th open problem in the book by Barenboim and Elkin. We give two optimal parallel algorithms for constructing the arrangement ofn lines in the plane. The first nethod is quite simple and runs inO(log2n) time usingO(n2) work, and the second method, which is more sophisticated, runs inO(logn) time usingO(n2) work. This second result solves a well-known open problem in parallel computational geometry, and involves the use of a new algorithmic technique, the construction of an ?-pseudocutting. Our results immediately imply that the arrangement ofn hyperplanes in ?d inO(logn) time usingO(nd) work, for fixedd, can be optimally constructed. Our algorithms are for the CREW PRAM. In this paper, we study the geometric RAC simultaneous drawing problem: Given two planar graphs that share a common vertex set but have disjoint edge sets, a geometric RAC simultaneous drawing is a straight-line drawing in which (i) each graph is drawn planar, (ii) there are no edge overlaps, and, (iii) crossings between edges of the two graphs occur at right-angles. We first prove that two planar graphs admitting a geometric simultaneous drawing may not admit a geometric RAC simultaneous drawing. We further show that a cycle and a matching always admit a geometric RAC simultaneous drawing, which can be constructed in linear time. Given a planar graph G together with a planar representation P, a region preserving grid embedding of G is a planar embedding of G in the rectilinear grid that has planar representation isomorphic to P. In this paper, an algorithm is presented that computes a region preserving grid embedding with the minimum number of bends in edges. This algorithm makes use of network flow techniques, and runs in time @math , where n is the number of vertices of the graph. Constrained versions of the problem are also considered, and most results are extended to k-gonal graphs, i.e., graphs whose edges are sequences of segments with slope multiple of @math degrees. Applications of the above results can be found in several areas: VLSI circuit layout, architectural design, communication by light or microwave, transportation problems, and automatic layout of graphlike diagrams. We present a novel dynamic graph visualization technique based on node-link diagrams. The graphs are drawn side-byside from left to right as a sequence of narrow stripes that are placed perpendicular to the horizontal time line. The hierarchically organized vertices of the graphs are arranged on vertical, parallel lines that bound the stripes; directed edges connect these vertices from left to right. To address massive overplotting of edges in huge graphs, we employ a splatting approach that transforms the edges to a pixel-based scalar field. This field represents the edge densities in a scalable way and is depicted by non-linear color mapping. The visualization method is complemented by interaction techniques that support data exploration by aggregation, filtering, brushing, and selective data zooming. Furthermore, we formalize graph patterns so that they can be interactively highlighted on demand. A case study on software releases explores the evolution of call graphs extracted from the JUnit open source software project. In a second application, we demonstrate the scalability of our approach by applying it to a bibliography dataset containing more than 1.5 million paper titles from 60 years of research history producing a vast amount of relations between title words. In this paper, we present a two-level generative model for representing the images and surface depth maps of drapery and clothes. The upper level consists of a number of folds which will generate the high contrast (ridge) areas with a dictionary of shading primitives (for 2D images) and fold primitives (for 3D depth maps). These primitives are represented in parametric forms and are learned in a supervised learning phase using 3D surfaces of clothes acquired through photometric stereo. The lower level consists of the remaining flat areas which fill between the folds with a smoothness prior (Markov random field). We show that the classical ill-posed problem-shape from shading (SFS) can be much improved by this two-level model for its reduced dimensionality and incorporation of middle-level visual knowledge, i.e., the dictionary of primitives. Given an input image, we first infer the folds and compute a sketch graph using a sketch pursuit algorithm as in the primal sketch (, 2003). The 3D folds are estimated by parameter fitting using the fold dictionary and they form the "skeleton" of the drapery cloth surfaces. Then, the lower level is computed by conventional SFS method using the fold areas as boundary conditions. The two levels interact at the final stage by optimizing a joint Bayesian posterior probability on the depth map. We show a number of experiments which demonstrate more robust results in comparison with state-of-the-art work. In a broader scope, our representation can be viewed as a two-level inhomogeneous MRF model which is applicable to general shape-from-X problems. Our study is an attempt to revisit Marr's idea (Marr and Freeman, 1982) of computing the 2frac12D sketch from primal sketch. In a companion paper (Barbu and Zhu, 2005), we study shape from stereo based on a similar two-level generative sketch representation. Cognitive experiments show that humans can read graph drawings in which all edge crossings are at right angles equally well as they can read planar drawings; they also show that the readability of a drawing is heavily affected by the number of bends along the edges. A graph visualization whose edges can only cross perpendicularly is called a RAC (Right Angle Crossing) drawing . This paper initiates the study of combinatorial and algorithmic questions related with the problem of computing RAC drawings with few bends per edge. Namely, we study the interplay between number of bends per edge and total number of edges in RAC drawings. We establish upper and lower bounds on these quantities by considering two classical graph drawing scenarios: The one where the algorithm can choose the combinatorial embedding of the input graph and the one where this embedding is fixed. Abstract A drawing of a graph is 1-planar if each edge is crossed at most once. A graph is 1-planar if it has a 1-planar drawing. A k-bend RAC (Right Angle Crossing) drawing of a graph is a polyline drawing where each edge has at most k bends and edges cross only at right angles. A graph is k-bend RAC if it has a k -bend RAC drawing. A 0-bend RAC graph (drawing) is also called a straight-line RAC graph (drawing) . The relationships between 1-planar and k -bend RAC graphs have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC and straight-line RAC graphs that are not 1-planar. The existence of 1-planar straight-line RAC drawings has been proven only for restricted families of 1-planar graphs. Two of the main questions still open are: ( i ) What is the complexity of deciding whether a graph has a drawing that is both 1-planar and straight-line RAC? ( i i ) Does every 1-planar graph have a drawing that is both 1-planar and 1-bend RAC? In this paper we answer these two questions. Namely, we prove an NP-hardness result for the first question, and we positively answer the second question by describing a drawing algorithm for 1-planar graphs. We present a novel method for recovering the 3D-line structure of a scene from multiple widely separated views. Traditional optimization-based approaches to line-based structure from motion minimize the error between measured line segments and the projections of corresponding 3D lines. In such a case, 3D lines can be optimized using a minimum of 4 parameters. We show that this number of parameters can be further reduced by introducing additional constraints on the orientations of lines in a 3D scene. In our approach, 2D-lines are automatically detected in images with the assistance of an EM-based vanishing point estimation method which assumes the existence of edges along mutally orthogonal vanishing directions. Each detected line is automatically labeled with the orientation (e.g. vertical, horizontal) of the 3D line which generated the measurement, and it is this additional knowledge that we use to reduce the number of degrees of freedom of 3D lines during optimization. We present 3D reconstruction results for urban scenes based on manually established feature correspondences across images. Abstract In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given two-dimensional signal of size N × N , the proposed algorithm's complexity is O ( N 2 log N ) , just like in a Cartesian 2D-FFT. A special feature of our approach is that it involves only 1D equispaced FFT's and 1D interpolations. A central tool in our method is the pseudo-Polar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudo-Polar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT's, the pseudo-Polar FFT plays the role of a halfway point—a nearly-Polar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT's and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesian-based unequally-sampled FFT method to ours, both algorithms using a small-support interpolation and no pre-compensating, and show marked advantage to the use of the pseudo-Polar initial grid.