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Register a new userStudy of Neural Network Algorithm for Straight-Line Drawings of Planar Graphs
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Added by
Mohamed A. El-Sayed
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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No Arabic abstract
Graph drawing addresses the problem of finding a layout of a graph that satisfies given aesthetic and understandability objectives. The most important objective in graph drawing is minimization of the number of crossings in the drawing, as the aesthetics and readability of graph drawings depend on the number of edge crossings. VLSI layouts with fewer crossings are more easily realizable and consequently cheaper. A straight-line drawing of a planar graph G of n vertices is a drawing of G such that each edge is drawn as a straight-line segment without edge crossings. However, a problem with current graph layout methods which are capable of producing satisfactory results for a wide range of graphs is that they often put an extremely high demand on computational resources. This paper introduces a new layout method, which nicely draws internally convex of planar graph that consumes only little computational resources and does not need any heavy duty preprocessing. Here, we use two methods: The first is self organizing map known from unsupervised neural networks which is known as (SOM) and the second method is Inverse Self Organized Map (ISOM).
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We consider straight line drawings of a planar graph $G$ with possible edge crossings. The emph{untangling problem} is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let $fix(G)$ denote the maximum number of vertices that can be left fixed in the worst case. In the emph{allocation problem}, we are given a planar graph $G$ on $n$ vertices together with an $n$-point set $X$ in the plane and have to draw $G$ without edge crossings so that as many vertices as possible are located in $X$. Let $fit(G)$ denote the maximum number of points fitting this purpose in the worst case. As $fix(G)le fit(G)$, we are interested in upper bounds for the latter and lower bounds for the former parameter. For each $epsilon>0$, we construct an infinite sequence of graphs with $fit(G)=O(n^{sigma+epsilon})$, where $sigma<0.99$ is a known graph-theoretic constant, namely the shortness exponent for the class of cubic polyhedral graphs. To the best of our knowledge, this is the first example of graphs with $fit(G)=o(n)$. On the other hand, we prove that $fix(G)gesqrt{n/30}$ for all $G$ with tree-width at most 2. This extends the lower bound obtained by Goaoc et al. [Discrete and Computational Geometry 42:542-569 (2009)] for outerplanar graphs. Our upper bound for $fit(G)$ is based on the fact that the constructed graphs can have only few collinear vertices in any crossing-free drawing. To prove the lower bound for $fix(G)$, we show that graphs of tree-width 2 admit drawings that have large sets of collinear vertices with some additional special properties.
We investigate straight-line drawings of topological graphs that consist of a planar graph plus one edge, also called almost-planar graphs. We present a characterization of such graphs that admit a straight-line drawing. The characterization enables a linear-time testing algorithm to determine whether an almost-planar graph admits a straight-line drawing, and a linear-time drawing algorithm that constructs such a drawing, if it exists. We also show that some almost-planar graphs require exponential area for a straight-line drawing.
A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For k=0 and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of layered 1-bend drawings.
Symmetry is an important factor in human perception in general, as well as in the visualization of graphs in particular. There are three main types of symmetry: reflective, translational, and rotational. We report the results of a human subjects experiment to determine what types of symmetries are more salient in drawings of graphs. We found statistically significant evidence that vertical reflective symmetry is the most dominant (when selecting among vertical reflective, horizontal reflective, and translational). We also found statistically significant evidence that rotational symmetry is affected by the number of radial axes (the more, the better), with a notable exception at four axes.
This paper studies the problem of computing quasi-upward planar drawings of bimodal plane digraphs with minimum curve complexity, i.e., drawings such that the maximum number of bends per edge is minimized. We prove that every bimodal plane digraph admits a quasi-upward planar drawing with curve complexity two, which is worst-case optimal. We also show that the problem of minimizing the curve complexity in a quasi-upward planar drawing can be modeled as a min-cost flow problem on a unit-capacity planar flow network. This gives rise to an $tilde{O}(m^frac{4}{3})$-time algorithm that computes a quasi-upward planar drawing with minimum curve complexity; in addition, the drawing has the minimum number of bends when no edge can be bent more than twice. For a contrast, we show bimodal planar digraphs whose bend-minimum quasi-upward planar drawings require linear curve complexity even in the variable embedding setting.
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