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74
2Layer Straightline Crossing Minimization: Performance of Exact and Heuristic Algorithms
, 1997
"... We present algorithms for the two layer straightline crossing minimization problem that are able to compute exact optima. Our computational results lead us to the conclusion that there is no need for heuristics if one layer is fixed, even though the problem is NPhard, and that for the general probl ..."
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Cited by 64 (6 self)
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We present algorithms for the two layer straightline crossing minimization problem that are able to compute exact optima. Our computational results lead us to the conclusion that there is no need for heuristics if one layer is fixed, even though the problem is NPhard, and that for the general problem with two variable layers, true optima can be computed for sparse instances in which the smaller layer contains up to 15 nodes. For bigger instances, the iterated barycenter method turns out to be the method of choice among several popular heuristics whose performance we could assess by comparing their results to optimum solutions.
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 31 (19 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
personal communication
, 1997
"... these abstracts from the Eighteenth Annual Scientific Meeting may not present completed work nor were they formally peerreviewed for technical content. Individuals wishing to reference or quote from these abstracts in whole or part should obtain the author's permission. Abstracts were optically sca ..."
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Cited by 26 (1 self)
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these abstracts from the Eighteenth Annual Scientific Meeting may not present completed work nor were they formally peerreviewed for technical content. Individuals wishing to reference or quote from these abstracts in whole or part should obtain the author's permission. Abstracts were optically scanned and then edited for entry into this compilation. But since the process is not perfect, errors may have been introduced, for which we apologize.
Crossing reduction in circular layouts
 PROC. WORKSHOP ON GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE, WG 2004, VOLUME 3353 OF LNCS
, 2004
"... We propose a twophase heuristic for crossing reduction in circular layouts. While the first algorithm uses a greedy policy to build a good initial layout, an adaptation of the sifting heuristic for crossing reduction in layered layouts is used for local optimization in the second phase. Both phase ..."
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Cited by 22 (4 self)
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We propose a twophase heuristic for crossing reduction in circular layouts. While the first algorithm uses a greedy policy to build a good initial layout, an adaptation of the sifting heuristic for crossing reduction in layered layouts is used for local optimization in the second phase. Both phases are conceptually simpler than previous heuristics, and our extensive experimental results indicate that they also yield fewer crossings. An interesting feature is their straightforward generalization to the weighted case.
A Practical Approach to Drawing Undirected Graphs
, 1994
"... Although there is extensive research on drawing graphs, none of the published methods are satisfactory for drawing general undirected graphs. Generating drawings which are optimal with respect to several aesthetic criteria is known to be NPhard, so all published approaches to the problem have used ..."
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Cited by 21 (2 self)
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Although there is extensive research on drawing graphs, none of the published methods are satisfactory for drawing general undirected graphs. Generating drawings which are optimal with respect to several aesthetic criteria is known to be NPhard, so all published approaches to the problem have used heuristics. These heuristics are too slow to be practical for graphs of moderate size, and they do not produce consistently good drawings for general graphs. Moreover, they rely on general optimization methods, because problemspecific methods require a deeper theoretical understanding of the graph drawing problem. This paper presents an algorithm to generate twodimensional drawings of undirected graphs. The algorithm uses a combination of heuristics to obtain drawings which are nearoptimal with respect to an aesthetic cost function. The algorithm is incremental in nature, but preprocesses the graph to determine an order for node placement. The algorithm uses a local optimization strategy...
On the Parameterized Complexity of Layered Graph Drawing
 PROC. 5TH ANNUAL EUROPEAN SYMP. ON ALGORITHMS (ESA '01
, 2001
"... We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for ..."
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Cited by 20 (8 self)
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We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a lineartime algorithm to decide if a graph has a crossingfree hlayer drawing (for fixed h). This algorithm is extended to solve a large number of related problems, including allowing at most k crossings, or removing at most r edges to leave a crossingfree drawing (for fixed k or r). If the number of crossings or deleted edges is a nonfixed parameter then these problems are NPcomplete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case the total span or the maximum span of edges can be minimized. In contrast to the socalled Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.
An Algorithm For Drawing A Hierarchical Graph
, 1995
"... this paper we present a method for drawing "hierarchical directed graphs", which are digraphs in which each node is assigned a layer, as in Figure 1. ..."
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Cited by 18 (7 self)
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this paper we present a method for drawing "hierarchical directed graphs", which are digraphs in which each node is assigned a layer, as in Figure 1.
On Bipartite Drawings and the Linear Arrangement Problem
"... The bipartite crossing number problem is studied, and a connection between this problemand the linear arrangement problem is established. It is shown that when the arboricity is close ..."
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Cited by 17 (0 self)
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The bipartite crossing number problem is studied, and a connection between this problemand the linear arrangement problem is established. It is shown that when the arboricity is close
A Radial Adaptation of the Sugiyama Framework for Visualizing Hierarchical Information
, 2007
"... In radial drawings of hierarchical graphs the vertices are placed on concentric circles rather than on horizontal lines and the edges are drawn as outwards monotone segments of spirals rather than straight lines as it is both done in the standard Sugiyama framework. This drawing style is well suite ..."
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Cited by 16 (6 self)
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In radial drawings of hierarchical graphs the vertices are placed on concentric circles rather than on horizontal lines and the edges are drawn as outwards monotone segments of spirals rather than straight lines as it is both done in the standard Sugiyama framework. This drawing style is well suited for the visualisation of centrality in social networks and similar concepts. Radial drawings also allow a more flexible edge routing than horizontal drawings, as edges can be routed around the center in two directions. In experimental results this reduces the number of crossings by approximately 30 percent on average. Few crossings are one of the major criteria for human readability. This paper is a detailed description of a complete framework for visualizing hierarchical information in a new radial fashion. Particularly, we briefly cover extensions of the level assignment step to benefit by the increasing perimeters of the circles, present three heuristics for crossing reduction in radial level drawings, and also show how to visualize the results.
Simple and Efficient Bilayer Cross Counting
, 2002
"... We consider the problem of counting the interior edge crossings when a bipartite graph G = (V, E) with node set V and edge set E is drawn such that the nodes of the two shores of the bipartition are on two parallel lines and the edges are straight lines. The efficient solution of this problem is imp ..."
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Cited by 15 (2 self)
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We consider the problem of counting the interior edge crossings when a bipartite graph G = (V, E) with node set V and edge set E is drawn such that the nodes of the two shores of the bipartition are on two parallel lines and the edges are straight lines. The efficient solution of this problem is important in layered graph drawing. Our main observation is that it can be reduced to counting the inversions of a certain sequence. This leads to an O(E + C) algorithm, where C denotes the set of pairwise interior edge crossings, as well as to a simple O(E log V) algorithm, where V small is the smaller cardinality node set in the bipartition of the node set V of the graph. We present the algorithms and the results of computational experiments with these and other algorithms on a large collection of instances.