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An SDP Approach to Multilevel Crossing Minimization
"... We present an approach based on semidefinite programs (SDP) to tackle the multilevel crossing minimization problem. Thereby, we are given a layered graph (i.e., the graph’s vertices are assigned to multiple parallel levels) and ask for an ordering of the nodes on their levels such that, when drawin ..."
Abstract

Cited by 7 (5 self)
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We present an approach based on semidefinite programs (SDP) to tackle the multilevel crossing minimization problem. Thereby, we are given a layered graph (i.e., the graph’s vertices are assigned to multiple parallel levels) and ask for an ordering of the nodes on their levels such that, when drawing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in the probably most widely used graph drawing scheme, the socalled Sugiyama framework. The problem has received a lot of attention both in the field of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable at least to small graphs. Recently, SDP formulations for the special case of two levels were proposed
Global kLevel Crossing Reduction
, 2011
"... Directed graphs are commonly drawn by a four phase framework introduced by Sugiyama et al. in 1981. The vertices are placed on parallel horizontal levels. The edge routing between consecutive levels is computed by solving onesided 2level crossing minimization problems, which are repeated in up and ..."
Abstract

Cited by 4 (2 self)
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Directed graphs are commonly drawn by a four phase framework introduced by Sugiyama et al. in 1981. The vertices are placed on parallel horizontal levels. The edge routing between consecutive levels is computed by solving onesided 2level crossing minimization problems, which are repeated in up and down sweeps over all levels. Crossing minimization problems are generally N Phard. We introduce a global crossing reduction, which at any particular time considers all crossings between all levels. Our approach is based on the sifting technique. It yields an improvement of 5 – 10 % in the number of crossings over the levelbylevel onesided 2level crossing reduction heuristics. In addition, it avoids type 2 conflicts which are crossings between edges whose endpoints are dummy vertices. This helps straightening long edges spanning many levels. Finally, the global crossing reduction approach can directly be extended to cyclic, radial, and clustered level graphs achieving similar improvements. The running time is quadratic in the size of the input graph, whereas the common levelbylevel approaches are faster but operate on larger graphs with many dummy vertices for long edges. Submitted:
Grid Sifting: Leveling and Crossing Reduction
, 2011
"... Directed graphs are commonly drawn by the Sugiyama algorithm where first vertices are placed on distinct hierarchical levels and second the vertices on the same level are permuted to reduce the overall number of crossings. Separating these two phases simplifies the algorithms but diminishes the qual ..."
Abstract

Cited by 3 (1 self)
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Directed graphs are commonly drawn by the Sugiyama algorithm where first vertices are placed on distinct hierarchical levels and second the vertices on the same level are permuted to reduce the overall number of crossings. Separating these two phases simplifies the algorithms but diminishes the quality of the result. We introduce a combined leveling and crossing reduction algorithm based on sifting, which prioritizes few crossings over few levels. It avoids type 2 conflicts which help to straighten the edges, and has a running time, which is roughly quadratic in the size of the input graph independent of dummy vertices.
Grid Shifting: Leveling . . .
, 2012
"... Directed graphs are commonly drawn by the Sugiyama algorithm where first vertices are placed on distinct hierarchical levels, and second the vertices on the same level are permuted to reduce the overall number of crossings. Separating these two phases simplifies the algorithms but diminishes the qua ..."
Abstract
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Directed graphs are commonly drawn by the Sugiyama algorithm where first vertices are placed on distinct hierarchical levels, and second the vertices on the same level are permuted to reduce the overall number of crossings. Separating these two phases simplifies the algorithms but diminishes the quality of the result. We introduce a combined leveling and crossing reduction algorithm based on sifting, which prioritizes few crossings over few levels. It avoids type 2 conflicts, which are crossings of edges whose endpoints are dummy vertices. This helps straightening long edges spanning many levels. The obtained running time is roughly quadratic in the size of the input graph and independent of dummy vertices.
RealTime and Embedded Systems Group, ChristianAlbrechtsUniversität zu Kiel
"... Abstract. Many practical applications for drawing graphs are modeled by directed graphs with domain specific constraints. In this paper, we consider the problem of drawing directed hypergraphs with (and without) port constraints, which cover multiple realworld graph drawing applications like data f ..."
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Abstract. Many practical applications for drawing graphs are modeled by directed graphs with domain specific constraints. In this paper, we consider the problem of drawing directed hypergraphs with (and without) port constraints, which cover multiple realworld graph drawing applications like data flow diagrams and electric schematics. Most existing algorithms for drawing hypergraphs with port constraints are adaptions of the framework originally proposed by Sugiyama et al. in 1981 for simple directed graphs. Recently, a practical approach for upward crossing minimization of directed graphs based on the planarization method was proposed [7]. With respect to the number of arc crossings, it clearly outperforms prior (mostly layeringbased) approaches. We show how to adopt this idea for hypergraphs with given port constraints, obtaining an upwardplanar representation (UPR) of the input hypergraph where crossings are modeled by dummy nodes. Furthermore, we present the new problem of computing an orthogonal upward drawing with minimal number of crossings from such an UPR, and show that it can be solved efficiently by providing a simple method. 1
Global kLevel Crossing Reduction
, 2011
"... Directed graphs are commonly drawn by a four phase framework introduced by Sugiyama et al. in 1981. The vertices are placed on parallel horizontal levels. The edge routing between consecutive levels is computed by solving onesided 2level crossing minimization problems, which are repeated in up and ..."
Abstract
 Add to MetaCart
Directed graphs are commonly drawn by a four phase framework introduced by Sugiyama et al. in 1981. The vertices are placed on parallel horizontal levels. The edge routing between consecutive levels is computed by solving onesided 2level crossing minimization problems, which are repeated in up and down sweeps over all levels. Crossing minimization problems are generally N Phard. We introduce a global crossing reduction, which at any particular time considers all crossings between all levels. Our approach is based on the sifting technique. It yields an improvement of 5 – 10 % in the number of crossings over the levelbylevel onesided 2level crossing reduction heuristics. In addition, it avoids type 2 conflicts which are crossings between edges whose endpoints are dummy vertices. This helps straightening long edges spanning many levels. Finally, the global crossing reduction approach can directly be extended to cyclic, radial, and clustered level graphs achieving similar improvements. The running time is quadratic in the size of the input graph, whereas the common levelbylevel approaches are faster but operate on larger graphs with many dummy vertices for long edges. Submitted: