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30
A Technique for Drawing Directed Graphs
 IEEE Transactions on Software Engineering
, 1993
"... We describe a fourpass algorithm for drawing directed graphs. The first pass finds an optimal rank assignment using a network simplex algorithm. The second pass sets the vertex order within ranks by an iterative heuristic incorporating a novel weight function and local transpositions to reduce cros ..."
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Cited by 222 (19 self)
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We describe a fourpass algorithm for drawing directed graphs. The first pass finds an optimal rank assignment using a network simplex algorithm. The second pass sets the vertex order within ranks by an iterative heuristic incorporating a novel weight function and local transpositions to reduce crossings. The third pass finds optimal coordinates for nodes by constructing and ranking an auxiliary graph. The fourth pass makes splines to draw edges. The algorithm makes good drawings and runs fast. 1.
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...
An Alternative Method to Crossing Minimization on Hierarchical Graphs
 SIAM J. Optimization
, 1997
"... . A common method for drawing directed graphs is, as a first step, to partition the vertices into a set of k levels and then, as a second step, to permute the vertices within the levels such that the number of crossings is minimized. We suggest an alternative method for the second step, namely, remo ..."
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Cited by 28 (0 self)
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. A common method for drawing directed graphs is, as a first step, to partition the vertices into a set of k levels and then, as a second step, to permute the vertices within the levels such that the number of crossings is minimized. We suggest an alternative method for the second step, namely, removing the minimal number of edges such that the resulting graph is klevel planar. For the final diagram the removed edges are reinserted into a klevel planar drawing. Hence, instead of considering the klevel crossing minimization problem, we suggest solving the klevel planarization problem. In this paper we address the case k = 2. First, we give a motivation for our approach. Then, we address the problem of extracting a 2level planar subgraph of maximum weight in a given 2level graph. This problem is NPhard. Based on a characterization of 2level planar graphs, we give an integer linear programming formulation for the 2level planarization problem. Moreover, we define and investigate t...
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...
A Polyhedral Approach to the MultiLayer Crossing Minimization Problem
 PROCEEDINGS OF THE INTERNATIONAL SYMPOSIUM ON GRAPH DRAWING, LECTURE NOTES IN COMPUTER SCIENCE 1353
, 1997
"... We study the multilayer crossing minimization problem from a polyhedral point of view. After the introduction of an integer programming formulation of the multilayer crossing minimization problem, we examine the 2layer case and derive several classes of facets of the associated polytope. Prelimin ..."
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Cited by 20 (2 self)
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We study the multilayer crossing minimization problem from a polyhedral point of view. After the introduction of an integer programming formulation of the multilayer crossing minimization problem, we examine the 2layer case and derive several classes of facets of the associated polytope. Preliminary computational results for 2 and 3layer instances indicate, that the usage of the corresponding facetdefining inequalities in a branchandcut approach may only lead to a practically useful algorithm, if deeper polyhedral studies are conducted.
Heuristics and Experimental Design for Bigraph Crossing Number Minimization
 IN ALGORITHM ENGINEERING AND EXPERIMENTATION (ALENEX’99), NUMBER 1619 IN LECTURE NOTES IN COMPUTER SCIENCE
, 1999
"... The bigraph crossing problem, embedding the two vertex sets of a bipartite graph G = (V0;V1;E) along two parallel lines so that edge crossings are minimized, has application to circuit layout and graph drawing. We consider the case where both V0 and V1 can be permuted arbitrarily  both this and ..."
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Cited by 14 (9 self)
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The bigraph crossing problem, embedding the two vertex sets of a bipartite graph G = (V0;V1;E) along two parallel lines so that edge crossings are minimized, has application to circuit layout and graph drawing. We consider the case where both V0 and V1 can be permuted arbitrarily  both this and the case where the order of one vertex set is fixed are NPhard. Two new heuristics that perform well on sparse graphs such as occur in circuit layout problems are presented. The new heuristics outperform existing heuristics on graph classes that range from applicationspecific to random. Our experimental design methodology ensures that differences in performance are statistically significant and not the result of minor variations in graph structure or input order.
A FixedParameter Approach to TwoLayer Planarization
, 2002
"... A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar ..."
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Cited by 12 (4 self)
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A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar. This problem is NPcomplete, as is the 1Layer Planarization problem in which the permutation of the vertices in one layer is fixed. We give the following fixed parameter tractability results: an O(k ·6 k +G) algorithm for 2Layer Planarization and an O(3 k ·G) algorithm for 1Layer Planarization, thus achieving linear time for fixed k.
A Fast Layout Algorithm for kLevel Graphs
 PROC. GRAPH DRAWING: 8TH INTERNATIONAL SYMPOSIUM (GD'00), VOLUME 1984 OF LECTURE NOTES IN COMPUT. SCI
, 1999
"... In this paper, we present a fast layout algorithm for klevel graphs with given permutations of the vertices on each level. The algorithm can be used in particular as a third phase of the Sugiyama algorithm [STT81]. The Sugiyama algorithm computes a layout for an arbitrary graph by (1) converting it ..."
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Cited by 12 (3 self)
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In this paper, we present a fast layout algorithm for klevel graphs with given permutations of the vertices on each level. The algorithm can be used in particular as a third phase of the Sugiyama algorithm [STT81]. The Sugiyama algorithm computes a layout for an arbitrary graph by (1) converting it into a klevel graph, (2) reducing the number of edge crossings by permuting the vertices on the levels, and (3) assigning ycoordinates to the levels and xcoordinates to the vertices. In the layouts generated by our algorithm, every edge will have at most two bends, and will be drawn vertically between these bends.
Drawing Planar Partitions I: LLDrawings and LHDrawings
, 1998
"... Let a planar graph G = (V; E) and a partition V = A[B of the vertices be given. Can we draw G without edge crossings such that the partition is clearly visible? Such drawings aid to display partitions and cuts as they arise in various applications. In this paper, we first review a number of mode ..."
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Cited by 11 (1 self)
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Let a planar graph G = (V; E) and a partition V = A[B of the vertices be given. Can we draw G without edge crossings such that the partition is clearly visible? Such drawings aid to display partitions and cuts as they arise in various applications. In this paper, we first review a number of models of displaying the partition. Studying two of these models in detail, we provide necessary and sufficient conditions for the existence of a straightline planar drawing, and algorithms to create such drawings, if possible, with area O(n²).
TextFlow: Towards better understanding of evolving topics in text
 Proc. IEEE Symp. Information Visualization (InfoVis), 17(12):2412–2421, 2011
"... Abstract—Understanding how topics evolve in text data is an important and challenging task. Although much work has been devoted to topic analysis, the study of topic evolution has largely been limited to individual topics. In this paper, we introduce TextFlow, a seamless integration of visualization ..."
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Cited by 10 (3 self)
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Abstract—Understanding how topics evolve in text data is an important and challenging task. Although much work has been devoted to topic analysis, the study of topic evolution has largely been limited to individual topics. In this paper, we introduce TextFlow, a seamless integration of visualization and topic mining techniques, for analyzing various evolution patterns that emerge from multiple topics. We first extend an existing analysis technique to extract threelevel features: the topic evolution trend, the critical event, and the keyword correlation. Then a coherent visualization that consists of three new visual components is designed to convey complex relationships between them. Through interaction, the topic mining model and visualization can communicate with each other to help users refine the analysis result and gain insights into the data progressively. Finally, two case studies are conducted to demonstrate the effectiveness and usefulness of TextFlow in helping users understand the major topic evolution patterns in timevarying text data. Index Terms—Text visualization, Topic evolution, Hierarchical Dirichlet process, Critical event. 1