## Drawing Directed Graphs Using One-Dimensional Optimization (2001)

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Venue: | Proc. Graph Drawing 2002, LNCS 2528 |

Citations: | 9 - 6 self |

### BibTeX

@INPROCEEDINGS{Carmel01drawingdirected,

author = {Liram Carmel and David Harel and Yehuda Koren},

title = {Drawing Directed Graphs Using One-Dimensional Optimization},

booktitle = {Proc. Graph Drawing 2002, LNCS 2528},

year = {2001},

pages = {193--206},

publisher = {Springer-Verlag}

}

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### Abstract

We present an algorithm for drawing directed graphs, which is based on rapidly solving a unique one-dimensional optimization problem for each of the axes. The algorithm results in a clear description of the hierarchy structure of the graph. Nodes are not restricted to lie on fixed horizontal layers, resulting in layouts that convey the symmetries of the graph very naturally. The algorithm can be applied without change to cyclic or acyclic digraphs, and even to graphs containing both directed and undirected edges. We also derive a hierarchy index from the input digraph, which quantitatively measures its amount of hierarchy.

### Citations

9061 | Introduction to Algorithms
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- 2001
(Show Context)
Citation Context ...gure 1 of [10]. There, the authors obtained the drawing of this digraph using four different variations of standard layering algorithms. Figure 15(b) shows an example that was manually illustrated in =-=[1]-=-, page 487. In Figure 15(c) we bring our drawing of the digraph shown in Figure 9.1 of [3]. All these examples demonstrate the way our algorithm captures the structure of the digraphs. A salient diffe... |

521 | An algorithm for drawing general undirected graphs
- Kamada, Kawai
(Show Context)
Citation Context ...graph drawing. Probably, the majority of the undirected graph drawing algorithms are of this type, would it be by directly assigning forces between the nodes [5, 6], or by minimizing energy functions =-=[11, 2]-=-. We are aware of only one other occasion where a force directed model was suggested for the benefit of digraph drawing [15], forcing directionality by applying a homogeneous magnetic field and favori... |

395 | How to draw a graph
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- 1963
(Show Context)
Citation Context ...ighted cyclic digraph; (b) its optimal arrangement. The idea of using energy minimization to determine a vector of coordinates on one axis was already exploited in undirected graph drawing by � Tutt=-=e [12] a-=-nd Hall [5], n i,j=1 wij(yi − yj) 2 = both utilizing the same quadratic energy function, ETH = 1 2 y T Ly. Comparing this energy with the hierarchy energy (1), it is clear that they become identical... |

361 |
Methods for Visual Understanding of Hierarchical Systems Structure
- SUGIYAMA, TAGAWA, et al.
- 1989
(Show Context)
Citation Context ...opriate for digraphs. Consequently, algorithms for digraph drawing usually adopt different strategies from their undirected counterparts. The dominant strategy, rooted in the work of Sugiyama et. al. =-=[11]-=-, is based on separating the axes, where the y-axis represents the directional information, or hierarchy, and the x-axis allows for additional aesthetic considerations, such as shortening edge lengths... |

191 | Drawing Graphs Nicely Using Simulated Annealing
- Davison, Harel
- 1996
(Show Context)
Citation Context ...aph drawing. Probably, the majority of the undirected graph drawing algorithms are of this type, would it be by 3sdirectly assigning forces between the nodes [5, 6], or by minimizing energy functions =-=[9, 2]-=-. We are aware of only one other occasion where a force directed model was suggested for digraph drawing, [14], forcing directionality by applying a homogeneous magnetic field and favoring edges that ... |

170 |
An R-Dimensional Quadratic Placement Algorithm
- Hall
- 1970
(Show Context)
Citation Context ...e real eigenvalues. Moreover, when G is a connected digraph, L has exactly one zero eigenvalue, corresponding to the eigenvector c · 1n, where 1n = (1, . . . , 1) T and c any constant. Proof See Hall=-= [8]-=-, or Koren et al. [12]. Naturally, a drawing algorithm has to deal only with connected digraphs. When a digraph is disconnected, one should draw each of its connected sub-digraphs separately. Unless o... |

129 | A survey of graph layout problems
- Diaz, Petit, et al.
(Show Context)
Citation Context ...ind the minimizer of ET H using ACE [12] — an extremely fast multiscale algorithm for undirected graph drawing. • Minimizing edge lengths: This is the well known problem of minimum linear arrangem=-=ent [4]. Th-=-e solution is obtained by minimizing the energy function ELA = 1 2 � n i,j=1 wij|xi − xj|, where (x1, . . . , xn) is a permutation of (1, . . . , n). This is an NP-hard problem, and hence we shoul... |

124 |
Graph Drawing by Force-Directed Placement. Software-Practice & Experience
- Fruchterman, Reingold
- 1991
(Show Context)
Citation Context ...r in undirected graph drawing than in digraph drawing. Probably, the majority of the undirected graph drawing algorithms are of this type, would it be by 3sdirectly assigning forces between the nodes =-=[5, 6]-=-, or by minimizing energy functions [9, 2]. We are aware of only one other occasion where a force directed model was suggested for digraph drawing, [14], forcing directionality by applying a homogeneo... |

102 |
A heuristic for graph drawing. Congressus Numerantium
- Eades
- 1984
(Show Context)
Citation Context ...r in undirected graph drawing than in digraph drawing. Probably, the majority of the undirected graph drawing algorithms are of this type, would it be by 3sdirectly assigning forces between the nodes =-=[5, 6]-=-, or by minimizing energy functions [9, 2]. We are aware of only one other occasion where a force directed model was suggested for digraph drawing, [14], forcing directionality by applying a homogeneo... |

72 |
eds. Drawing Graphs: Methods and Models
- Kaufmann, Wagner
- 2001
(Show Context)
Citation Context ... stage determines the exact locations of the nodes along the x-axis, taking into account various parameters, such as the finite size of the nodes and the smoothness of the edges. For more details see =-=[3, 10]-=-. Such digraph drawing algorithms have evolved to produce nice and useful layouts for many different types of digraphs. Nevertheless, we would like to point out two inherent properties of the standard... |

64 | Ace: A fast multiscale eigenvectors computation for drawing huge graphs
- Koren, Carmel, et al.
(Show Context)
Citation Context ...nce the Laplacian is independent of ∆, the definition does not distinguish between digraphs and undirected graphs. The Laplacian has a key role in some undirected graph drawing algorithms, see, e.g.=-=, [12], -=-and will be shown to play a fundamental role here, too. One of its most important properties is the following: Lemma 2.1 Let G(V, W, ∆) be a digraph. Its Laplacian is a positive semi-definite matrix... |

60 | Specifying and executing behavioral requirements: the play in/playout approach
- Harel, Marelly
- 2003
(Show Context)
Citation Context ...ique hierarchical status. Finally, we bring in Figure 16 two digraphs that describe partial order of system execution, as generated by the play engine tool for requirement specification and execution =-=[9]-=-. These examples are interesting, because they describe common execution constructs, such as loops and conditions. 6 Discussion We have presented a digraph drawing algorithm that uses a sequence of th... |

27 | A multi-scale algorithm for the linear arrangement problem
- Koren, Harel
- 2002
(Show Context)
Citation Context ...ined by minimizing the energy function ELA = 1 2 n∑ wij|xi − xj|, i,j=1 where (x1, . . . , xn) is a permutation of (1, . . . , n). We find the minimizer of ELA using another fast multiscale algorithm =-=[14]-=-, designed especially for this problem. In most of the cases we have studied, the Fiedler vector was inferior with respect to the final result. The reason for this is that nothing in the Tutte-Hall en... |

20 |
A Simple and Unified Method for Drawing Graphs: Magnetic-Spring Algorithm
- Sugiyama, Misue
- 1995
(Show Context)
Citation Context ...3sdirectly assigning forces between the nodes [5, 6], or by minimizing energy functions [9, 2]. We are aware of only one other occasion where a force directed model was suggested for digraph drawing, =-=[14]-=-, forcing directionality by applying a homogeneous magnetic field and favoring edges that are parallel to its field lines. Yet, we are under the impression that the inferred energy function is complic... |

7 |
How to layer a directed acyclic graph
- Healy, Nikolov
- 2002
(Show Context)
Citation Context ...hs that have appeared in literature, and the reader is invited to compare our results with those shown in the original works. Figure 15(a) shows an example that was analyzed in details in Figure 1 of =-=[10]-=-. There, the authors obtained the drawing of this digraph using four different variations of standard layering algorithms. Figure 15(b) shows an example that was manually illustrated in [1], page 487.... |

3 | Fast layout methods for timetable graphs
- Brandes, Shubina, et al.
- 1984
(Show Context)
Citation Context ...(yi−yj −δij) 2 be small. The larger this quantity, the smaller (yi − yj − δij) 2 should be, in order to keep the contribution to the energy small. A similar energy function was used by Brand=-=es et al. [1]-=- for a completely different drawing application (edge routing of timetable graphs). Using the previously defined notions of Laplacian and balance, the hierarchy energy can be written in a compact form... |

3 |
On Spectral Graph Drawing”, manuscript
- Koren
- 2002
(Show Context)
Citation Context ...ethod: • Minimizing edge-squared lengths: This means minimizing the already familiar Tutte-Hall energy function, ET H = 1 2 � n i,j=1 wij(xi − xj) 2 = x T Lx. As discussed in Subsection 3.1, see=-= also [11]-=-, the non-trivial minimizer of this energy function is the Fiedler vector, which is the eigenvector of the Laplacian associated with the smallest positive eigenvalue. We find the minimizer of ET H usi... |

2 |
A Survey on Graph Layout Problems”, Technical report LSI-0061-R, Universitat Politècnica de Catalunya, Departament de Llenguatges i Sistemes Informàtics
- Diaz, Petit, et al.
(Show Context)
Citation Context ... fast multiscale algorithm for undirected graph drawing which computes, among other things, the Fiedler vector. – Minimizing edge lengths: This is the well known problem of minimum linear arrangement =-=[4]-=-. The solution is obtained by minimizing the energy function ELA = 1 2 n∑ wij|xi − xj|, i,j=1 where (x1, . . . , xn) is a permutation of (1, . . . , n). We find the minimizer of ELA using another fast... |

1 |
A Simple and Unified Method fro Drawing Graphs: MagneticSpring Algorithm
- Sugiyama, Misue
- 1995
(Show Context)
Citation Context ...gning forces between the nodes [5, 6], or by minimizing energy functions [11, 2]. We are aware of only one other occasion where a force directed model was suggested for the benefit of digraph drawing =-=[15]-=-, forcing directionality by applying a homogeneous magnetic field and favoring edges that are parallel to its field lines. Yet, we are under the impression that the inferred energy function is complic... |