## Heuristics and Experimental Design for Bigraph Crossing Number Minimization (1999)

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Venue: | IN ALGORITHM ENGINEERING AND EXPERIMENTATION (ALENEX’99), NUMBER 1619 IN LECTURE NOTES IN COMPUTER SCIENCE |

Citations: | 14 - 9 self |

### BibTeX

@INPROCEEDINGS{Stallmann99heuristicsand,

author = {Matthias Stallmann and Franc Brglez and Debabrata Ghosh},

title = {Heuristics and Experimental Design for Bigraph Crossing Number Minimization},

booktitle = {IN ALGORITHM ENGINEERING AND EXPERIMENTATION (ALENEX’99), NUMBER 1619 IN LECTURE NOTES IN COMPUTER SCIENCE},

year = {1999},

pages = {74--93},

publisher = {Springer Verlag}

}

### Years of Citing Articles

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### Abstract

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 NP-hard. 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 application-specific 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.

### Citations

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Citation Context ... design of experiments 1 Introduction The minimization of the crossing number in a specific graph embedding has often been motivated by factors such as (1) improving the appearance of a graph drawing =-=[5, 8, 11, 22, 28]-=-, (2) reducing the wiring congestion and crosstalk in VLSI circuits, which in turn may reduce the total wire length and the layout area [18, 21, 24, 25]. This paper is about bigraph crossing defined a... |

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Citation Context ...s that can be extracted from VLSI designs such as shown in Figure 6. This particular example is based on the largest connected component found in a dag, referenced as a classifier controller or cc in =-=[23]-=-. Since the circuit cc implements the controller, there are a few nodes with a large out-degree (or fanout) that introduce unavoidable edge crossings; note however, that there is also a small biplanar... |

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Citation Context ... design of experiments 1 Introduction The minimization of the crossing number in a specific graph embedding has often been motivated by factors such as (1) improving the appearance of a graph drawing =-=[5, 8, 11, 22, 28]-=-, (2) reducing the wiring congestion and crosstalk in VLSI circuits, which in turn may reduce the total wire length and the layout area [18, 21, 24, 25]. This paper is about bigraph crossing defined a... |

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Citation Context ...) improving the appearance of a graph drawing [5, 8, 11, 22, 28], (2) reducing the wiring congestion and crosstalk in VLSI circuits, which in turn may reduce the total wire length and the layout area =-=[18, 21, 24, 25]-=-. This paper is about bigraph crossing defined as follows for any bipartite graph (bigraph) G = (V 0 ; V 1 ; E) [15]: Let G be embedded in the plane so that the nodes in V i occupy distinct positions ... |

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Citation Context ... design of experiments 1 Introduction The minimization of the crossing number in a specific graph embedding has often been motivated by factors such as (1) improving the appearance of a graph drawing =-=[5, 8, 11, 22, 28]-=-, (2) reducing the wiring congestion and crosstalk in VLSI circuits, which in turn may reduce the total wire length and the layout area [18, 21, 24, 25]. This paper is about bigraph crossing defined a... |

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Citation Context ...) improving the appearance of a graph drawing [5, 8, 11, 22, 28], (2) reducing the wiring congestion and crosstalk in VLSI circuits, which in turn may reduce the total wire length and the layout area =-=[18, 21, 24, 25]-=-. This paper is about bigraph crossing defined as follows for any bipartite graph (bigraph) G = (V 0 ; V 1 ; E) [15]: Let G be embedded in the plane so that the nodes in V i occupy distinct positions ... |

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Citation Context ...istics for both fixed-layer and general bigraph crossing has mostly been theoretical [5, 10, 9, 19, 20]. Experimental evaluation has focused on dense graphs, for which good lower bounds are available =-=[6, 17]-=-, and sparse random graphs [17]. Graphs arising in circuit design are very sparse and highly structured. Moreover, the dot package [11], used widely for graph drawing, has not been * F. Brglez and D. ... |

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Citation Context ...g prohibitive execution time. First, a connected graph is biplanar iff it is a comb (tree that becomes a path when its leaves are removed) [14] and its embedding can be computed easily in linear time =-=[7]-=-: the trick is to embed the spine (path that remains when leaves are deleted from the comb) from left to right, zig-zagging from layer to layer, and inserting any leaves attached to a spine node to th... |

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Citation Context ...istics for both fixed-layer and general bigraph crossing has mostly been theoretical [5, 10, 9, 19, 20]. Experimental evaluation has focused on dense graphs, for which good lower bounds are available =-=[6, 17]-=-, and sparse random graphs [17]. Graphs arising in circuit design are very sparse and highly structured. Moreover, the dot package [11], used widely for graph drawing, has not been * F. Brglez and D. ... |

33 |
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Citation Context ...ect to the constraint that the permutation of V 0 on y = 0 must stay fixed. Even that is NP-hard [10]. Work on heuristics for both fixed-layer and general bigraph crossing has mostly been theoretical =-=[5, 10, 9, 19, 20]-=-. Experimental evaluation has focused on dense graphs, for which good lower bounds are available [6, 17], and sparse random graphs [17]. Graphs arising in circuit design are very sparse and highly str... |

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Citation Context ...sentations imposes a distribution on C(h(hG; �� 0 ; �� 1 i)). Recently the distribution of C(h(hG 0 ; �� 0 ; �� 1 i)) using a specific heuristic h became an important factor in charact=-=erizing a graph [13]: ��-=- 0 and �� 1 were chosen randomly and G 0 was chosen from a well-defined set of perturbations of G. In this paper, we demonstrate the converse: sets of graph presentations have an important role in... |

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Citation Context ...erimental evaluation of heuristics by (1) creating a presentation equivalence class and (2) repeating the experiments for each member in the class. The basic abstractions for such experiments include =-=[2]-=-: 1. an equivalence class of experimental subjects, eligible for a treatment; 2. application of a specific treatment; 3. statistical evaluation of treatment effectiveness. Here, a treatment is synonym... |

25 |
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Citation Context ...mean of the set of neighboring positions, rather than the median, as a key for sorting (the name comes from the fact that it's a one-dimensional analog of Tutte's barycenter method for drawing graphs =-=[26, 27]-=-). It performs poorly on paths, cycles, and other very sparse highlystructured graphs (since these require decisive movement of degree-2 nodes). On graphs that are more random and/or have several node... |

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Citation Context ...) improving the appearance of a graph drawing [5, 8, 11, 22, 28], (2) reducing the wiring congestion and crosstalk in VLSI circuits, which in turn may reduce the total wire length and the layout area =-=[18, 21, 24, 25]-=-. This paper is about bigraph crossing defined as follows for any bipartite graph (bigraph) G = (V 0 ; V 1 ; E) [15]: Let G be embedded in the plane so that the nodes in V i occupy distinct positions ... |

16 | Design of experiments in BDD variable ordering: Lessons learned
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Citation Context ...ms that rely on hashing the input data may unknowingly undo the randomization of input presentations and confound the experiments. An important lesson on this subject has been learned and reported in =-=[16]-=-. In addition to using the cc graph directly, we use three types of reference graphs in sizes that are increasing powers of 2: (1) the biplanar type of size q is a comb with 2q + 1 net nodes and 2q ce... |

13 |
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Citation Context |

13 |
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Citation Context |

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Citation Context ...mean of the set of neighboring positions, rather than the median, as a key for sorting (the name comes from the fact that it's a one-dimensional analog of Tutte's barycenter method for drawing graphs =-=[26, 27]-=-). It performs poorly on paths, cycles, and other very sparse highlystructured graphs (since these require decisive movement of degree-2 nodes). On graphs that are more random and/or have several node... |

5 |
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Citation Context ...(G) subject to the constraint that the permutation of V0 on y =0must stay xed. Even that is NP-hard [9]. Work on heuristics for both xed-layer and general bigraph crossing has mostly been theoretical =-=[4, 8, 9, 20, 21]-=-. Experimental evaluation has focused on dense graphs, for which good lower bounds are available [5, 18], and sparse random graphs [18]. Graphs arising in circuit design are very sparse and highly str... |

4 |
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Citation Context ... the crossing number of the presentation. All classes of input data are being organized for easy access through the Web, similarly to the variety of illustrative experimental designs reported earlier =-=[4]. As in [4], web-posting will include complete 5 EXPERIMENTAL RESULTS 9 �� �-=-��������������� ���� ������ ���� �� �� �� �� �� 19 11 18 17 15 16 13 12 10 14 9 8 7 5 4 6 2 3 1 0 40 100 900 Crossing Number T... |

3 |
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Citation Context .... The (bigraph) crossing number C(G) = min f C f (G). Garey and Johnson [12] proved that it is NP-hard to compute C(G). Detection of a biplanar graph, a bigraph G for which C(G) = 0, however, is easy =-=[14]-=-. Previous work on bigraph crossing has focused primarily on the fixed-layer (version of the) problem, namely computing C(G) subject to the constraint that the permutation of V 0 on y = 0 must stay fi... |

2 | First steps towards experimental design in evaluating layout algorithms: Wire length versus wire crossing in linear placement optimization
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(Show Context)
Citation Context ...esting convergence properties (they converge quickly to a "local optimum"). -- Relationship between bigraph crossing and other objective functions for layout of VLSI circuits: preliminary ex=-=periments [12, 13]-=- indicate a high correlation with wire length in the final routing obtained by at least two different design automation tools. In fact, our bigraph crossing heuristics appear to achieve better wire le... |

1 |
A CBL PosterNote: Wire Crossing Problem and Benchmarking
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(Show Context)
Citation Context ...(C(G) is known for this graph, which we call cyclic), (c) perturbed classes (presentations have the form hG 0 ; �� 0 ; �� 1 i where G 0 is a well-defined perturbation of G) based on the graphs=-= in (b) [3]-=-. It quickly became apparent that dot performs far from optimal, even for graphs with as few as 9 nodes. An example of the observed performance with dot on the three classes is shown in Figure 1. All ... |

1 |
akinen, A Note on the Median Heuristic for Drawing Bipartite Graphs
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Citation Context ...ect to the constraint that the permutation of V 0 on y = 0 must stay fixed. Even that is NP-hard [10]. Work on heuristics for both fixed-layer and general bigraph crossing has mostly been theoretical =-=[5, 10, 9, 19, 20]-=-. Experimental evaluation has focused on dense graphs, for which good lower bounds are available [6, 17], and sparse random graphs [17]. Graphs arising in circuit design are very sparse and highly str... |

1 |
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Citation Context ...mean of the set of neighboring positions, rather than the median, as a key for sorting (the name comes from the fact that it's a one-dimensional analog of Tutte's barycenter method for drawing graphs =-=[27, 28]-=-). It performs poorly on paths, cycles, and other very sparse highlystructured graphs (since these require decisive movement of degree-2 nodes). On graphs that are more random and/or have several node... |