## Drawing with Fat Edges

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Venue: | INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE |

Citations: | 21 - 7 self |

### BibTeX

@MISC{Duncan_drawingwith,

author = {Christian A. Duncan and Alon Efrat and Stephen Kobourov and Carola Wenk},

title = {Drawing with Fat Edges },

year = {}

}

### OpenURL

### Abstract

Traditionally, graph drawing algorithms represent vertices as circles and edges as curves connecting the vertices. We introduce the problem of drawing with “fat ” edges, i.e., with edges of variable thickness. The thickness of an edge is often used as a visualization cue, to indicate importance, or to convey some additional information. We present a model for drawing with fat edges and a corresponding polynomial time algorithm that uses the model. We focus on a restricted class of graphs that occur in VLSI wire routing and show how to extend the algorithm to general planar graphs. We show how to convert an arbitrary wire routing into a homotopically equivalent routing that maximizes the distance between any two wires. Among such, we obtain the routing with minimum total wire length. A homotopically equivalent routing that maximizes the distance between any two wires yields a graph drawing which maximizes edge thickness. Finally, our algorithm also allows for different edge weights, that is, the requirement for unit wire thickness can be removed.

### Citations

76 |
A theorem on polygon cutting with applications
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- 1982
(Show Context)
Citation Context ... of our algorithm we use a geometric shortest path algorithm. For triangulated polygons, the Euclidean shortest path between two points can be computed in linear time using the algorithms of Chazelle =-=[1]-=- or Lee and Preparata [8]. The latter algorithm is known as the funnel algorithm and it can be extended to river routing [2, 4, 9]. In our setting, the shortest paths can be found in optimal O(nk) tim... |

74 | Computing minimum length paths of a given homotopy class
- Hershberger, Snoeyink
- 1994
(Show Context)
Citation Context ... known as the funnel algorithm and it can be extended to river routing [2, 4, 9]. In our setting, the shortest paths can be found in optimal O(nk) time using the algorithm of Hershberger and Snoeyink =-=[5]-=-. 1.2 Our Results We show how to solve the FED problem in O(nk + n3 ) time and O(n + k) space. We describe the algorithm in the tradition of the homotopic wire routing where n is the number of wires a... |

30 | Embedding planar graphs at fixed vertex locations
- Pach, Wenger
(Show Context)
Citation Context ...by Mehlhorn and Kaufmann [6, 7], and by Schrijver [16, 17]. Work in this area is related to the river routing problem in VLSI chips [2, 3, 12, 14]. A related problem was considered by Pach and Wenger =-=[13]-=-. They address the problem of laying out a planar graph at predefined locations in the plane. Given a planar graph on n vertices and a point set with n points, we want to draw the graph subject to the... |

26 |
River routing every which way, but loose
- Cole, Siegel
- 1984
(Show Context)
Citation Context ...t seem to be any previous work on drawing graphs with edges of varying thickness. Some related work has been done in addressing a classic VLSI problem, the continuous homotopic routing problem (CHRP) =-=[2, 9]-=-. For the CHRP problem, we need to route wires with fixed terminals among fixed obstacles when a sketch of the wires is given, i.e., each wire is given a specified homotopy class. If the wiring sketch... |

25 |
Algorithms for routing and testing routability of planar VLSI layouts
- Leiserson, Maley
- 1985
(Show Context)
Citation Context ...t seem to be any previous work on drawing graphs with edges of varying thickness. Some related work has been done in addressing a classic VLSI problem, the continuous homotopic routing problem (CHRP) =-=[2, 9]-=-. For the CHRP problem, we need to route wires with fixed terminals among fixed obstacles when a sketch of the wires is given, i.e., each wire is given a specified homotopy class. If the wiring sketch... |

24 | Computing homotopic shortest paths efficiently - Efrat, Kobourov, et al. |

20 | Testing homotopy for paths in the plane - Cabello, Liu, et al. |

17 |
Optimal Placement for River Routing
- Leiserson, Pinter
- 1983
(Show Context)
Citation Context ...among fixed obstacles when a sketch of the wires is given, i.e., each wire is given a specified homotopy class. If the wiring sketch is not given or the terminals are not fixed, the problem is NPhard =-=[10, 14, 15]-=-. In the CHRP problem we are given a wiring layout and some constant ɛ and we want to find if this wiring can be continuously transformed to a new wiring in which the wire separation is at least ɛ. In... |

14 | Disjoint homotopic paths and trees in a planar graph, Discrete Comput - Schrijver - 1991 |

11 | On continuous homotopic one layer routing
- Gao, Jerrum, et al.
- 1988
(Show Context)
Citation Context ...hich k is arbitrarily large. More surprisingly, even after shortest paths have been computed for each wire, k can be as large as k = Ω(2 n ), see Figure 1. The best result so far is due to Gao et al. =-=[4]-=- who present a O(kn 2 log(kn)) time and O(kn 2 ) space algorithm.sw4 w3 w2 1 w’ 4 w’ 2 w 3 w’ w’ 1 Fig. 1. An example with exponential complexity: k = Ω(2 n ): On the left is the initial wiring sketch... |

11 |
River routing in VLSI
- Mirzaian
- 1987
(Show Context)
Citation Context ...lgorithms for such restricted versions of the problem are presented by Mehlhorn and Kaufmann [6, 7], and by Schrijver [16, 17]. Work in this area is related to the river routing problem in VLSI chips =-=[2, 3, 12, 14]-=-. A related problem was considered by Pach and Wenger [13]. They address the problem of laying out a planar graph at predefined locations in the plane. Given a planar graph on n vertices and a point s... |

10 |
River-routing: Methodology and analysis
- Pinter
- 1983
(Show Context)
Citation Context ...among fixed obstacles when a sketch of the wires is given, i.e., each wire is given a specified homotopy class. If the wiring sketch is not given or the terminals are not fixed, the problem is NPhard =-=[10, 14, 15]-=-. In the CHRP problem we are given a wiring layout and some constant ɛ and we want to find if this wiring can be continuously transformed to a new wiring in which the wire separation is at least ɛ. In... |

9 | C.: Drawing planar graphs with large vertices and thick edges - Barequet, Goodrich, et al. - 2004 |

7 | Optimal Wiring between Rectangles
- Dolev, Karplus, et al.
- 1981
(Show Context)
Citation Context ...lgorithms for such restricted versions of the problem are presented by Mehlhorn and Kaufmann [6, 7], and by Schrijver [16, 17]. Work in this area is related to the river routing problem in VLSI chips =-=[2, 3, 12, 14]-=-. A related problem was considered by Pach and Wenger [13]. They address the problem of laying out a planar graph at predefined locations in the plane. Given a planar graph on n vertices and a point s... |

7 | Growing fat graphs - Efrat, Kobourov, et al. - 2002 |

5 |
Single-Layer Wire Routing
- Maley
- 1987
(Show Context)
Citation Context ...They show that in L∞ norm a solution can be found in O(k 3 log n) time and O(k 3 ) space, where n is the number of wires and k is the maximum of the input and output complexities of the wiring. Maley =-=[11]-=- shows how to extend the distance metric to arbitrary polygonal distance functions (including Euclidean distance) and presents a O(k 4 log n) time and O(k 4 ) space algorithm. Note that k cab be arbit... |

3 |
Complexity of single layer routing
- Richards
- 1984
(Show Context)
Citation Context ...among fixed obstacles when a sketch of the wires is given, i.e., each wire is given a specified homotopy class. If the wiring sketch is not given or the terminals are not fixed, the problem is NPhard =-=[10, 14, 15]-=-. In the CHRP problem we are given a wiring layout and some constant ɛ and we want to find if this wiring can be continuously transformed to a new wiring in which the wire separation is at least ɛ. In... |

2 |
On local routing of two-terminal nets
- Kaufmann, Mehlhorn
- 1992
(Show Context)
Citation Context ... been done for a special class of grid graphs: finite subgraphs of the planar rectangular grid. The first algorithms for such restricted versions of the problem are presented by Mehlhorn and Kaufmann =-=[6, 7]-=-, and by Schrijver [16, 17]. Work in this area is related to the river routing problem in VLSI chips [2, 3, 12, 14]. A related problem was considered by Pach and Wenger [13]. They address the problem ... |

1 |
Routing through a generalized switchbox
- Kaufmann, Mehlhorn
- 1986
(Show Context)
Citation Context ... been done for a special class of grid graphs: finite subgraphs of the planar rectangular grid. The first algorithms for such restricted versions of the problem are presented by Mehlhorn and Kaufmann =-=[6, 7]-=-, and by Schrijver [16, 17]. Work in this area is related to the river routing problem in VLSI chips [2, 3, 12, 14]. A related problem was considered by Pach and Wenger [13]. They address the problem ... |

1 |
homotopic paths and trees in a planar graph
- Disjoint
- 1991
(Show Context)
Citation Context ...class of grid graphs: finite subgraphs of the planar rectangular grid. The first algorithms for such restricted versions of the problem are presented by Mehlhorn and Kaufmann [6, 7], and by Schrijver =-=[16, 17]-=-. Work in this area is related to the river routing problem in VLSI chips [2, 3, 12, 14]. A related problem was considered by Pach and Wenger [13]. They address the problem of laying out a planar grap... |

1 |
Edge-disjoint homotopic paths in straight-line planar graphs
- Schrijver
- 1991
(Show Context)
Citation Context ...class of grid graphs: finite subgraphs of the planar rectangular grid. The first algorithms for such restricted versions of the problem are presented by Mehlhorn and Kaufmann [6, 7], and by Schrijver =-=[16, 17]-=-. Work in this area is related to the river routing problem in VLSI chips [2, 3, 12, 14]. A related problem was considered by Pach and Wenger [13]. They address the problem of laying out a planar grap... |

1 | Drawing graphs with fat edges - Duncan, Efrat, et al. - 2001 |