## A Fast Multi-Dimensional Algorithm for Drawing Large Graphs (2000)

### Cached

### Download Links

Venue: | In Graph Drawing’00 Conference Proceedings |

Citations: | 28 - 4 self |

### BibTeX

@INPROCEEDINGS{Gajer00afast,

author = {Pawel Gajer and Michael T. Goodrich and Stephen G. Kobourov},

title = {A Fast Multi-Dimensional Algorithm for Drawing Large Graphs},

booktitle = {In Graph Drawing’00 Conference Proceedings},

year = {2000},

pages = {211--221}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a novel hierarchical force-directed method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higher-dimensional embedding into a two or three dimensional subspace of E. Projecting high-dimensional drawings onto two or three dimensions often results in drawings that are "smoother" and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, e#cient memory management, and an intelligent initial placement of vertices. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a mid-range PC. 1 Introduction Graphs are common in many applications, from data structures to networks, from software engineering...

### Citations

8533 |
Introduction to Algorithms
- Cormen, Leiserson, et al.
- 1990
(Show Context)
Citation Context ...th time and space expensive. Using a binary heap implementation the all-pairs shortest paths problem can be solved in O(nm log n) time, and using Fibonacci heaps, in O(n 2 log n + nm) time, e.g., see =-=[6]-=-. The quadratic space complexity incurred by the matrix of distances between vertices of the graph also quickly becomes an obstacle for drawing large graphs. Other computationally expensive procedures... |

824 |
Generalized fisheye views, in
- Furnas
- 1986
(Show Context)
Citation Context ...thm which allows for drawing simple undirected graphs with tens of thousands of vertices in under a minute. Even larger graphs can be displayed using this algorithm in conjunction with a fisheye view =-=[20, 28, 36]-=- or multi-level display algorithm [11, 29] which would allow us to accomodate graphs with more vertices than the number of pixels of the display device. However, the e#ectiveness of the above algorith... |

515 | A heuristic for graph drawing
- Eades
- 1984
(Show Context)
Citation Context ...ehend increasingly complex structures if they are displayed as objects in three dimensional space [38, 39]. Some initial research on three dimensional information visualization applications has begun =-=[5, 24]-=-. Existing work in three dimensional (3D) graph drawing algorithms focuses on algorithms for special kinds of graphs, for example the algorithms of Cohen, Eades, Lin, and Ruskey [5]. Drawing general g... |

428 | Graph drawing by force-directed placement
- Fruchterman, Reingold
- 1991
(Show Context)
Citation Context ...hms for special kinds of graphs, for example the algorithms of Cohen, Eades, Lin, and Ruskey [5]. Drawing general graphs in 3D using the force-directed approach is studied by Fruchterman and Reingold =-=[19]-=-, and Monien, Ramme, and Salmen [31]. Other recent 3D drawing algorithms include Bru and Frick [4], Cruz and Twarog [7], and Ostry [33]. In the context of orthogonal drawings, 3D point-drawing algorit... |

329 |
Graph Drawing: Algorithms for the Visualization of Graphs
- Battista, Eades, et al.
- 1999
(Show Context)
Citation Context ...rawing graphs by hand becomes more challenging as the complexity of the graphs increases. Graph drawing tools have been the focus of the graph drawing community for at least the last two decades, see =-=[9, 10, 23]-=- for a comprehensive reviews of the field. Numerous algorithms have been developed for drawing special classes of graphs such as trees and planar graphs. There are fewer general purpose graph drawing ... |

261 |
Engineering psychology and human performance
- Wickens
- 1992
(Show Context)
Citation Context ... more natural representations and there is growing evidence which shows that the human brain can comprehend increasingly complex structures if they are displayed as objects in three dimensional space =-=[38, 39]-=-. Some initial research on three dimensional information visualization applications has begun [5, 24]. Existing work in three dimensional (3D) graph drawing algorithms focuses on algorithms for specia... |

232 |
Algorithms for drawing graphs: an annotated bibliography
- EADES, TAMASSIA
- 1989
(Show Context)
Citation Context ...rawing graphs by hand becomes more challenging as the complexity of the graphs increases. Graph drawing tools have been the focus of the graph drawing community for at least the last two decades, see =-=[9, 10, 23]-=- for a comprehensive reviews of the field. Numerous algorithms have been developed for drawing special classes of graphs such as trees and planar graphs. There are fewer general purpose graph drawing ... |

168 | H.: Graphical fisheye views
- SARKAR, BROWN
- 1994
(Show Context)
Citation Context ...thm which allows for drawing simple undirected graphs with tens of thousands of vertices in under a minute. Even larger graphs can be displayed using this algorithm in conjunction with a fisheye view =-=[20, 28, 36]-=- or multi-level display algorithm [11, 29] which would allow us to accomodate graphs with more vertices than the number of pixels of the display device. However, the e#ectiveness of the above algorith... |

101 | A fast adaptive layout algorithm for undirected graphs
- Frick, Ludwig, et al.
- 1995
(Show Context)
Citation Context ...lexible forcedirected algorithm. Fruchterman and Reingold [19] use a slightly di#erent heuristic which results in a faster algorithm. The algorithms of Bru and Frick [4] and Frick, Ludwig and Mehldau =-=[18]-=- add the notion of local temperature to further speed up the drawing process. A force-directed method can also be used to draw graphs with node labels as shown by Gansner and North [22]. The main prob... |

93 |
A Heuristic for Graph Drawing. Congressus Numerantium
- Eades
- 1984
(Show Context)
Citation Context ...by itself or as a preprocessing step to these large-graph layout methods. 2.3 Force-Directed Algorithms The force-directed placement algorithm of Quinn and Breur [35] and the spring embedder of Eades =-=[12]-=- are among thesrst practical algorithms for graph drawing. In the latter algorithm the graph is modeled as a physical system of rings and springs. Classical force-directed methods start from a random ... |

80 | Multilevel visualization of clustered graphs
- Eades, Feng
- 1996
(Show Context)
Citation Context ... a 3D drawing with each level drawn on a plane at a di#erent z-coordinate, and with the clustering structure drawn as a tree in 3D. The multi-level display algorithms are introduced by Eades and Feng =-=[13]-=- and Feng [16] in the context of visualization for clustered graphs. Compound and clustered graphs are studied by Sugiyama and Misue [30, 37], by North [32], by Eades, Feng and Lin [14], and Feng, Coh... |

79 | A fast multi-scale method for drawing large graphs
- Harel, Koren
- 2000
(Show Context)
Citation Context ... results of the previous computation. This strategy has been brought to the area of force-directed graph drawing from particle physics [2, 3] in the multi-scale algorithm of Hadany and Harel [25]. In =-=[26]-=- Harel and Koren introduce several simplifications to the algorithm resulting in faster drawings and allowing for larger graphs. With their beautiful drawings of graphs with 3,000 vertices they mark a... |

69 |
Visualization of structural information: Automatic drawing of compound digraphs
- Sugiyama, Misue
- 1991
(Show Context)
Citation Context ...ulti-level display algorithms are introduced by Eades and Feng [13] and Feng [16] in the context of visualization for clustered graphs. Compound and clustered graphs are studied by Sugiyama and Misue =-=[30, 37]-=-, by North [32], by Eades, Feng and Lin [14], and Feng, Cohen, and Eades [17]. Creating a graph clustering based on binary space partitions and using it to display large graphs was introduced by Dunca... |

58 | Straight-line drawing algorithms for hierarchical graphs and clustered graphs, in
- Eades, Feng, et al.
- 1996
(Show Context)
Citation Context ...Eades and Feng [13] and Feng [16] in the context of visualization for clustered graphs. Compound and clustered graphs are studied by Sugiyama and Misue [30, 37], by North [32], by Eades, Feng and Lin =-=[14]-=-, and Feng, Cohen, and Eades [17]. Creating a graph clustering based on binary space partitions and using it to display large graphs was introduced by Duncan, Goodrich, and Kobourov [11]. The quality ... |

50 |
A force directed component placement procedure for printed circuit boards
- Quinn, Breur
- 1979
(Show Context)
Citation Context ...e times; hence, it can be used either by itself or as a preprocessing step to these large-graph layout methods. 2.3 Force-Directed Algorithms The force-directed placement algorithm of Quinn and Breur =-=[35]-=- and the spring embedder of Eades [12] are among the first practical algorithms for graph drawing. In the latter algorithm the graph is modeled as a physical system of rings and springs. Classical for... |

47 |
Multilevel computations of integral transforms and particle interactions with oscillatory kernels
- Brandt
- 1991
(Show Context)
Citation Context ...roceeding to larger and larger graphs and using at each stage the results of the previous computation. This strategy has been brought to the area of force-directed graph drawing from particle physics =-=[2, 3]-=- in the multi-scale algorithm of Hadany and Harel [25]. In [26] Harel and Koren introduce several simplifications to the algorithm resulting in faster drawings and allowing for larger graphs. With the... |

46 |
How to draw a planar clustered graph
- Feng, Cohen, et al.
- 1995
(Show Context)
Citation Context ... in the context of visualization for clustered graphs. Compound and clustered graphs are studied by Sugiyama and Misue [30, 37], by North [32], by Eades, Feng and Lin [14], and Feng, Cohen, and Eades =-=[17]-=-. Creating a graph clustering based on binary space partitions and using it to display large graphs was introduced by Duncan, Goodrich, and Kobourov [11]. The quality of the resulting multi-level draw... |

39 |
Visualizing object oriented software in three dimensions
- Ware, Hui, et al.
- 1993
(Show Context)
Citation Context ... more natural representations and there is growing evidence which shows that the human brain can comprehend increasingly complex structures if they are displayed as objects in three dimensional space =-=[38, 39]-=-. Some initial research on three dimensional information visualization applications has begun [5, 24]. Existing work in three dimensional (3D) graph drawing algorithms focuses on algorithms for specia... |

35 | 3D graph drawing with simulated annealing
- Cruz, Twarog
(Show Context)
Citation Context ... in 3D using the force-directed approach is studied by Fruchterman and Reingold [19], and Monien, Ramme, and Salmen [31]. Other recent 3D drawing algorithms include Bru and Frick [4], Cruz and Twarog =-=[7]-=-, and Ostry [33]. In the context of orthogonal drawings, 3D point-drawing algorithms were developed by Eades, Symvonis, and Whitesides [15] and Papakostas and Tollis [34]. 3D orthogonal box-drawings w... |

33 | GRIP: Graph dRawing with Intelligent Placement
- Gajer, Kobourov
- 2002
(Show Context)
Citation Context ...ntelligent placement of vertices, and multi-dimensional drawing. In Section 4 we discuss possible modifications of the algorithm. Also included are several drawings obtained by the GRIP layout system =-=[21]-=- which is based on our algorithm. 2 Previous Work 2.1 Drawing in Three Dimensions Although the majority of the work in graph drawing is in two dimensional graph layout, there have been several algorit... |

26 |
Fast interactive 3-D graph visualization
- Bruss, Frick
- 1995
(Show Context)
Citation Context ...rawing general graphs in 3D using the force-directed approach is studied by Fruchterman and Reingold [19], and Monien, Ramme, and Salmen [31]. Other recent 3D drawing algorithms include Bru and Frick =-=[4]-=-, Cruz and Twarog [7], and Ostry [33]. In the context of orthogonal drawings, 3D point-drawing algorithms were developed by Eades, Symvonis, and Whitesides [15] and Papakostas and Tollis [34]. 3D orth... |

25 | Algorithms for drawing clustered graphs
- Feng
- 1997
(Show Context)
Citation Context ...with each level drawn on a plane at a di#erent z-coordinate, and with the clustering structure drawn as a tree in 3D. The multi-level display algorithms are introduced by Eades and Feng [13] and Feng =-=[16]-=- in the context of visualization for clustered graphs. Compound and clustered graphs are studied by Sugiyama and Misue [30, 37], by North [32], by Eades, Feng and Lin [14], and Feng, Cohen, and Eades ... |

22 |
A simple algorithm for drawing large graphs on small screens
- Kaugars, Reinfelds, et al.
- 1995
(Show Context)
Citation Context ...thm which allows for drawing simple undirected graphs with tens of thousands of vertices in under a minute. Even larger graphs can be displayed using this algorithm in conjunction with a fisheye view =-=[20, 28, 36]-=- or multi-level display algorithm [11, 29] which would allow us to accomodate graphs with more vertices than the number of pixels of the display device. However, the e#ectiveness of the above algorith... |

20 | Balanced aspect ratio trees and their use for drawing very large graphs
- DUNCAN, GOODRICH, et al.
- 1998
(Show Context)
Citation Context ... graphs with tens of thousands of vertices in under a minute. Even larger graphs can be displayed using this algorithm in conjunction with a fisheye view [20, 28, 36] or multi-level display algorithm =-=[11, 29]-=- which would allow us to accomodate graphs with more vertices than the number of pixels of the display device. However, the e#ectiveness of the above algorithms depends on a good recursive clustering,... |

20 | Two algorithms for three dimensional orthogonal graph drawing
- Eades, Symvonis, et al.
- 1996
(Show Context)
Citation Context ...drawing algorithms include Bru and Frick [4], Cruz and Twarog [7], and Ostry [33]. In the context of orthogonal drawings, 3D point-drawing algorithms were developed by Eades, Symvonis, and Whitesides =-=[15]-=- and Papakostas and Tollis [34]. 3D orthogonal box-drawings were studied by Biedl [1] and multi-dimensional orthogonal graph drawings are presented by Wood [40]. 2.2 Visualization of Large Graphs Visu... |

19 |
Multigrid methods in lattice field computations
- Brandt
- 1992
(Show Context)
Citation Context ...roceeding to larger and larger graphs and using at each stage the results of the previous computation. This strategy has been brought to the area of force-directed graph drawing from particle physics =-=[2, 3]-=- in the multi-scale algorithm of Hadany and Harel [25]. In [26] Harel and Koren introduce several simplifications to the algorithm resulting in faster drawings and allowing for larger graphs. With the... |

18 | Incremental orthogonal graph drawing in three dimensions, in
- Papakostas, Tollis
- 1997
(Show Context)
Citation Context ...and Frick [4], Cruz and Twarog [7], and Ostry [33]. In the context of orthogonal drawings, 3D point-drawing algorithms were developed by Eades, Symvonis, and Whitesides [15] and Papakostas and Tollis =-=[34]-=-. 3D orthogonal box-drawings were studied by Biedl [1] and multi-dimensional orthogonal graph drawings are presented by Wood [40]. 2.2 Visualization of Large Graphs Visualizing large graphs presents u... |

18 | GIOTTO3D: A system for visualizing hierarchical structures in 3D
- Garg, Tamassia
- 1997
(Show Context)
Citation Context ...ehend increasingly complex structures if they are displayed as objects in three dimensional space [38, 39]. Some initial research on three dimensional information visualization applications has begun =-=[5, 24]-=-. Existing work in three dimensional (3D) graph drawing algorithms focuses on algorithms for special kinds of graphs, for example the algorithms of Cohen, Eades, Lin, and Ruskey [5]. Drawing general g... |

17 |
Automatic display of network structures for human understanding
- Kamada, Kawai
- 1988
(Show Context)
Citation Context ... graph drawings. The main di#erences between force-directed algorithms are in the choice of energy function and the methods for its minimization. The energy minimization algorithm of Kamada and Kawai =-=[27]-=- uses the Newton-Raphson method for improved drawings. The simulated annealing method of Davidson and Harel [8] is another flexible forcedirected algorithm. Fruchterman and Reingold [19] use a slightl... |

16 | Fast Interactive 3-D Graph Visualization
- Brus, Frick
- 1996
(Show Context)
Citation Context ...rawing general graphs in 3D using the forcedirected approach is studied by Fruchterman and Reingold [19], and Monien, Ramme, and Salmen [31]. Other recent 3D drawing algorithms include Bruß and Frick =-=[4]-=-, Cruz and Twarog [7], and Ostry [33]. In the context of orthogonal drawings, 3D point-drawing algorithms were developed by Eades, Symvonis, and Whitesides [15] and Papakostas and Tollis [34]. 3D orth... |

15 | A parallel simulated annealing algorithm for generating 3D layouts of undirected graphs
- Monien, Ramme, et al.
- 1996
(Show Context)
Citation Context ... example the algorithms of Cohen, Eades, Lin, and Ruskey [5]. Drawing general graphs in 3D using the force-directed approach is studied by Fruchterman and Reingold [19], and Monien, Ramme, and Salmen =-=[31]-=-. Other recent 3D drawing algorithms include Bru and Frick [4], Cruz and Twarog [7], and Ostry [33]. In the context of orthogonal drawings, 3D point-drawing algorithms were developed by Eades, Symvoni... |

14 |
Three approaches to 3D-orthogonal box-drawings, in
- Biedl
- 1998
(Show Context)
Citation Context ...the context of orthogonal drawings, 3D point-drawing algorithms were developed by Eades, Symvonis, and Whitesides [15] and Papakostas and Tollis [34]. 3D orthogonal box-drawings were studied by Biedl =-=[1]-=- and multi-dimensional orthogonal graph drawings are presented by Wood [40]. 2.2 Visualization of Large Graphs Visualizing large graphs presents unique problems which require non-orthodox solutions. D... |

14 | Graph Drawing
- Tamassia
- 1997
(Show Context)
Citation Context ...rawing graphs by hand becomes more challenging as the complexity of the graphs increases. Graph drawing tools have been the focus of the graph drawing community for at least the last two decades, see =-=[9, 10, 23]-=- for a comprehensive reviews of the field. Numerous algorithms have been developed for drawing special classes of graphs such as trees and planar graphs. There are fewer general purpose graph drawing ... |

14 |
Fast interactive 3-d graph visualization
- Bru, Frick
- 1996
(Show Context)
Citation Context ...awing general graphs in 3D using the force-directed approach is studied by Fruchterman and Reingold [19], and Monien, Ramme, and Salmen [31]. Other recent 3D drawing algorithms include Bru and Frick [=-=4]-=-, Cruz and Twarog [7], and Ostry [33]. In the context of orthogonal drawings, 3D point-drawing algorithms were developed by Eades, Symvonis, and Whitesides [15] and Papakostas and Tollis [34]. 3D orth... |

13 |
Drawing graphics nicely using simulated annealing
- Davidson, Harel
- 1996
(Show Context)
Citation Context ...the methods for its minimization. The energy minimization algorithm of Kamada and Kawai [27] uses the Newton-Raphson method for improved drawings. The simulated annealing method of Davidson and Harel =-=[8]-=- is another flexible forcedirected algorithm. Fruchterman and Reingold [19] use a slightly di#erent heuristic which results in a faster algorithm. The algorithms of Bru and Frick [4] and Frick, Ludwig... |

13 |
Drawing ranked digraphs with recursive clusters, in
- North
- 1993
(Show Context)
Citation Context ...algorithms are introduced by Eades and Feng [13] and Feng [16] in the context of visualization for clustered graphs. Compound and clustered graphs are studied by Sugiyama and Misue [30, 37], by North =-=[32]-=-, by Eades, Feng and Lin [14], and Feng, Cohen, and Eades [17]. Creating a graph clustering based on binary space partitions and using it to display large graphs was introduced by Duncan, Goodrich, an... |

13 | Multi-dimensional orthogonal graph drawing with small boxes, in
- Wood
- 1999
(Show Context)
Citation Context ...oped by Eades, Symvonis, and Whitesides [15] and Papakostas and Tollis [34]. 3D orthogonal box-drawings were studied by Biedl [1] and multi-dimensional orthogonal graph drawings are presented by Wood =-=[40]-=-. 2.2 Visualization of Large Graphs Visualizing large graphs presents unique problems which require non-orthodox solutions. Drawings that display the entire graph have the advantage of showing the glo... |

12 |
A multi-scale algorithm for drawing graphs nicely
- Hadany
- 2001
(Show Context)
Citation Context ...stage the results of the previous computation. This strategy has been brought to the area of force-directed graph drawing from particle physics [2, 3] in the multi-scale algorithm of Hadany and Harel =-=[25]-=-. In [26] Harel and Koren introduce several simplifications to the algorithm resulting in faster drawings and allowing for larger graphs. With their beautiful drawings of graphs with 3,000 vertices th... |

11 | Some three-dimensional graph drawing algorithms
- Ostry
- 1996
(Show Context)
Citation Context ...e force-directed approach is studied by Fruchterman and Reingold [19], and Monien, Ramme, and Salmen [31]. Other recent 3D drawing algorithms include Bru and Frick [4], Cruz and Twarog [7], and Ostry =-=[33]-=-. In the context of orthogonal drawings, 3D point-drawing algorithms were developed by Eades, Symvonis, and Whitesides [15] and Papakostas and Tollis [34]. 3D orthogonal box-drawings were studied by B... |

8 |
SC: Improved force-directed layouts
- ER, North
- 1998
(Show Context)
Citation Context ...ig and Mehldau [18] add the notion of local temperature to further speed up the drawing process. A force-directed method can also be used to draw graphs with node labels as shown by Gansner and North =-=[22]-=-. The main problem with most standard force-directed algorithms is their inability to draw large graphs. Even the best classical algorithms can draw graphs with a maximum of only several hundred verti... |

4 |
Generalized views
- Furnas
- 1986
(Show Context)
Citation Context ...rithm which allows for drawing simple undirected graphs with tens of thousands of vertices in under a minute. Even larger graphs can be displayed using this algorithm in conjunction with assheye view =-=[20, 28, 36-=-] or multi-level display algorithm [11, 29] which would allow us to accomodate graphs with more vertices than the number of pixels of the display device. However, the eectiveness of the above algorith... |

3 |
GIOTT03D: A system for visualizing hierarchical structures
- Garg, Tamassia
- 1997
(Show Context)
Citation Context ...ehend increasingly complex structures if they are displayed as objects in three dimensional space [38, 39]. Some initial research on three dimensional information visualization applications has begun =-=[5, 24]-=-. Existing work in three dimensional (3D) graph drawing algorithms focuses on algorithms for special kinds of graphs, for example the algorithms of Cohen, Eades, Lin, and Ruskey [5]. Drawing general g... |

2 |
Visualizing Large Graphs
- Kobourov
- 2000
(Show Context)
Citation Context ... graphs with tens of thousands of vertices in under a minute. Even larger graphs can be displayed using this algorithm in conjunction with a fisheye view [20, 28, 36] or multi-level display algorithm =-=[11, 29]-=- which would allow us to accomodate graphs with more vertices than the number of pixels of the display device. However, the e#ectiveness of the above algorithms depends on a good recursive clustering,... |

2 |
An overview of diagram based idea organizer
- Misue, Sugiyama
- 1993
(Show Context)
Citation Context ...ulti-level display algorithms are introduced by Eades and Feng [13] and Feng [16] in the context of visualization for clustered graphs. Compound and clustered graphs are studied by Sugiyama and Misue =-=[30, 37]-=-, by North [32], by Eades, Feng and Lin [14], and Feng, Cohen, and Eades [17]. Creating a graph clustering based on binary space partitions and using it to display large graphs was introduced by Dunca... |

1 |
Multigrid methods in lattice computations
- Brandt
- 1992
(Show Context)
Citation Context ...roceeding to larger and larger graphs and using at each stage the results of the previous computation. This strategy has been brought to the area of force-directed graph drawing from particle physics =-=[2, 3-=-] in the multi-scale algorithm of Hadany and Harel [25]. In [26] Harel and Koren introduce several simplications to the algorithm resulting in faster drawings and allowing for larger graphs. With thei... |