## Graph Drawing by High-Dimensional Embedding (2002)

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### Other Repositories/Bibliography

Venue: | In GD02, LNCS |

Citations: | 61 - 10 self |

### BibTeX

@INPROCEEDINGS{Harel02graphdrawing,

author = {David Harel and Yehuda Koren},

title = {Graph Drawing by High-Dimensional Embedding},

booktitle = {In GD02, LNCS},

year = {2002},

pages = {207--219},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2-D plane using PCA. Experiments we have carried out show the ability of the method to draw graphs of 10 nodes in few seconds. The new method appears to have several advantages over classical methods, including a significantly better running time, a useful inherent capability to exhibit the graph in various dimensions, and an effective means for interactive exploration of large graphs.

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Citation Context ...ensions is given in Fig. 1. The graph theoretic distances are computed using breadth-first-search (BFS). (When edges are positively weighted, BFS should be replaced by Dijkstra’s algorithm; see e.g.=-=, [1].)-=- The set Centers is chosen as follows. The first member, p1, is chosen at random. For j =2,...,m, node pj is a node that maximizes the shortest distance from {p1,p2,...,pj−1}. This method is mention... |

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Citation Context ...The set Pivots is chosen as follows. The first member, p1, is chosen at random. For j =2,...,m,nodepj is a node that maximizes the shortest distance from {p1,p2,...,pj−1}. This method is mentioned in =-=[12]-=- as a 2-approximation 2 to the k-center problem, where we want to choose k vertices of V , such that the longest distance from V to these k centers is minimized. However, different approaches to selec... |

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Citation Context ...r approaches to this appear to be those that define a cost function (or a force model), whose minimization determines the optimal drawing. The resulting algorithms are known as force-directed methods =-=[1, 3, 5, 8, 14]-=-. We suggest a new approach to the problem of graph drawing, relying on the observation that laying out a graph in a high dimension is significantly easier than drawing it in a low dimension. Hence, t... |

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Citation Context ...r approaches to this appear to be those that define a cost function (or a force model), whose minimization determines the optimal drawing. The resulting algorithms are known as force-directed methods =-=[1, 4, 6, 8, 14]-=-. We suggest a new approach to the problem of graph drawing, relying on the observation that laying out a graph in a high dimension is significantly easier than drawing it in a low dimension. Hence, t... |

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Citation Context ...r approaches to this appear to be those that define a cost function (or a force model), whose minimization determines the optimal drawing. The resulting algorithms are known as force-directed methods =-=[1, 3, 5, 8, 14]-=-. We suggest a new approach to the problem of graph drawing, relying on the observation that laying out a graph in a high dimension is significantly easier than drawing it in a low dimension. Hence, t... |

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Citation Context ...ty of the resulting layout; that is, the drawing algorithm’s capability of conveying the meaning of the diagram quickly and clearly. Consequently, many approaches to graph drawing have been developed =-=[4, 15]-=-. We concentrate on the problem of drawing undirected graphs with straight-line edges, and the most popular approaches to this appear to be those that define a cost function (or a force model), whose ... |

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Citation Context ...r approaches to this appear to be those that define a cost function (or a force model), whose minimization determines the optimal drawing. The resulting algorithms are known as force-directed methods =-=[1, 3, 5, 8, 14]-=-. We suggest a new approach to the problem of graph drawing, relying on the observation that laying out a graph in a high dimension is significantly easier than drawing it in a low dimension. Hence, t... |

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Citation Context ...; see e.g., [1]). As to computing the first eigenvectors of the m × m covariance matrix S (i.e., those that correspond to the largest eigenvalues), we use the simple power-iteration method; see e.g.,=-= [12])-=-. Since m<<n, the running time is negligible (taking in practice only a few milliseconds) and there is no need for more complicated techniques. The basic idea is as follows. Say we are given an m × m... |

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Citation Context ...r approaches to this appear to be those that define a cost function (or a force model), whose minimization determines the optimal drawing. The resulting algorithms are known as force-directed methods =-=[1, 4, 6, 8, 14]-=-. We suggest a new approach to the problem of graph drawing, relying on the observation that laying out a graph in a high dimension is significantly easier than drawing it in a low dimension. Hence, t... |

105 |
Applied Multivariate Data Analysis
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Citation Context ...sible. By using only the first few principal components, PCA makes it possible to reduce the number of significant dimensions of the data, while maintaining the maximum possible variance thereof. See =-=[3]-=- for a comprehensive discussion of PCA. In our case, we have mn-dimensional variables X1 ,...,Xm , describing the n nodes in m dimensions. We want to represent the n nodes using only k dimensions (typ... |

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Citation Context ...ze of the graph (i.e., |V | and |E|), and is independent of the structure of the graph. This is unlike force-directed methods. graph |V| |E| total high dim. covariance time embedding comp. matrix 516 =-=[11] 516 7-=-29 0.00 0.00 0.00 4970 [11] 4970 7400 0.08 0.03 0.05 3elt † 4720 13722 0.09 0.05 0.05 Crack 10240 30380 0.30 0.14 0.08 4elt2 † 11143 32818 0.25 0.16 0.09 Sphere † 16386 49152 0.81 0.47 0.16 Sier... |

82 | A fast multi-scale method for drawing large graph
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(Show Context)
Citation Context ...e is still the second issue of getting out of the initial random placement. Both these issues receive adequate treatment by incorporating the multi-scale strategy as suggested by several authors; see =-=[9, 10, 11, 21]-=-. These methods considerably improve running times by rapidly constructing a simplified initial globally nice layout and then refining it locally. The fastest of them all, [21], draws a 10 5 -node gra... |

64 | ACE: a fast multiscale eigenvectors computation for drawing huge graphs
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Citation Context ...orted. In Fig. 6(b) we depict the result of zooming-in on the left part of the graph, revealing its nice structure. The layout of the 4elt2 graph, depicted in Fig. 6(c), resembles the one obtained by =-=[9]-=-. For a better understanding of its structure we may zoom-in on parts of the drawing. Fig. 6(d) shows the results of zooming-in on the bottom strip. In Fig. 6(e) we provide a drawing of the Ocean grap... |

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Citation Context ...undreds of nodes. Recently, several authors, including work done in our own group, have designed multi-scale enhancements of force directed methods, which considerably improve their running time; see =-=[5, 6, 11, 4, 10]-=-. The fastest of them all, [11], draws a 10 5 - node graph in a typical time of ten minutes. Our algorithm is considerably faster, being able to draw a 10 5 -node graph in less than 3 seconds. Moreove... |

37 |
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(Show Context)
Citation Context ...ty of the resulting layout; that is, the drawing algorithm’s capability of conveying the meaning of the diagram quickly and clearly. Consequently, many approaches to graph drawing have been develope=-=d [2, 8]-=-. We concentrate on the problem of drawing undirected graphs with straight-line edges, and the most popular approaches to this appear to be those that define a cost function (or a force model), whose ... |

26 |
Fast interactive 3-D graph visualization
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Citation Context |

19 | A Numerical Optimization Approach to General Graph Drawing
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(Show Context)
Citation Context ...ration of the optimization process very slow. Second, for large graphs the optimization process needs too many iterations for turning the initial random placement into a nice layout. Some researchers =-=[13, 12]-=- have improved these methods to some extent, by accelerating force calculation using quad-trees that reduce the complexity of the force model. This way, [13] reports drawing 1000-node graphs in around... |

19 | Robust linear dimensionality reduction
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(Show Context)
Citation Context ...| 2 . (2) a�=b∈P A good projection should maximize (2), thus, separating elements of P. In other words, the best projection is one that best preserves the pairwise squared distances of P. The work of =-=[17]-=- proves that the maximizer is exactly the PCA projection of the points in P. Therefore, we have an algorithm for finding the “best” viewpoint, whose running time is O(|P|). It is also possible to maxi... |

18 | Finding the best viewpoints for three-dimensional graph drawings
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Citation Context ...eated a need for algorithms that automatically compute the best 2-D projection of a 3-D layout. An exact algorithm that requires a preprocessing time of Ω((|V |+|E|) 4 log(|V |+|E|)) was suggested in =-=[6]-=-. Iterative heuristic algorithms, with a faster running time (but still, at least O(|V |·|E|)), were suggested in [13]. We suggest using PCA for calculating the best viewpoint, resulting in a much fas... |

16 |
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Citation Context ... set Centers is chosen as follows. The first member, p1, is chosen at random. For j =2,...,m, node pj is a node that maximizes the shortest distance from {p1,p2,...,pj−1}. This method is mentioned i=-=n [7]-=- as a 2-approximation to the kcenter problem, where we want to choose k vertices of V , such that the longest distance from V to these k centers is minimized. Function HighDimDraw (G(V = {1,...,n},E),... |

9 |
Graph drawing by subspace optimization
- KOREN
(Show Context)
Citation Context ...tes a novel technique for interactive exploration of the graph, by focusing on selected portions thereof, showing them in a way that is not possible in a 2-D drawing of the entire graph. A newer work =-=[18]-=- suggests an alternative approach to the PCA projection. Thus, the second step of our algorithm is replaced by a new projection method that maximizes the scatter of the nodes while keeping edge length... |

8 |
A Multi-Scale Method for Drawing Graphs
- Hadany, Harel
- 2001
(Show Context)
Citation Context ...undreds of nodes. Recently, several authors, including work done in our own group, have designed multi-scale enhancements of force directed methods, which considerably improve their running time; see =-=[5, 6, 11, 4, 10]-=-. The fastest of them all, [11], draws a 10 5 - node graph in a typical time of ten minutes. Our algorithm is considerably faster, being able to draw a 10 5 -node graph in less than 3 seconds. Moreove... |

7 |
A multi-scale method for drawing graphs nicely
- Hadany, Hardel
- 2001
(Show Context)
Citation Context ...undreds of nodes. Recently, several authors, including work done in our own group, have designed multi-scale enhancements of force directed methods, which considerably improve their running time; see =-=[5, 6, 11, 4, 10]-=-. The fastest of them all, [11], draws a 10 5 - node graph in a typical time of ten minutes. Our algorithm is considerably faster, being able to draw a 10 5 -node graph in less than 3 seconds. Moreove... |

5 | Approximation algorithms for finding best viewpoints
- Houle, Webber
- 1998
(Show Context)
Citation Context ...at requires a preprocessing time of Ω((|V |+|E|) 4 log(|V |+|E|)) was suggested in [6]. Iterative heuristic algorithms, with a faster running time (but still, at least O(|V |·|E|)), were suggested in =-=[13]-=-. We suggest using PCA for calculating the best viewpoint, resulting in a much faster exact algorithm. A good projection should prevent overlaps between the graph elements, avoiding vertex-vertex, ver... |

3 |
On Spectral Graph Drawing”, manuscript
- Koren
- 2002
(Show Context)
Citation Context ... we call ACE [10]. ACE draws a graph by quickly calculating eigenvectors of the Laplacian matrix associated with it, using a special algebraic multigrid technique (for spectral graph drawing see also =-=[11]-=-). ACE can draw 105-node graphs in about 2 seconds. However, the running-time of ACE (like that of force-directed methods) depends on the graph’s structure, unlike our algorithm, where it depends only... |

3 |
A Heuristic for Graph Drawing”,Congressus Numerantium 42
- Eades
- 1984
(Show Context)
Citation Context ...r approaches to this appear to be those that define a cost function (or a force model), whose minimization determines the optimal drawing. The resulting algorithms are known as force-directed methods =-=[1, 4, 6, 8, 14]-=-. We suggest a new approach to the problem of graph drawing, relying on the observation that laying out a graph in a high dimension is significantly easier than drawing it in a low dimension. Hence, t... |