Journal of Graph Algorithms and Applications
http://jgaa.info/ vol. 11, no. 2, pp. 345–369 (2007)
Large-Graph Layout Algorithms at Work:
An Experimental Study
Stefan Hachul
Michael Jünger
Institut für Informatik
Universität zu Köln
Pohligstraße 1, 50969 Köln, Germany
{hachul,mjuenger}@informatik.uni-koeln.de
Abstract
In the last decade several algorithms that generate straight-line drawings of general large graphs have been invented. In this paper we investigate some of these methods that are based on force-directed or algebraic
approaches in terms of running time and drawing quality on a big variety of artificial and real-world graphs. Our experiments indicate that
there exist significant differences in drawing qualities and running times
depending on the classes of tested graphs and algorithms.
Article Type
Regular paper
Communicated by
P. Eades and P. Healy
Submitted
January 2006
Revised
January 2007
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 346
1
Introduction
What do biochemical reactions of proteins in baker’s yeast, the ecosystem of
plankton, sea perch, and anchovy, the American electricity network, and the
international air traffic have in common? They can be modeled as graphs.
In general a graph G = (V, E) is used to model information that can be
described as objects (the node set V ) and connections between those objects
(the edge set E).
One fundamental tool for analyzing such graphs is the automatic generation
of layouts that visualize the graphs and are easy to understand. A popular
class of algorithms that is used to visualize general graphs are force-directed
graph-drawing methods. Those methods generate drawings of a given general
graph G = (V, E) in the plane in which each edge is represented by a straight
line connecting its two adjacent nodes. The computation of the drawings is
based on associating G with a physical model. Then, an iterative algorithm
tries to find a placement of the nodes so that the total energy in the underlying
physical system is minimal. Important esthetic criteria are uniformity of edge
length, few edge crossings, non-overlapping nodes and edges, and the display of
symmetries if some exist.
Classical force-directed algorithms like [5, 16, 7, 4, 6] are used successfully in
practice (see e.g., [2]) for drawing general graphs containing few hundreds of vertices. However, in order to generate drawings of graphs that contain thousands
or hundreds of thousands of vertices more efficient force-directed techniques
have been developed [21, 20, 9, 8, 13, 23, 12, 11]. Besides fast force-directed
algorithms other very fast methods for drawing large graphs (see e.g., [14, 17])
have been invented. These methods are based on techniques of linear algebra
instead of physical analogies. But they strive for the same esthetic drawing
criteria.
Previous experimental tests of these methods are mainly restricted to regular
graphs with grid-like structures (see e.g., [14, 17, 9, 23, 13]). Since general graphs
share these properties quite seldom, and since the test environments of these
experiments are different, a standardized comparison of the methods on a wider
range of graphs is needed.
In this study we experimentally compare some of the fastest state-of-the-art
algorithms for straight-line drawing of general graphs on a big variety of graph
classes. In particular, we investigate the force-directed algorithm GRIP of Gajer
and Kobourov [9] and Gajer et al. [8], the Fast Multi-scale Method (FMS) of
Harel and Koren [13], and the Fast Multipole Multilevel Method (FM3 ) of Hachul
and Jünger [12, 11]. We did not test other efficient force-directed methods like
FADE of Quigley and Eades [20] and JIGGLE of Tunkelang [21] since the original
implementations were not available to us or the implementations did not allow
to import test graphs, respectively.
The examined algebraic methods are the algebraic multigrid method ACE of
Koren et al. [17] and the high-dimensional embedding approach (HDE) of Harel
and Koren [14]. Additionally, one of the fastest classical force-directed algorithms, namely the grid-variant algorithm (GVA) of Fruchterman and Rein-
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 347
gold [7], is tested as a benchmark.
After a short description of the tested algorithms in Section 2 and a presentation of the experimental framework in Section 3, our results will be presented
in Section 4.
2
2.1
The Algorithms
The Grid-Variant Algorithm (GVA)
The grid-variant algorithm (GVA) of Fruchterman and Reingold [7] is based on a
model of pairwise repelling charged particles (the nodes) and attracting springs
(the edges), similar to the model of the Spring Embedder of Eades [5]. Since a
naive exact calculation of the repulsive forces acting between all pairs of charges
needs Θ(|V 2 |) time per iteration, GVA only calculates the repulsive forces acting
between nodes that are placed relatively near to each other. Therefore, the
rectangular drawing area is subdivided into a regular square grid. The repulsive
forces that act on a node v that is contained in a grid box B are approximated by
summing up only the repulsive forces that are induced by the nodes contained
in B and the nodes in the grid boxes that are neighbors of B. If the number
of iterations is assumed to be constant, the best-case running time of GVA is
Θ(|V | + |E|). The worst-case running time, however, remains Θ(|V |2 + |E|).
2.2
The Method GRIP
Gajer and Kobourov [9] and Gajer et al. [8] developed the force-directed multilevel algorithm GRIP. In general, multilevel algorithms are based on two phases.
A coarsening phase, in which a sequence of coarse graphs with decreasing sizes is
computed and a refinement phase in which successively drawings of finer graphs
are computed, using the drawings of the next coarser graphs and a variant of a
suitable force-directed single-level algorithm.
The coarsening phase of GRIP is based on the construction of a maximum
independent set filtration or MIS filtration of the node set V . A MIS filtration
is a family of sets {V =: V0 , V1 , . . . , Vk } with ∅ ⊂ Vk ⊂ Vk−1 . . . ⊂ V0 so that
each Vi with i ∈ {1, . . . , k} is a maximal subset of Vi−1 for which the graphtheoretic distance between any pair of its elements is at least 2i−1 + 1. Gajer
and Kobourov [9] use a Spring Embedder-like method as a single-level algorithm
at each level. The used force vector is similar to that used in the method of
Kamada and Kawai [16], but is restricted to a suitable chosen subset of Vi .
Other notable specifics of GRIP are that it computes the MIS filtration only
and no edge sets of the coarse graphs G0 , . . . , Gk that are induced by the filtrations. Furthermore, it is designed to place the nodes in an n-dimensional space
(n ≥ 2), to draw the graph in this space, and to project it into two or three
dimensions.
The asymptotic running time of the algorithm, excluding the time that is
needed to construct the MIS filtration, is Θ(|V |(log diam(G)2 )) for graphs with
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 348
bounded maximum node degree, where diam(G) denotes the diameter of G.
2.3
The Fast Multi-scale Method (FMS)
In order to create the sequence of coarse graphs in the force-directed multilevel
method FMS, Harel and Koren [13] use an O(k|V |) algorithm that finds a 2approximative solution of the N P-hard k-center problem [10]. The node set Vi
of a graph Gi in the sequence G0 , . . . , Gk is determined by the approximative
solution of the ki -center problem on G with ki > ki+1 for all i ∈ {0, . . . , k − 1}.
The authors use a variation of the algorithm of Kamada and Kawai [16] as
a force-directed single-level algorithm. In order to speed up the computation of
this method, they modify the energy function of Kamada and Kawai [16] that
is associated with a graph Gi with i ∈ {0, . . . , k − 1}. The difference to the
original energy of Kamada and Kawai [16] is that only some of the |V (Gi )| − 1
springs that are connected with a node v ∈ V (Gi ) are considered.
The asymptotic running time of FMS is Θ(|V ||E|). Additionally, Θ(|V |2 )
memory is needed to store the distances between all pairs of nodes.
2.4
The Fast Multipole Multilevel Method (FM3 )
The force-directed multilevel algorithm FM3 has been introduced by Hachul and
Jünger [12, 11]. It is based on a combination of an efficient multilevel technique
with an O(|V | log |V |) approximation algorithm to obtain the repulsive forces
between all pairs of nodes.
In the coarsening step subgraphs with a small diameter (called solar systems)
are collapsed to obtain a multilevel representation of the graph. In the used
single-level algorithm, the bottleneck of calculating the repulsive forces acting
between all pairs of charged particles in the Spring Embedder-like force model
is overcome by rapidly evaluating potential fields using a novel multipole-based
tree code. The worst-case running time of FM3 is O(|V | log |V | + |E|) with linear
memory requirements.
2.5
The Algebraic Multigrid Method ACE
In the description of their method ACE, Koren et al. [17] define the quadratic
optimization problem
(P )
min xT Lx so that
xT x = 1
in the subspace xT 1n = 0 .
Here, n = |V | and L is the Laplacian matrix of G.
The minimum of (P) is obtained by the eigenvector that corresponds to the
smallest positive eigenvalue of L. The problem of drawing the graph G in two
dimensions is reduced to the problem of finding the two eigenvectors of L that
are associated with the two smallest eigenvalues.
Instead of calculating the eigenvectors directly, an algebraic multigrid algorithm is used. Similar to the force-directed multilevel ideas, the idea is to
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 349
express the originally high-dimensional problem in lower and lower dimensions,
solving the problem at the lowest dimension, and progressively solving a highdimensional problem by using the solutions of the low-dimensional problems.
The authors do not give an upper bound on the asymptotic running time of
ACE in the number of nodes and edges.
2.6
High-Dimensional Embedding (HDE)
The method HDE of Harel and Koren [14] is based on a two phase approach
that, first, generates an embedding of the graph in a very high-dimensional
vector space and, then, projects this drawing into the plane.
The high-dimensional embedding of the graph is computed by, first, using
a linear time algorithm for approximatively solving the k-center problem [10].
A fixed value of k = 50 is chosen, and k is also the dimension of the highdimensional vector space. Then, breadth-first search starting from each of the k
center nodes is performed resulting in k |V |-dimensional vectors that store the
graph-theoretic distances of each v ∈ V to each of the k centers. These vectors
are interpreted as a k-dimensional embedding of the graph.
In order to project the high-dimensional embedding of the graph into the
plane, the k vectors are used to define a covariance matrix S. The x- and ycoordinates of the two-dimensional drawing are obtained by calculating the two
eigenvectors of S that are associated with its two largest eigenvalues. HDE runs
in Θ(|V | + |E|) time.
3
3.1
The Experiments
Test-Environment, Implementations, and Parameter
Settings
All experiments were performed on a 2.8 GHz Intel Pentium 4 PC with one
gigabyte of memory running Linux.
We tested an implementation of GVA by S. Näher and D. Alberts that is
part of the AGD [15] library, an implementation of GRIP by R. Yusufov that
is available from [24], and implementations of FMS, ACE, HDE by Y. Koren that
are available from [18]. Finally, we tested our own implementation of FM3 .
The provided executables of ACE, HDE, and FMS are Microsoft Windows executables. They were tested on the same machine but using Windows as operating system, instead.
In order to obtain a fair comparison, we ran each algorithm with the same
set of standard-parameter settings (given by the authors) on each tested graph.
However, we are aware that in some cases it might be possible to obtain better
results by spending a considerable amount of time with trial-and-error searching
for an optimal set of parameters for each algorithm and graph.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 350
3.2
The Set of Test Graphs
Since only few implementations can handle disconnected and weighted graphs,
we restrict our attention to connected unweighted graphs, here.
We generated several classes of artificial graphs to examine the scaling of
the algorithms on graphs with predefined structures but different sizes.
These are random grid graphs that were obtained by, first, creating regular
square grid graphs and, then, randomly deleting 3% of the nodes. The sierpinski graphs were created by associating the Sierpinski Triangles with graphs.
Furthermore, we generated complete 6-nary trees.
The next two classes of artificial graphs were designed to test how well the
algorithms can handle highly non-uniform distributions of the nodes and high
node degrees. Therefore, we created these graphs in a way so that one can expect
that an energy-minimal configuration of the nodes in a drawing that relies on a
Spring Embedder-like force model induces a tiny subregion of the drawing area
which contains Θ(|V |) nodes. In particular, we constructed trees that contain
a root node r with |V |/4 neighbors. The other nodes were subdivided into six
subtrees of equal size rooted at r. We called these graphs snowflake graphs.
Additionally we created spider graphs by constructing a circle C containing
25% of the nodes. Each node of C is also adjacent to 12 other nodes of the
circle. The remaining nodes were distributed on 8 paths of equal length that
were rooted at one node of C. In contrast to the snowflake graphs, the spider
graphs have bounded maximum degree.
The last kind of artificial graphs are graphs with a relatively high edge
density |E|/|V | ≥ 14. We call them flower graphs. They are constructed by
joining 6 circles of equal length at a single node before replacing each of the
nodes by a complete subgraph with 30 nodes (K30 ).
The rest of the test graphs are taken from real-world applications. In particular, we selected graphs from the AT&T graph library [1], from C. Walshaw’s
graph collection [22], and a graph that describes a social network of 2113 people
that we obtained from C. Lipp.
We partitioned the artificial and real-world graphs into two sets. The first
set are graphs that consist of few biconnected components, have a constant maximum node degree, and have a low edge density. Furthermore, one can expect
that an energy-minimal configuration of the nodes in a Spring Embedder drawing of such a graph does not contain Θ(|V |) nodes in an extremely tiny subregion of the drawing area. Since one can anticipate from previous experiments [14, 17, 9, 13] that graphs contained in this set do not cause problems
for many of the tested algorithms, we call the set of these graphs kind. The
second set is the complement of the first one, and we call the set of these graphs
challenging.
Table 1 gives an overview of the structures of the tested graphs.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 351
Type
Kind
Artificial
Kind
Real
World
Challenging
Artificial
Challenging
Real
World
Name
|V |
|E|
|B|
|E|
|V |
rnd grid 032
rnd grid 100
rnd grid 320
sierpinski 06
sierpinski 08
sierpinski 10
crack
fe pwt
finan 512
fe ocean
tree 06 04
tree 06 05
tree 06 06
snowflake A
snowflake B
snowflake C
spider A
spider B
spider C
flower A
flower B
flower C
ug 380
esslingen
add 32
dg 1087
bcsstk 33
bcsstk 31
985
9497
97359
1095
9843
88575
10240
36463
74752
143437
1555
9331
55987
971
9701
97001
1000
10000
100000
930
9030
90030
1104
2075
4960
7602
8738
35586
1834
17849
184532
2187
19683
177147
30380
144794
261120
409593
1554
9330
55986
970
9700
97000
2200
22000
220000
13521
131241
1308441
3231
5530
9462
7601
291583
572913
2
6
2
1
1
1
1
55
1
39
1554
9330
55986
970
9700
97000
801
8001
80001
1
1
1
27
867
951
7601
1
48
1.8
1.8
1.9
2.0
2.0
2.0
2.9
3.9
3.4
2.8
1.0
1.0
1.0
1.0
1.0
1.0
2.2
2.2
2.2
14.5
14.5
14.5
2.9
2.6
1.9
1.0
33.3
16.1
max.
degree
4
4
4
4
4
4
9
15
54
6
7
7
7
256
2506
25006
18
18
18
30
30
30
856
97
31
6566
140
188
Table 1: The test graphs and their structures
3.3
The Criteria of Evaluation
The natural criteria to evaluate a graph-drawing algorithm in practice are the
needed running times and the quality of the drawings.
In order to evaluate the quality of the drawings, we focus on the aesthetic
criteria that all tested algorithms strive for (i.e., uniformity of edge length, few
edge crossings, non-overlapping nodes and edges, and the display of symmetries).
Suppose G = (V, E) is a graph and Γ is a drawing of G. Let lΓ (e) denote
the length of an edge e ∈ E in Γ and lΓav denote the average edge length of an
edge in Γ. To measure the uniformity of the edges, we calculated the normalized
standard deviation of the edge length that is
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 352
v
uX
2
u
(lΓ (e) − lΓav )
σΓ := t
.
av
|E| · (lΓ )2
e∈E
We counted the number of edge crossings in Γ (denoted by ecn Γ ) as well as
the number of pairs of edges that completely overlap each other (denoted by
eon Γ ). To compare these measures on graphs with different sizes, we define the
relative edge-crossing number (recn Γ ) and the relative edge-overlapping number
(reon Γ ) of a drawing Γ of G as follows:
recn Γ :=
|ecn Γ |
|E|
reon Γ :=
|eon Γ |
|E|
We did not measure the number of (partially) overlapping nodes. The reason
is that per default some implementations represent nodes by points (ACE, HDE),
whereas the others draw them as circles that occupy nonempty area. Since
symmetry-detection is N P-hard [19], we did not explicitly measure the symmetry of a given drawing and printed the drawing, instead. One of the most
important implicit goals of a graph-drawing algorithm is that an individual user
is satisfied with a drawing. Therefore, we provide printouts of the computed
drawings, too.
4
4.1
The Results
Comparison of the Running Times
The running times of the methods GVA, FM3 , GRIP, FMS, ACE, and HDE for the
tested graphs are presented in Table 2.
As expected, in most cases GVA is the slowest method among the forcedirected algorithms. The largest graph fe ocean is drawn by GVA in 5 hours and
20 minutes.
The method FM3 is significantly faster than GVA for all tested graphs. The
running times range from less than 2 seconds for the smallest graphs to less than
6 minutes for the largest graph fe ocean. The sub-quadratic scaling of FM3 can
be experimentally confirmed for all classes of tested graphs.
Except for the dense graphs flower B and bcsstk 33 GRIP is faster than
FM3 (up to a factor of 9). Unfortunately, we could not examine the scaling
of GRIP for the largest graphs due to an error in the executable.
Since the memory requirement of FMS is quadratic in the size of the graph,
the implementation of FMS is restricted to graphs that contain at most 10, 000
nodes. The running times of FMS are comparable with those of FM3 for the
smallest and the medium sized kind graphs. In contrast to this, the CPU times of
FMS increase drastically for several challenging graphs, in particular for graphs
that either contain nodes with a very high degree or have a high edge density.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 353
Type
Kind
Artificial
Kind
Real
World
Challenging
Artificial
Challenging
Real
World
Name
rnd grid 032
rnd grid 100
rnd grid 320
sierpinski 06
sierpinski 08
sierpinski 10
crack
fe pwt
finan 512
fe ocean
tree 06 04
tree 06 05
tree 06 06
snowflake A
snowflake B
snowflake C
spider A
spider B
spider C
flower A
flower B
flower C
ug 380
esslingen
add 32
dg 1087
bcsstk 33
bcsstk 31
GVA
12.5
203.4
6316.1
13.1
171.7
3606.4
317.5
1869.1
6319.8
19247.0
14.3
130.3
1769.2
8.0
143.2
14685.7
17.6
189.0
4568.3
61.7
595.1
11841.5
23.1
43.8
80.6
624.8
1494.6
4338.4
CPU Time
FM3 GRIP
1.9
0.3
19.1
4.4
215.4 (E)
1.8
0.3
16.8
4.8
162.0 (E)
23.0
6.8
69.0 (E)
158.2 (E)
355.9 (E)
2.6
0.3
17.7
2.4
121.3 (E)
1.6
0.4
17.4
6.1
166.5 (E)
1.9
0.4
17.7
7.2
177.2 (E)
1.2
0.7
11.9 19.3
121.4 (E)
2.1
0.4
4.0
0.5
12.1
1.6
18.1
3.6
23.8 29.1
83.6 (E)
in Seconds
FMS
ACE
1.0 < 0.1
32.0
0.5
(M )
4.1
1.0 < 0.1
33.0
1.0
(M )
23.4
(M )
0.4
(M )
(T )
(M )
7.5
(M )
4.0
2.0 < 0.1
43.0
0.5
(M )
4.5
73.0
0.4
3320.0
(T )
(M )
(T )
1.0
1.1
47.0
8.9
(M ) 280.7
1.0 < 0.1
46.0
1.4
(M )
(T )
1.0 < 0.1
404.0
1.0
17.0
0.5
5402.0 108.4
6636.0
0.4
(M )
1.9
HDE
< 0.1
0.1
1.3
< 0.1
0.1
1.0
0.2
0.5
1.0
3.4
< 0.1
< 0.1
0.5
< 0.1
< 0.1
0.8
< 0.1
0.1
1.3
< 0.1
0.2
1.4
< 0.1
< 0.1
< 0.1
< 0.1
0.3
0.7
Table 2: The test graphs and the running times that are needed by the tested
algorithms to draw them. Explanations: (E) No drawing was computed due to
an error in the executable. (M ) No drawing was computed because the memory
is restricted to graphs with ≤ 10, 000 nodes. (T ) No drawing was computed
within 10 hours of CPU time. B denotes the sets of biconnected components of
the graphs. Best values are printed bold. Worst values are underlined.
The algorithm ACE is much faster than the force-directed algorithms for
nearly all kind graphs. However, the running times grow extremely if ACE is
used to draw several of the challenging graphs.
The linear time method HDE is by far the fastest algorithm. It needs less
than 3.4 seconds for drawing even the largest tested graph.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 354
4.2
Comparison of the Drawings
4.2.1
Uniformity of Edge Lengths
5
4
GVA
FM3
FMS
ACE
HDE
3
2
1
Kind Graphs
Norm. Std. Dev. of Edge Length
Norm. Std. Dev. of Edge Length
Figure 1 shows the normalized standard deviations of the edge lengths σΓ of the
drawings Γ for the tested kind and challenging graphs.
25
20
15
GVA
FM3
FMS
ACE
HDE
10
5
Challenging Graphs
Figure 1: The normalized standard deviations of the edge lengths of the drawings for the kind graphs (left) and challenging graphs (right).
For the kind graphs the σΓ values of the drawings that are computed by GVA,
FM3 , FMS, and HDE are below 1 while the σΓ values associated with ACE are still
below 2. This indicates that for those graphs the goal of generating drawings
with uniform edge lengths has been reached well by all algorithms.
The σΓ values of some drawings of challenging graphs that are generated by
HDE and ACE exceed values of 5 and 25, respectively. In contrast to this, the
σΓ values of GVA and FMS remain below 1 for the majority of the drawings of
the challenging graphs. All σΓ values associated with drawings computed by
FM3 are smaller than 1.
Notice that it is not possible to obtain the measures σΓ , recn Γ , and reon Γ for
the drawings generated by GRIP, since the used implementation of GRIP provides screen output only. Hence, in this case we had to restrict ourselves to
printing screenshots for visual comparisons.
4.2.2
Edge Crossings and Overlapping Edges
The relative edge-crossing numbers recn Γ and the relative edge-overlapping
numbers reon Γ of the drawings Γ generated by the algorithms are listed in
Tables 3 and 4, respectively.
For the kind graphs, the multilevel and algebraic methods generate comparable and significantly smaller numbers of edge crossings than the classical
force-directed method GVA.
With two exceptions, the recn Γ values of the drawings of the challenging
graphs that are generated by FM3 are smaller than the corresponding values of
GVA. The recn Γ values of ACE are much smaller than those of FM3 for several
challenging artificial graphs, while they are comparable with those of FM3 for the
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 355
Type
Kind
Artificial
Kind
Real
World
Challenging
Artificial
Challenging
Real
World
Name
rnd grid 032
rnd grid 100
rnd grid 320
sierpinski 06
sierpinski 08
sierpinski 10
crack
fe pwt
finan 512
fe ocean
tree 06 04
tree 06 05
tree 06 06
snowflake A
snowflake B
snowflake C
spider A
spider B
spider C
flower A
flower B
flower C
ug 380
esslingen
add 32
dg 1087
bcsstk 33
bcsstk 31
GVA
3.82
14.75
181.51
2.00
9.49
99.97
30.82
150.70
301.25
622.48
2.21
9.33
70.68
0.63
1.46
15.53
15.62
154.70
2522.89
46.71
64.90
578.22
22.93
47.52
8.65
1.74
720.94
708.69
FM3
0
0
0
0.05
0.07
0.09
<0.01
2.45
18.81
7.13
1.16
1.89
3.31
0
0
0
16.55
132.96
1029.64
49.08
51.57
53.39
19.55
23.71
1.69
< 0.01
376.18
94.26
recn Γ
FMS
0
0
(N )
<0.01
0.01
(N )
(N )
(N )
(N )
(N )
7.89
11.48
(N )
0.10
8.18
(N )
1.17
1.64
(N )
5.63
1.90
(N )
13.67
28.42
5.75
37.07
4171.05
(N )
ACE
0
0
<0.01
0
0.02
0.27
0
(N )
12.27
9.07
0.01
0
4.16
<0.01
(N )
(N )
6.60
0
0
0.26
0.06
(N )
20.99
20.81
0.89
5.92
413.56
63.00
HDE
<0.01
<0.01
<0.01
0.02
0.08
0.01
0.07
1.61
21.27
8.24
0
22.92
128.82
0.62
6.92
195.87
1.25
0
0
0.55
0.34
0.30
1.35
3.89
5.80
6.49
113.86
611.21
Table 3: The relative edge-crossing numbers (recn Γ ) of the drawings Γ computed by the tested algorithms. The entry (N ) indicates that no drawing was
computed. Best values are printed bold. Worst values are underlined.
challenging real-world graphs. Depending on the classes of tested challenging
graphs, the recn Γ values of the drawings computed by FMS and HDE vary a lot:
FMS and HDE generate many crossings for the 6-nary trees and the snowflake
graphs but comparatively few crossings for the spider and flower graphs.
No drawing of a kind graph generated by GVA or FM3 contains overlapping
edges. In contrast to this, few pairs of overlapping edges exist in many drawings
of kind graphs computed by FMS and ACE, and in all drawings computed by HDE.
The challenging graphs spider A and esslingen are the only graphs in the test
set that contain parallel edges, resulting in minimum relative edge-overlapping
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 356
Type
Kind
Artificial
Kind
Real
World
Challenging
Artificial
Challenging
Real
World
Name
rnd grid 032
rnd grid 100
rnd grid 320
sierpinski 06
sierpinski 08
sierpinski 10
crack
fe pwt
finan 512
fe ocean
tree 06 04
tree 06 05
tree 06 06
snowflake A
snowflake B
snowflake C
spider A
spider B
spider C
flower A
flower B
flower C
ug 380
esslingen
add 32
dg 1087
bcsstk 33
bcsstk 31
GVA
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.27
0
0
0
0
0
0
0.14
0
0
0
0
FM3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.27
0
0
0
0
0
0
0.14
0
0
0
0
reon Γ
FMS
ACE
0.03
0
0
0
(N )
0
0.03
0
0.14
<0.01
(N )
0.98
(N )
0
(N )
(N )
(N )
3.68
(N )
0.22
0.38
59.09
0.86
504.27
(N )
2929.74
42.02
32.11
169.77
(N )
(N )
(N )
0.67
19.57
6903.67
4248.23
(N ) 44554.09
5.14
149.31
21.63
167.48
(N )
(N )
1.29
0.79
0.97
1.93
1.21
0.19
109.23
1882.14
3416.70
0.13
(N )
2.50
HDE
<0.01
0.03
0.08
0.01
0.11
1.43
0.12
34.25
9.14
1.67
78.08
466.83
2030.40
85.34
398.06
16818.82
63.50
297.13
40163.33
160.36
210.83
340.19
10.69
2.54
5.73
1390.30
329.17
2386.76
Table 4: The relative edge-overlapping numbers (reon Γ ) of the drawings Γ computed by the tested algorithms. The entry (N ) indicates that no drawing was
computed. Best values are printed bold. Worst values are underlined.
numbers of 0.27 and 0.14, respectively. Hence, GVA and FM3 are the only algorithms that generate drawings with the minimum number of overlapping edges
for all tested graphs. In contrast to this, the recn Γ values of all drawings of challenging graphs generated by FMS, ACE, and HDE are positive and reach extremely
high values in many cases.
For FMS an explanation of this behavior might be that the positions have
integer values only, and that the underlying grid-resolution is too small. For
the algebraic methods an explanation of the many overlapping nodes and edges
can be found in [3, 17] and [11].
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 357
Since overlapping edges are at least as undesirable as crossing edges we
additionally compared the sums of the relative edge-crossing numbers and the
dedicated relative edge-overlapping numbers (see Figure 2).
recn + reon
10000
100
GVA
3
FM
FMS
ACE
HDE
1e+06
recn + reon
1e+06
1
0.01
10000
GVA
3
FM
FMS
ACE
HDE
100
1
0.01
Kind Graphs
Challenging Graphs
Figure 2: The sums of the relative edge-crossing and dedicated relative edgeoverlapping numbers of the drawings for the tested kind graphs (left) and challenging graphs (right).
For the kind graphs, the recn Γ values of the drawings computed by GVA are
significantly larger than the sum of recn Γ and reon Γ of all other algorithms. For
those graphs FM3 and ACE have smaller values than HDE. FM3 reaches the lowest
added values of recn Γ and reon Γ for the majority of the challenging graphs. The
other multilevel and algebraic methods have frequently higher added recn Γ and
reon Γ values than the classical force-directed algorithm GVA.
4.2.3
The Overall Picture
In the remainder of this section, we will discuss the quality of the drawings of
the tested graphs that are presented in Figures 3 to 10 by keeping the modeled
esthetic criteria in mind.
Note that due to the very restricted drawing area, not all details of each
large graphs can be displayed, here. However, those details can be examined by
creating large printouts or by using simple zoom and pan techniques.
For all kind graphs the classical method GVA does not untangle the drawings
that were induced by the random initial placements. In contrast to this, nearly
all algorithms computed relatively pleasing drawings of the kind graphs (see
Figure 3, Figure 4, and Figure 5(a)-(d)).
None of the drawings of the complete 6-nary trees (see Figure 5(e)-(j)) is
really convincing, since the algorithms either produce many unnecessary edge
crossings or they place many nodes at the same coordinates.
Except FM3 none of the tested algorithms displays the global structure of
the snowflake graphs. Even the drawings of the smallest snowflake graph (see
Figure 6(a)-(f)) leave room for improvement. However, GVA and GRIP visualize
parts of its structure in an appropriate way.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 358
The drawings of the spider A graph (see Figure 6(g)-(l)) that are generated
by GRIP, FMS, and HDE are not as symmetric as the drawing computed by FM3 .
But they display the global structure of the graph. The drawing generated by
GVA shows the dense subregion, but GVA does not untangle the 8 paths. The
paths in the drawing of ACE are not displayed in the same length. The drawings
of the larger spider graphs are of comparable quality.
The drawings of the flower B graph (see Figure 7(a)-(f)) that are computed
by FMS and HDE display the global structure of the graph but the symmetries
are not as clear as in the drawing generated by FM3 . The drawings of the other
flower graphs are of comparable quality.
We concentrate on the challenging real-world graphs now. The graphs ug 380
and dg 1087 both contain one node with a very high degree. Furthermore,
dg 1087 has many biconnected components, since it is a tree. Only the drawings
that are computed by GVA, FM3 , and GRIP (see Figure 7(g)-(l) and Figure 8(a)(f)) clearly display the central regions of these graphs.
The social network esslingen (see Figure 8(g)-(l)) consists of two big wellconnected subgraphs. This can be visualized by FM3 , GRIP, and HDE. But the
drawings contain several edge crossings.
Since add 32 that describes a 32 bit adder contains many biconnected components, we expect that the drawings have a tree-like shape. This structure is
visualized by GVA, FM3 , GRIP, and ACE (see Figure 9(a)-(f)). The drawings of
GVA and GRIP contain comparatively many edge crossings, while the drawing
of ACE displays the global structure, but hides local details.
Finally, we discuss the drawings of the dense graphs bcsstk 31 and bcsstk 33.
The drawings of bcsstk 33 (see Figure 9(g)-(l)) that are generated by FM3 , GRIP,
and ACE are comparable and visualize the regular structure of the graph. The
car body that is modeled by the graph bcsstk 31 (see Figure 10) is visualized
by FM3 and ACE only.
5
Conclusion
We can summarize that only GVA, FM3 , and HDE generate drawings of all tested
graphs. The force-directed multilevel methods and the algebraic methods are
– except the methods FMS and ACE for some graphs – much faster than the
comparatively slow classical algorithm GVA. HDE, FM3 and GRIP scale well on all
tested graphs. FM3 needs a few minutes to draw the largest graphs. GRIP is up
to factor 9 faster than FM3 but it could not be tested on the largest graphs. All
tested methods are much slower than HDE that needs only few seconds to draw
even the largest graphs.
As expected, all algorithms, except GVA, generate pleasing drawings of the
kind graphs with relatively uniform edge length and few edge crossings. In
contrast to this, the quality of the computed drawings varies a lot depending
on the structures of the tested challenging graphs. In particular, FMS, HDE, and
ACE frequently generate drawings with many overlapping edges. Unlike this,
FM3 generates pleasing drawings for the majority of the challenging graphs. But
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 359
there still remain classes of tested graphs (e.g., the complete trees and the social
network graph esslingen) for which the drawing quality of all tested algorithms
leaves much room for improvement.
A practical advice for developers of graph-drawing systems is as follows:
First use HDE followed by ACE, since they are the fastest methods in all or
many cases, respectively. If the drawings are not satisfactory or one supposes
that important details of the graph’s structure are hidden, use FM3 to obtain
comparable or better results in reasonable time. If those drawings are still not
nice, one should experiment with other methods or try to improve the existing
ones with new concepts.
Acknowledgments
We would like to thank David Alberts, Steven Kobourov, Yehuda Koren, Stefan
Näher, and Roman Yusufov for making the implementations of their algorithms
available to us. We thank Ulrik Brandes, Carola Lipp and Chris Walshaw for
the access to the real-world test graphs.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 360
References
[1] The AT&T graph collection: www.graphdrawing.org.
[2] F. Brandenburg, M. Himsolt, and C. Rohrer. An Experimental Comparison
of Force-Directed and Randomized Graph Drawing Methods. In Graph
Drawing 1995, volume 1027 of LNCS, pages 76–87. Springer-Verlag, 1996.
[3] U. Brandes and D. Wagner. In Graph Drawing Software, volume XII of
Mathematics and Visualization, chapter visone - Analysis and Visualization
of Social Networks, pages 321–340. Springer-Verlag, 2004.
[4] R. Davidson and D. Harel. Drawing Graphs Nicely Using Simulated Annealing. ACM Transactions on Graphics, 15(4):301–331, 1996.
[5] P. Eades. A heuristic for graph drawing. Congressus Numerantium, 42:149–
160, 1984.
[6] A. Frick, A. Ludwig, and H. Mehldau. A Fast Adaptive Layout Algorithm
for Undirected Graphs. In Graph Drawing 1994, volume 894 of LNCS,
pages 388–403. Springer-Verlag, 1995.
[7] T. Fruchterman and E. Reingold. Graph Drawing by Force-directed Placement. Software–Practice and Experience, 21(11):1129–1164, 1991.
[8] P. Gajer, M. Goodrich, and S. Kobourov. A Multi-dimensional Approach to
Force-Directed Layouts of Large Graphs. In Graph Drawing 2000, volume
1984 of LNCS, pages 211–221. Springer-Verlag, 2001.
[9] P. Gajer and S. Kobourov. GRIP: Graph Drawing with Intelligent Placement. In Graph Drawing 2000, volume 1984 of LNCS, pages 222–228.
Springer-Verlag, 2001.
[10] T. Gonzalez. Clustering to Minimize the Maximum Inter-Cluster Distance.
Theoretical Computer Science, 38:293–306, 1985.
[11] S. Hachul. A Potential-Field-Based Multilevel Algorithm for Drawing Large
Graphs. PhD thesis, Institut für Informatik, Universität zu Köln, Germany,
2005. http://kups.ub.uni-koeln.de/volltexte/2005/1409.
[12] S. Hachul and M. Jünger. Drawing Large Graphs with a PotentialField-Based Multilevel Algorithm (Extended Abstract). In Graph Drawing
2004, volume 3383 of Lecture Notes in Computer Science, pages 285–295.
Springer-Verlag, 2005.
[13] D. Harel and Y. Koren. A Fast Multi-scale Method for Drawing Large
Graphs. In Graph Drawing 2000, volume 1984 of LNCS, pages 183–196.
Springer-Verlag, 2001.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 361
[14] D. Harel and Y. Koren. Graph Drawing by High-Dimensional Embedding.
In Graph Drawing 2002, volume 2528 of LNCS, pages 207–219. SpringerVerlag, 2002.
[15] M. Jünger, G. Klau, P. Mutzel, and R. Weiskircher. In Graph Drawing
Software, volume XII of Mathematics and Visualization, chapter AGD - A
Library of Algorithms for Graph Drawing, pages 149–172. Springer-Verlag,
2004.
[16] T. Kamada and S. Kawai. An Algorithm for Drawing General Undirected
Graphs. Information Processing Letters, 31:7–15, 1989.
[17] Y. Koren, L. Carmel, and D. Harel. Drawing Huge Graphs by Algebraic
Multigrid Optimization. Multiscale Modeling and Simulation, 1(4):645–673,
2003.
[18] Y. Koren’s algorithms: research.att.com/~yehuda/index_programs.
html.
[19] J. Manning. Computational complexity of geometric symmetry detection
in graphs. In Great Lakes Computer Science Conference, volume 507 of
Lecture Nodes in Computer Science, pages 1–7. Springer-Verlag, 1990.
[20] A. Quigley and P. Eades. FADE: Graph Drawing, Clustering, and Visual
Abstraction. In Graph Drawing 2000, volume 1984 of LNCS, pages 197–210.
Springer-Verlag, 2001.
[21] D. Tunkelang. JIGGLE: Java Interactive Graph Layout Environment. In
Graph Drawing 1998, volume 1547 of LNCS, pages 413–422. SpringerVerlag, 1998.
[22] C. Walshaw’s graph collection: staffweb.cms.gre.ac.uk/~c.walshaw/
partition.
[23] C. Walshaw. A Multilevel Algorithm for Force-Directed Graph Drawing.
In Graph Drawing 2000, volume 1984 of LNCS, pages 171–182. SpringerVerlag, 2001.
[24] R. Yusufov’s implementation of GRIP: www.cs.arizona.edu/~kobourov/
GRIP.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 362
(a) GVA
(d) FMS
(g) GVA
(j) FMS
(b) FM3
(e) ACE
(h) FM3
(k) ACE
(c) GRIP
(f) HDE
(i) GRIP
(l) HDE
Figure 3: (a)-(f) Drawings of rnd grid 100 and (g)-(l) sierpinski 08 generated
by different algorithms.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 363
(a) GVA
(d) ACE
(g) FM3
(j) FM3
(b) FM3
(e) HDE
(h) HDE
(k) ACE
(c) GRIP
(f) GVA
(i) GVA
(l) HDE
Figure 4: (a)-(e) Drawings of crack, (f)-(h) fe pwt, and (i)-(l) finan 512 generated by different algorithms.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 364
(b) FM3
(a) GVA
(c) ACE
2861
2866
6887
7207
7208
7211
7239
6888
7220
9246
7210
7240
7241
7212
6897
6884
6910
6907
(d) HDE
2859
2860 2857
2883
2858
2867 2879
6899
2875
6900
6885
1201
2862
2864
6914
6908
6886
7209
7221
6912
6911
7222
9230
2878
2868
6895
477
6916
476
2871
2852
2876
2865
2863
479
2050
2885
2854
2886
2049
2855
2873
2874
2870
2869
7217
2851
6891
2881
480
6892
9233
7223
7229
475
7238
6889
478
7214
9231
6883
2853
7215
2872
1147
6894
2452
9232
2453
2880
9229
2048
2943
2884
2877
9249
7224
6913
4680
2056
6915
7232
7233
2038 2882
9234
1152
2944
7237
2053
9245
4678
2454
2856
341
6896
9241
6898
9252
1148
9237
20472033
7231
2422
4675
92517216 9228
6902
9225
1203
29322942
2039
1538
2419
7219
2433
6909
6917
2428
7242
2030
24512450
1149
7235
9248
4677
6893
2035
2423
2036
1151
2945
7225
9240
2052
2952
2424
408
1206
9247
9238
7213
2031
1202
7534
2948
3754
7230
4682
2951
9224
9150
9243
46614679
7543
2057
2430
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1541
4659 4676
7531
4683
2444
7227 1205
9235
2954
2034
1745
2448
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6905 2058
2431 6890
9227
490
2941
9236
2032
2044
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6179
9146
6176
7236
2446
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20292023
1540
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2040339
2958
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2421
9226 9250
9255
2946
9148
2955
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7218
7551 24439124
2929
342
7536
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1204
9258 3796
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2054
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2427
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338
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7228
4674
2043
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403 7540
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37852425
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488
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9253 1537
72472437
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4651
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6029
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9223
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2025
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2024
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337
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776
1524
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7043 1255
7538
5900
404
2927
5881
7407
3756 1257
2026
7190
7189
2957
3798
3780405
6041
2045 3827
6906
3818
6033
7559
5898
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7261 7270
5901
3775
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3826
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5894 4670
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6901
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9128
6024
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2041
3772 7541
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2928
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632
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1207
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5897
3836
3761
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2027
7260
2924
9119
70417034
406
9130
9122
7553
290 7267
7251
6019
3834
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3752
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7556
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6027
487
7263
3765
3766 630
7408
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1005 6035
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2426 625
2936
9123
3825
7244 7245
6036
3763
7254
1258
6028
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633
3776 9140
5893
9131
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7204 3803
9147
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627
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3847
17591763
629
6037
293
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9127
3778
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9144
3828
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1004
9125
2090
1753
7268
1741
3678
5885
637
4652
3835
3759
200
3849
4671
626
6030
1757
1209
982
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1211
7194
639 3840
7051
9135
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7272
5902
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9133
2937
9118
1259
7257
9121
9815883
5896
1198
6048
1766 775
2926
60426025
7045 2442
7030
2624
3857
7255
7250
3779
5887
3830
2499
7396 7042
7422
5880
7266 5882
5875
7199
1210
980
628
3802
6049
3672 1256
6022
2939
38233773
7037 1521
6439
36743677
5886
6038
2634
5904
9116 704470401173
7395
2628
3784
7265
56
8255
1765
9139
2089
3646
7253
7203
6044
292
72567258
9141
1199
3769
6039
9129
9134
7269
8257
7054
5891
191
1234
7264
7048
6043
9115
7028
5907
8277
3670
7049
3757
8260
5884
7566
3817
8284
3783
7191
8256
3643
1758
26322633
7201 256
2627
2623
2501
7420
7047
7417
983
2440
82623843
6020
1768
4663
6054
3762
7542
2626
70551519
8252
6575
6051
7195
7419
9138
1523
979
2498
2630
3668
1175
1008
8251
6045
3819
6387
6021
5908
1003
17481770
7046 7402
437
7035
1767
7262
81 3855
7052
4668
6442 8681
1749
7205
7394
2629 2631
1208
636
3667
7557
4654
1007
7185
67
4665
3858
7555
3645
638
7412
7188
1736
1174
3650
1522
6443
7404
1006
2611
82618282
984
6423
7562
3647
1260 7561
3850
6046
6421
489
6405
3644
3851
1760
1752
438
1172
8278
6440
6574
6571
3832
7183
641
294
7187
8662
6444
8659
3660
7038
1375
3848
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6564
777
9143
7418
3841
3669
2940
7415
5876
6925
7715
2603
8283
8502
36522712
8679
6446
8259
5879
3671
8650
3842
2502
7196
1376
7032
7252
1955
3845
7200
3813 7397
6052
9136
3856
607
1739
5899
7393
2613
3653
642 7565
611
5175
5795
8254
4667
5775
9126
3833
2061
6053
9137
7564
2709
6565
3649
2938
612
5157
3658
2708 8274
7478
6930
7711
3852
1171
7713
8663 209
5155
7413 5178
57787400
3790
69397053
6385
7198
2071
8280
1232
7479
343
6568
7184 5177
7563
8647
1200
6573 6040
65786577
3659 3661
6424
3829
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1989
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1984
171
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1965
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1981
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8846
88418842
8840
8839
8843
(e) GVA
(h) FMS
(f) FM3
(i) ACE
(g) GRIP
(j) HDE
Figure 5: (a)-(d) Drawings of fe ocean and (e)-(j) tree 06 05 generated by different algorithms.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 365
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313
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753
683
697
657
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769
849
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719
310
309
733
877
321
903
291
884
415
478
790
848
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638
682
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252
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616
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398
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311
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203
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219
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120
194
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74
205
208
164
211
86
249
150
241
133
41
(b) FM3
(a) GVA
(d) FMS
340
(c) GRIP
(e) ACE
(f) HDE
339
404
403
341
405
401
402
406
338
400
337
342
407
623
336
335
408
334
622
343
674
14
15
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333
13
51
409
624
16
332
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50
52
620
673
675
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48
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330
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344
56
627
36
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32
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329
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638
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669
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6
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323
248
935
931
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836
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161
432
513
463
247
943
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829
4
754
734
536
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156
162
255
293
979
923
304
306
535
752
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154
308
509
319
291
907
399
88
518
745
166
508
520
685
780
228
507
176
270
474 94
779
777
421
711
473
596
555
423
353
424
425
556
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709
281
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282
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188
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179
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286
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123
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565
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213
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362
217
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352
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564
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467
151
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505
195
196
280
597
140
650
493
178
725
287
233
132
498
241
422
490
491
96
492
189
214
775
216
363
268
506
242
267
729
523
551
552
553
194
572
649
776
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269
232
312
313
314
687
351
488
279
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193
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215
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522
226
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131
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311
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111
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109
586
110
585
581
584
583
582
(g) GVA
(j) FMS
(h) FM3
(i) GRIP
(k) ACE
(l) HDE
Figure 6: (a)-(f) Drawings of snowflake A and (g)-(l) spider A generated by
different algorithms.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 366
(b) FM3
(a) GVA
(d) FMS
(c) GRIP
(e) ACE
(f) HDE
842
845
846
836
841
837
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847
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272
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108
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465
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306
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911
500
499
498
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1078
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194
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105
466
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505
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106
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503
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132
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131
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176
547
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212 502
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110
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263
171
261
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48
520
193
597
1004
1003
484
1000
550
431
963
521
523
1010
332
585
594
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528
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548
564
490
1002
876
513
1011
274
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266
198
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600
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279
288
270
568
588
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488
232
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231
107
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535
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559
334
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219
432
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536
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183
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567
214
964
77
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199
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308
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329
123
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14
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1100
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281
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47
282
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314
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615
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173
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5 1036
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633
17
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363
365
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80
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422
1090
238
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416
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83
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850
38
130
1080
127
496
125
519
126
129
884
1014
(h) FM3
(g) GVA
(j) FMS
(k) ACE
(i) GRIP
(l) HDE
Figure 7: (a)-(f) Drawings of flower B and (g)-(l) ug 380 generated by different
algorithms.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 367
(b) FM3
(a) GVA
(d) FMS
(e) ACE
(c) GRIP
(f) HDE
1177
1181
1179
1015
1735
1182
1180
1379
1380
971
790
1178
399
788
1088
970
160
1388
1383
969
1203
1382
677
1158
1207
1247
1243
1211
968
1245
746
1441
1618
634
711
715
1433
1030
941
1490
520
912
1979
534
180
561
171
1259
1419
1408
1791
498
1866
1402
1420
2066
457
1948
963
2018
1725
1633
962
1629
265
1068
552
1479
1256
170
524
1671
184
666
1636
1067
458
1527
198
494
191
2064
1957
495
493
1921
282
185
491
1474
1374
551
1073
1727
230
921
310
501
1303
1899
508
398
316
507
154
309
1398
1084
1360
955
1786
1660
1900
1871
1951
957
672
589
656
401
406
43
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42
1410
41
1466
405
2048
1222
1220
1295
925
2047
542
541
546
547
1574
543
2050
116
544
539
1486
1228
1255
204
1294
545
210
376
1426
1505
217
220
934
1227
1578
218
1270
1286
120
690
935
212
59
58
931
576
219
377
2038
113 122
112
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540
119
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123
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2020
223
1425
1273
114
586
109
928
581
209
216
110
929
575
205
966
1803
1940
1939
2054
930
1120
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358
1744
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965
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933
601
239
1756
1755
1750
1377
1758
1751
1391
594
598
592
1376
1743
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1959
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272
269
626
1035
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1026
1395
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463
1980
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670 724
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1858
884
514
713
722
949
693
942
1622
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525
380
999
991
1094
1508
1509
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2013
1276
1814 1960
1829
1992 2051
1049
1821
716
911
1485
1577
379
924
1507
1520
1522
997
995
937
1272
1983
1172
2017
1993
1962
2005
1042
1861
856
1961
1738
803 813
1715
228
1511
1859
1903
1373
1223
1943
1239
1982
2002
1967
1171
667
61
1226
18
16
1483
1215
1468
1523
1521
1907688
240
1971
624
889
888
989
1
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1639
1516
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1197
800
855
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733
461
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793
1438
1729
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45
1970
1863
802
1143
692 9
1000
1645
662
1841
1200
1739
1086
1095
382
2522
1241
1984
1234
1198
1500
1249
1152
13
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1412
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227
1190
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27
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1935
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368
369
1278
359
772
360
1311
1847
2032
(g) GVA
(j) FMS
(h) FM3
(k) ACE
(i) GRIP
(l) HDE
Figure 8: (a)-(f) Drawings of dg 1087 and (g)-(l) esslingen generated by different
algorithms
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 368
(a) GVA
(b) FM3
(d) FMS
(e) ACE
(h) FM3
(g) GVA
(j) FMS
(k) ACE
(c) GRIP
(f) HDE
(i) GRIP
(l) HDE
Figure 9: (a)-(f) Drawings of add 32 and (g)-(l) bcsstk 33 generated by different
algorithms.
Hachul and Jünger, Large-Graph Layout Alg., JGAA, 11(2) 345–369 (2007) 369
(a) GVA
(c) ACE
(b) FM3
(d) HDE
Figure 10: (a)-(d) Drawings of bcsstk 31 generated by different algorithms.