Node Overlap Removal by Growing a Tree
Lev Nachmanson1 , Arlind Nocaj2 , Sergey Bereg3 , Leishi Zhang4 , and
Alexander Holroyd1
1
Microsoft Research, Redmond, USA
University of Konstanz, Konstanz, Germany
3
The University of Texas at Dallas, Richardson, USA
4
Middlesex University, London,UK
[email protected], [email protected], [email protected],
[email protected], [email protected]
2
Abstract. Node overlap removal is a necessary step in many scenarios
including laying out a graph, or visualizing a tag cloud. Our contribution
is a new overlap removal algorithm that iteratively builds a Minimum
Spanning Tree on a Delaunay triangulation of the node centers and removes the node overlaps by ”growing” the tree. The algorithm is simple
to implement yet produces high quality layouts. According to our experiments it runs several times faster than the current state-of-the-art
methods.
1
Introduction
Removing node overlap after laying out a graph is a common task in network
visualization. Most graph layout algorithms [23] consider nodes as points that do
not occupy any geometrical space. In practice, nodes often have shapes, labels,
and so on. These shapes and labels may overlap in the visualization and affect
the visual readability. To remove such overlaps a specialized algorithm is usually
applied.
The main contribution of this paper is a new node overlap removal algorithm
that we call Growing Tree, or GTree further on. The basic idea is to first capture
most of the overlap and the local structure with a specific spanning tree on top
of a proximity graph, and then resolve the overlap by letting the tree ”grow”.
We compare GTree with PRISM [6], which is widely used for the same purpose. Needing more area than PRISM, our method preserves the original layout
well and is up to eight times faster than PRISM. To compare the two algorithms we implemented GTree in the open source graph visualization software
Graphviz 1 , where PRISM is the default overlap removal algorithm. On the other
side, GTree is the default in MSAGL2 , where we also have an implementation
of PRISM. We ran comparisons by using both tools.
1
2
http://www.graphviz.org/
https://github.com/Microsoft/automatic-graph-layout
2
Related Work
There is vast research on node overlap removal. Some methods, including hierarchical layouts [4], incorporate the overlap removal with the layout step. Likewise,
force-directed methods [5] have been extended to take the node sizes into account [17, 16, 24], but it is difficult to guarantee overlap-free layouts without
increasing the repulsive forces extensively. Dwyer et al. [2] show how to avoid
node overlaps with Stress Majorization [7]. The method can remove node overlaps during the layout step, but it needs an initial state that is overlap free;
sometimes such a state is not given.
Another approach, which we also choose, is to use a post-processing step. In
Cluster Busting [18, 8] the nodes are iteratively moved towards the centers of
their Voronoi cells. The process has the disadvantage of distributing the nodes
uniformly in a given bounding box.
Imamichi et al. [15] approximate the node shapes by circles and minimize a
function penalizing the circle overlaps.
Starting from the center of a node, RWorldle [22] removes the overlaps by
discovering the free space around a node by using a spiral curve and then utilizing
this space. The approach requires a large number of intersection queries that are
time consuming. This idea is extended by Strobelt et al. [21] to discover available
space by scanning the plane with a line or a circle.
Another set of node overlap removal algorithms focus on the idea of defining
pairwise node constraints and translating the nodes to satisfy the constraints [20,
11, 19, 13]. These methods consider horizontal and vertical problems separately,
which often leads to a distorted aspect ratio [6]. A Force-transfer-algorithm is
introduced by Huang et al. [14]; horizontal and vertical scans of overlapped nodes
create forces moving nodes vertically and horizontally; the algorithm takes O(n2 )
steps, where n is the number of the nodes. Gomez et al. [9] develop Mixed Integer
Optimization for Layout Arrangement to remove overlaps in a set of rectangles.
The paper discusses the quality of the layout, which seems to be high, but not
the effectiveness of the method, which relies on a mixed integer problem solver.
Dwyer et al. [3] reduce the overlap removal to a quadratic problem and solve it
efficiently in O(n log n) steps. According to Gansner and Hu [6], the quality and
the speed of the method of Dwyer et al. [3] is very similar to the ones of PRISM.
The ProjSnippet method [10] generates good quality layouts. The method
requires O(n2 ) amount of memory, at least if applied directly as described in the
paper, and the usage of a nonlinear problem solver.
In PRISM [6, 12], a Delaunay triangulation on the node centers is used as the
starting point of an iterative step. Then a stress model for node overlap removal
is built on the edges of the triangulation and the stress function of the model
is minimized. GTree also starts with building this Delaunay triangulation, but
then the algorithms diverge.
3
GTree Algorithm
An input to GTree is a set of nodes V , where each node i ∈ V is represented
by an axis-aligned rectangle Bi with the center pi . We assume that for different
i, j ∈ V the centers pi , pj are different too. If this is not the case, we randomly
shift the nodes by tiny offsets. We denote by D a Delaunay triangulation of the
set {pi : i ∈ V }, and let E be the set of edges of D.
On a high level, our method proceeds as follows. First we calculate the triangulation D, then we define a cost function on E and build a minimum cost
spanning tree on D for this cost function. Finally, we let the tree “grow”. The
steps are repeated until there are no more overlaps. The last several steps are
slightly modified. Now we explain the algorithm in more detail.
We define the cost function c on E in such a way that the larger the overlap
on an edge becomes, the smaller the cost of this edge comes to be. Let (i, j) ∈ E.
If the rectangles Bi and Bj do not overlap then c(i, j) = dist(Bi , Bj ), that is
the distance between Bi and Bj . Otherwise, for a real number t let us denote by
Bj (t) a rectangle with the same dimensions as Bj and with the same orientation,
but with the center at pi + t(pj − pi ). We find tij > 1 such that the rectangles Bi
and Bj (tij ) touch each other. Let s = kpj − pi k, where kk denotes the Euclidean
norm. We set c(i, j) = −(tij − 1)s. See Figure 1 for an illustration.
s pj
pi
overlapping nodes
Bj
d
d = tij s
cij = s − d
dist(Bi , Bj )
Bi
cij = dist(Bi , Bj )
non overlapping nodes
Fig. 1: Cost function cij for edges of the Delaunay triangulation. For overlapping
nodes −cij is equal to the minimal distance that is necessary to shift the boxes
along the edge direction so they touch each other.
Having the cost function ready, we compute a minimum spanning tree T on
D.
the set of vertices V for which the cost,
P Remember that it is a tree with
0
c(e),
is
minimal,
where
E
is
the
set of edges of the tree. We use Prim’s
0
e∈E
algorithm to find T .
The algorithm proceeds by growing T , similar to the growth of a tree in
nature. It starts from the root of T . For each child of the root overlapping with
the root, it extends the edge connecting the root and the child to remove the
overlap. To achieve this, it keeps the root fixed but translates the sub-tree of the
child. The edges between the root and other children remain unchanged. The
algorithm recursively processes the children of the root in the same manner. This
process is described in Algorithm 1.
The number tij in line 5 of Algorithm 1 is the same as in the definition of
the cost of the edge (i, j) when Bi and Bj overlap, and is 1 otherwise.
Algorithm 1: Growing T
1
2
3
4
5
6
Input: Current center positions p and root r
Output: New center positions p0
p0r = pr
GrowAtNode (r)
function GrowAtNode (i)
foreach j ∈ Children(i) do
p0j = p0i + tij (pj − pi )
GrowAtNode (j);
The algorithm does not update all positions for the child sub-tree nodes
immediately, but updates only the root of the sub-tree. Using the initial positions
of a parent and a child, and the new position of the parent, the algorithm obtains
the new position of the child in line 5. In total, Algorithm 1 works in O(|V |)
steps. The choice of the root of the tree does not matter. Different roots produce
the same results modulo a translation of the plane by a vector. Indeed it can be
(a) iteration 1
(d) iteration 4
(b) iteration 2
(e) iteration 5
(c) iteration 3
(f) final overlap free graph
with original shapes
Fig. 2: Overlap removal process with the minimum spanning tree on the proximity graph, where the latter here is a Delaunay triangulation on rectangle centers.
The blue edges form a tree; there are four different trees in the figure. The tree
edges connecting overlapped nodes are thick and solid. In each iteration the
thick edges are elongated and the dashed tree edges shift accordingly. Overlap
is completely resolved in four iterations.
shown that after applying the algorithm, for any i, j ∈ V the vector p0j − p0i is
defined uniquely by the path from i to j in T .
While an overlap along any edge of the triangulation exists, we iterate, starting from finding a Delaunay triangulation, then building a minimum spanning
tree on it, and finally running Algorithm 1. See Figure 2 for an example.
When there are no overlaps on the edges of the triangulation, as noticed
by Gansner and Hu [6], overlaps are still possible. We follow the same idea as
PRISM and modify the iteration step. In addition to calculating the Delaunay
triangulation we run a sweep-line algorithm to find all overlapping node pairs
and augment the Delaunay graph D with each such a pair. As a consequence,
the resulting minimum spanning tree contains non-Delaunay edges catching the
overlaps, and the rest of the overlaps are removed. This stage usually requires
much less time than the previous one.
It is possible to create an example where the algorithm will not remove all
overlaps. However, such examples are extremely rare and have not been seen yet
in practice of using MSAGL or in our experiments. MSAGL applies random tiny
changes to the initial layout which prevents GTree from cycling.
4
Comparing PRISM and GTree by Measuring Layout
Similarity, Quality, and Run Time
Our data includes the same set of graphs that was used by the authors of PRISM
to compare it with other algorithms [6]. The set is available in the Graphviz
open source package3 . We also used a small collection of random graphs and
a collection of about 10,000 files residing here4 . For the experiments we use a
modified version of Dot, where we can invoke either GTree or Prism for the overlap removal step, and we also used MSAGL, where we implemented PRISM and
GTree. MSAGL was used only to obtain the quality measures. We ran the experiments on a PC with Linux, 64bit and an Intel Core i7-2600K [email protected]
with 16GB RAM.
Some of resulting layouts can be seen in Figures 3, 5, 6.
One can try to resolve overlap by scaling the node centers of the original
layout. If there are no two coincident node centers this will work, but the resulting
layout may require a huge area if some centers are close to each other. We
consider the area of the final layout as one of the quality measures, and usually
PRISM produces a smaller area than GTree, see Table 1.
In addition to comparing the areas, we compare some other layout properties.
Following Gansner and Hu [6], we look at edge length dissimilarity, denoted as
σedge . This measure reflects the relative change of the edge lengths of a Delaunay
Triangulation on the node centers of the original layout.
The other measure, which is denoted by σdisp , is the Procrustean similarity [1]. It shows how close the transformation of the original graph is to a com3
4
http://www.graphviz.org
https://www.dropbox.com/sh/4q0k89yrv4x3ae3/AAA3xyKFRhLyyHXcG9jpcgata?dl=0
PRISM
original layout
GTree
Fig. 3: Comparison between PRISM, original, and GTree layouts. In four top
rows the initial layouts were generated randomly. At the bottom are the drawings of nodes of graph ”root” which was initially laid out by the Multi Dimensional Scaling algorithm of MSAGL. In our opinion, the initial structure is more
preserved in the right column, containing the results of GTree.
bination of a scale, a rotation, and a shift transformation. PRISM and GTree
performs similar in the last two measures as Table 1 shows.
To distinguish the methods further, we measure the change in the set of k
closest neighbors of the nodes. Namely, let p1 , . . . , pn be the positions of the
node centers, and let k be an integer such that 0 < k ≤ n. Let I = {1, . . . , n}
be the set of node indices. For each i ∈ I we define Nk (i) ⊂ I \ {i}, such
that |Nk (p, i)| = k, and for every j ∈ I \ Nk (p, i) and for every j 0 ∈ Nk (p, i)
holds kpj − pi k ≥ kpj 0 − pi k. In other words, Nk (p, i) represents a set of k
closest neighbors of i, excluding i. Let p01 , . . . , p0n be transformed node centers.
To see how much the layout is distorted nearby node i, we intersect Nk (p, i)
and Nk (p0 , i). We measure the distortion as (k − m)2 , where m is the number of
elements in the intersection. One can see that if the node preserves its k closest
neighbors then the distortion is zero.
Our experiments for k from 8 to 12 show that under this measure GTree
produced a smaller error, showing less distortion, on 8 graphs from 14, and on
the rest PRISM produced a better result, see Table 2. GTree produced a smaller
error on all small random graphs from other collections5 .
Table 1: Similarity to the initial layout (left) and number of iterations for different graph sizes and different initialization methods (right). PR stands for
PRISM
Graph
dpd
unix
rowe
size
ngk10 4
NaN
b124
b143
mode
b102
xx
root
badvoro
b100
σedge
σdisp
area
PR GTree
PR GTree
PR GTree
0.34
0.22
0.29
0.39
0.30
0.56
0.55
0.67
0.54
0.71
0.75
1.09
0.88
0.84
0.28
0.19
0.26
0.37
0.30
0.44
0.53
0.70
0.50
0.77
0.70
1.19
0.92
0.98
0.37
0.24
0.23
0.24
0.27
0.73
0.97
1.12
0.59
1.43
1.65
2.89
2.27
3.08
0.36
0.20
0.24
0.26
0.30
0.51
0.83
0.93
0.53
1.27
1.42
2.45
2.42
3.14
0.82 0.84
2.38 2.38
0.68 0.73
1.09 1.28
0.00 0.00
4.03 4.34
5.52 6.22
3.62 3.88
1.53 2.29
4.50 6.62
6.21 9.57
34.58 91.87
25.68 47.43
20.64 37.38
init. layout:
Graph |V |
dpd
36
unix
41
rowe
43
size
47
ngk10 4 50
NaN
76
b124
79
b143
135
mode
213
b102
302
xx
302
root
1054
badvoro1235
b100
1463
neato
|E|
108
49
68
55
100
121
281
366
269
611
611
1083
1616
5806
PR GTree
4
3
5
7
6
8
14
21
37
60
83
95
40
80
7
4
4
3
3
3
4
6
8
24
18
18
20
24
SFDP
PR GTree
3
12
13
9
14
24
30
37
11
113
50
99
50
136
6
5
7
5
7
6
12
12
6
19
19
22
23
28
We ran tests on the graphs from a subdirectory of the same site called
“dot files”, let us call this set of graphs collection A. Each graph from A represents the control flow of a method from a version of the .NET framework. A
contains 10077 graphs. The graph sizes do not exceed several thousands. We
used the Multi Dimensional Scaling algorithms of MSAGL for the initial layout
in this test. The results of the run are summarized in Table 3.
5
https://www.dropbox.com/sh/4q0k89yrv4x3ae3/AAA3xyKFRhLyyHXcG9jpcgata?dl=0
Table 2: k closest neighbors error, the Multi Dimensional Scaling algorithm of
MSAGL was used for the initial layout. PR stands for PRISM.
Graph
dpd
unix
rowe
size
ngk10 4
NaN
b124
b143
mode
b102
xx
root
badvoro
b100
k=8
k=9
k = 10
k = 11
k = 12
PR GTree
PR GTree
PR GTree
PR GTree
PR GTree
7.75 6.06
8.56 7.05
6.28 8.09
4.68 6.09
6.76 7.4
11.83 8.95
11.03 11.44
13.49 12.39
16.91 11.46
15.99 14.62
15.68 15.65
17.09 15.7
16.18 15.15
18 19.25
9.61 7.36
10.51 8.8
7.09 9.95
5.47 6.47
7.52 9.26
14.46 11.5
13.22 13.56
16.31 14.99
20.58 13.95
19.61 18.78
19.01 19.45
20.89 19.36
20.16 18.98
22.11 23.65
9.5
8
10.95 10.02
7.49 10.49
6.28 7.57
8.28 11.38
17.32 13.88
14.76 15.54
19.49 17.93
24.68 16.85
23.38 22.77
23.05 23.37
25.48 23.3
24.37 23.28
26.79 28.69
10.14 8.5
11.66 10.54
9.12 11.4
6.89 8.13
10.72 13.74
19.88 16.37
15.91 17.32
23.11 21.04
29.54 19.92
27.28 26.77
26.98 27.35
30.48 27.66
29.18 28.03
32.03 34.46
9.97 7.64
13 11.41
11.05 12.51
8.26 10.02
11.92 14.66
22.17 19.7
18.23 20.04
26.53 24.43
34.48 22.56
32.15 31.45
31.29 32.47
35.74 32.83
34.29 33.29
37.44 40.5
Table 3: Statistics on collection A. Here k-cn stands for k-closest neighbors, and
“iters” stands for the number of iterations. Each cell contains the number of
graphs for the measure on which the method performed better. We can see that
PRISM produced a layout of smaller area than the one of GTree on 8498 graph,
against 1579 graphs where GTree required less area. From the other side, GTree
gives better results on all other measures. The columns of k-cn and “iters” do
not sum to 10077, the number of graphs in A, because some of the results were
equal for PRISM and GTree.
Method
PRISM
GTree
k-cn σedge σdisp
area
3237 4741 4114 8498
4088 5336 5963 1579
iters time
46
7
9986 10070
Runtime Comparison
Both methods remove the overlappiteratively using the proximity graph. However, while PRISM needs O(|V | · |V |) time to solve the stress model, GTree
needs only O(|V |) time per iteration with the growing tree procedure. Therefore, GTree is asymptotically faster in a single iteration. In addition, as Table 1
(right) shows, GTree usually needs fewer iterations than PRISM, especially on
larger graphs. The overall runtime can be seen in Figure 4. It shows that GTree
outperforms PRISM on larger graphs.
In Figure 5 we experiment with the way we expand the edges. Instead of the
formula p0j = p0i +tij (pj −pi ), which resolves the overlap between the nodes i and
j immediately, we use the update p0j = p0i + min(tij , 1.5)(pj − pi ). As a result,
the algorithm runs a little bit slower but produces layouts with smaller area.
Overlap Removal Method
Graph
|V|
|E|
dpd
36
108
0.01
0.00
unix
41
49
0.00
0.00
rowe
43
68
0.01
0.01
size
47
55
0.00
0.01
ngk10_4
50
100
0.00
0.00
NaN
76
121
0.01
0.00
b124
79
281
0.01
0.01
b143
135
366
0.03
0.00
mode
213
269
0.08
0.02
b102
302
611
0.19
0.07
xx
302
611
0.27
0.05
root
1054 1083
1.19
0.21
badvoro
1235 1616
0.58
0.26
b100
1463 5806
1.46
0.37
Time in s
2
1
PRISM
GTree
●
PRISM GTree
●
●
●
●
●●
0
●
●
●
●
●
●
●
●
●
●
●
0
500
1000
1500
Graph size |V|
Fig. 4: Runtimes for PRISM and GTree.
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9c9e2e0f2da8758e436c
(a) initial layout
(b) PRISM
(c) GTree
Fig. 5: root graph with 1054 nodes and 1083 edges. (a) initial layout with
NEATO, (b) applying PRISM, (c) applying GTree.
5
Conclusion & Future Work
We proposed a new overlap removal algorithm that uses the minimum spanning
tree. The algorithm is simple and easy to implement, and yet it preserves the
initial layout well and is efficient.
Although we introduced our approach in the context of graph visualization,
our method can also be used for any other purpose where overlap needs to be
resolved while maintaining the initial layout. Finding a measure of how well an
overlap removal algorithm preserves clusters of the initial layout seems to be an
interesting challenge.
GTree
GTree
original
original
PRISM
PRISM
Fig. 6: Results for GTree and PRISM initialized with SFDP. From top to bottom
and left to right: b100, b102, b124, b143, badvoro, dpd, mode, - NaN, ngk10 4,
root, rowe, size, unix, and xx. To make the original drawings more readable
they have been changed; In most cases the nodes were diminished and the edges
removed. The drawings were scaled differently.
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