Efficient, Proximity-Preserving Node Overlap Removal · For example, the Voronoi cluster busting...

Journal of Graph Algorithms and Applicationshttp://jgaa.info/ vol. 14, no. 1, pp. 53–74 (2010)

E!cient, Proximity-Preserving Node Overlap

Removal

Emden R. Gansner Yifan Hu

AT&T Labs,Shannon Laboratory,

180 Park Ave.,Florham Park, NJ 07932.

Abstract

When drawing graphs whose nodes contain text or graphics, the non-trivial node sizes must be taken into account, either as part of the initiallayout or as a post-processing step. The core problem in avoiding or

removing overlaps is to retain the structural information inherent in alayout while minimizing the additional area required. This paper presentsa new node overlap removal algorithm that does well at retaining a graph’sshape while using little additional area and time. As part of the analysis,

we consider and evaluate two measures of dissimilarity for two layouts ofthe same graph.

Submitted:November 2008

Reviewed:March 2009

Revised:May 2009

Accepted:November 2009

Final:November 2009

Published:January 2010

Article type:Regular paper

Communicated by:I. G. Tollis and M. Patrignani

E-mail addresses: [email protected] (Emden R. Gansner) [email protected] (Yifan

Hu)

mailto:[email protected]

mailto:[email protected]

54 Gansner and Hu E!cient Node Overlap Removal

1 Introduction

Most existing symmetric graph layout algorithms treat nodes as points. Inpractice, nodes usually contain labels or graphics that need to be displayed.Naively incorporating this can lead to nodes that overlap, causing informationof one node to occlude that of others. If we assume that the original layoutconveys significant aggregate information such as clusters, the goal of any layoutthat avoids overlaps should be to retain the “shape” of the layout based on pointnodes.

The simplest and, in some sense, the best solution is to scale up the drawingwhile preserving the node size until the nodes no longer overlap [30]. This hasthe advantage of preserving the shape of the layout exactly, but can lead toinconveniently large drawings. In general, overlap removal is typically a trade-o! between preserving the shape and limiting the area, with scaling at oneextreme.

Many techniques to avoid overlapping nodes have been devised. One ap-proach is to make the node size part of the model of the layout algorithm. Itis assumed that whatever structure that would have been exposed using pointnodes will still be evident in these more general layouts. For hierarchical lay-outs, the node size can be naturally incorporated into the algorithm [11, 35].For symmetric layouts, various authors [3, 19, 28, 37] have extended the spring-electrical model [7, 12] to take into account node sizes, usually as increased re-pulsive forces. Node overlap removal can also be built into the stress model [26]by specifying the ideal edge length to avoid overlap along the graph edges. Suchheuristics, however, cannot guarantee all overlaps will be removed, so they typ-ically rely on overly large repulsive forces, or the type of post-processing stepconsidered below. This problem was addressed by Dwyer et al. [5], who showedhow to encode overlap avoidance, as well as many other layout features, as lin-ear constraints in a stress function, and then optimize the stress function usingstress majorization [13].

An alternative approach is to remove node overlaps as a post-processingstep after the graph is laid out. Here the trade-o! between layout size andpreserving the graph’s shape is more explicit. In addition to many of the al-gorithms mentioned above that can be adapted for this use, a number of otheralgorithms have been proposed. For example, the Voronoi cluster busting algo-rithm [15, 29] works by iteratively forming a Voronoi diagram from the currentlayout and moving each node to the center of its Voronoi cell until no overlapsremain. The idea is that restricting each node to its corresponding Voronoicell should preserve the relative positions of the nodes. In practice, because ofthe number of iterations often required and the use of a rectangular boundingbox,1 the nodes in the final drawing can be homogeneously distributed withina rectangle, the graph bearing little resemblance to the original layout.

Another group of post-processing algorithms is based on setting up pairwisenode constraints to remove overlaps and then performing some procedure to

1The latter might be improved by using something like !-shapes [8] to provide a moreaccurate boundary.

JGAA, 14(1) 53–74 (2010) 55

generate a feasible solution. The force scan algorithm [31] and its later vari-ants [20, 24] proceed along these lines. More recently, Marriott et al. [30, 31]have presented quadratic programming algorithms which remove node overlapswhile, if desired, minimizing node displacement. All of these techniques relyon solving separate horizontal and vertical problems. They behave di!erentlyif the layout is rotated by a degree that is not a multiple of !/2. We have alsofound that, in practice, this asymmetry often results in a layout with a distortedaspect ratio (e.g., Figure 4, bottom right).

One desideratum proposed [31] for overlap removal is the preservation oforthogonal ordering, i.e., the relative ordering of the x and y coordinates oftwo nodes should be the same in the original layout and the one with overlapsremoved. Thus, if a node is above and to the left of another node in the originallayout, it is above and to the left in the derived one. The idea is that, inthis manner, a user’s “mental map” of the graph is better preserved. Many ofthe algorithms described above either inherently preserve orthogonal ordering,or have simple variations that do. While preserving orthogonal ordering canbe important, it alone cannot ensure that the relative proximity relations [23]between nodes are preserved. At other times, it is too restrictive, causing twohighly-related nodes to be moved far apart due to some largely unrelated thirdnode. For these reasons, our approach is to concentrate on proximity relationsand not deal with orthogonal ordering.

In this paper, we focus on the problem of removing overlaps rather thanavoiding them. To this end, we discuss (Section 3) metrics for the similaritybetween two layouts which we believe better quantify the desired outcome ofoverlap removal than minimized displacement or such simpler measures as as-pect ratio or edge ratio (the ratio of the longest and the shortest edge lengths).We then present (Section 4) a node overlap removal algorithm based on a prox-imity graph of the nodes in the original layout. Using this graph as a guide,it iteratively moves the nodes, particularly those that overlap, while keepingthe relative positions between them as close to those in the original layout aspossible. The algorithm is similar to the stress model [26] used for graph layout,except that the stress function involves only a sparse selection of all possiblenode pairs. Because of this sparse stress function, the algorithm is e"cient andis able to handle very large graphs. In Section 5, we evaluate our algorithmand others using the proposed similarity measures. Finally, Section 6 presentsa summary and topics for further study.

2 Background

We use G = (V,E) to denote an undirected graph, with V the set of nodes(vertices) and E edges. We use |V | and |E| for the number of vertices andedges, respectively. We let xi represent the current coordinates of vertex iin Euclidean space. For this paper we are interested in 2D layout, thereforexi ! R2.

The aim of graph drawing is to find xi for all i ! V so that the resulting


drawing gives a good visual representation of the information in the graph. Twopopular methods, the spring-electrical model [7, 12], and the stress model [26],both convert the problem of finding an optimal layout to that of finding aminimal energy configuration of a physical system. We shall describe the stressmodel in more detail since we will use a similar model for the purpose of nodeoverlap removal in Section 4.

The stress model assumes that there are springs connecting all nodes ofthe graph, with the ideal spring length equal to the graph theoretical distancebetween nodes. The energy of this spring system is

!

i!=j

wij ("xi # xj" # dij)2, (1)

where dij is the graph theoretical distance between vertices i and j, and wij is aweight factor, typically 1/dij2. The layout that minimizes the above stress en-ergy is an optimal layout of the graph. There are several ways to find a solutionof the minimization problem. An iterative approach can be employed. Startingfrom a random layout, the total spring force on each vertex is calculated, andthe vertex is moved along the direction of the force for a certain step length.This process is repeated, with the step length decreasing every iteration, untilthe layout stabilizes. Alternatively, a stress majorization technique can be em-ployed, where the cost function (1) is bounded by a series of quadratic functionsfrom above, and the process of finding an optimum becomes that of solving aseries of linear systems [13].

In the stress model, the graph theoretical distance between all pairs of ver-tices has to be calculated, leading to quadratic complexity in the number ofvertices. There have been attempts (e.g., [2, 13]) to simplify the stress functionby considering only a sparse portion of the graph. Our experience, however,with real-life graphs is that these techniques may fail to yield good layouts.Therefore, algorithms based on a spring-electrical model employing a multi-level approach and an e"cient approximation scheme for long range repulsiveforces [18, 22, 36] are still the most e"cient choices to lay out large graphswithout consideration of the node size.

3 Measuring Layout Similarity

The outcome of an overlap removal algorithm should be measured in two aspects.The first aspect is the overall bounding box area: we want to minimize the areataken by the drawing after overlap removal. The second aspect is the changein relative positions. Here we want the new drawing to be as “close” to theoriginal as possible. It is this aspect that is hard to quantify.

When total node movement is optimized [6, 24], comparison is reduced tomeasuring the amount of displacement of the vertices in the new layout fromthose of the original graph. This measurement does not take into account pos-sible shifts, scalings or rotations, nor the importance of maintaining the relativeposition among vertices.

JGAA, 14(1) 53–74 (2010) 57

As far as we are aware, there is no definitive way to measure similarity oftwo layouts of the same graph. We adopt two approaches. The first approachis based on measuring changes in lengths of edges. The second approach, whichis a modification of the metric of Dwyer et al. [6], is based on measuring thedisplacement of vertices, after discounting shift, scaling and rotation.

Before defining these two measures of similarity, we first introduce the con-cept of a proximity graph. A proximity graph is a graph derived from a set ofpoints in space: points that are “neighbors” to each other in the space forman edge in the proximity graph. There are many ways to create a proximitygraph [25]. In this paper, we shall work with the Delaunay triangulation (DT),which is an approximation to the proximity graph, and has the advantage beingrigid. Two points are neighbors in DT if and only if there exists a sphere passingthrough these two points, and no other points lie in the interior of this sphere.

One way to measure the similarity of two layouts is to measure the distancebetween all pairs of vertices in the original and the new layout. If the twolayouts are similar, then these distances should match, subject to scaling. Thisis known as Frobenius metric in the sensor localization problem [9]. Calculatingall pairwise distances is expensive for large graphs, both in CPU time and inthe amount of memory. As we want a metric feasible for very large graphs, weinstead form a DT of the original graph, then measure the distance betweenvertices along the edges of the triangulation for the original and new layouts.2

If x0 and x denote the original and the new layout, and EP is the set of edgesin the triangulation, we calculate the ratio of the edge length

rij ="xi # xj"

"x0i # x0

j", {i, j} ! EP ,

then define a measure of the dissimilarity as the normalized standard deviation

"dist(x0, x) =

"!{i,j}!EP

(rij"r)2

|EP |

r,

where

r =1

|EP |

!

{i,j}#EP

rij

is the mean ratio.The reason we measure the edge length ratio along edges of the proximity

graph, rather than along edges of the original graph, is that if the originalgraph is not rigid, then even if two layouts of the same graph have the sameedge lengths, they could be completely di!erent. For example, think of thegraph of a square, and a new layout of the same graph in the shape of a non-square rhombus. These two layouts may have exactly the same edge lengths,

2 Another possibility would be to sample O(|V |) out of the |V |(|V |! 1)/2 possible vertexpairs. Some care is needed in making sure that the resulting graph is rigid.


1 2

34

1 2

34

Figure 1: The edge lengths of these two layouts of a non-rigid graph are exactlythe same, but the layouts are clearly di!erent.

but are clearly di!erent (see Figure 1). The rigidity of the triangulation avoidsthis problem.

Notice that "dist(x0, x) is not symmetric with regard to which layout comesfirst. Furthermore, in theory, this non-symmetric version could class a layoutand a foldover of it (e.g., a square grid with one half folded over the other) asthe same. We can symmetrize it by defining the dissimilarity between layout xand x0 as ("dist(x0, x) + "dist(x, x0))/2. This also resolves the “foldover prob-lem”. The symmetric version may be more appropriate if we are comparing twounrelated layouts. Since, however, we are comparing a layout derived from anexisting layout, we feel that the asymmetric version is adequate.

An alternative measure of similarity is to calculate the displacement of ver-tices of the new layout from the original layout [6]. Clearly a new layout derivedfrom a shift, scaling and rotation should be considered identical. Therefore wemodify the straight displacement calculation by discounting the aforementionedtransformations. This is achieved by finding the optimal scaling, shift and ro-tation that minimize the displacement. The optimal displacement is then ameasure of dissimilarity.

We denote by scalars r and # the scaling and rotation,

T =

#

cos(#) sin(#)# sin(#) cos(#)

$

(2)

the rotation matrix, p ! R2 the translation, and define the displacement dissim-ilarity as

"disp(x0, x) = minp#R2,!,r#R

!

i#V

"rTxi + p# x0i "

2, (3)

This is a known problem in the Procrustes analysis [1, 16] and the solution (theProcrustes statistic) is

"disp(x0, x) = tr(X0X0T )# (tr((XTX0X0TX)

12 )2tr(XTX), (4)

where tr(A) is the trace of a matrix A, X is a matrix with columns xi # x, X0

is a matrix with columns x0i # x0, and x and x0 are the centers of gravity of the

new and original layout.

JGAA, 14(1) 53–74 (2010) 59

i

j

i

j

Figure 2: Nodes i and j overlap (left). The overlap factor, according to (5), ismin((2+ 2)/3, (1+ 2)/2) = 1.33. Expanding the edge i# j by 33% removes theoverlap (right).

In the above we do not consider shearing, since we believe a layout derivedfrom shearing of the original should not be considered identical to the latter.

The quality of an overlap removal algorithm is a combination of how similarthe new layout is to the original, and how small an area it occupies. Thesimplest overlap removal algorithm is that of scaling the layout until all overlapsare removed. This has a dissimilarity of 0, but usually occupies a very largearea. The alternative extreme is to pack the nodes as close to each other aspossible while ignoring the original layout. This will have the smallest area, buta large dissimilarity. A good solution should be a compromise between thesetwo extremes. We now describe one candidate.

4 A Proximity Stress Model for Node Overlap

Removal

Our goal is to remove overlaps while preserving the shape of the initial layout bymaintaining the proximity relations [23] among the nodes. To do this, we firstset up a rigid “sca!olding” structure so that while vertices can move around,their relative positions are maintained. This sca!olding is constructed using anapproximate proximity graph, in the form of a Delaunay triangulation (DT).

Once we form a DT, we check every edge in it and see if there are any nodeoverlaps along that edge. Let wi and hi denote the half width and height ofthe node i, and x0

i (1) and x0i (2) the current X and Y coordinates of this node.

If i and j form an edge in the DT, we calculate the overlap factor of these twonodes

tij = max

%

min

%

wi + wj

|x0i (1)# x0

j (1)|,

hi + hj

|x0i (2)# x0

j (2)|

&

, 1

&

. (5)

For nodes that do not overlap, tij = 1. For nodes that do overlap, such overlapscan be removed if we expand the edge by this factor, see Figure 2. Thereforewe want to generate a layout such that an edge in the proximity graph has the


ideal edge length close to tij"x0i #x0

j". In other words, we want to minimize thefollowing stress function

!

(i,j)#EP

wij ("xi # xj" # dij )2. (6)

Here dij = sij"x0i # x0

j" is the ideal distance for the edge {i, j}, sij is a scalingfactor related to the overlap factor tij (see (7)), wij = 1/"dij"2 is a weightingfactor, and EP is the set of edges of the proximity graph. We call (6) theproximity stress model in obvious analogy.

Because DT is a planar graph, which has no more than 3|V |# 6 edges, theabove stress function has no more than 3|V | # 6 terms. Furthermore, becauseDT is rigid, it provides a good sca!olding that constrains the relative positionof the vertices and helps to preserve the global structure of the original layout.

It is important that we do not attempt to remove overlaps in one iterationby using the above model with sij = tij. Imagine the situation of a regular meshgraph, with one node i of particularly large size that overlaps badly with itsnearby nodes, but the other nodes do not overlap with each other. Supposenodes i and j form an edge in the proximity graph, and they overlap. If wetry to make the length of the edge equal tij"x0

i # x0j", we will find that tij is a

number much larger than 1, and the optimum solution to the stress model is tokeep all the other vertices at or close to their current positions, but move thelarge node i outside of the mesh, at a position that does not cause overlap. Thisis not desirable because it destroys the original layout. Therefore we damp theoverlap factor by setting

sij = min(tij, smax) (7)

and try to remove overlap a little at a time. Here smax > 1 is a number limitingthe amount of overlap we are allowed to remove in one iteration. We found thatsmax = 1.5 works well.

After minimizing (6), we arrive at a layout that may still have node overlaps.We then regenerate the proximity graph using DT and calculate the overlapfactor along the edges of this graph, and redo the minimization. This forms aniterative process that ends when there are no more overlaps along the edges ofthe proximity graph.

For many graphs, the above algorithm yields a drawing that is free of nodeoverlaps. For some graphs, however, especially those with nodes having extremeaspect ratios, node overlaps may still occur. Such overlaps happen for pairs ofnodes that are not near each other, and thus do not constitute edges of theproximity graph. Figure 3(a) shows the drawing of a graph after minimizing(6) iteratively, so that no more node overlap is found along the edges of theDelaunay triangulation. Clearly, node 2 and node 4 still overlap. If we plotthe Delaunay triangulation (Figure 3(b)), it is seen that nodes 2 and 4 are notneighbors in the proximity graph, which explains the overlap.

To overcome this situation, once the above iterative process has convergedso that no more overlaps are detected over the DT edges, we apply a scan-

JGAA, 14(1) 53–74 (2010) 61

1

2 with a tall

label3

4 with an extremely looong label

5

678

9

10

11

1

2 with a tall

label3

4 with an extremely looong label

5

678

9

10

11

Figure 3: (a): A graph layout where nodes 2 and 4 overlap. (b): the proximitygraph (Delaunay triangulation) of the current layout. No two nodes linked byan edge of the proximity graph overlap.

line algorithm [6] to find all overlaps, and augment the proximity graph withadditional edges, where each edge consists of a pair of nodes that overlap. Wethen re-solve (6). This process is repeated until the scan-line algorithm findsno more overlaps. We call our algorithm PRISM (PRoxImity Stress Model).Algorithm 1 gives a detailed description of this algorithm.

Algorithm 1 Proximity stress model based overlap removal algorithm (PRISM)

Input: coordinates for each vertex, x0i , and bounding box width and height

{wi, hi}, i = 1, 2, . . . , |V |.repeat

Form a proximity graph GP of x0 by Delaunay triangulation.Find the overlap factors (5) along all edges in GP .Solve the proximity stress model (6) for x. Set x0 = x.

until (no more overlaps along edges of GP )repeat

Form a proximity graph GP of x0 by Delaunay triangulation.Find all node overlaps using a scan-line algorithm. AugmentGP with edgesfrom node pairs that overlap.Find the overlap factor (5) along all edges of GP .Solve the proximity stress model (6) for x. Set x0 = x.

until (no more overlaps)

We now discuss some of the main computational steps in the above algo-rithm. Delaunay triangulation can be computed in O(|V | log |V |) time [10, 17,27]. We used the mesh generator Triangle [33, 34] for this purpose.

The scan-line algorithm can be implemented to find all the overlaps inO(l|V |(log |V | + l)) time [6], where l is the number of overlaps. Because weonly apply the scan-line algorithm after no more node overlaps are found alongedges of the proximity graph, l is usually a very small number, hence this stepcan be considered as taking time O(|V | log |V |).

The proximity stress model (6), like the spring model (1), can be solved


using the stress majorization technique [13] which is known to be a robustprocess for finding the minimum of (1). The technique works by bounding(6) with a series of quadratic functions from above, and the process of findingan optimum becomes that of finding the optimum of the series of quadraticfunctions, which involves solving linear systems with a weighted Laplacian of theproximity graph. We solve each linear system using a preconditioned conjugategradient algorithm. Because we use DT as our proximity graph and it has nomore than 3|V | # 6 edges, each iteration of the conjugate gradient algorithmtakes a time of O(|V |).

Overall, therefore, Algorithm 1 takes O(t(mk|V |+ |V | log |V |)) time, wheret is the total number of iterations in the two main loops in Algorithm 1, m isthe average number of stress majorization iterations, and k the average numberof iterations for the conjugate gradient algorithm.

In practice, we found that the majority of CPU time is spent in repeatedlysolving the linear systems (which takes a total time of O(tmk|V |)). We ter-minate the conjugate gradient algorithm if the relative 2-norm residual for thelinear system involved in the stress majorization process is less than 0.01. Atighter tolerance is not necessary because the solution of each linear system con-stitutes an intermediate step of the stress majorization. Furthermore, solutionof the proximity stress model (6) is an intermediate step itself in Algorithm 1,so we do not need to solve (6) accurately either. Hence we set a limit of mmax

iterations. By experimentation, we found that a smaller value mmax gives afaster algorithm, and that, in terms of quality, a smaller mmax is often just asgood as, if not better than, a larger value of mmax. Therefore, we set mmax = 1.

5 Numerical Results

To evaluate the PRISM algorithm and other overlap removal algorithms, we ap-ply them as a post-processing step to a selection of graphs from the Graphviz [14]test suite. This suite, part of the Graphviz source distribution, contains manygraphs from users. As such, these are good examples of the kind of graphsactually being drawn.

Our baseline algorithm is Scalable Force Directed Placement (SFDP) [22], amultilevel, spring-electrical algorithm. Using the layout of SFDP, we then applyone of the overlap removal algorithms to get a new layout that has no nodeoverlaps, and compare the new layout with the original in terms of dissimilarityand area.

In Table 1, we list the 14 test graphs, the number of vertices and edges, aswell as CPU time3 for PRISM and three other overlap removal algorithms. Thegraphs are selected randomly with the criteria that a graph chosen should beconnected, and is of relatively large size. We compared PRISM with an imple-mentation of the solve VPSC algorithm [6] provided by its authors. We alsoevaluated the companion algorithm satisfy VPSC. This o!ered, at best, a 2%

3All timings were derived on a 4 processor, 3.2 GHz Intel Xeon CPU, with 8.16 GB ofmemory, running Linux.

JGAA, 14(1) 53–74 (2010) 63

Table 1: Comparing the CPU time (in seconds) of several overlap removal al-gorithms. Initially the layout is scaled to an average edge length of 1 inch.

Graph |V | |E| PRISM VPSC VORO ODNLSb100 1463 5806 1.44 14.85 350.7 258.9b102 302 611 0.14 0.10 4.36 5.7b124 79 281 0.03 0.01 0.02 0.5b143 135 366 0.04 0.01 0.47 1.3

badvoro 1235 1616 0.54 71.15 351.51 73.6mode 213 269 0.09 0.09 2.15 2.1

ngk10 4 50 100 0.01 0.00 0.02 0.14NaN 76 121 0.01 0.01 0.11 0.27dpd 36 108 0.01 0.01 0.02 0.1root 1054 1083 0.89 7.81 398.49 46.9rowe 43 68 0.00 0.00 0.04 0.1size 47 55 0.01 0.00 0.06 0.09unix 41 49 0.01 0.00 0.04 0.07xx 302 611 0.13 0.10 8.19 5.67

reduction in time, while producing almost identical results concerning displace-ment, dissimilarity and area. For this reason, in the sequel, we only considerthe solve VPSC algorithm, hereafter denoted as VPSC.4 We also evaluated theVoronoi cluster busting algorithm [15, 29], denoted by VORO, as well as theODNLS algorithm of Li et al. [28], which relies on varied edge lengths in a springembedder. We note, in passing, that we also tested scaling using the algorithmof Marriott et al. [30]. As expected, it outperformed all other algorithms interms of speed and dissimilarity, but at an unacceptably high cost in area. Forexample, on the b143 graph, the time and dissimilarity were essentially 0 butthe area was 82.2. Overall, scaling produced areas at least 4.5 times larger thanPRISM and, more typically, at least an order of magnitude larger. We thereforeremove scaling from further consideration.

The initial layout by SFDP is scaled so that the average edge length is 1 inch.From the table, it is seen that PRISM is usually faster, particularly for largegraphs on which it scales much better. The others are slow for large graphs,with VORO the slowest.

Table 2 compares the dissimilarities and drawing area of the four overlapremoval algorithms. The smaller the dissimilarities and area, the better.

The ODNLS algorithm performs best in terms of smaller dissimilarity, fol-lowed by PRISM, VPSC and VORO. In terms of area, PRISM and VPSC arepretty close, and both are better than ODNLS and VORO, which can give ex-

4 Both versions of the VPSC algorithm can be extended to support the preservation oforthogonal ordering. We did not have an opportunity to check these versions.


tremely large drawings. Indeed, in terms of area, scaling outperformed ODNLSand VORO in 20%-30% of the examples.

Comparing PRISM with VPSC, Table 2 shows that PRISM gives smallerdissimilarities most of the time. The two dissimilarity measures, "dist and "disp,are generally correlated, except for ngk10 4 and root. Based on "dist, VPSCis better for these two graphs, while based on "disp, PRISM is better. Thefirst row in Figure 4 shows the original layout of ngk10 4, as well as the resultafter applying PRISM and VPSC. Through visual inspection, we can see thatPRISM preserved the proximity relations of the original layout well. VPSC“packed” the labels more tightly, but it tends to line up vertices horizontallyand vertically, and also produces a layout with aspect ratio quite di!erent fromthe original graph. It seems that "dist is not as sensitive in detecting di!erencesin aspect ratio. This is evident in drawings of the root graph (Figure 4, secondrow): VPSC clearly produced a drawing that is overly stretched in the verticaldirection, but its "dist is actually smaller! Consequently, we conclude that "disp

may be a better dissimilarity measure.The fact that VPSC can produce very tall and thin, or very short and wide,

layouts is not surprising, and has been observed often in practice. VPSC worksin the vertical and horizontal directions alternatively, each time trying to removeoverlaps while minimizing displacement. As a result, when starting from a layoutwith severe node overlaps, it may move vertices significantly along one directionto resolve the overlaps, creating drawings with extreme aspect ratios. In fact,for 9 out of 14 test graphs, VPSC produces layouts with extreme aspect ratios.PRISM does not su!er from this problem.

When starting from a layout that is scaled su"ciently so that relative fewernodes overlap, VPSC’s performance can be improved. Table 3 compares thefour overlap removal algorithms, starting from layouts that are scaled to givean average edge length that equals 4 times the average node size. Here the sizeof a node is calculated as the average of its width and height.

From the table, we can see that in terms of dissimilarity, PRISM and VPSCare now similar, closer to the better performing ODNLS. In terms of drawingarea, PRISM is better than VPSC, with VORO and ODNLS much larger. Whenvisually inspected, VPSC again su!ers from extreme aspect ratio issue on atleast 5 out of the 14 graphs (b100, b143, badvoro, mode, root). Figure 5shows the layout of badvoro (first row), on which VPSC performed badly basedon the two similarity measures. In the same figure, we also show b124 (secondrow), on which PRISM is rated worse than VPSC based on the same measures.On badvoro it is clear that VPSC performed badly, as the similarity measuressuggest. On the other hand, if we look at b124, VPSC perhaps performed betterthan PRISM, but not as clearly as the similarity measures suggest. Overall,visual inspection of the drawings of these 14 graphs, as well as drawings forgraphs in the complete Graphviz test suite (a total of 204 graphs in March2008), shows that PRISM performs very well, and is overall better and fasterthan VPSC and VORO. The ODNLS algorithm preserves similarity somewhatbetter than PRISM, but at much higher costs in term of speed and area.

Considering a larger collection of graphs, Table 4 compares PRISM with

JGAA, 14(1) 53–74 (2010) 65

Table 2: Comparing the dissimilarities and area of overlap removal algorithms.Results shown are "dist, "disp and area. Area is measured with a unit of 106

square points. Initially the layout is scaled to an average length of 1 inch.

Graph PRISM VPSC"dist "disp area "dist "disp area

b100 0.74 0.38 14.05 0.76 0.72 18.91b102 0.44 0.25 2.45 0.58 0.8 2.71b124 0.65 0.37 1.04 0.78 0.73 0.91b143 0.59 0.35 1.5 0.78 0.83 2.16

badvoro 0.34 0.15 12.58 0.61 0.75 13.85mode 0.59 0.37 0.79 1.02 0.77 1.29

ngk10 4 0.41 0.16 0.33 0.39 0.3 0.25NaN 0.4 0.2 0.72 0.54 0.65 0.71dpd 0.34 0.18 0.25 0.51 0.4 0.18root 0.71 0.3 16.99 0.6 0.75 17.68rowe 0.33 0.14 0.22 0.44 0.31 0.19size 0.37 0.2 0.47 0.77 0.74 0.4unix 0.39 0.23 0.39 0.51 0.67 0.36xx 0.42 0.25 3.96 0.57 0.82 3.9

Graph VORO ODNLS"dist "disp area "dist "disp area

b100 - - - 0.33 0.20 1.02E3b102 0.8 0.3 31.79 0.30 0.16 53.13b124 0.86 0.39 13.42 0.33 0.19 14.79b143 0.99 0.45 22.91 0.49 0.34 23.79

badvoro 2.29 0.65 3.01E3 0.31 0.26 318.66mode 0.97 0.54 10.84 0.38 0.27 49.45

ngk10 4 0.48 0.26 0.52 0.22 0.13 2.30NaN 0.56 0.28 5.04 0.26 0.15 5.10dpd 0.48 0.32 0.45 0.37 0.29 1.30root 4.09 0.94 6.93E9 0.29 0.22 950.01rowe 0.49 0.26 0.95 0.27 0.12 2.10size 0.62 0.35 1.27 0.32 0.20 4.14unix 0.6 0.35 0.85 0.26 0.13 2.35xx 0.97 0.34 58.83 0.29 0.14 74.00


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root root+PRISM root+VPSC

Figure 4: Divergence of dissimilarity measures: for both graphs, "dist estimatesthat VPSC gives layout closer to the original, while "disp predicts the opposite.

akrnoh80

akrnoh60akrnoh61akrnoh62

albyny80

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albyny63albyny64albyny65

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wpbhfl65

badvoro badvoro+PRISM badvoro+VPSC

Roger So

L. Sassaman

Martin Schulze

Douglas F. Calvert

Hunter Matthews (Duke Biology Unix Sysadmin)

Theodore Ts’o [SIGNATURE]

Dave Del Torto

Nik Clayton

Daniel Stone (NEC Business Solutions)

Stephen C. Tweedie (Key signing key)

Linus Torvalds

Werner Koch (gnupg sig)

Bradley M. Kuhn

Dan Nowacki

Jeffrey I. Schiller

Wichert Akkerman

Douglas F. Calvert

Leonard D. Rosenthol

M. Drew Streib

Paul S. Traina

David Elson (Del)

Daniel Jacobowitz

Gregory L. Pratt

Andrew G. Malis

Robert van der Meulen

Peter Jones

Clint Adams (Debian)

Eric S. Raymond

LaMont Jones

H. Peter Anvin (hpa)

CERT Coordination Center

Citizen Gold (MedeMail - Babel Client)

[email protected]

Bill Scannell

Werner Koch

J. Daniel Backman

Andrew S. Nagy

Red Hat, Inc


K. Reid Wightman

Mark Crichton (preferred key)

Emily ManheimTheodore Y. Ts’o

Charles Gallo at WorkJack M Twilley

Mike Fratto

Wichert Akkerman

Cory F. Cohen

Vinnie Moscaritolo

Daniel Jacobowitz

Dr. Michael Meskes

Noah Meyerhans

Charles Gallo

Martin Schulze

Philip R. Zimmermann

Tim O’Shea [tmo]

AUSCERT

Matt Taggart

Miro Jurisic

Peter Nai Wan

Linux Kernel Archives Verification Key

Christoph Martin

Woz

Douglas F. Calvert (High Security Key Use 0x13300731 For Everyday Communication)

Michael Stone

Stanton McCandlish (POP acct. for file attachments)

Darxus

Tim Holmes

Peter Nai Wan

mauri stott


Eugene H. Spafford

Jamie Wilkinson

Leland Wallace

Brian D. WebbDavid Walter

FreeBSD Security Officer

dann frazier

Matt Crawford

Roger So

L. Sassaman

Martin Schulze

Douglas F. Calvert



Dave Del Torto

Nik Clayton



Linus Torvalds


Bradley M. Kuhn

Dan Nowacki

Jeffrey I. Schiller

Wichert Akkerman

Douglas F. Calvert


M. Drew Streib

Paul S. Traina

David Elson (Del)

Daniel Jacobowitz

Gregory L. Pratt

Andrew G. Malis


Peter Jones


Eric S. Raymond

LaMont Jones




[email protected]

Bill Scannell

Werner Koch

J. Daniel Backman

Andrew S. Nagy

Red Hat, Inc


K. Reid Wightman



Charles Gallo at WorkJack M Twilley

Mike Fratto

Wichert Akkerman

Cory F. Cohen

Vinnie Moscaritolo

Daniel Jacobowitz

Dr. Michael MeskesNoah Meyerhans

Charles Gallo

Martin Schulze

Philip R. Zimmermann

Tim O’Shea [tmo]

AUSCERT

Matt Taggart

Miro Jurisic

Peter Nai Wan


Christoph Martin

Woz


Michael Stone


Darxus

Tim Holmes

Peter Nai Wan

mauri stott


Eugene H. Spafford

Jamie Wilkinson

Leland Wallace

Brian D. WebbDavid Walter


dann frazier

Matt Crawford

Roger So

L. Sassaman

Martin Schulze

Douglas F. Calvert



Dave Del Torto

Nik Clayton



Linus Torvalds


Bradley M. KuhnDan Nowacki

Jeffrey I. Schiller

Wichert Akkerman

Douglas F. Calvert


M. Drew Streib

Paul S. Traina

David Elson (Del)

Daniel Jacobowitz

Gregory L. Pratt

Andrew G. Malis


Peter Jones


Eric S. Raymond

LaMont Jones




[email protected]

Bill Scannell

Werner Koch

J. Daniel Backman

Andrew S. Nagy

Red Hat, Inc


K. Reid Wightman



Charles Gallo at Work

Jack M Twilley

Mike Fratto

Wichert Akkerman

Cory F. Cohen

Vinnie Moscaritolo

Daniel Jacobowitz

Dr. Michael Meskes

Noah Meyerhans

Charles Gallo

Martin Schulze

Philip R. ZimmermannTim O’Shea [tmo]

AUSCERT

Matt Taggart

Miro Jurisic

Peter Nai Wan


Christoph Martin

Woz


Michael Stone


Darxus

Tim Holmes

Peter Nai Wan

mauri stott


Eugene H. Spafford

Jamie Wilkinson

Leland Wallace

Brian D. Webb

David Walter


dann frazier

Matt Crawford

b124 b124+PRISM b124+VPSC

Figure 5: Comparing PRISM and VPSC on two graphs. Original layouts arescaled to have an average edge length that equals 4 times the label size.

JGAA, 14(1) 53–74 (2010) 67

Table 3: Comparing the dissimilarities and area of overlap removal algorithms.Results shown are "dist, "disp and area. Area is measured with a unit of 106

square points. Initially the layout is scaled to an average edge length that equals4 times the average label size.

Graph PRISM VPSC"dist "disp area "dist "disp area

b100 0.74 0.36 14.59 0.95 0.62 25.33b102 0.41 0.18 2.62 0.44 0.33 2.77b124 0.52 0.12 3.39 0.28 0.06 3.04b143 0.5 0.22 2.59 0.67 0.41 4.26

badvoro 0.35 0.15 11.85 0.65 0.58 11.71mode 0.57 0.35 0.78 0.84 0.59 1.45

ngk10 4 0.25 0.05 0.54 0.16 0.03 0.53NaN 0.25 0.06 1.15 0.15 0.04 1.15dpd 0.14 0.03 0.54 0.1 0.03 0.53root 0.73 0.28 16.02 0.73 0.7 25.93rowe 0.16 0.04 0.38 0.11 0.03 0.38size 0.35 0.11 0.54 0.28 0.08 0.57unix 0.25 0.07 0.6 0.13 0.04 0.59xx 0.39 0.18 4.23 0.45 0.34 4.54

Graph VORO ODNLS"dist "disp area "dist "disp area

b100 - - - 0.49 0.24 1.09E3b102 1.18 0.47 131.55 0.33 0.15 41.01b124 0.69 0.36 19.09 0.43 0.26 14.64b143 0.84 0.39 19.35 0.57 0.36 29.39

badvoro 3.8 0.9 4.45E5 0.34 0.10 166.53mode 1.09 0.58 20.02 0.44 0.28 45.21

ngk10 4 0.46 0.2 0.79 0.23 0.12 2.21NaN 0.49 0.22 2.28 0.33 0.18 4.86dpd 0.43 0.23 0.6 0.38 0.29 1.45root 4.14 0.94 3.68E8 0.39 0.14 1.52E3rowe 0.41 0.2 0.68 0.25 0.11 2.01size 0.62 0.34 2.07 0.28 0.16 3.35unix 0.55 0.24 0.64 0.20 0.09 2.01xx 1.13 0.44 185.7 0.30 0.11 45.18


Table 4: Comparing dissimilarity and area of overlap removal algorithms on theRome suite of test graphs. # stands for the number of graphs within the sizerange specified.

sizes # PRISM VPSC VORO ODNLS"disp area "disp area "disp area "disp area

10–19 1407 0.009 0.16 0.09 0.061 0.14 0.069 0.07 0.3420–29 839 0.02 0.27 0.11 0.12 0.14 0.15 0.078 0.9230–39 2036 0.03 0.36 0.13 0.19 0.13 0.26 0.078 1.8540–49 1800 0.035 0.43 0.14 0.25 0.12 0.36 0.074 2.7150–59 1045 0.042 0.51 0.15 0.32 0.11 0.53 0.074 4.5360–69 1172 0.046 0.58 0.17 0.39 0.11 0.70 0.074 6.1870–79 1008 0.051 0.64 0.17 0.46 0.11 0.94 0.073 8.4480–89 788 0.05 0.71 0.17 0.52 0.10 1.20 0.07 9.9990–99 1296 0.054 0.78 0.18 0.59 0.10 1.49 0.069 13.26100–109 140 0.055 0.82 0.18 0.61 0.10 1.7 0.068 14.14

VPSC, VORO and ODNLS on the Rome test suite of graphs [4]. This suite hasa total of 11534 graphs5 of relatively small size. Due to space limitation, we onlygive the similarity measure "disp and the area, and we average the results overgraphs of similar sizes. Again, PRISM achieves the best compromise betweenbeing close to the original drawing, and having a smaller drawing area.

We note that while there is no theoretical result guaranteeing that PRISMalgorithm converges to an overlap free layout in a finite number of iterations, inpractice, out of thousands of graphs tested, some as large as tens of thousandsof vertices, PRISM always converges within a few hundred total number ofiterations in the two main loops in Algorithm 1. In our implementation we seta limit of 1000 iterations, even though this limit has never been observed to bereached. Table 5 gives the number of iterations taken for the 14 test cases inTables 2-3. As can be seen, for these graphs, the maximum number of iterationsis 122.

As a demonstration of the scalability of PRISM, we consider its applicationto a large graph. This is a tree from the Mathematics Genealogy Project [32].Each node is a mathematician, and an edge from node i to node j means that jis the first supervisor of i. The graph is disconnected and consists of thousandsof components. Here we consider the second largest component with 11766vertices. This graph took 31 seconds to lay out using SFDP, and 15 secondspost-processing using PRISM for overlap removal. PRISM converges in 81 itera-tions. Important mathematicians (those with the most o!spring) and importantedges (those that lead to the largest subtrees) are highlighted with larger nodes

5Three graphs were dropped from the test because they were disconnected.

JGAA, 14(1) 53–74 (2010) 69

Table 5: Number of iterations taken in the two main loops in Algorithm 1. A:initially the layout is scaled to an average edge length of 1 inch. B: initially thelayout is scaled to an average edge length that equals 4 times the average labelsize.

Graph |V | |E| A Bb100 1463 5806 122 75b102 302 611 66 44b124 79 281 43 19b143 135 366 46 26

badvoro 1235 1616 53 32mode 213 269 43 26

ngk10 4 50 100 29 8NaN 76 121 34 9dpd 36 108 26 6root 1054 1083 103 119rowe 43 68 28 7size 47 55 29 14unix 41 49 32 8xx 302 611 53 46

and thicker edges. Figure 6(top) gives the overall layout, which shows thatPRISM preserved the tree structure of the layout very well after node overlapremoval. Figure 6(bottom) gives a close up view of the details of a small area inthe center-left part of Figure 6(top) with many famous mathematicians of earlygenerations. Additional drawings of this and other components of the Mathe-matics Genealogy Project graph, including that of the largest component, areavailable [21].

6 Conclusions and Future Work

A number of algorithms have been proposed for removing node overlaps inundirected graph drawings. For graphs that are relatively large with nontrivialconnectivities, these algorithms often fail to produce satisfactory results, eitherbecause the resulting drawing is too large (e.g., scaling, VORO, ODNLS), orthe drawing becomes highly skewed (e.g., VPSC). In addition, many of themdo not scale well with the size of the graph in terms of computational costs.The main contributions of this paper is a new algorithm for removing overlapsthat is both highly e!ective and e"cient. The algorithm is shown to producelayouts that preserve the proximity relations between vertices, and scales wellwith the size of the graph. It has been applied to graphs of tens of thousands ofvertices, and is able to give aesthetic, overlap-free drawings with compact area


in seconds, which is not feasible with any algorithm known to us.It is possible that algorithms such as VPSC, which rely on separate passes in

the X and Y directions, might be improved by randomizing which overlaps areremoved in which pass or by gradually removing overlaps using many alternatingX and Y passes. This would, however, further increase their computational cost,which is already much higher than the algorithm proposed in this paper.

For future work, we would like to extend the overlap removal algorithm todeal with edge-node overlaps. We would also like to explore the possibility ofusing the proximity stress model for packing disconnected components.

Acknowledgments

We would like to thank Tim Dwyer and Wanchun Li for making their imple-mentations of VPSC and ODNLS, respectively, available to us. We would liketo thank Yehuda Koren and Stephen North for helpful discussions, and StephenKobourov for bringing our attention to the term “Frobenius metric”. Finally,the reviewers’ comments were very helpful, especially in making the expositionclearer and more accurate.

JGAA, 14(1) 53–74 (2010) 71

Bolie

Benson

Bunn

Mundt

Jones

Darken

Hildebrand

Sims

Robinson

BoessenrothSmith

IrudayanathanCurtis

McCoy

BakerKlasi

Fischer

Girolo

Riley

Maehara

Lehman

Neufeld

Ling

CliftonSmith

Keremedis

Miliaras

Nimmo

JonesHsieh

Bondari

Haliloglu

Wang

Cornette

Hentzel

Johnston

Sanderson

Wright

Moore

Kline

Hallett

Mullikin

Gehman

Wilder

Lubben

Whyburn

Roberts

Cleveland

Dorroh

Vickery

Klipple Basye

Jones

Swain

Sorgenfrey

Miller

Young

Bing

Moise

Anderson

Rudin

BurgessBall

Dyer

Hamstrom

SlyeMohatPearson

Armentrout

Mahavier

Treybig

Younglove

Henderson

Worrell

Cook

Reed

Baker

Baker

Davis

Rogers

Secker

Cook

Hinrichsen

O

Green

Proffitt

Purifoy

JacksonKroeger

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Gough

Montgomery

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Tidd

Uschold

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Anderson

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III

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Samulewicz

Gucht

Fatemi

D

Donno

Chinchuluun

McKenna

Fulina

Juteau

Chen

Loomis

Mueller

Yang

Cautis

Maruskin

Grant

Boginski

Hilali

Stadnicki

Juels

Steenberg

Fletcher

Takeda

Ferguson

Wang

Spraktes

Stern

Zemke

Glaeser

Coatmelec

Méhauté

Tessera

Bouhamidi

Chang

Choi

Jau

Shaikh

Porumbescu

Dalvi

Bisgaard

Laubinger

AlurBurckhardt

Singla

Mundt

BoessenrothSmithBaker

KlasiLing

Wright

Chewning

Brady

BuzbyCoutantCox

Crampton

Davis

FrancoDokken

Federighi

Geissinger

Glass

Graham

GrantGraves Hane

Hou

Howe

Hu

JamesLeitzel

Lin

Moore

Parr

PenazziLeitzel

Rim

Seo

Yang

Zhang

Briggs

Ginsburg

Komm

Fan

Maddaus

Phillips

Getchell

Kaltenborn

Dushnik

Ingraham

Stong

TullerScott

Newton

Stafford

Bruton

Conwell Pollard

Wegner

Trump

Wolf

Wolf

Larsen

Trimble

Whaples

Varino

Bell

Pothoven

Nadathur

Sims

Bendat

Higgins

Klein

Park

Kim

Byers Burdick

Lameier Cariola

Berkey

Castro

Coffey

Ortega

Wang

Ellis

Hlavacek

Fiedler

Kurepa

Unsurangsie Ali

Cossio Gadam

HassanpourNeuberger

Huneke

MerkesThron

Pree

Lazer

Riggs

Thale

Lange

Aull

Cook

Hillam

Drake

Hursch

Callas

DooherReed

Stevenson

Larson

Andima

Warren Morez

Valent

Ori

Diesto

Lin

Mullins

Ostrand

Burton

Benjelloun

Nemeth

Bernard Hook

Rowlett

Beem

Wheeler

Hansen

Nelson

Pettus Bryce

Frank

Alliston

Mader

Goggins

Young

Young

Singh

Anderson

Edwards Torrence

Saiers

Lustfield

Sanders

Lagrange Poisson

Bauschke

Dirichlet Fourier

EisensteinLipschitz

Wine

Cameron

Little

Yim

PaulsColdingLiu

Vitulli

Borwein

Croke

Cole

Hashtroodi

FabbriMantellini

FrancaZampogni

Sawyer

Markley

Wilson

Auslander

Chasles

Wang

Toure

GuediriLabbi

Dennis

Madden

Fryman

Kinch

Molnar

Brook

Fink

II

Ozeki

Huang

Euler

Silberger

Horelick

Eisenberg

BrookHuang

II

Bourgeois

Ferrari

Sun

IIICarr

Chang

GarciaGuerin

Beale

WiskottTrudinger

Palmer

Chan

Dempster

Plana

Pickrell

Dai

Dang

Brilleslyper

Menguy

Abbud

Dorey

Nyman

Rhie

Solodovnikov

EvansRawat

Ward

Strojwas Zhuang

Huang

Bernoulli

Bernoulli Habermas

Konstantinou

Liouville

Apostolov

Finan

Ambrose

ChavesBarrosBryan

Leibniz

Zhang

PerrizoPanda

McCann

Amirsoleymani

Haraty

Ye

Wu

Shi

Wituski

Gauduchon

Wang

Cholewa

Domanska

Schneider

Zhu

Czyba

Roethig

Johnson Yi

Dong

Pang

Li

Zhang

Ykhlef

Bidoit

KesselerHorowitzCahen

Gutt

Chattopadhyay

KurtzmanGrumbach

Waller

Abiteboul

Chandler

Connors

Giuliano

Kiel

Mager

Rovan

Vianu

ZaiddanMiles

Tian KottmanCobena

Segoufin

MignetNguyen

Ousmane

Coughlin

Aduen

Schneider

Hindawi

Anderson

Hong

Eswaran

Leach

Anderson

Ertekin

QianNeill

Xu LiuPadmanabhan

SundararaghavanYu

Hermann

Villaverde

Calisal

Ohn ParkerSleewagen

Xia

Port

SmedtBieliavskyDhooge

Chaibi

Young

CantonHamer

Krebsbach

Wang

Pan

Kelton

Chen

Anthappan

Brzenk

Phillips

Leitzel

Swinden

König

Brugmans

RossijnMoll

Jaworski Schoenfeld

Caine

Blassiere

SuiDenninghoff

Hennert

Angelkot

Barnaart

Calkoen

Damen Göde

OskampFagaras

Ondaatje

Tonis

Calkoen

Mensing

Wit

Simons Suerman

Swart Vrolik

Wenckebach Galen

Alew

Schröder

Schröder

Hoorn

Ekama

Ermerins

Chaudoir

Walraven

Allievi

ChanAkinturkWong

Senellart

Chen

JiangMao

TyralaJelgersma

Calle

Wiger

Cautis

Figure 6: The second largest component from the Mathematics GenealogyProject. Top: overall layout with node overlap removed. Bottom: close upview of a small area of the center-left part of above.


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Efficient, Proximity-Preserving Node Overlap Removal · For example, the Voronoi cluster busting...

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