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)
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
56 Gansner and Hu E!cient Node Overlap Removal
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.
58 Gansner and Hu E!cient Node Overlap Removal
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
60 Gansner and Hu E!cient Node Overlap Removal
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
62 Gansner and Hu E!cient Node Overlap Removal
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.
64 Gansner and Hu E!cient Node Overlap Removal
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
66 Gansner and Hu E!cient Node Overlap Removal
1
30 40
24
44
47
228
461639
50
3210
2519
33
1227
3515
43
9
48
36 1721
13 38
29
49 1418
37
34
2
42
11
26
41
31 28
320
6
45
4
23 5
7
1
30 40
8
46
1610
25
19
33
12
36
17
1338
24
49
47
27
14
29
44
35
2
42
11
18
26
41
31
39
37
50
32
3
20
28
6
45
9
15
48
4
21
43
23
34
5
7
22
1
30
40
8
46
16 1025
19
33
12
36
17
13
38
24
49
47
27
14
29
44
35
2
42
11
18
26
41
31
39
37
50
32
3
20
28
6
45
9
15 48
4
21
43
23
34
5
7
22
ngk10 4 ngk10 4+PRISM ngk10 4+VPSC
02f5daf56e299b8a8ecea892ca5af2 b4dfef6 647
171192dc1f8e6ea551548a910c00629e42 5c609b12c6bce02baf91781a831e1b951c08373 002656236a67933a619a6a3d48be8f4199f 0476750962c93b4cb293f5beb59ebbe8f4199f f0d99f1605d4b1ed6a6135eec3abd3f2199E
200E08769f73d31c1a99be2d9363f629e42 6e186ba6a196a504c3a7657d1fa41cd856f d382
837ebf4bde22e1f1535cb662d0eb84
dd2ba36 351dd0aefe480c
5f865c374cb3fe976dd376b823ad1 4d22e1
8be752bc95d436a90493bec9ee91c97828
01938827969a58db14386cb9d2f51ec7c7c 9
da24f74aad2ff519009d1f38c460aed10cc9 3124d3a6ed3381a6341c6bbe0a8f93dc1 71512ec7d43f958f2b6da3f0a2b4eb62f
3828a2c682419423cf2
784E 56e1a89679b69c612
aa868f65c34cdb64f1fad19a3089106e3b
1aaaab063
8cd4d
3a0ff0
dca32af03698c988b22eb8 3a8173d6c
d8f4a9e463a1e89217f4c6c8c 3dcf8f454
c96782ef56711c5d6a3f696a8f5bafb1 8cbae42a39004f04c39708fa49284e9
226E97284d4c3a5d499853f0e
53069e384a2
af9c489df53
c4d32527b670afb370d643e851f5ddd920 56292d076643
5e9156098c064233E 234E
3d475ea3aeca51b60212dd4280833ef80172
966d271c22e75c7538cab04b7c14a 8a
b630e1af6ae1997f0e8ba750bb828f1a326 499f6985db294c
248df40dae
ebd8ffc2ac3a90efb8af91ebeec c0b727 9fe65061641
69fdd1a1f4768c5efe735b8742610 244E
d93a80739fc1edb41a11b7294e03b8bc0435a 9be88247
bf65cfddeb00ff847feae0c8df 3703059dbc5a8
916c686a1e82dba72524aa849f9d352e
74e86
f496bcf0889b301d77819cf29dfb9 3dafb9a29c0076889f7d35ee7ef998 254E668d6360024379b5ed
256E
e1e4c23db39d8bd633c3a1ed5d7f63b8c6 4f05b
842bc5775657c1e0d67a387210a27b 47ebc3f17e4e2f4e6d1f4f0fdf
262E
04390dec6f1779353c07f5bac77c3f414a 6c93f24516f01d
69f2611acc42c36ed7cccab04b7c14a e8b
1562abef0d82416a8f5bafb1 fc73
e49aaa5cc4e44355d6a0cc3f63d
34d06d1ed6e8ebe1bf5f421c122396325ea
713db1c12759e82e30d6dca5af2
274E
23c1ec53358d237c1cab04b7c14a f
5838586c293d45583c397b8bf7f
f841118350a27b7ea29a9c9d69f4ecb77d efb52b499303115c33fd658d208447d8ec5d6de8f7b22b9640 cf5a6049ad
11180ae7706510211bc4052bb6e3
90dccb18c05807acd8d58e006f43285E
e17fea65
fe4e848cb5291ee59a2e3aefac763
c4f31ea3844e12da27ad47c6fb16636aae 8b09200cbeb87c182ca0785f3089106e3b c3bbf4
11f088bfd86a80cbe
294E3c2a62e0e5e9f7ae32701 296E
dd84fe6a65cfac7bca03ebd297E 298E
b06bbfa920aa95dd07 300E
6b5aaa4bdf44b2c8988544c6c8c
c5acd20cad2
855d26296eda4eb753069e384a2
8a46e6
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86569bffb49adf6b3d0ebac660ffeb76fc59 6331b3f
a96e47ff37983425a3e452095cab04b7c14a
1b34fb150
71a48d11b2e7e56b1df128bdbe8f4199f c6b5321a
a0befe6dd1ca7b1657868353cfae 616d8a7b
f33ec11db496f7bfcb024f71e6b
316Efe6be3206549f5b5564acde84783317E 318E
e4dba079d5fcb1f165920a3bf319E 320E16c508ab98483d430bbecab04b7c14a a7d2
9c9e2e0f2da8758e436ccd0d985a366cad7e
6aae8d25951 fb039d7a2a9fe73b5f468eba981dabfaba8 c0f34a600
2ef949c4a39b617809d979f 332E
a9497e0757b0969bde707ed5541ab86a2e 9dd5bf47f
230cc6bbc66b24eae94fa03d335E 336E
1d163eac141def176461c0acc5bb8ca4 24dfe1a997a
32979f8cf86a7e89580 340E
37d80ae421dba4a70730338860341E 342E
fbba7215e7c13173a60206617809d979f f556702dd8cc4d693415f93c0f8fc94da691e20e3
1ed67841
00880e6f50c765ebc1f85d3e9e7ef998 07283 ef13d45b1277ac9a0444adba7fe7
e3a76d31ca59a2573e1bf51f1b307f464084e4ede82074
64ef1d545
162d8039483d8a8e9 354E
f490de272a7f6e4af346d40460aed10cc9
391256c872
678bf739c344b9ad41da1396b16a892fe 358E
876d120b38b0e88817e5 360E
503737b64d432c60d6ac557e0e69937ccba1469 7e74e1587f3a4d208
b36e0be6f67fc2528612745687a7e69a72412
c8fc17180bea86
4cc20a0b7651e486e079d2c 7e494b
08dade990b228245827dbdd8 65a1f8128d574c356631b8a9369E
370E
88a4f0337c2189c3fc7b31da0d7bbcf30
65694ca6d575
1b13908a9f0763c0ae54af90620808b06a67a ffd1b1af3b6864078f3
e2a5d11499b7e66abc181ac4
73ba1714ee
90cc275011c2013c61eb11375E 376E
1927c743a0d440a5a0b12441ecff15fa12c
27709106
155d892827c33ed3cae371e6b 380E
9f24ba80192c339a64c0381E 382E
3e814305b42beb41b8c7061c08373 aeb8eccfe5ff0af70fe9fbec8b2360f90be8f4199f 2e53009d4a375
8fa622d9f842c5572a545ed729824dccb 3711ad9142a65f5eab78b4ca5ef36cce089
28533c
20f234fdcd0e1fc50261ce867219ef689f0146b544 85a34bc9616ff
e06cc38155ff6781cf944d74587a7e69a72412
09e64744536c5e1
cfdf1932665dcb4cd3c964b86fc1bba0e
8e24e9b4e
6d4a4a5a5af91b895272c30b5e86c73d1198f 398Ee0ad365c2fb444358201bb5e89c8963 400E
b07bbdc8cca5985d4c450023f6f88 df5dba74c75b228de48c7e493ee44b28 0b8694c9ef9b27b9c3d82342b759c03 81e20155999fa64e0ae6fd4280833ef80172 3ef07ae75d29a7074280833ef80172
4a36db80f1ab1e97460aed10cc9 16da5f1301b36df4df0f460aed10cc9
6b3f3fa236bb90592d23a83c397b8bf7f
f2a57e4d4f0cec516891e3bd2484 418E
deb3089920548bf1ecb23f0d87a7e69a72412
af4a1fac3e2076
bf01c8a26201 422E
23dc3a52fed9c119610b5e871e6b c
78cc16f965adc5f712ea2372c623ad1
e65185ca5be631dff7b97697be7dc0a2f07f2427E 421 428E
48398d080dfcccced48da1980866808df
d13da6273c9b4da03716a2c341e5edaa3121407f8a6d7
15d44ab97ddfeabe456a9de5f5784aac615ae78 8d5137b16a
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4c82921f4ad3f07066540a7fe7
b3cadc253f7
0bc7f8f513e0e74b270a849f9d352e a5a
3b1563a23eb9a8e9 444E
be233fafa38d931d894a849f9d352e
bce
e7a887d88c2318beba519d8988c0945d6
be6b73bd46a7a5183e8c91aee91c97828 444189d179b5db71fe
1e1fbbe14ac24e0518 b473716644f112bb0aa452ee7040a52f247fc3b
93ea0
010957669f3770aac78 454E
0a185946ee443342b07d8e187a7e69a72412
238805e2194c3
f66fe4df3d189e69ce10c9c21407f8a6d7 78b2d75d00166
247e407f45b353f8459E
460E84907547f36d0ff7e920b915087
462E
805004328dad9d315d4280833ef80172
4f0cbd3fbf0cb1e8c403126
fb58e11
4869e993f2bb10fff
f9d09 665b76844ff78fc2cf66ca2af0268dddd c338481d79773
3f16509139c7dad5163b917993089106e3b c5255d
01db23a60422ba93a68611cc0473E 474E
46125fcc583c0f494a3a1d3db6c4213a717bc
731857fe189fb398e80a05943089106e3b 5c6a2
6fb7a84e370ef70feac5cb396b16a892fe
e343cea291b79a2ed4e88d8b220746882d a3781072b
5f2592b20f13356b7fc8b42483E 484E
275a0407e33e9b8aa9cdd051731E
37f57e81ebed95
173fd00917644f0f1f3e30acc5bb8ca4 59a8b435ccd
c72df69b40156a3254fff03efcd a738ba39
6c632ad9c42228bb337eb8 86e9b0428
bbb13dc62adf2de2a42b669ce90c9b2 a4c7d0 6282bc21f6de34214b4c258c9333ec3
16c3ae29c0bc713
71cf45dd4e91bcca945137b40e65fd8495 a8f0fe2eb7bc1471
a3b6df27179b175c88fa4c9cf9f6577 4f6f7f
284f14a259991806654e744280833ef80172
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7528dd86b
c32d2697e852f247fc3b 508E
d12bd75c24b110ef90cdd35d30668 32b98f11f3d01d6
1c07453d584f3d14b1876fdb460aed10cc9
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0da9135e3bdbca0e2256fffa8a5901881dabfaba8 fe821bce
75ba8d840070942eb4e73784981dabfaba8 e64f22a31fbdc3ca37406f66635c8b226e8cbcf5cb5 46e412a3
40b49a5a9bb256c7a3286e56f72564578be
1d792d81
3b2f08d52e4bca3f9ca7bbbd681dabfaba8 99da1f8a54a38abc630c82b0c48dfbf5271f0bd1521
0f1672802d7b7fb6c9ad6821752651f747b2da3d 82d201910b00285f11bb90d0a1564181dabfaba8 1d529eb4
24431c3eb075102f07cc2c1be533E 534E
07f8a9e55a16beddb3c9153b081dabfaba8 bf141dbcec1c30f30d40c4f1f84924622fc5d5be3942 e3fd0c7b3
86276bb1e23f2c7ffcbe82a00f940646 c96cb3f78e145a127014eb43345a0cd370c12dbc 0fabab47
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5a0b4b906a 30cc206b187848523ad1 550E
5d69639a5e3bdd3d6139fa6adc88d 552E
b656f0ed2202b8e46ebf6e6236b48bc3 2a47a6a27
3b566eaa70ed401479d43a94c6c8c 6c998bf2a5eddd6125ef42bd99584c6c8c 4b683
dd12f26f8d9bb5583c397b8bf7f
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7c13bf753da
666f11bb45c3a8dcf26e1ed79c90f755c8b6612d 82a65ed4b69
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30c3f3bf8463d3843dc57d8e983089106e3b 05fed5e
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4e4a79111d
3eecb304bab2136a76deda8df cd1e9af3dec2d886e4b76537a99bc71b8a9331c941172dca23
bb92d1da1f38d52f8ff
dcc5d5e9d6c4ea8e9
582E
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a141a557a60912051f3c135587E 588E
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3dafe1619016463f521fb9 8d52f183ec
0f5db6ce12751ddcc64ebb828f1a326 67513d
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3b4fdad8e0429d112cab04b7c14a a
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626E
86633c26be93ada8b08500a6044 cef12b6
3f9ddf1ffbc0d38b07
630E
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644E
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431430c49
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38b3b0d9
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aebb7b91b
b082f7a5ff
cd6
7c0ab977f5a3c4ab6d625f503320949455f573f
7e0b91491c8c8566892cd9a0889de9efa12873949
c71043819b6a79d58478d9c273ad4f4b2e0913241a220eb692c 30417faf916
8be0efdd94a6383e87fbfded4fc8a6c26d4fd9f
3aeb78ea51020a44f2d2615436dae96deede0c6b44119 6bbd5b422edb8e358dcc20eecf94f2de229621272
d495de0b35f6 e618df70814a
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ff9d64ddf49a20f
7c1767485953d9c2086 76E
15b3ad672cd2f713a
9c1305d59c37e9be9f13d7d049c817
efe092824916a5637ee35d439589
49E
214E 216E
236E
278E 402E 404E
406E
408E 412E 438E 448E 476E
504E 634E 768E
792fd6f9df1fa1e33
70815f0352b43dc1562133ab6ebef2d4636934472
22bd92e302816
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06eff1db45cdf
2ced414a91575a48f2dd29a85221d5e9e 97a7eea3f 38f162cf917ce7298663a1f1c607a031c9192ae8e75
062fc905b9eb35
549fa15d68f0b3bee6192f888cd8d17f8f4eeb8e63d cca7040e47789
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ea920d9f5b295119
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593caebf2037317648bb451aa79
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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
albyny60albyny61albyny62
albyny63albyny64albyny65
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brhmal82
brhmal66brhmal67brhmal68brhmal69
brhmal70brhmal71
brhmal72brhmal73brhmal74brhmal75
brhmal76brhmal77
nworla81
cdknnj80
cdknnj60
cdknnj61
cdknnj62cdknnj63
cdknnj64cdknnj65
cdknnj81
cdknnj82
desmia80
hrfrct80
noc30k80nwrknj81
phlapa81
cdknnj66cdknnj67
cdknnj68cdknnj69
cdknnj70cdknnj71
rcpknj80
cdknnj72cdknnj73cdknnj74cdknnj75
cdknnj76
cdknnj77
chcgcg80
chcgcg60chcgcg61
chcgcg62
chcgcg63
chcgcg64chcgcg65
chcgcg81
dnvrco82
milwwi81
chcgcg66chcgcg67chcgcg68chcgcg69
chcgcg70chcgcg71
mplsmn81
chcgil08chcgil80
chcgil09chcgil29
chcgil81chcgil30
chcgil32
chcgil33
chcgil34
chcgil35 chcgil36
chcgil37
chcgil38chcgil39
chcgil70chcgil71
chcgil72
kscymo80lsanca80
chcgil67chcgil68
chcgil69
dllstx81
nwrknj82
nycmny81
chcgil58
chcgil59
chcgil60
chcgil61
chcgil62
chcgil63
chcgil64chcgil65
chcgil79
chcgil73chcgil74chcgil75
chcgil76chcgil77
chcgil78
kscymo83
chrlnc60chrlnc61
chrlnc62chrlnc63
chrlnc64
chrlnc65
chrlnc69chrlnc70chrlnc71
chrlnc81cncnoh81
ojusfl80
pitbpa80
rlghnc80
tampfl80
chrlnc66
chrlnc67chrlnc68
clevoh80
clevoh60clevoh61clevoh62clevoh63
clevoh64clevoh65
clmboh81
dtrtmi80
dytnoh80
hrfrct81iplsin80
lsvlky80
milwwi80
rlmdil80
clevoh81
clevoh66clevoh67clevoh68
washdc83
clmasc60clmasc61
clmasc62
clmboh80
clmboh60
clmboh61clmboh62clmboh63clmboh64
clmboh65
iplsin81
clmboh66clmboh67clmboh68clmboh69
clmboh70
clmboh71
cmbrma13cmbrma80
cmbrma16cmbrma67
cmbrma68cmbrma69
cmbrma82 cmbrma83
nybwny80
cmbrma81cmbrma60cmbrma61
cmbrma62cmbrma63
cmbrma64cmbrma65
phlapa80
washdc81
cmbrma70cmbrma71
cmbrma72cmbrma73
cmbrma74
cmbrma75
cmbrma52
cmbrma53
cmbrma54
cmbrma55cmbrma56
cmbrma57
cmbrma58cmbrma59cmbrma76cmbrma77
cmbrma78
cmbrma79
whplny81cmdnnj60cmdnnj61
cmdnnj62
cncnoh13cncnoh14
cncnoh15cncnoh16cncnoh65
cncnoh66cncnoh67
cncnoh82
cncnoh60
cncnoh61cncnoh62
cncnoh63cncnoh64
nsvltn80
scrmca80
cncnoh68
cncnoh69
cncnoh70cncnoh71
cncnoh72cncnoh73
desmia60desmia61
desmia62
desmia63
desmia64desmia65desmia81
dnvrco80
mplsmn80
okbril80
ptldor81
snfpca80
sttlwa80
desmia66
desmia67desmia68
omahne80
rlmdil81
stplmn82
dllstx21dllstx22
dllstx24
dllstx25
dllstx28dllstx29
dllstx31
dllstx70dllstx71
dllstx72ftwotx80
dllstx67dllstx68dllstx69
dllstx55dllstx56
dllstx57 dllstx58dllstx59
dllstx60dllstx61
dllstx62
dllstx63dllstx65dllstx66
dllstx79
kscymo81
lsanca81
dllstx73dllstx74
dllstx75
dllstx76dllstx77
dllstx78
dnvrco60
dnvrco61dnvrco62
dnvrco63
dnvrco64
dnvrco81
slkcut80
dnvrco65
dnvrco66dnvrco67
dnvrco68
dnvrco69dnvrco70dnvrco77dnvrco78
dnvrco79
okcyok80
dnvrco71dnvrco72
dnvrco73dnvrco74
dnvrco75dnvrco76
dtrtmi60dtrtmi61 dtrtmi62
dtrtmi63dtrtmi64
dtrtmi82
stlsmo81
stplmn81
dtrtmi65
dtrtmi66
dtrtmi67dtrtmi68
dtrtmi69dtrtmi70
dtrtmi71dtrtmi72
dtrtmi73dtrtmi74
dtrtmi75
dtrtmi76dtrtmi77dtrtmi78
dtrtmi79dytnoh60
dytnoh61dytnoh62dytnoh63
dytnoh64dytnoh65
ftwotx60ftwotx61ftwotx62ftwotx63
ftwotx64
ftwotx65
gnbonc60gnbonc61gnbonc62
grcyny80
grcyny60
grcyny61grcyny62grcyny63grcyny64
grcyny65
grdnca80
grdnca60grdnca61
grdnca62 hmsqnj80
hmsqnj60hmsqnj61hmsqnj62
hrbgpa80
hrbgpa60hrbgpa61
hrbgpa62
hrfrct03
hrfrct04hrfrct05
hrfrct02
hrfrct63hrfrct64hrfrct65
hrfrct82
hrfrct60hrfrct61hrfrct62
hrfrct66hrfrct67
hrfrct68
whplny80
hrfrct69
hrfrct70hrfrct71hrfrct72
hrfrct73hrfrct74
hstntx16
hstntx80
hstntx17 hstntx22hstntx23
hstntx67hstntx68hstntx69
hstntx82
hstntx81
hstntx60hstntx61hstntx62hstntx63hstntx64hstntx65
snantx80
hstntx70hstntx71
hstntx72hstntx73
hstntx74
hstntx75
iplsin60iplsin61
iplsin62iplsin63iplsin64
iplsin65
iplsin66iplsin67
iplsin68
jcvlfl60jcvlfl61
jcvlfl62jcvlfl63
jcvlfl64jcvlfl65
jcvlfl81 wpbhfl80
jcvlfl66
jcvlfl67
jcvlfl68
kscymo17
kscymo18kscymo21
kscymo22
kscymo23
kscymo24
kscymo67
kscymo68kscymo69
kscymo60
kscymo61kscymo62
kscymo63kscymo64kscymo65
kscymo70
kscymo71
kscymo72
kscymo73
kscymo74kscymo75
sttlwa82
kscymo58kscymo59
kscymo76kscymo77kscymo78
kscymo79
lsanca13
lsanca19
lsanca23
lsanca26
lsanca27
lsanca28
lsanca29
lsanca30lsanca31
lsanca32
lsanca33
lsanca70
lsanca71lsanca72
lsanca83
lsanca67lsanca68
lsanca69
lsanca60
lsanca61lsanca62
lsanca63lsanca64lsanca65
nycmny80
lsanca58
lsanca59lsanca73lsanca74lsanca75
lsanca76
lsanca77
lsanca78
lsanca79
slkcut81
lsvlky60lsvlky61lsvlky62
lsvlky63
lsvlky64lsvlky65
lsvlky66lsvlky67lsvlky68lsvlky69
lsvlky70lsvlky71
miamfl60
miamfl61miamfl62
milwwi60
milwwi61milwwi62milwwi63milwwi64milwwi65
milwwi66milwwi67milwwi68milwwi69milwwi70
milwwi71
mmphtn60mmphtn61mmphtn62mmphtn63 mmphtn64mmphtn65stlsmo82
mmphtn66mmphtn67mmphtn68
mmphtn69
mmphtn70mmphtn71
mplsmn60mplsmn61mplsmn62
mplsmn63mplsmn64mplsmn65
okbril81
mplsmn66mplsmn67mplsmn68
mplsmn69
mplsmn70
mplsmn71
ptldor82
noc30k60
noc30k61
nwrknj83
nsvltn60nsvltn61nsvltn62nsvltn63
nsvltn64nsvltn65
nsvltn81
nsvltn66nsvltn67nsvltn68
nworla80
nworla60nworla61
nworla62nworla63nworla64nworla65
nworla66nworla67nworla68
nwrknj16nwrknj80 nwrknj20
nwrknj21nwrknj22nwrknj23
nwrknj67nwrknj68
nwrknj69washdc80
nwrknj60nwrknj61
nwrknj62
nwrknj63
nwrknj64nwrknj65
nwrknj70
nwrknj71
nwrknj72
nwrknj73nwrknj74
nwrknj75
nwrknj55nwrknj56
nwrknj57nwrknj58
nwrknj59nwrknj76
nwrknj77
nwrknj78
nwrknj79
nybwny60
nybwny61nybwny62
nybwny63
nybwny64nybwny65
nybwny81
nybwny82
nybwny66
nybwny67
nybwny68
nybwny69nybwny70
nybwny71nybwny75
nybwny76
nybwny77
nybwny72
nybwny73nybwny74
nycmny09
nycmny28
nycmny29
nycmny31nycmny32nycmny33
nycmny35nycmny36
nycmny37nycmny38
nycmny70nycmny71nycmny72
nycmny67nycmny68
nycmny69
nycmny60
nycmny61
nycmny62nycmny63
nycmny64nycmny65
nycmny55nycmny56
nycmny57
nycmny58
nycmny59nycmny73
nycmny74nycmny75nycmny76nycmny77nycmny78
nycmny79
ojusfl60ojusfl61
ojusfl62
ojusfl63
ojusfl64
ojusfl65
ojusfl81
ojusfl66ojusfl67
ojusfl68ojusfl69ojusfl70
ojusfl71
okbril60okbril61
okbril62
okbril63okbril64
okbril65
okbril66okbril67
okbril68okbril69okbril70okbril71
okcyok60
okcyok61okcyok62
okldca80okldca60okldca61
okldca62
okldca63
okldca64
okldca65
okldca81
okldca66okldca67
okldca68okldca69
okldca70
okldca71sndgca81
omahne60omahne61omahne62
orlnfl08
orlnfl80orlnfl13
orlnfl15orlnfl16orlnfl17
orlnfl67orlnfl68
orlnfl69
orlnfl82
orlnfl60
orlnfl61
orlnfl62
orlnfl63orlnfl64orlnfl65
orlnfl76orlnfl77orlnfl78
orlnfl70orlnfl71
orlnfl72orlnfl73orlnfl74orlnfl75
phlapa18phlapa19
phlapa26phlapa27phlapa60
phlapa61phlapa62
phlapa63
phlapa65phlapa66
phlapa22
phlapa23
phlapa67
phlapa68
phlapa69
phlapa70phlapa71
phlapa72
phlapa73phlapa74phlapa75
phlapa58
phlapa59
phlapa76phlapa77phlapa78
phlapa79
phnxaz10
phnxaz80
phnxaz11phnxaz12phnxaz13
phnxaz67
phnxaz68phnxaz69
phnxaz82
phnxaz60
phnxaz61phnxaz62
phnxaz63phnxaz64
phnxaz65
phnxaz70
phnxaz71phnxaz72phnxaz73phnxaz74phnxaz75
snbrca80
pitbpa60
pitbpa61
pitbpa62
pitbpa63pitbpa64
pitbpa65
pitbpa66
pitbpa67pitbpa68
pitbpa69
pitbpa70pitbpa71
ptldor06
ptldor80
ptldor12ptldor13
ptldor14ptldor07
ptldor65ptldor66
ptldor67
ptldor60ptldor61
ptldor62
ptldor63ptldor64
ptldor68
ptldor69
ptldor70
spknwa80
rcmdva80
rcmdva60
rcmdva61
rcmdva62
rcmdva63rcmdva64rcmdva65
rcmdva66rcmdva67
rcmdva68rcpknj60
rcpknj61rcpknj62
rlghnc60
rlghnc61rlghnc62
rlghnc63rlghnc64
rlghnc65
rlghnc81
rlghnc66
rlghnc67
rlghnc68
rlghnc69rlghnc70
rlghnc71
rlmdil60rlmdil61rlmdil62
rlmdil63rlmdil64
rlmdil65
rlmdil66rlmdil67rlmdil68rlmdil69rlmdil70rlmdil71
scrmca60
scrmca61scrmca62
scrmca63
scrmca64scrmca65
scrmca81
scrmca66
scrmca67
scrmca68scrmca69
scrmca70scrmca71snjsca81
shokca60shokca61shokca62shokca63
shokca64
shokca65
shokca81
shokca66
shokca67shokca68
shokca69
shokca70shokca71
slkcut60slkcut61
slkcut62slkcut63
slkcut64slkcut65
slkcut66
slkcut67slkcut68
slspmd60slspmd61
slspmd62
snantx60snantx61
snantx62snantx63snantx64snantx65
sndgca60sndgca61sndgca62
sndgca63sndgca64
sndgca65
sndgca66sndgca67sndgca68
snfcca10
snfcca80
snfcca12
snfcca20
snfcca21
snfcca25snfcca26snfcca27snfcca28
snfcca29snfcca30
snfcca31snfcca32
snfcca70snfcca71
snfcca72
snfcca83
sttlwa81
snfcca67
snfcca68snfcca69
snfcca60snfcca61snfcca62
snfcca63snfcca64snfcca65
snfcca73snfcca74
snfcca75snfpca60snfpca61
snfpca62snfpca63
snfpca64
snfpca65
snfpca66snfpca67snfpca68
snfpca69snfpca70snjsca60
snjsca61snjsca62
snjsca63snjsca64snjsca65
snjsca66snjsca67
snjsca68snjsca69
snjsca70
snjsca71snjsca72snjsca73
snjsca74
spknwa60spknwa61
spknwa62
stlsmo11
stlsmo80
stlsmo12
stlsmo13stlsmo14
stlsmo65stlsmo66
stlsmo67 stlsmo60stlsmo61
stlsmo62stlsmo63stlsmo64
stlsmo68stlsmo69stlsmo70stlsmo71
stlsmo72stlsmo73stlsmo74
stlsmo75
stlsmo76
stplmn06
stplmn80
stplmn07stplmn08stplmn09
stplmn67stplmn68
stplmn69
stplmn60stplmn61stplmn62
stplmn63
stplmn64
stplmn65
stplmn70stplmn71stplmn72
sttlwa60sttlwa61sttlwa62
sttlwa63sttlwa64
sttlwa65
sttlwa66
sttlwa67sttlwa68sttlwa69
sttlwa70
sttlwa71sttlwa72
sttlwa73sttlwa74
sttlwa75sttlwa76
syrcny60syrcny61syrcny62
tampfl60
tampfl61tampfl62tampfl63
tampfl64tampfl65
tampfl66
tampfl67tampfl68tampfl69
tampfl70tampfl71
tcsnaz60tcsnaz61
tcsnaz62tcsnaz63tcsnaz64tcsnaz65
tulsok60
tulsok61
tulsok62tulsok63
tulsok64tulsok65
tulsok66
tulsok67
tulsok68
washdc11
washdc12washdc15washdc16
washdc17washdc18
washdc65
washdc66washdc67
washdc60
washdc61
washdc62washdc63washdc64
washdc68washdc69
washdc70washdc71washdc72washdc73
washdc74
washdc75washdc76
washdt60
washdt61
washdt62
waynpa60
waynpa61waynpa62
waynpa63waynpa64waynpa65waynpa66
waynpa67
waynpa68
whplny60
whplny61whplny62
whplny63
whplny64whplny65
whplny66
whplny67whplny68
whplny69
whplny70whplny71
wpbhfl60wpbhfl61
wpbhfl62wpbhfl63
wpbhfl64wpbhfl65
akrnoh80
akrnoh60
akrnoh61akrnoh62
albyny80
albyny60
albyny61albyny62
albyny63
albyny64albyny65
anhmca80
anhmca60anhmca61
anhmca62
anhmca63
anhmca64 anhmca81anhmca82
lsanca82
phnxaz81
shokca80
sndgca80
snjsca80
tulsok80
anhmca65
anhmca66
anhmca67
anhmca68
anhmca69
anhmca70
anhmca71
anhmca72
anhmca73
anhmca74
anhmca75
anhmca76
anhmca77
anhmca78
anhmca79
snfpca81
artnva80
artnva60
artnva61
artnva62 artnva63
artnva64
artnva65artnva66
artnva67
artnva68
pitbpa81
washdc82
atlnga09
atlnga81
atlnga16
atlnga25
atlnga80
atlnga30
atlnga31atlnga32
atlnga33
atlnga34
atlnga35
atlnga36
atlnga37
atlnga75
atlnga76
atlnga77
gnbonc80
lsvlky81
miamfl80
mmphtn80
phlapa83
tampfl81
atlnga72 atlnga73
atlnga74
bltmmd81
brhmal80
chcgil83
clmasc80
jcvlfl80
atlnga82
atlnga60
atlnga61
atlnga62
atlnga66
atlnga67
atlnga68
atlnga83
chcgil82
chrlnc80
cncnoh80
dllstx82
kscymo82
nycmny82
orlnfl81
snfcca82
atlnga56
atlnga57
atlnga58
atlnga59
atlnga63
atlnga64
atlnga65
atlnga69
atlnga70 atlnga71
atlnga78
atlnga79
dllstx80
dllstx83
nycmny83
snfcca81
austtx80
austtx60
austtx61
austtx62
austtx63
austtx64
austtx65
mmphtn81
tcsnaz80
bflony80
bflony60
bflony61bflony62
bflony63
bflony64bflony65
dtrtmi81
syrcny80
bltmmd80
bltmmd60
bltmmd61
bltmmd62
bltmmd63
bltmmd64
bltmmd65
phlapa82
waynpa80
bltmmd66
bltmmd67
bltmmd68
bltmmd69bltmmd70
bltmmd71
cmdnnj80
slspmd80
washdt80
brhmal60
brhmal61
brhmal62brhmal63brhmal64
brhmal65
brhmal81
brhmal82
brhmal66
brhmal67
brhmal68
brhmal69
brhmal70
brhmal71
brhmal72
brhmal73brhmal74
brhmal75
brhmal76
brhmal77
nworla81
cdknnj80
cdknnj60
cdknnj61
cdknnj62cdknnj63
cdknnj64
cdknnj65
cdknnj81
cdknnj82
desmia80
hrfrct80
noc30k80
nwrknj81
phlapa81
cdknnj66
cdknnj67
cdknnj68
cdknnj69
cdknnj70
cdknnj71
rcpknj80
cdknnj72
cdknnj73
cdknnj74cdknnj75
cdknnj76
cdknnj77
chcgcg80
chcgcg60
chcgcg61
chcgcg62
chcgcg63
chcgcg64chcgcg65
chcgcg81
dnvrco82
milwwi81
chcgcg66
chcgcg67chcgcg68
chcgcg69
chcgcg70
chcgcg71
mplsmn81
chcgil08
chcgil80
chcgil09chcgil29
chcgil81
chcgil30
chcgil32
chcgil33
chcgil34
chcgil35chcgil36
chcgil37
chcgil38
chcgil39
chcgil70
chcgil71
chcgil72
kscymo80
lsanca80
chcgil67
chcgil68
chcgil69
dllstx81
nwrknj82
nycmny81
chcgil58
chcgil59
chcgil60
chcgil61
chcgil62
chcgil63
chcgil64chcgil65
chcgil79
chcgil73chcgil74
chcgil75
chcgil76
chcgil77
chcgil78
kscymo83
chrlnc60
chrlnc61
chrlnc62chrlnc63
chrlnc64
chrlnc65
chrlnc69
chrlnc70
chrlnc71
chrlnc81cncnoh81
ojusfl80
pitbpa80
rlghnc80
tampfl80
chrlnc66
chrlnc67
chrlnc68
clevoh80
clevoh60
clevoh61clevoh62
clevoh63
clevoh64clevoh65
clmboh81
dtrtmi80
dytnoh80
hrfrct81
iplsin80
lsvlky80
milwwi80
rlmdil80
clevoh81
clevoh66
clevoh67
clevoh68
washdc83
clmasc60
clmasc61
clmasc62
clmboh80
clmboh60
clmboh61clmboh62
clmboh63clmboh64
clmboh65
iplsin81
clmboh66
clmboh67clmboh68
clmboh69
clmboh70
clmboh71
cmbrma13
cmbrma80
cmbrma16
cmbrma67
cmbrma68
cmbrma69
cmbrma82
cmbrma83
nybwny80
cmbrma81
cmbrma60cmbrma61
cmbrma62
cmbrma63
cmbrma64
cmbrma65
phlapa80
washdc81
cmbrma70
cmbrma71
cmbrma72
cmbrma73
cmbrma74
cmbrma75
cmbrma52
cmbrma53
cmbrma54
cmbrma55cmbrma56
cmbrma57
cmbrma58
cmbrma59 cmbrma76cmbrma77
cmbrma78
cmbrma79
whplny81
cmdnnj60cmdnnj61
cmdnnj62
cncnoh13cncnoh14
cncnoh15cncnoh16 cncnoh65
cncnoh66
cncnoh67
cncnoh82
cncnoh60
cncnoh61
cncnoh62
cncnoh63
cncnoh64
nsvltn80
scrmca80
cncnoh68
cncnoh69
cncnoh70
cncnoh71
cncnoh72cncnoh73
desmia60
desmia61
desmia62
desmia63
desmia64
desmia65
desmia81
dnvrco80
mplsmn80
okbril80
ptldor81
snfpca80
sttlwa80
desmia66
desmia67
desmia68
omahne80
rlmdil81
stplmn82
dllstx21
dllstx22
dllstx24
dllstx25dllstx28
dllstx29
dllstx31
dllstx70dllstx71
dllstx72
ftwotx80
dllstx67
dllstx68dllstx69
dllstx55
dllstx56
dllstx57 dllstx58
dllstx59
dllstx60
dllstx61
dllstx62
dllstx63
dllstx65dllstx66dllstx79
kscymo81
lsanca81
dllstx73
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dllstx78
dnvrco60
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okcyok80
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stplmn81
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dtrtmi76dtrtmi77
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grcyny65
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hrfrct69
hrfrct70
hrfrct71hrfrct72
hrfrct73hrfrct74
hstntx16
hstntx80
hstntx17 hstntx22
hstntx23
hstntx67
hstntx68
hstntx69
hstntx82
hstntx81
hstntx60
hstntx61hstntx62 hstntx63
hstntx64hstntx65
snantx80
hstntx70
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hstntx75
iplsin60
iplsin61
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jcvlfl60
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jcvlfl64
jcvlfl65
jcvlfl81
wpbhfl80
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kscymo17
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kscymo75
sttlwa82
kscymo58
kscymo59
kscymo76kscymo77
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lsanca83
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lsanca79slkcut81
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mmphtn60mmphtn61mmphtn62mmphtn63
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mmphtn65stlsmo82
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okbril81
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nworla66nworla67
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nwrknj80 nwrknj20
nwrknj21
nwrknj22
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nwrknj67
nwrknj68
nwrknj69
washdc80
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nwrknj61
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nwrknj55nwrknj56
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nwrknj58
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nwrknj76
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nwrknj78
nwrknj79
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nycmny33
nycmny35nycmny36
nycmny37
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nycmny67nycmny68
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nycmny55
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nycmny57
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nycmny59nycmny73
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nycmny75
nycmny76 nycmny77
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nycmny79
ojusfl60
ojusfl61
ojusfl62
ojusfl63
ojusfl64
ojusfl65
ojusfl81
ojusfl66
ojusfl67
ojusfl68
ojusfl69
ojusfl70
ojusfl71
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okbril64
okbril65
okbril66
okbril67
okbril68okbril69
okbril70
okbril71
okcyok60
okcyok61
okcyok62
okldca80
okldca60 okldca61
okldca62
okldca63
okldca64
okldca65
okldca81
okldca66okldca67
okldca68
okldca69
okldca70
okldca71
sndgca81
omahne60 omahne61
omahne62
orlnfl08
orlnfl80orlnfl13
orlnfl15orlnfl16
orlnfl17
orlnfl67
orlnfl68
orlnfl69
orlnfl82
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orlnfl61
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orlnfl63
orlnfl64
orlnfl65
orlnfl76
orlnfl77
orlnfl78
orlnfl70orlnfl71
orlnfl72
orlnfl73 orlnfl74
orlnfl75
phlapa18
phlapa19
phlapa26
phlapa27phlapa60
phlapa61
phlapa62
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phlapa65
phlapa66
phlapa22
phlapa23
phlapa67
phlapa68
phlapa69
phlapa70
phlapa71
phlapa72
phlapa73phlapa74
phlapa75
phlapa58
phlapa59
phlapa76 phlapa77phlapa78
phlapa79
phnxaz10
phnxaz80
phnxaz11
phnxaz12
phnxaz13
phnxaz67
phnxaz68
phnxaz69
phnxaz82
phnxaz60
phnxaz61
phnxaz62
phnxaz63phnxaz64
phnxaz65
phnxaz70
phnxaz71 phnxaz72
phnxaz73
phnxaz74
phnxaz75
snbrca80
pitbpa60
pitbpa61
pitbpa62
pitbpa63
pitbpa64
pitbpa65
pitbpa66
pitbpa67
pitbpa68
pitbpa69
pitbpa70
pitbpa71
ptldor06
ptldor80
ptldor12
ptldor13
ptldor14ptldor07
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ptldor66
ptldor67
ptldor60
ptldor61
ptldor62
ptldor63
ptldor64
ptldor68
ptldor69
ptldor70
spknwa80
rcmdva80
rcmdva60
rcmdva61
rcmdva62
rcmdva63
rcmdva64rcmdva65
rcmdva66
rcmdva67
rcmdva68
rcpknj60
rcpknj61
rcpknj62
rlghnc60
rlghnc61
rlghnc62
rlghnc63
rlghnc64
rlghnc65rlghnc81
rlghnc66
rlghnc67
rlghnc68
rlghnc69
rlghnc70
rlghnc71
rlmdil60
rlmdil61
rlmdil62
rlmdil63
rlmdil64
rlmdil65
rlmdil66
rlmdil67rlmdil68
rlmdil69
rlmdil70
rlmdil71
scrmca60
scrmca61
scrmca62
scrmca63
scrmca64
scrmca65
scrmca81
scrmca66
scrmca67
scrmca68
scrmca69
scrmca70
scrmca71snjsca81
shokca60
shokca61
shokca62shokca63
shokca64
shokca65
shokca81
shokca66
shokca67
shokca68
shokca69
shokca70 shokca71
slkcut60
slkcut61
slkcut62
slkcut63
slkcut64
slkcut65
slkcut66
slkcut67
slkcut68
slspmd60
slspmd61
slspmd62
snantx60
snantx61
snantx62
snantx63
snantx64
snantx65
sndgca60
sndgca61sndgca62
sndgca63
sndgca64
sndgca65
sndgca66
sndgca67
sndgca68
snfcca10
snfcca80
snfcca12
snfcca20
snfcca21
snfcca25snfcca26
snfcca27snfcca28
snfcca29
snfcca30
snfcca31
snfcca32
snfcca70
snfcca71
snfcca72
snfcca83
sttlwa81
snfcca67
snfcca68
snfcca69
snfcca60
snfcca61snfcca62
snfcca63snfcca64
snfcca65
snfcca73
snfcca74
snfcca75snfpca60
snfpca61
snfpca62
snfpca63
snfpca64
snfpca65
snfpca66snfpca67
snfpca68
snfpca69snfpca70
snjsca60
snjsca61
snjsca62
snjsca63
snjsca64
snjsca65
snjsca66
snjsca67
snjsca68
snjsca69
snjsca70
snjsca71
snjsca72
snjsca73
snjsca74
spknwa60
spknwa61
spknwa62
stlsmo11
stlsmo80
stlsmo12
stlsmo13
stlsmo14
stlsmo65
stlsmo66
stlsmo67 stlsmo60
stlsmo61
stlsmo62
stlsmo63 stlsmo64
stlsmo68
stlsmo69
stlsmo70
stlsmo71
stlsmo72
stlsmo73
stlsmo74
stlsmo75
stlsmo76
stplmn06
stplmn80
stplmn07
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stplmn09
stplmn67
stplmn68
stplmn69
stplmn60
stplmn61
stplmn62
stplmn63
stplmn64
stplmn65
stplmn70stplmn71
stplmn72
sttlwa60
sttlwa61
sttlwa62
sttlwa63
sttlwa64
sttlwa65
sttlwa66
sttlwa67
sttlwa68
sttlwa69
sttlwa70
sttlwa71 sttlwa72
sttlwa73
sttlwa74
sttlwa75
sttlwa76
syrcny60syrcny61
syrcny62
tampfl60
tampfl61
tampfl62
tampfl63
tampfl64
tampfl65
tampfl66
tampfl67
tampfl68tampfl69
tampfl70
tampfl71
tcsnaz60
tcsnaz61
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tcsnaz63
tcsnaz64
tcsnaz65
tulsok60
tulsok61
tulsok62
tulsok63
tulsok64
tulsok65
tulsok66
tulsok67
tulsok68
washdc11
washdc12
washdc15washdc16
washdc17
washdc18
washdc65
washdc66
washdc67
washdc60
washdc61
washdc62
washdc63
washdc64
washdc68washdc69
washdc70washdc71
washdc72
washdc73
washdc74
washdc75
washdc76
washdt60
washdt61
washdt62
waynpa60
waynpa61
waynpa62
waynpa63 waynpa64
waynpa65waynpa66
waynpa67
waynpa68
whplny60
whplny61
whplny62
whplny63
whplny64
whplny65
whplny66
whplny67whplny68
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whplny70whplny71
wpbhfl60
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akrnoh80
akrnoh60
akrnoh61
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lsanca82
phnxaz81
shokca80
sndgca80
snjsca80
tulsok80
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anhmca66
anhmca67
anhmca68
anhmca69
anhmca70
anhmca71
anhmca72
anhmca73
anhmca74
anhmca75
anhmca76
anhmca77
anhmca78
anhmca79
snfpca81
artnva80
artnva60
artnva61
artnva62
artnva63
artnva64
artnva65
artnva66
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artnva68
pitbpa81
washdc82
atlnga09
atlnga81
atlnga16
atlnga25
atlnga80
atlnga30
atlnga31
atlnga32
atlnga33
atlnga34
atlnga35
atlnga36
atlnga37
atlnga75
atlnga76
atlnga77
gnbonc80
lsvlky81
miamfl80
mmphtn80
phlapa83
tampfl81
atlnga72
atlnga73
atlnga74
bltmmd81
brhmal80
chcgil83
clmasc80
jcvlfl80
atlnga82
atlnga60
atlnga61
atlnga62
atlnga66
atlnga67
atlnga68
atlnga83
chcgil82
chrlnc80
cncnoh80
dllstx82
kscymo82
nycmny82
orlnfl81
snfcca82
atlnga56
atlnga57
atlnga58
atlnga59
atlnga63
atlnga64
atlnga65
atlnga69
atlnga70
atlnga71
atlnga78
atlnga79
dllstx80
dllstx83
nycmny83
snfcca81
austtx80
austtx60
austtx61
austtx62
austtx63
austtx64
austtx65
mmphtn81
tcsnaz80
bflony80
bflony60
bflony61
bflony62
bflony63
bflony64
bflony65
dtrtmi81
syrcny80
bltmmd80
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bltmmd61
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bltmmd64
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phlapa82
waynpa80
bltmmd66
bltmmd67
bltmmd68
bltmmd69
bltmmd70
bltmmd71
cmdnnj80
slspmd80
washdt80
brhmal60
brhmal61
brhmal62
brhmal63
brhmal64
brhmal65
brhmal81
brhmal82
brhmal66
brhmal67
brhmal68
brhmal69
brhmal70
brhmal71
brhmal72
brhmal73
brhmal74
brhmal75
brhmal76
brhmal77
nworla81
cdknnj80
cdknnj60
cdknnj61
cdknnj62
cdknnj63
cdknnj64
cdknnj65
cdknnj81
cdknnj82
desmia80
hrfrct80
noc30k80
nwrknj81
phlapa81
cdknnj66
cdknnj67
cdknnj68
cdknnj69
cdknnj70
cdknnj71
rcpknj80
cdknnj72
cdknnj73
cdknnj74
cdknnj75
cdknnj76
cdknnj77
chcgcg80
chcgcg60
chcgcg61
chcgcg62
chcgcg63
chcgcg64
chcgcg65
chcgcg81
dnvrco82
milwwi81
chcgcg66
chcgcg67
chcgcg68
chcgcg69
chcgcg70
chcgcg71
mplsmn81
chcgil08
chcgil80
chcgil09
chcgil29
chcgil81
chcgil30
chcgil32
chcgil33
chcgil34
chcgil35 chcgil36
chcgil37
chcgil38
chcgil39
chcgil70
chcgil71
chcgil72
kscymo80
lsanca80
chcgil67
chcgil68
chcgil69
dllstx81
nwrknj82
nycmny81
chcgil58
chcgil59
chcgil60
chcgil61
chcgil62
chcgil63
chcgil64
chcgil65
chcgil79
chcgil73
chcgil74
chcgil75
chcgil76
chcgil77
chcgil78
kscymo83
chrlnc60
chrlnc61
chrlnc62
chrlnc63
chrlnc64
chrlnc65
chrlnc69
chrlnc70
chrlnc71
chrlnc81
cncnoh81
ojusfl80
pitbpa80
rlghnc80
tampfl80
chrlnc66
chrlnc67
chrlnc68
clevoh80
clevoh60
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clevoh62
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clevoh64
clevoh65
clmboh81
dtrtmi80
dytnoh80
hrfrct81
iplsin80
lsvlky80
milwwi80
rlmdil80
clevoh81
clevoh66
clevoh67
clevoh68
washdc83
clmasc60
clmasc61
clmasc62
clmboh80
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clmboh61
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clmboh63
clmboh64
clmboh65
iplsin81
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clmboh67
clmboh68
clmboh69
clmboh70
clmboh71
cmbrma13
cmbrma80
cmbrma16
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cmbrma69
cmbrma82
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nybwny80
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cmbrma60
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cmbrma64
cmbrma65
phlapa80
washdc81
cmbrma70
cmbrma71
cmbrma72
cmbrma73
cmbrma74
cmbrma75
cmbrma52
cmbrma53
cmbrma54
cmbrma55
cmbrma56
cmbrma57
cmbrma58
cmbrma59
cmbrma76
cmbrma77
cmbrma78
cmbrma79
whplny81
cmdnnj60
cmdnnj61
cmdnnj62
cncnoh13
cncnoh14
cncnoh15
cncnoh16
cncnoh65
cncnoh66
cncnoh67
cncnoh82
cncnoh60
cncnoh61
cncnoh62
cncnoh63
cncnoh64
nsvltn80
scrmca80
cncnoh68
cncnoh69
cncnoh70
cncnoh71
cncnoh72
cncnoh73
desmia60
desmia61
desmia62
desmia63
desmia64
desmia65
desmia81
dnvrco80
mplsmn80
okbril80
ptldor81
snfpca80
sttlwa80
desmia66
desmia67
desmia68
omahne80
rlmdil81stplmn82
dllstx21
dllstx22
dllstx24
dllstx25
dllstx28
dllstx29
dllstx31
dllstx70
dllstx71
dllstx72
ftwotx80 dllstx67
dllstx68
dllstx69
dllstx55
dllstx56
dllstx57
dllstx58
dllstx59
dllstx60
dllstx61
dllstx62
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dllstx65
dllstx66
dllstx79
kscymo81
lsanca81
dllstx73
dllstx74
dllstx75
dllstx76
dllstx77
dllstx78
dnvrco60
dnvrco61
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dnvrco64
dnvrco81
slkcut80
dnvrco65
dnvrco66
dnvrco67
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dnvrco70
dnvrco77
dnvrco78
dnvrco79
okcyok80
dnvrco71
dnvrco72
dnvrco73
dnvrco74
dnvrco75
dnvrco76
dtrtmi60
dtrtmi61
dtrtmi62
dtrtmi63
dtrtmi64
dtrtmi82
stlsmo81
stplmn81
dtrtmi65
dtrtmi66
dtrtmi67
dtrtmi68
dtrtmi69
dtrtmi70
dtrtmi71
dtrtmi72
dtrtmi73
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dtrtmi75
dtrtmi76
dtrtmi77
dtrtmi78
dtrtmi79
dytnoh60
dytnoh61
dytnoh62
dytnoh63
dytnoh64
dytnoh65
ftwotx60
ftwotx61
ftwotx62
ftwotx63
ftwotx64
ftwotx65
gnbonc60
gnbonc61
gnbonc62
grcyny80
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grcyny64
grcyny65
grdnca80
grdnca60
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hmsqnj80
hmsqnj60
hmsqnj61
hmsqnj62
hrbgpa80
hrbgpa60
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hrbgpa62
hrfrct03
hrfrct04
hrfrct05
hrfrct02
hrfrct63
hrfrct64
hrfrct65
hrfrct82
hrfrct60
hrfrct61
hrfrct62
hrfrct66
hrfrct67
hrfrct68
whplny80
hrfrct69
hrfrct70
hrfrct71
hrfrct72 hrfrct73
hrfrct74
hstntx16
hstntx80
hstntx17 hstntx22
hstntx23
hstntx67
hstntx68
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hstntx82
hstntx81
hstntx60
hstntx61
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hstntx64
hstntx65
snantx80
hstntx70
hstntx71
hstntx72
hstntx73
hstntx74
hstntx75
iplsin60
iplsin61
iplsin62
iplsin63
iplsin64
iplsin65
iplsin66
iplsin67
iplsin68
jcvlfl60
jcvlfl61
jcvlfl62
jcvlfl63
jcvlfl64
jcvlfl65
jcvlfl81
wpbhfl80
jcvlfl66
jcvlfl67
jcvlfl68
kscymo17
kscymo18
kscymo21
kscymo22
kscymo23
kscymo24
kscymo67
kscymo68
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kscymo60
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kscymo64
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sttlwa82
kscymo58
kscymo59
kscymo76
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kscymo78
kscymo79
lsanca13
lsanca19
lsanca23
lsanca26
lsanca27
lsanca28
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lsanca70
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lsanca83
lsanca67
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lsanca60
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nycmny80
lsanca58
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lsanca73
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lsanca78
lsanca79slkcut81
lsvlky60
lsvlky61
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lsvlky65
lsvlky66
lsvlky67
lsvlky68
lsvlky69
lsvlky70
lsvlky71
miamfl60
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milwwi69
milwwi70
milwwi71
mmphtn60mmphtn61
mmphtn62
mmphtn63
mmphtn64
mmphtn65
stlsmo82
mmphtn66
mmphtn67
mmphtn68
mmphtn69
mmphtn70
mmphtn71
mplsmn60
mplsmn61
mplsmn62
mplsmn63
mplsmn64
mplsmn65
okbril81
mplsmn66
mplsmn67
mplsmn68
mplsmn69
mplsmn70
mplsmn71
ptldor82
noc30k60
noc30k61
nwrknj83
nsvltn60
nsvltn61
nsvltn62 nsvltn63
nsvltn64
nsvltn65
nsvltn81
nsvltn66
nsvltn67
nsvltn68
nworla80
nworla60
nworla61
nworla62
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nworla64
nworla65
nworla66
nworla67
nworla68
nwrknj16
nwrknj80
nwrknj20
nwrknj21
nwrknj22
nwrknj23
nwrknj67
nwrknj68
nwrknj69
washdc80
nwrknj60
nwrknj61
nwrknj62
nwrknj63
nwrknj64
nwrknj65
nwrknj70
nwrknj71
nwrknj72
nwrknj73
nwrknj74
nwrknj75
nwrknj55
nwrknj56
nwrknj57
nwrknj58
nwrknj59
nwrknj76
nwrknj77
nwrknj78
nwrknj79
nybwny60
nybwny61
nybwny62
nybwny63
nybwny64
nybwny65
nybwny81
nybwny82
nybwny66
nybwny67
nybwny68
nybwny69
nybwny70
nybwny71
nybwny75
nybwny76
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nycmny09
nycmny28
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nycmny77nycmny78
nycmny79
ojusfl60
ojusfl61
ojusfl62
ojusfl63
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ojusfl81
ojusfl66 ojusfl67
ojusfl68
ojusfl69
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ojusfl71
okbril60
okbril61
okbril62
okbril63
okbril64
okbril65
okbril66
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okcyok60
okcyok61
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okldca80
okldca60
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okldca81
okldca66
okldca67
okldca68
okldca69
okldca70
okldca71
sndgca81
omahne60
omahne61
omahne62
orlnfl08
orlnfl80
orlnfl13
orlnfl15
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orlnfl67
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phlapa18
phlapa19
phlapa26
phlapa27
phlapa60
phlapa61
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phlapa22
phlapa23
phlapa67
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phlapa58
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phlapa76 phlapa77
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phnxaz10
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phnxaz11
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phnxaz67
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phnxaz82
phnxaz60
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phnxaz70
phnxaz71
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phnxaz75
snbrca80
pitbpa60
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pitbpa63
pitbpa64
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pitbpa66
pitbpa67 pitbpa68
pitbpa69
pitbpa70
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ptldor06
ptldor80
ptldor12
ptldor13
ptldor14
ptldor07
ptldor65
ptldor66
ptldor67
ptldor60
ptldor61
ptldor62
ptldor63
ptldor64
ptldor68
ptldor69
ptldor70
spknwa80
rcmdva80
rcmdva60
rcmdva61
rcmdva62
rcmdva63
rcmdva64
rcmdva65
rcmdva66
rcmdva67
rcmdva68
rcpknj60
rcpknj61
rcpknj62
rlghnc60
rlghnc61
rlghnc62
rlghnc63
rlghnc64
rlghnc65
rlghnc81
rlghnc66
rlghnc67
rlghnc68
rlghnc69
rlghnc70
rlghnc71
rlmdil60
rlmdil61
rlmdil62
rlmdil63rlmdil64
rlmdil65
rlmdil66
rlmdil67
rlmdil68
rlmdil69
rlmdil70
rlmdil71
scrmca60
scrmca61
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scrmca63
scrmca64
scrmca65
scrmca81
scrmca66
scrmca67
scrmca68
scrmca69
scrmca70
scrmca71
snjsca81
shokca60
shokca61
shokca62
shokca63
shokca64
shokca65
shokca81
shokca66
shokca67
shokca68
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shokca71
slkcut60
slkcut61
slkcut62
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slkcut64
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slkcut67
slkcut68
slspmd60
slspmd61
slspmd62
snantx60
snantx61
snantx62
snantx63
snantx64
snantx65
sndgca60
sndgca61
sndgca62
sndgca63
sndgca64
sndgca65
sndgca66
sndgca67
sndgca68
snfcca10
snfcca80
snfcca12
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snfcca25snfcca26
snfcca27snfcca28
snfcca29
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snfcca31
snfcca32
snfcca70
snfcca71
snfcca72
snfcca83
sttlwa81
snfcca67
snfcca68
snfcca69
snfcca60
snfcca61
snfcca62
snfcca63
snfcca64
snfcca65
snfcca73
snfcca74
snfcca75
snfpca60snfpca61
snfpca62
snfpca63
snfpca64
snfpca65
snfpca66
snfpca67
snfpca68
snfpca69
snfpca70
snjsca60
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snjsca63
snjsca64
snjsca65
snjsca66
snjsca67
snjsca68
snjsca69
snjsca70
snjsca71
snjsca72
snjsca73
snjsca74
spknwa60
spknwa61
spknwa62
stlsmo11
stlsmo80
stlsmo12
stlsmo13
stlsmo14
stlsmo65
stlsmo66
stlsmo67 stlsmo60
stlsmo61
stlsmo62
stlsmo63
stlsmo64
stlsmo68
stlsmo69
stlsmo70
stlsmo71
stlsmo72
stlsmo73
stlsmo74
stlsmo75
stlsmo76
stplmn06
stplmn80
stplmn07
stplmn08
stplmn09
stplmn67
stplmn68
stplmn69
stplmn60
stplmn61
stplmn62
stplmn63
stplmn64
stplmn65
stplmn70
stplmn71
stplmn72
sttlwa60 sttlwa61
sttlwa62
sttlwa63
sttlwa64
sttlwa65
sttlwa66
sttlwa67
sttlwa68
sttlwa69
sttlwa70
sttlwa71
sttlwa72
sttlwa73
sttlwa74
sttlwa75
sttlwa76
syrcny60
syrcny61
syrcny62
tampfl60
tampfl61
tampfl62
tampfl63
tampfl64
tampfl65
tampfl66
tampfl67
tampfl68
tampfl69
tampfl70
tampfl71
tcsnaz60
tcsnaz61
tcsnaz62
tcsnaz63
tcsnaz64
tcsnaz65
tulsok60
tulsok61
tulsok62
tulsok63
tulsok64
tulsok65
tulsok66
tulsok67
tulsok68
washdc11
washdc12
washdc15
washdc16
washdc17
washdc18
washdc65
washdc66washdc67
washdc60
washdc61
washdc62
washdc63
washdc64
washdc68
washdc69
washdc70
washdc71
washdc72
washdc73
washdc74
washdc75
washdc76
washdt60
washdt61
washdt62
waynpa60
waynpa61
waynpa62
waynpa63
waynpa64
waynpa65
waynpa66
waynpa67
waynpa68
whplny60
whplny61
whplny62
whplny63
whplny64
whplny65
whplny66
whplny67
whplny68
whplny69
whplny70
whplny71
wpbhfl60
wpbhfl61
wpbhfl62
wpbhfl63
wpbhfl64
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)
Bill Scannell
Werner Koch
J. Daniel Backman
Andrew S. Nagy
Red Hat, Inc
CERT Coordination Center
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
CERT Coordination Center
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
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)
Bill Scannell
Werner Koch
J. Daniel Backman
Andrew S. Nagy
Red Hat, Inc
CERT Coordination Center
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 MeskesNoah 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
CERT Coordination Center
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
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. KuhnDan 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)
Bill Scannell
Werner Koch
J. Daniel Backman
Andrew S. Nagy
Red Hat, Inc
CERT Coordination Center
K. Reid Wightman
Mark Crichton (preferred key)
Emily ManheimTheodore Y. Ts’o
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
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
CERT Coordination Center
Eugene H. Spafford
Jamie Wilkinson
Leland Wallace
Brian D. Webb
David Walter
FreeBSD Security Officer
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
68 Gansner and Hu E!cient Node Overlap Removal
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
70 Gansner and Hu E!cient Node Overlap Removal
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
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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.
72 Gansner and Hu E!cient Node Overlap Removal
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