Efficient Node Overlap Removal Using a Proximity Stress Model

Efﬁcient Node Overlap Removal Using a Proximity Stress Model

Emden R. Gansner and Yifan Hu

AT&T Labs, Shannon Laboratory, 180 Park Ave., Florham Park, NJ 07932 {erg,yifanhu}@research.att.com

Abstract. When drawing graphs whose nodes contain text or graphics, the nontrivial node sizes must be taken into account, either as part of the initial layout or as a post-processing step. The core problem is to avoid overlaps while retaining the structural information inherent in a layout using little additional area. This paper presents a new node overlap removal algorithm that does well by these measures.

1 Introduction

Most existing symmetric graph layout algorithms treat nodes as points. In practice, nodes usually contain labels or graphics that need to be displayed. Naively incorporating this can lead to nodes that overlap, causing information of one node to occlude that of others. If we assume that the original layout conveys signiﬁcant aggregate information such as clusters, the goal of any layout that avoids overlaps should be to retain the “shape” of the layout based on point nodes. The simplest and, in some sense, the best solution is to scale up the drawing [23] while preserving the node size until the nodes no longer overlap. This has the advantage of preserving the shape of the layout exactly, but can lead to inconveniently large drawings. In general, overlap removal is typically a trade-off between preserving the shape and limiting the area, with scaling at one extreme. Many techniques to avoid overlapping nodes have been devised. One approach is to make the node size part of the model of the layout algorithm. It is assumed that whatever structure that would have been exposed using point nodes will still be evident in these more general layouts. Various authors [2,13,21,26] have extended the spring-electrical model [4,7] to take into account node sizes, usually as increased repulsive forces. Node overlap removal can also be built into the stress model [19] by specifying the ideal edge length to avoid overlap along the graph edges. Such heuristics, however, cannot guarantee all overlaps will be removed, so they rely on overly large repulsive forces, or the type of post-processing step considered next. An alternative approach is to remove overlaps as a post-processing step after the graph is laid out. Here the trade-off between layout size and preserving the graph’s shape is more explicit. A number of such algorithms have been proposed. For example, the Voronoi cluster busting algorithm [10, 22] works by iteratively forming a Voronoi diagram from the current layout and moving each node to the center of its Voronoi cell

I.G. Tollis and M. Patrignani (Eds.): GD 2008, LNCS 5417, pp. 206–217, 2009. c Springer-Verlag Berlin Heidelberg 2009 Efﬁcient Node Overlap Removal Using a Proximity Stress Model 207 until no overlaps remain. Although roughly maintaining relative node positions, the overall affect is to lose much of the layout structure. Another group of post-processing algorithms is based on maintaining the orthogonal ordering [25] of the initial layout as a way to preserve its shape. A force scan algorithm and variants were proposed [14, 17, 21, 25] based on these constraints. More recently, Marriott et al. [3,23] have presented a quadratic programming algorithm which removes node overlaps while minimizing node displacement and keeping the orthogonal ordering. An orthogonal ordering invariant is fairly effective at preserving structure, but it still cannot ensure that relative proximity relations between nodes are preserved, while at other times, it is too restrictive. Also, some of these algorithm require, in practice, separate horizontal and vertical passes which often results in a layout with a distorted aspect ratio (e.g., Fig. 2, bottom right). In this paper, we discuss (Sect. 2) metrics for the similarity between two layouts which we believe better quantiﬁes the desired outcome of overlap removal than min- imized displacement or such simpler measures as aspect ratio or edge ratio. We then present (Sect. 3) a node overlap removal algorithm based on a proximity graph of the nodes in the original layout. In Sect. 4, we evaluate our algorithm and others using the proposed similarity measures. In the following, 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 and edges, respectively. We let xi represent the current coordinates of vertex i in Euclidean space.

2 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 area taken by the drawing after overlap removal. The second aspect is the change in relative positions. Here we want the new drawing to be as “close” to the original as possible. It is this aspect that is hard to quantify. One way to measure the similarity of two layouts is to measure the distance between all pairs of vertices in the original and the new layout. If the two layouts are similar, then these distances should match, subject to scaling. This is known as Frobenius metric in the sensor localization problem [5]. However, calculating all pairwise distances is expensive for large graphs, both in CPU time and in the amount of memory, so instead we form a Delaunay triangulation (DT) of the original graph, then measure the distance between vertices along the edges of the triangulation for the original and new layouts. 0 If x and x denote the original and the new layout, and EP is the set of edges in the triangulation, we calculate the ratio of the edge length

xi − xj rij = 0 0 , {i, j}∈EP , xi − xj then deﬁne a measure of the dissimilarity as the normalized standard deviation 208 E.R. Gansner and Y. Hu

(r −r¯)2 {i,j}∈EP ij |EP | σ (x0,x)= , dist r¯ where 1 r¯ = rij |EP | {i,j}∈EP 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 original graph is not rigid, then even if two layouts of the same graph have the same edge lengths, they could be completely different. For example, think of the graph 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, but are clearly different. The rigidity of the triangulation avoids this problem. 0 Notice that σdist(x ,x) is not symmetric with regard to which layout comes first. Fur- thermore, in theory, this non-symmetric version could class a layout and a foldover of it (e.g., a square grid with one half folded over the other) as the same. We can symmetrize 0 0 0 it by defining the dissimilarity between layout x and x as (σdist(x ,x)+σdist(x, x ))/2. This also resolves the “foldover problem”. The symmetric version may be more appro- priate if we are comparing two unrelated layouts. Since, however, we are comparing a layout derived from an existing layout, we feel that the asymmetric version is adequate. An alternative measure of similarity is to calculate the displacement of vertices of the new layout from the original layout [3]. Clearly a new layout derived from a shift, scaling and rotation should be considered identical. Therefore we modify the straight displacement calculation by discounting the aforementioned transformations. This is achieved by finding the optimal scaling, shift and rotation that minimize the displacement. The optimal displacement is then a measure of dissimilarity. We define the displacement dissimilarity as 0 0 2 σdisp(x ,x)=minp∈R2,θ,r∈R rT xi + p − xi , (1) i∈V where r is the scaling, θ the rotation with T = T (θ) its rotation matrix, and p ∈ R2 is the translation. Solving this is a known problem in Procrustes analysis [1, 11] and the solution (the Procrustes statistic) is

T T 1 0 0 0 T 0 0 2 2 T σdisp(x ,x)=Tr(X X ) − (Tr((X X X X) ) Tr(X X), (2) 0 0 0 where X is a matrix with columns xi − x¯, X is a matrix with columns xi − x¯ ,and x¯ and x¯0 are the centers of gravity of the new and original layout. In the above we do not consider shearing, since we believe a layout derived from shearing of the original should not be considered identical to the latter.

3 A Proximity Stress Model for Node Overlap Removal

Our goal now is to remove overlaps while preserving the shape of the initial layout by maintaining the proximity relations. To do this, we ﬁrst set up a rigid “scaffolding” Efﬁcient Node Overlap Removal Using a Proximity Stress Model 209 structure so that while vertices can move around, their relative positions are maintained. This scaffolding is constructed using a proximity graph [18]. Here again, we work with the Delaunay triangulation. Once we form a DT, we check every edge in it and see if there are any node overlaps 0 along that edge. Let wi and hi denote the half width and height of the node i,andxi (1) 0 and xi (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 two nodes wi + wj hi + hj tij = max min 0 0 , 0 0 , 1 . (3) |xi (1) − xj (1)| |xi (2) − xj (2)|

For nodes that do not overlap, tij =1. For nodes that do overlap, such overlaps can be removed if we expand the edge by this factor. Therefore we want to generate a layout 0 0 such that an edge in the proximity graph has the ideal edge length close to tijxi − xj . In other words, we want to minimize the following stress function 2 wij (xi − xj−dij ) . (4)

(i,j)∈EP

0 0 Here dij = sijxi − xj is the ideal distance for the edge {i, j}, sij is a scaling factor 2 related to the overlap factor tij (see (6)), wij =1/||dij|| is a scaling factor, and EP is the set of edges of the proximity graph. We call (4) the proximity stress model in obvious analogy with the standard stress model [19] 2 wij (xi − xj−dij) , (5) i=j where dij is the graph theoretical distance between vertices i and j,andwij is a weight 2 factor, typically 1/dij . Because DT is a planar graph, which has no more than 3|V |−3 edges, the above stress function has no more than 3|V |−3 terms. Furthermore, because DT is rigid, it provides a good scaffolding that constrains the relative position of 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 iteration by using the above model with sij = tij. Imagine the situation of a regular mesh graph, with one node i of particularly large size that overlaps badly with its nearby nodes, but the other nodes do not overlap with each other. Suppose nodes i and j form an edge in the proximity 0 0 graph, and they overlap. If we try to make the length of the edge equal tijxi − xj ,we will ﬁnd that tij is a number much larger than 1, and the optimum solution to the stress model is to keep all the other vertices at or close to their current positions, but move the large node i outside of the mesh, at a position that does not cause overlap. This is not desirable because it destroys the original layout. Therefore we damp the overlap factor by setting sij = min(tij,smax) (6) and try to remove overlaps a little at a time. Here smax > 1 is a number limiting the amount of overlap we are allowed to remove in one iteration. We found that smax =1.5 works well. 210 E.R. Gansner and Y. Hu

8 7 6 8 7 6

9 5 9 5

11 11

10 4 with an extremely looong label 10 4 with an extremely looong label 2 with a 2 with a 1 tall 3 1 tall 3 label label

Fig. 1. (a): A graph layout where nodes 2 and 4 overlap. (b): the proximity graph (Delaunay triangulation) of the current layout. No two nodes linked by an edge of the proximity graph overlap.

After minimizing (4), we arrive at a layout that may still have node overlaps. We then regenerate the proximity graph using DT and calculate the overlap factor along the edges of this graph, and redo the minimization. This forms an iterative process that ends when there are no more overlaps along the edges of the proximity graph. For many graphs, the above algorithm yields a drawing that is free of node overlaps. For some graphs, however, especially those with nodes having extreme aspect ratios, node overlaps may still occur. Such overlaps happen for pairs of nodes that are not near each other, and thus do not constitute edges of the proximity graph. Fig. 1(a) shows the drawing of a graph after minimizing (4) iteratively, so that no more node overlap is found along the edges of the Delaunay triangulation. Clearly, node 2 and node 4 still overlap. If we plot the Delaunay triangulation (Fig. 1(b)), it is seen that nodes 2 and 4 are not neighbors in the proximity graph, which explains the overlap. To overcome this situation, once the above iterative process has converged so that no more overlaps are detected over the DT edges, we apply a scan-line algorithm [3] to find all overlaps, and augment the proximity graph with additional edges, where each edge consists of a pair of nodes that overlap. We then re-solve (4). This process is repeated until the scan-line algorithm finds no more overlaps. We call this algorithm PRISM (PRoxImity Stress Model). Concerning its complexity, Delaunay triangulation can be computed in O(|V |log(|V |)) time [6, 12, 20]. The scan- line algorithm can be implemented to find all the overlaps in O(l|V |(log|V | + l)) time [3], where l is the number of overlaps. Because we only apply the scan-line algorithm after no more node overlaps are found along edges of the proximity graph, l is usually a very small number, hence this step can be considered as taking time O(|V |log|V |). The proximity stress model (4), like the standard stress model (5), can be solved using the stress majorization technique [8] with a conjugate gradient algorithm. Because we use DT as our proximity graph and it has no more than 3|V |−3 edges, each iteration of the conjugate gradient algorithm takes a time of O(|V |). Overall, therefore, PRISM takes O(t(mk|V | + |V |log|V |)) time, where t is the total number of iterations in the two main loops, m is the average number of stress majorization iterations, and k the average number of iterations for the conjugate gradient algorithm. Efficient Node Overlap Removal Using a Proximity Stress Model 211

4 Numerical Results

To evaluate the PRISM algorithm and other overlap removal algorithms, we apply them as a post-processing step to a selection of graphs from the Graphviz [9] test suite. This suite, part of the Graphviz source distribution, contains many graphs from users. As such, these are good examples of the kind of graphs actually being drawn. Our baseline algorithm is Scalable Force Directed Placement (SFDP) [16], a multi- level, spring-electrical algorithm. Using the layout of SFDP, we then apply one of the overlap removal algorithms to get a new layout that has no node overlaps, and compare the new layout with the original in terms of dissimilarity and area. In Table 1, we list the 14 test graphs, the number of vertices and edges, as well as CPU time1 for PRISM and three other overlap removal algorithms. The graphs are selected randomly with the criteria that a graph chosen should be connected, and is of relatively large size. We compared PRISM with an implementation in Graphviz of the solve VPSC algorithm [3]2, hereafter denoted as VPSC, as well as VORO, the Voronoi cluster busting algorithm [10,22]. The ﬁnal algorithm is the ODNLS algorithm of Li et al. [21], which relies on varied edge lengths in a spring embedder. 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 large graphs 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 overlap removal algorithms. The smaller the dissimilarities and area, the better. The ODNLS algorithm performs best in terms of smaller dissimilarity, followed by PRISM, VPSC and VORO. In terms of area, PRISM and VPSC are pretty close, and both are better than ODNLS and VORO, which can give extremely large drawings. Indeed, in terms of area, scaling outperformed ODNLS and VORO in 20%-30% of the examples. Comparing PRISM with VPSC, Table 2 shows that PRISM gives smaller dissimilarities most of the time. The two dissimilarity measures, σdist and σdisp, are generally correlated, except for ngk10 4 and root. Based on σdist, VPSC is better for these two graphs, while based on σdisp, PRISM is better. The ﬁrst row in Fig. 2 shows the original layout of ngk10 4, as well as the result after applying PRISM and VPSC. Through visual inspection, we can see that PRISM preserved the proximity relations of the original layout well. VPSC “packed” the labels more tightly, but it tends to line up vertices horizontally and vertically, and also produces a layout with aspect ratio quite different from the original graph. It seems that σdist is not as sensitive in detecting dif- ferences in aspect ratio. This is evident in drawings of the root graph (Fig. 2, second row). VPSC clearly produced a drawing that is overly stretched in the vertical direction, but its σdist is actually smaller than that of PRISM! Consequently, we conclude that σdisp 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 works in the vertical and 1 All timings were derived on a 4 processor, 3.2 GHz Intel Xeon CPU, with 8.16 GB of memory, running Linux. 2 A stand alone version of solve VPSC by the authors of this algorithm has also been tried but was found to offer no advantage over VPSC. VPSC itself was also contributed originally by the same authors to Graphviz. 212 E.R. Gansner and Y. Hu

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

Graph |V | |E| PRISM VPSC VORO ODNLS b100 1463 5806 1.44 14.85 350.7 258.9 b102 302 611 0.14 0.10 4.36 5.7 b124 79 281 0.03 0.01 0.02 0.5 b143 135 366 0.04 0.01 0.47 1.3 badvoro 1235 1616 0.54 71.15 351.51 73.6 mode 213 269 0.09 0.09 2.15 2.1 ngk10 4 50 100 0.01 0.00 0.02 0.14 NaN 76 121 0.01 0.01 0.11 0.27 dpd 36 108 0.01 0.01 0.02 0.1 root 1054 1083 0.89 7.81 398.49 46.9 rowe 43 68 0.00 0.00 0.04 0.1 size 47 55 0.01 0.00 0.06 0.09 unix 41 49 0.01 0.00 0.04 0.07 xx 302 611 0.13 0.10 8.19 5.67

Table 2. Comparing the dissimilarities and area of overlap removal algorithms. Results shown 6 are σdist, σdisp and area. Area is measured with a unit of 10 square points. Initially the layout is scaled to an average length of 1 inch.

Graph PRISM VPSC VORO ODNLS σdist σdisp area σdist σdisp area σdist σdisp area σdist σdisp area b100 0.74 0.38 14.05 0.76 0.72 18.91 - - - 0.33 0.20 1.02E3 b102 0.44 0.25 2.45 0.58 0.8 2.71 0.8 0.3 31.79 0.30 0.16 53.13 b124 0.65 0.37 1.04 0.78 0.73 0.91 0.86 0.39 13.42 0.33 0.19 14.79 b143 0.59 0.35 1.5 0.78 0.83 2.16 0.99 0.45 22.91 0.49 0.34 23.79 badvoro 0.34 0.15 12.58 0.61 0.75 13.85 2.29 0.65 3.01E3 0.31 0.26 318.66 mode 0.59 0.37 0.79 1.02 0.77 1.29 0.97 0.54 10.84 0.38 0.27 49.45 ngk10 4 0.41 0.16 0.33 0.39 0.3 0.25 0.48 0.26 0.52 0.22 0.13 2.30 NaN 0.4 0.2 0.72 0.54 0.65 0.71 0.56 0.28 5.04 0.26 0.15 5.10 dpd 0.34 0.18 0.25 0.51 0.4 0.18 0.48 0.32 0.45 0.37 0.29 1.30 root 0.71 0.3 16.99 0.6 0.75 17.68 4.09 0.94 6.93E9 0.29 0.22 950.01 rowe 0.33 0.14 0.22 0.44 0.31 0.19 0.49 0.26 0.95 0.27 0.12 2.10 size 0.37 0.2 0.47 0.77 0.74 0.4 0.62 0.35 1.27 0.32 0.20 4.14 unix 0.39 0.23 0.39 0.51 0.67 0.36 0.6 0.35 0.85 0.26 0.13 2.35 xx 0.42 0.25 3.96 0.57 0.82 3.9 0.97 0.34 58.83 0.29 0.14 74.00 horizontal directions alternatively, each time trying to remove overlaps while minimizing displacement. As a result, when starting from a layout with severe node overlaps, it may move vertices signiﬁcantly along one direction to resolve the overlaps, creating Efﬁcient Node Overlap Removal Using a Proximity Stress Model 213

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81dabfaba8 c1c30f30d40c4f1f84924622f

81dabfaba8 81dabfaba8

9937ccba1469 08dade990b2282 6c632ad9c42228bb337 11bb8ed3ae227d3acefc 84e4f1c582427a98d7b cd0d985a366cad7e 0f940646

ea890ccca2f7c2107351 f0bd1521 dca32af03698c988b22 3b2f08d52e4bca3f9ca7bbbd6

6c632ad9c42228bb337

1d163eac141def176461c

f78e145a127014eb43345a0c 503737b64d432c60d6ac557e0e6 dca32af03698c988b22

84e4f1c582427a98d7b

18115fa32ff1cb99 ea890ccca2f7c2107351 18115fa32ff1cb99

fb039d7a2a9fe73b5f468eba9

a27037332d9fa5c43bcfe94c0

0c72a426929600000f5 0c72a426929600000f5

3cdc90c57243373efaba65a

173fd00917644f0f1f3e3

24431c3eb075102f07cc2c1be

173fd00917644f0f1f3e3 e3bdbca0e2256fffa8a59018

75ba8d840070942eb4e737849

07f8a9e55a16beddb3c9153b0

9c9e2e0f2da8758e436c 4a38abc630c82b0c48dfbf5271

86276bb1e23f2c7ffcbe82a0 root root+PRISM root+VPSC

Fig. 2. Divergence of dissimilarity measures: for both graphs, σdist estimates that VPSC gives layout closer to the original, while σdisp predicts the opposite

hstntx72 hstntx67

hstntx75 hstntx69

hstntx70 hstntx23 brhmal71

hstntx16 hstntx68 brhmal67 tampfl66

hstntx73 hstntx17 hstntx22 brhmal66 tampfl70

hstntx71 snantx62 nsvltn66 nsvltn68 tampfl71 ojusfl71

hstntx74 brhmal72 nworla65 nsvltn67 tampfl69 ojusfl66 ojusfl67

snantx64 brhmal73 nworla64 brhmal68 orlnfl72 ojusfl70

snantx63 brhmal74 nworla62 brhmal70 orlnfl74 ojusfl69

snantx65 brhmal75 nworla63 brhmal69 tampfl68 miamfl60 ojusfl68

hstntx65 brhmal77 nworla68 tampfl67 orlnfl73 orlnfl70 ojusfl81

hstntx64 hstntx80 nworla60 orlnfl71 miamfl62

hstntx60 nworla61 nworla66 orlnfl75 miamfl61

hstntx62 nsvltn81 nworla67 jcvlfl68 jcvlfl66 ojusfl64

dllstx24 brhmal76 brhmal64 brhmal60 tampfl81 orlnfl82 ojusfl61

dllstx72 hstntx61 lsvlky66 brhmal81 tampfl60 miamfl80

snantx80 hstntx63 lsvlky68 brhmal63 tampfl64 jcvlfl67

dllstx70 lsvlky69 brhmal62 tampfl62

ftwotx64 hstntx82 brhmal82 brhmal65 tampfl63

ftwotx61 snantx61 nworla80 lsvlky67 brhmal61 wpbhfl64

ftwotx60 snantx60 lsvlky71 nworla81 orlnfl61 wpbhfl65

stlsmo72 stlsmo76 dllstx71 lsvlky70 orlnfl65 wpbhfl60 hstntx72hstntx75 ojusfl66ojusfl71 ftwotx80 stlsmo69 dllstx25 dllstx67 orlnfl64 ojusfl60 hstntx73 ojusfl67 austtx61 stlsmo73 dllstx22 dllstx28 nsvltn61 orlnfl63 orlnfl78 tampfl65 ojusfl65 wpbhfl61 hstntx70 ojusfl70 austtx64 stlsmo68 hstntx81 nsvltn62 nsvltn63 orlnfl77 tampfl61 ojusfl63 wpbhfl63 hstntx74hstntx71 ftwotx62 austtx60 stlsmo70 dllstx68 dllstx21 nsvltn64 brhmal80 jcvlfl81 ojusfl80 ojusfl62 wpbhfl62 hstntx67 tampfl70tampfl66 ojusfl68ojusfl69 ftwotx63 austtx65 stlsmo74 dllstx29 nsvltn60 nsvltn65 atlnga33 jcvlfl61 wpbhfl80 hstntx16 hstntx69 orlnfl72 ftwotx65 austtx63 stlsmo75 dllstx69 mmphtn65 atlnga32 jcvlfl63 hstntx23 dnvrco79 okcyok61 stlsmo71 dllstx80 mmphtn62 atlnga76 jcvlfl62 orlnfl16 hstntx68 brhmal71 tampfl69tampfl71tampfl68orlnfl73 hstntx17 hstntx22 brhmal67 orlnfl74miamfl60 dnvrco77 dnvrco66 austtx62 stlsmo82 mmphtn63 orlnfl60 atlnga31 jcvlfl60 orlnfl17 brhmal68 tampfl67 dnvrco70 dnvrco67 okcyok62 austtx80 dllstx31 stlsmo64 atlnga25 atlnga75 rlghnc67 nsvltn66nsvltn68brhmal66brhmal70 ojusfl81 orlnfl71 dnvrco69 okcyok60 stlsmo12 stlsmo63 orlnfl76 tampfl80 jcvlfl64 rlghnc70 nsvltn67 orlnfl75 hstntx82 brhmal69 dnvrco78 sndgca60 dllstx76 mmphtn64 orlnfl62 atlnga30 jcvlfl65 orlnfl80 ftwotx64 snantx64snantx62 orlnfl70miamfl62 snantx63 brhmal72nworla65 phnxaz10 sndgca65 stlsmo11 stlsmo62 mmphtn61 mmphtn60 chrlnc64 atlnga77 rlghnc69 ftwotx62ftwotx61ftwotx60 snantx61 brhmal73nworla64 jcvlfl68 miamfl61 dnvrco65 sndgca62 tcsnaz63 stlsmo14 dllstx74 lsvlky81 clmasc61 jcvlfl80 rlghnc71 orlnfl15 snantx60snantx65 brhmal74 nworla68 jcvlfl66 phnxaz13 sndgca61 tcsnaz62 stlsmo67 stlsmo60 atlnga71 chrlnc69 chrlnc71 rlghnc66 orlnfl68 nworla66 anhmca69 phnxaz12 tcsnaz64 dllstx77 atlnga56 atlnga70 orlnfl81 atlnga80 chrlnc60 rlghnc61 orlnfl67 ftwotx63ftwotx65 brhmal75nworla63nworla62nworla60nworla67 tampfl81 ojusfl64 wpbhfl64 hstntx80brhmal77 ojusfl61 wpbhfl60wpbhfl65 anhmca70 dnvrco68 phnxaz67 tcsnaz65 stlsmo61 atlnga64 atlnga79 clmasc60 chrlnc70 rlghnc62 orlnfl13 nworla61 anhmca67 phnxaz11 sndgca64 okcyok80 stlsmo65 dllstx81 atlnga69 chrlnc65 rlghnc81 orlnfl69 brhmal76 jcvlfl67orlnfl82miamfl80 ojusfl60 hstntx65 anhmca65 phnxaz69 stlsmo13 dllstx73 atlnga59 atlnga78 atlnga16 chrlnc61 rlghnc68 stlsmo76hstntx64 brhmal81brhmal60tampfl64tampfl60 ojusfl65ojusfl63wpbhfl61wpbhfl63 stlsmo72 hstntx60 nsvltn81 brhmal63brhmal62 tampfl62 wpbhfl62 anhmca66 phnxaz68 tcsnaz61 tulsok60 stlsmo66 dllstx78 atlnga65 atlnga58 atlnga34 clmasc62 rlghnc64 hstntx61 brhmal64 anhmca63 sndgca63 tcsnaz60 tulsok67 dllstx55 nsvltn80 atlnga35 atlnga74 rlghnc63 orlnfl08 stlsmo73stlsmo69hstntx62dllstx24hstntx63 brhmal65 tampfl63tampfl65 ftwotx80austtx61stlsmo68 dllstx72 brhmal61 tampfl61 ojusfl62 anhmca60 dnvrco81 tulsok64 dllstx59 dllstx75 mmphtn80 clmasc80 rlghnc60 nybwny75 snantx80dllstx70 anhmca62 sndgca80 dllstx62 atlnga57 atlnga63 atlnga09 rlghnc65 hmsqnj61 nybwny71 stlsmo70stlsmo74 lsvlky69lsvlky66lsvlky68 jcvlfl81 wpbhfl80 dllstx71 anhmca73 anhmca61 phnxaz80 dllstx56 stlsmo81 cncnoh63 chrlnc66 chrlnc62 washdc65 hmsqnj60 nybwny69 austtx60austtx64 dllstx25 lsvlky67 orlnfl16 hstntx67 brhmal82 orlnfl61 anhmca72 anhmca64 tulsok61 tulsok65 stlsmo80 dllstx58 atlnga62 atlnga36 chrlnc63 washdc12 nybwny67 nworla80 ojusfl80 orlnfl15 hstntx69 stlsmo75 lsvlky71lsvlky70nworla81 orlnfl17orlnfl68 hstntx16 austtx65stlsmo71 orlnfl65 atlnga33 hstntx23 anhmca78 anhmca81 tulsok68 dllstx61 atlnga83 chrlnc80 rlghnc80 washdc15 hmsqnj62 nybwny81 hstntx68 austtx63 jcvlfl61 hstntx72 brhmal67 tampfl66 hstntx75 brhmal71 tampfl70 orlnfl72 dllstx67 orlnfl78atlnga32 orlnfl08 hstntx17 anhmca75 anhmca68 shokca65 tcsnaz80 dllstx79 cncnoh64 chrlnc68 gnbonc60 washdc16 nybwny73 dllstx22 orlnfl64 atlnga76jcvlfl63 orlnfl67 hstntx22 austtx62 dllstx28 orlnfl63 nsvltn68 okcyok61 orlnfl77 jcvlfl62 anhmca76 sndgca67 sndgca81 kscymo21 dllstx57 dllstx83 cncnoh61 atlnga67 atlnga72 phlapa72 phlapa70 washdc67 washdc66 atlnga31 brhmal66 tampfl68 orlnfl73 orlnfl60 jcvlfl60 hstntx73 hstntx82 hstntx70 nsvltn66 tampfl69 tampfl71 orlnfl74 miamfl60 ojusfl81 dllstx68 orlnfl13 snantx62 dnvrco79dnvrco66 hstntx81 nsvltn62nsvltn61nsvltn63 tampfl80atlnga25 brhmal72 brhmal68 anhmca77 sndgca68 shokca60 tulsok63 kscymo18 dllstx65 atlnga68 chrlnc67 gnbonc61 phlapa71 hmsqnj80 washdc18 dnvrco67 dllstx29 dllstx21 atlnga75 rlghnc67 snantx64 nsvltn67 brhmal70 okcyok62 dllstx69 rlghnc70 orlnfl71 orlnfl75 dnvrco77 brhmal80 hstntx71 nworla65 tampfl67 orlnfl70 anhmca71 anhmca82 shokca63 tulsok66 phnxaz63 dllstx66 atlnga61 atlnga81 atlnga73 gnbonc62 phlapa75 washdc11 dnvrco70 dllstx80 orlnfl76 atlnga30 hstntx74 brhmal74 nworla64 nsvltn60nsvltn64 jcvlfl64jcvlfl65 orlnfl69 snantx61 brhmal73 dnvrco69 okcyok60 stlsmo82 orlnfl62 snantx63 brhmal69 miamfl62 nsvltn65 nworla68 anhmca79 sndgca66 anhmca80 tulsok62 kscymo67 kscymo63 dllstx60 atlnga60 atlnga37 phlapa73 nwrknj76 mmphtn65 orlnfl80 ojusfl66 dllstx31 atlnga77 jcvlfl68 ojusfl71 rlghnc69 brhmal75 nworla62 nworla60 dnvrco78 sndgca60sndgca65 snantx60 snantx65 nworla63 tampfl81 anhmca74 snfpca66 lsanca29 phnxaz64 kscymo23 mmphtn71 dllstx63 washdc61 phlapa78 phlapa74 nwrknj78 phnxaz10 rlghnc71 nworla66 nworla67 jcvlfl66 miamfl61 anhmca73 austtx80 mmphtn62 mmphtn64 chrlnc64 ftwotx64 ojusfl70 anhmca72 tcsnaz63 stlsmo63stlsmo64mmphtn63mmphtn61lsvlky81mmphtn60 ojusfl67 okldca67 snfpca67 shokca62 lsanca71 phnxaz60 tulsok80 kscymo64 atlnga66 chrlnc81 phlapa76 phlapa77 nwrknj58 anhmca69 dnvrco65 hstntx65 hstntx80 brhmal77 nworla61 tcsnaz62 stlsmo12 rlghnc66 tampfl60 wpbhfl64 dnvrco68 sndgca62 clmasc61 stlsmo76 brhmal60 anhmca70 phnxaz13 tcsnaz64 stlsmo72 hstntx64 brhmal81 snfpca68 grdnca62 lsanca27 phnxaz65 kscymo71 mmphtn70 dllstx82 cncnoh62 cncnoh60 gnbonc80 nwrknj72 washdc17 anhmca78 sndgca61 stlsmo11 dllstx76stlsmo62 chrlnc69jcvlfl80chrlnc71 orlnfl82 ojusfl64 orlnfl81 rlghnc61rlghnc68 tampfl64 jcvlfl67 ojusfl61 ojusfl69 ftwotx61 hstntx60 brhmal76 grdnca60 lsanca72 kscymo68 kscymo65 kscymo61 cncnoh81 phlapa58 washdc80 nybwny82 nybwny74 anhmca75 stlsmo14 dllstx74 atlnga71 atlnga80chrlnc60 nsvltn81 tampfl62 wpbhfl60 wpbhfl65 phnxaz12 stlsmo67 stlsmo60 stlsmo69 brhmal64 brhmal63 brhmal62 anhmca65anhmca67 phnxaz67 atlnga56atlnga64atlnga70 rlghnc62 stlsmo73 tcsnaz65okcyok80 chrlnc70 hstntx62 hstntx61 miamfl80 ojusfl68 snfpca65 shokca61 shokca80 kscymo73 kscymo62 kscymo79 atlnga82 phlapa79 nwrknj59 nybwny72 phnxaz11 sndgca64 stlsmo65 clmasc60 nybwny75 ftwotx62 hstntx63 tampfl63 ojusfl60 anhmca76 tcsnaz61 dllstx77 atlnga79 ftwotx80 stlsmo68 anhmca77 anhmca68anhmca66 stlsmo13 stlsmo61 rlghnc81 austtx61 dllstx24 lsvlky66 lsvlky68 tampfl61 tampfl65 dllstx81 brhmal65 brhmal61 wpbhfl61 okldca81 shokca64 lsanca19 kscymo69 kscymo81 lsvlky62 washdc64 phlapa82 nwrknj75 nwrknj57 nwrknj69 phnxaz69 dllstx73 atlnga69atlnga78 chrlnc65 rlghnc64 dllstx72 lsvlky69 anhmca71 tulsok60 dllstx78 atlnga59 clmasc62chrlnc61 rlghnc63 nybwny71 lsvlky67 ojusfl65 ojusfl63 tcsnaz60 stlsmo66 ftwotx63 ftwotx60 wpbhfl63 grdnca61 grdnca80 lsanca26 phnxaz62 kscymo80 kscymo76 cncnoh69 washdc63 nwrknj70 nwrknj79 nwrknj21 nybwny76 phnxaz68 atlnga65nsvltn80 atlnga16 jcvlfl81 ojusfl80 sndgca67 sndgca63 dllstx75 atlnga58 atlnga34 nybwny69 snantx80 orlnfl61 anhmca63 chrlnc62 rlghnc60 nybwny67 dllstx71 brhmal82 wpbhfl62 snjsca61 phnxaz70 dnvrco60 kscymo24 kscymo60 kscymo77 lsvlky60 washdc62 nwrknj73 nwrknj77 nwrknj67 nybwny70 ftwotx65 stlsmo74 dllstx70 lsvlky70 nworla81 anhmca79 dnvrco81 tulsok67 dllstx55 mmphtn80 austtx60 ojusfl62 anhmca60 atlnga63atlnga35 austtx64 stlsmo70 lsvlky71 orlnfl65 anhmca74 anhmca62 atlnga74 rlghnc65 nybwny76 dnvrco79 atlnga33 jcvlfl61 snfpca81 lsanca70 dnvrco62 phnxaz61 phnxaz81 mmphtn81 kscymo78 lsvlky61 cncnoh73 phlapa59 bltmmd66 nwrknj22 nybwny66 sndgca68 tulsok64 atlnga57 chrlnc63 hmsqnj61 dnvrco66 dllstx25 orlnfl78 okldca66 sndgca80 tulsok65 dllstx59 clmasc80 hmsqnj60 dllstx67 nworla80 nsvltn63 anhmca61 dllstx62 nybwny70 dnvrco67 dllstx22 dllstx28 atlnga76 jcvlfl63 wpbhfl80 orlnfl16 anhmca82 dllstx56 dnvrco77 austtx63 stlsmo75 atlnga32 snjsca60 phnxaz75 lsanca80 kscymo22 mmphtn69 kscymo58 chcgil63 lsvlky63 cncnoh72 washdc60 bltmmd64 nwrknj68 nybwny77 sndgca66 phnxaz80 atlnga09 austtx65 stlsmo71 orlnfl64 orlnfl15 okldca71 anhmca64 tulsok61 stlsmo81 dnvrco70 okcyok61 orlnfl77 atlnga31 jcvlfl62 dnvrco69 nsvltn61 rlghnc67 anhmca81 chrlnc66 orlnfl60 phnxaz74 dnvrco63 kscymo17 kscymo82 chcgil58 cncnoh71 phlapa83 nwrknj74 nwrknj55 nwrknj16 nybwny68 atlnga36 nsvltn62 okldca68 tulsok68tcsnaz80 dllstx58cncnoh63 chrlnc80 nybwny66 dllstx68 hstntx81 orlnfl63 jcvlfl60 rlghnc70 orlnfl17 orlnfl68 stlsmo80 atlnga62 rlghnc80 hmsqnj62 okcyok62 atlnga25 shokca65 dllstx57 dllstx61 washdc65washdc12 austtx62 dllstx29 atlnga75 dllstx79 dllstx69 dllstx21 orlnfl76 tampfl80 orlnfl08 lsanca28 dnvrco61 kscymo75 mmphtn67 kscymo83 lsvlky80 washdc81 phlapa61 nwrknj56 nwrknj23 kscymo21 dllstx83 atlnga83 nybwny81 nybwny77 dnvrco78 nsvltn64 jcvlfl65 okldca69 tulsok63 kscymo18 chrlnc68 sndgca60 atlnga30 sndgca81 tulsok66 cncnoh64 atlnga72 gnbonc60 anhmca70 phnxaz10 okcyok60 dllstx80 nsvltn60 mmphtn65 nsvltn65 okldca67 jcvlfl64 rlghnc69 orlnfl80 orlnfl67 dllstx65 washdc15 nybwny68 lsanca13 kscymo72 mmphtn68 chcgil60 cncnoh68 nycmny60 phlapa19 bltmmd68 nwrknj20 okldca66 shokca60 cncnoh61 atlnga67 atlnga73 phlapa72phlapa70 sndgca65 stlsmo82 brhmal80 snfpca70 snjsca62 snfpca67snfpca66 shokca63 dllstx66 washdc16 nybwny73 anhmca69 dnvrco65 austtx80 dllstx31 mmphtn64 orlnfl62 atlnga77 orlnfl13 tcsnaz63 snjsca80 phnxaz82 mmphtn66 kscymo59 chcgil79 cncnoh70 phlapa62 nwrknj83 bltmmd71 nwrknj80 rcmdva61 shokca62 tulsok62phnxaz63kscymo67 dllstx60 chrlnc64 snfpca69 snjsca64 anhmca80 atlnga37 anhmca65 dnvrco68 phnxaz13 stlsmo63 stlsmo64 mmphtn63 mmphtn62 mmphtn61 rlghnc61 rlghnc71 atlnga68 chrlnc67 gnbonc61 sndgca62 mmphtn60 rlghnc66 orlnfl69 okldca70 kscymo63dllstx63 atlnga66 atlnga81 phlapa71 anhmca73 tcsnaz64 tcsnaz62 stlsmo12 lsvlky81 okldca71 shokca61 mmphtn71 atlnga61 nybwny74 phnxaz73 dnvrco64 snfcca69 kscymo74 chcgil08 chcgil62 cncnoh82 phlapa18 bltmmd65 nybwny62 lsanca29 phnxaz64 gnbonc62phlapa75 sndgca61 dllstx76 stlsmo62 orlnfl81 slkcut63 okldca65 okldca68snfpca68 lsanca27lsanca71 cncnoh62 washdc67washdc66 stlsmo11 kscymo64 atlnga60 anhmca72 anhmca67 phnxaz67 dllstx74 kscymo23 hmsqnj80 stlsmo14 jcvlfl80 chrlnc71 rlghnc62 rlghnc68 nybwny75 snjsca63 snfcca29 dnvrco80 chcgil80 lsvlky65 nycmny61 phlapa23 nwrknj82 bltmmd63 bltmmd70 nybwny64 grdnca62 phnxaz60 tulsok80 dllstx82 phlapa73 washdc18 phnxaz12 chrlnc69 shokca66 okldca62 chcgil32 shokca64 phnxaz65kscymo71 kscymo61 chrlnc81 tcsnaz65 okcyok80 stlsmo67 stlsmo60 atlnga56 atlnga64 atlnga70 atlnga71 grdnca60 lsanca72 mmphtn70 cncnoh81cncnoh60 nybwny72 clmasc61 rlghnc64 shokca80 kscymo65 anhmca68 phnxaz11 dllstx77 snjsca65 phnxaz71 snfcca68 chcgil72 lsvlky64 nycmny64 phlapa63 phlapa68 bltmmd81 bltmmd69 nybwny61 rcmdva65 okldca69 phlapa74 sndgca64 tcsnaz61 shokca70 slkcut62 anhmca78 phlapa78 anhmca66 stlsmo61 hmsqnj61 hmsqnj60 snfpca65okldca81 kscymo68 atlnga82 washdc61 washdc11 phnxaz69 stlsmo65 atlnga79 clmasc60 chrlnc60 chrlnc70 rlghnc63 nybwny71 kscymo62 tulsok60 stlsmo13 dllstx78 atlnga59 atlnga69 okldca64 okldca80 phnxaz72 snfcca30 kscymo70 chcgil34 chcgil82 chcgil37 nycmny82 phlapa66 noc30k61 nybwny65 rcmdva64 lsanca19 phnxaz62 atlnga80 rlghnc60 nybwny81 shokca71 slkcut65 kscymo73 kscymo79 phlapa77 washdc17 dllstx81 dllstx73 rlghnc81 okldca70 kscymo69 phlapa76 nwrknj76nwrknj78 tcsnaz60 atlnga78 lsanca26 anhmca75 anhmca63 slkcut80 scrmca63 snfcca21 snfcca81 lsanca64 lsanca82 chcgil59 chcgil81 phlapa80 bltmmd62 rcmdva67 grdnca61 nybwny69 nybwny76 slkcut64 kscymo81 lsvlky62 washdc64 nwrknj69 phnxaz68 sndgca63 nsvltn80 atlnga58 rlghnc65 snjsca61snfpca81grdnca80phnxaz70 kscymo24kscymo80 kscymo76 cncnoh69 gnbonc80 nwrknj72nwrknj58 nybwny82 anhmca77 tulsok67 stlsmo66 dllstx55 atlnga65 chrlnc65 nybwny67 snfpca70 kscymo60 washdc63 atlnga16 atlnga34 clmasc62 washdc12 hmsqnj62 dnvrco60 okldca61 lsanca76 scrmca61 scrmca80 lsanca65 snfcca82 chcgil64 chcgil30 phlapa81 nwrknj71 bltmmd67 rcmdva63 phnxaz81 dllstx75 chrlnc61 shokca68 snfpca69 phnxaz61 mmphtn81 kscymo77lsvlky60 phlapa58 washdc80 dnvrco81 sndgca80 tulsok64 washdc65 lsvlky61 washdc62 phlapa79 nwrknj59 nwrknj21nwrknj67 anhmca76 sndgca67 anhmca62 anhmca60 atlnga63 nybwny73 dnvrco62 kscymo22 nwrknj75nwrknj57 anhmca61 tulsok65 dllstx56 dllstx62 atlnga57 mmphtn80 atlnga35 nybwny70 okldca60 lsanca78 lsanca83 snfcca67 snfcca32 lsanca63 chcgil33 chcgil65 nycmny65 phlapa65 bltmmd80 nybwny80 rcmdva80 rcmdva66 snjsca62snjsca60 lsanca70 lsanca80 kscymo78 cncnoh73 phlapa82 dllstx59 atlnga74 chrlnc63 shokca69 phnxaz75 mmphtn69 clmasc80 washdc16 washdc15 kscymo58 lsvlky63cncnoh72 anhmca71 phnxaz80 tulsok61 tulsok68 stlsmo81 rlghnc80 chrlnc62 kscymo82 shokca81 snbrca80 snfcca31 lsanca61 chcgil09 chcgil70 chcgil38 nycmny62 phlapa69 nwrknj64 bltmmd61 nybwny63 phnxaz74 kscymo75kscymo17 washdc60 nwrknj79 nwrknj68nwrknj22 sndgca68 anhmca64 anhmca81 washdc66 nybwny66 shokca67 dnvrco63 kscymo72 nwrknj77 anhmca82 tcsnaz80 dllstx61 cncnoh63 atlnga09 lsanca28 chcgil63 nwrknj70 dllstx57 stlsmo80 dllstx58 chrlnc66 chrlnc80 atlnga36 phlapa70 snjsca64 chcgil58 nwrknj73bltmmd66 sndgca66 kscymo21 atlnga62 washdc67 slkcut61 lsanca58 snfpca62 snfpca80 dnvrco82 lsanca81 chcgil71 chcgil61 chcgil69 nycmny63 washdc82 noc30k80 noc30k60 cmbrma61 rcpknj60 dnvrco61 mmphtn67 kscymo83 phlapa59 tulsok63 gnbonc60 phlapa72 lsvlky80 nwrknj16 tulsok66 kscymo18 dllstx79 hmsqnj80 nybwny77 okldca65 lsanca13 mmphtn68kscymo59 phlapa83 shokca65 sndgca81 anhmca80 washdc18 nybwny74 slkcut63 snjsca80 phnxaz82 cncnoh68cncnoh71washdc81 dllstx83 atlnga83 okldca63 lsanca75 scrmca65 snfcca20 lsanca68 lsanca62 chcgil35 chcgil29 chcgil36 phlapa27 nwrknj81 nycmny37 cmbrma60 rcmdva68 waynpa60 nycmny60phlapa61 bltmmd64 anhmca79 dllstx66 dllstx65 chrlnc68 phnxaz73 kscymo74mmphtn66 chcgil60 nwrknj74nwrknj56nwrknj55 nwrknj23 shokca60 cncnoh64 atlnga72 atlnga73 phlapa71 washdc11 okldca62 snjsca63 chcgil79 phlapa19 anhmca74 lsanca29 phnxaz63 atlnga67 phlapa75 snjsca65 snfcca69 chcgil08 bltmmd68 nwrknj20 tulsok62 cncnoh61 nwrknj76 slkcut60 lsanca74 snfpca63 scrmca62 dnvrco71 lsanca60 snfcca65 cncnoh80 phlapa60 phlapa67 nwrknj65 nybwny60 rcpknj61 waynpa62 dnvrco80 cncnoh70 shokca63 lsanca27 lsanca71 kscymo67 mmphtn71 gnbonc62 dnvrco64 nwrknj83 nwrknj80 rcmdva61 phnxaz64 gnbonc61 nwrknj78 washdc17 nybwny68 nybwny72 cncnoh82 bltmmd71 snfpca66 dllstx63 dllstx60 shokca62 kscymo63 atlnga81 slkcut62 kscymo70 chcgil62lsvlky65 phlapa18phlapa62 nybwny62 atlnga68 chrlnc67 atlnga37 phlapa73 slkcut81 lsanca73 lsanca79 scrmca64 snfcca75 lsanca67 snfcca62 cncnoh14 chcgil39 phlapa22 nwrknj62 cmbrma81 rcpknj80 rcmdva60 waynpa68 okldca64 okldca80phnxaz71phnxaz72 bltmmd65 cncnoh62 atlnga66 atlnga61 shokca66slkcut65 snfcca29 chcgil32 nwrknj82 snfpca67 phnxaz60 kscymo23 snfcca81 chcgil72chcgil80 lsvlky64 nybwny64 shokca61 lsanca72 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snfpca64 snfcca80 lsanca30 snfcca60 chcgil83 cncnoh66 washdc73 hrbgpa80 nycmny35 cmbrma65 waynpa65 whplny66 chcgil30 phlapa80 rcmdva67rcmdva63 kscymo79 gnbonc80 phlapa79 shokca81 lsanca61lsanca63 chcgil33 phlapa81 bltmmd80 okldca71 snfpca65 okldca68 nwrknj79 slkcut61 lsanca78 scrmca80snfcca32 chcgil64 nycmny65 rcmdva80 snjsca61 snfpca81 grdnca80 washdc63 nwrknj75 bltmmd66 nwrknj77 chcgil65 phnxaz70 nwrknj68 snjsca68 scrmca68 snfcca71 dnvrco75 lsanca69 snfcca64 cncnoh16 washdc71 nwrknj61 nycmny36 cmbrma62 waynpa66 whplny70 kscymo60 lsvlky62 cncnoh69 nwrknj22 lsanca76snbrca80lsanca83scrmca61snfcca67snfcca31 chcgil70 nybwny80 rcpknj60 dnvrco60 phnxaz61 kscymo24 kscymo80 kscymo81 kscymo76 phlapa82 chcgil09 nybwny63rcmdva66 grdnca61 phnxaz81 lsvlky60 lsanca81 phlapa65nwrknj64 cmbrma61 okldca81 kscymo77 lsvlky61 washdc62 snjsca69 scrmca66 snfcca12 dnvrco76 chcgil76 cncnoh65 washdc68 washdc72 nwrknj63 washdt80 waynpa80 waynpa64 whplny71 shokca68 dnvrco82 chcgil38nycmny62 cmbrma60 waynpa60 snjsca60 nwrknj70 nwrknj73 nwrknj56 nwrknj55 okldca63lsanca58 snfpca80 lsanca62 chcgil69 phlapa69 phnxaz75 lsanca70 cncnoh73 cncnoh72 washdc60 nwrknj16 snfpca62scrmca65 chcgil29 nycmny63 noc30k80 bltmmd61 kscymo22 mmphtn81 phlapa83 phlapa59 slkcut60 chcgil61cncnoh80 washdc82 noc30k60 snjsca62 dnvrco62 lsanca80 mmphtn69 kscymo78 nwrknj23 snjsca71 scrmca70 scrmca81 snfcca73 dnvrco73 desmia80 cncnoh67 nycmny83 nwrknj60 hrbgpa62 cmbrma64 waynpa63 lsanca60 chcgil71 nycmny37 nybwny60rcmdva68 kscymo58 lsanca75 scrmca62 snfcca20 lsanca68 chcgil35 nwrknj81 okldca69 kscymo82 chcgil63 lsvlky80 lsvlky63 bltmmd68 shokca69 chcgil36 phlapa27 rcpknj80 rcpknj61 snfpca70 phnxaz74 lsanca28 bltmmd64 nwrknj74 nwrknj80 rcmdva61 nwrknj20 snjsca73 snjsca81 ptldor81 lsanca33 chcgil77 iplsin67 cncnoh15 pitbpa80 washdc69 washdt60 nycmny28 rcmdva62 shokca67 phlapa60 kscymo75 cncnoh71 washdc81 phlapa61 nwrknj65 cmbrma63 waynpa62 snfpca69 phlapa19 bltmmd71 nybwny62 lsanca74 dnvrco71 phlapa67 cmbrma81 rcmdva60 waynpa68 snjsca64 kscymo17 mmphtn67 nycmny60 snfcca65 chcgil39 okldca65 dnvrco63 kscymo72 scrmca64 nycmny38 dnvrco61 kscymo59 kscymo83 chcgil58 snjsca72 ptldor69 snfcca72 snfcca74 omahne80 rlmdil80 dtrtmi80 nycmny59 cdknnj80 cmdnnj80 washdt62 cmbrma69 lsanca73lsanca79snfpca63 snfcca83lsanca67 snfcca62snfcca61 phlapa22 mmphtn68 cncnoh68 snfpca60 lsanca31 cncnoh14 nwrknj62nycmny81bltmmd60 okldca70 lsanca13 nwrknj83 bltmmd65 bltmmd70 rcmdva65 rcmdva64 snfcca75 chcgil68 snjsca80 cncnoh70 nwrknj82 bltmmd63 nybwny64 lsanca59 dnvrco72 cncnoh13iplsin81 washdc70 cmbrma65 chcgil79 phlapa62 bltmmd69 snjsca74 snfcca70 sttlwa82 dnvrco74 chcgil73 iplsin66 nycmny73 washdc83 nycmny68 cmbrma80 cdknnj67 snjsca63 phlapa18 nybwny61 snfpca61 phlapa26 nycmny35 cmbrma62waynpa80rcpknj62 waynpa61 slkcut63 okldca62 snjsca65 phnxaz73 phnxaz82 kscymo74 chcgil60 cncnoh82 scrmca60snfcca80dnvrco75 chcgil67 hrbgpa80 snfcca69 dnvrco80 mmphtn66 chcgil08 lsanca23lsanca32 snfcca63 cncnoh66 washdc73 nycmny36 waynpa67 lsvlky65 phlapa23 bltmmd81 snjsca67 snfcca26 snfcca25 sttlwa80 desmia64 chcgil74 iplsin68 pitbpa60 nycmny57 nycmny67 cdknnj66 lsanca77 snfcca60snfcca64 chcgil83 washdt80 dnvrco64 nybwny65 rcmdva67 rcmdva63 washdc71 nycmny61 bltmmd62 snfpca64 hrbgpa62 cmbrma64 chcgil80 chcgil62 phlapa63 bltmmd67 rcmdva80 lsanca30 cncnoh16 washdc72 okldca80 phnxaz72 chcgil37 nycmny64 rcpknj60 ptldor63 snfcca28 snfcca27 desmia60 chcgil75 dtrtmi81 pitbpa62 nycmny79 slspmd80 cmbrma13 slkcut81 nwrknj61 slkcut62 okldca64 phnxaz71 kscymo70 chcgil32 lsvlky64 waynpa60 washdc68 rcmdva62 waynpa65 snfcca29 snfcca68 chcgil34 phlapa68 nycmny83 waynpa66 chcgil72 phlapa66 noc30k61 nwrknj71 slkcut67 dnvrco76 cncnoh65 nycmny28 slkcut80 bltmmd80 nybwny63 rcmdva66 washdt60 snfcca21 lsanca64 nybwny80 scrmca71 ptldor61 snfcca10 omahne62 desmia62 chcgil78 dtrtmi61 pitbpa64 nycmny56 nycmny29 cdknnj81 cdknnj71 cmbrma68 dnvrco73 cncnoh67cncnoh15 washdc69 shokca66 slkcut65 snfcca30 chcgil81 nycmny82 cmbrma61 cmbrma60 snfcca71 snfcca73 pitbpa80 nwrknj63 okldca60 okldca61 scrmca63 snfcca81 chcgil82 lsanca69 dtrtmi80 cmdnnj80 washdt62 lsanca65 lsanca82 chcgil59 phlapa80 phlapa81 waynpa68 waynpa62 ptldor81 nycmny68 spknwa80 ptldor82 ptldor64 sttlwa75 omahne60 desmia61 rlmdil65 dtrtmi62 nycmny80 hrbgpa61 washdt61 cmbrma16 whplny81 slkcut68 omahne80lsanca33 nycmny67 waynpa63waynpa64 snfcca82 nycmny65 rcmdva68 rcpknj61 snfcca12 desmia80chcgil76 nwrknj60 cmbrma80 lsanca78 scrmca80 lsanca63 nwrknj64 scrmca67 snfcca70snfcca72 shokca71 slkcut64 snbrca80 lsanca61 chcgil30 nycmny37 bltmmd61 noc30k60 nybwny60 cdknnj67cmbrma69 shokca70 snfcca32 chcgil33 chcgil65 phlapa65 rcpknj80 snjsca70 ptldor65 ptldor60 sttlwa73 omahne61 desmia65 rlmdil60 dtrtmi64 nycmny76 hrbgpa60 nycmny69 cmdnnj62 whplny68 scrmca81 dnvrco74 iplsin67 nycmny59 cdknnj80 slspmd80 slkcut61 shokca81 lsanca76 lsanca83 scrmca61 noc30k80 cmbrma63 slkcut66 snfcca74 nycmny73 nycmny57washdc83 snfcca67 chcgil70 chcgil64 phlapa69 rcmdva60 waynpa61 scrmca69 snfcca26 sttlwa82 chcgil77chcgil73rlmdil80 nycmny79 cdknnj81 nycmny62 washdc82 snfcca25 chcgil74 spknwa62 ptldor68 ptldor62 sttlwa74 desmia63 desmia81 dtrtmi60 iplsin80 nycmny78 nycmny77 cdknnj61 artnva80 slspmd60 whplny67 pitbpa60 nycmny29 whplny69 lsanca58 snfcca31 lsanca81 cmbrma81 iplsin66iplsin68 hrbgpa61 shokca68 snfpca62 dnvrco82 chcgil09 chcgil38 nycmny63 nwrknj81 nwrknj65 pitbpa62 washdt61 cdknnj66 chcgil71 chcgil29 chcgil69 nycmny38 waynpa67 cmdnnj62 cmbrma13 okldca63 lsanca75 scrmca65 snfpca80 cmbrma65 cmbrma62 waynpa80 rcpknj62 snfcca28 nycmny56nycmny80 cmbrma68 slkcut60 chcgil61 cncnoh80 whplny69 ptldor70 ptldor80 sttlwa76 rlmdil61 dtrtmi67 dtrtmi63 nycmny55 clmboh81 hrfrct81 grcyny80 cdknnj68 cmbrma58 scrmca68 snfcca10snfcca27 nycmny69 cdknnj71 whplny66 scrmca62 chcgil35 snfcca20 phlapa27 phlapa67 nycmny35 ptldor63 chcgil75 nycmny76 hrbgpa60 lsanca60 lsanca62 nycmny81 bltmmd60 chcgil78 dtrtmi81 nycmny77 snfpca63 lsanca68 chcgil36 phlapa60 nwrknj62 whplny66 snjsca66 ptldor66 sttlwa81 sttlwa71 rlmdil64 rlmdil63 dtrtmi65 clevoh80 nycmny74 cmdnnj61 cdknnj70 cmbrma67 slspmd60 shokca69 snfcca65 chcgil39 scrmca66 nycmny78 cdknnj61 whplny81 shokca67 lsanca73 lsanca74 nycmny36 waynpa66 waynpa65 desmia64 phlapa22 cmbrma64 rcmdva62 scrmca70 sttlwa80desmia60 pitbpa64nycmny55 cmbrma16 whplny70 scrmca64 dnvrco71 ptldor61 dtrtmi61 cncnoh14 iplsin81 spknwa61 ptldor13 sttlwa72 desmia67 rlmdil62 dtrtmi68 pitbpa61 nycmny75 cmdnnj60 slspmd61 grcyny61 desmia62 artnva80grcyny80 cdknnj68 whplny71 slkcut81 snfpca60 cncnoh13 washdt80 snjsca68 clmboh81hrfrct81 cmdnnj61 cdknnj70 cmbrma67 lsanca79 snfcca75 snfcca83 lsanca67 snfcca62 chcgil68 washdc73 hrbgpa80 omahne62 rlmdil65 nycmny74 cmdnnj60 whplny68 lsanca31 snfcca61 dtrtmi62 slkcut67 washdc70 hrbgpa62 whplny71 spknwa60 ptldor14 ptldor12 sttlwa63 okbril80 milwwi80 nycmny58 washdc75 cdknnj69 cmbrma83 ptldor64 dtrtmi64 clevoh80nycmny75 washdc75 slspmd61 lsanca59 dnvrco72 phlapa26 washdc71 nycmny28 whplny70 dtrtmi60 snfcca63 chcgil83 chcgil67 snjsca81scrmca71 sttlwa75 desmia61 nycmny58 cdknnj69 scrmca60 cncnoh66 washdt60 sttlwa73 desmia65 iplsin80 whplny80 nwrknj61 cmbrma69 ptldor07 sttlwa70 sttlwa60 sttlwa61 chcgcg80 dtrtmi70 pitbpa63 cdknnj64 whplny80 cmbrma82 grcyny62 cmbrma76 ptldor82 pitbpa63 cdknnj64 whplny67 snfpca61 snfcca80 snfcca64 washdc72 cdknnj67 ptldor60 rlmdil60 washdc76 slkcut68 lsanca23 lsanca32 snfcca60 washdt62 omahne60 desmia81 dtrtmi63 pitbpa61 grcyny61 snfpca64 waynpa63 snjsca69 ptldor62 omahne61 pitbpa65 cdknnj60cdknnj62 scrmca67 lsanca77 dnvrco75 washdc68 nwrknj63 waynpa64 slspmd62 scrmca69 cncnoh16 cncnoh65 cncnoh67 pitbpa80 ptldor67 sttlwa66 sttlwa62 okbril62 desmia68 dtrtmi69 pitbpa65 washdc76 slspmd62 grcyny63 cmbrma72 cmbrma59 snjsca71snjsca73 ptldor69 ptldor80 sttlwa74 rlmdil61 cmbrma82 cncnoh15 nycmny83 washdc69 nycmny68 desmia63 dtrtmi67 grcyny62 cmbrma83 cmbrma58 slkcut66 nwrknj60 cmdnnj80 nycmny67 rlmdil63 dnvrco73 lsanca30 desmia80 chcgil76 cmbrma80 whplny68 sttlwa76 cdknnj63 cdknnj80 ptldor06 sttlwa69 sttlwa64 desmia66 dtrtmi66 cdknnj62 hrfrct66 artnva60 grcyny64 cmbrma77 rlmdil64 washdc74 snfcca71 ptldor81 dtrtmi80 nycmny31 cdknnj65 artnva60grcyny63 snfcca73 dnvrco76 cdknnj66 whplny67 nycmny70 scrmca68 lsanca69 iplsin67 nycmny73 nycmny59 clmboh80 snfcca12 nycmny79 slspmd80 hrfrct66 cmbrma76 snjsca68 nycmny57 cdknnj81 mplsmn81 okbril65 stplmn81 cdknnj60 washdc74 whplny60 cmbrma73 cmbrma53 dtrtmi65dtrtmi68 nycmny71 pitbpa81 hrfrct68 scrmca81 snfcca72 cdknnj71 omahne80 iplsin66 hrbgpa61 cmbrma68 snjsca72snjsca74spknwa80 sttlwa71sttlwa72 rlmdil62 grcyny64 cmbrma59 snfcca70 lsanca33 chcgil77 chcgil73 rlmdil80 washdc83 nycmny29 cmbrma13 sttlwa81 dtrtmi70milwwi80 clmboh71 artnva67whplny60 cmbrma72 cmbrma77 chcgil74 iplsin68 pitbpa60 cmdnnj62 ptldor65 nycmny32nycmny09 hrfrct67 scrmca66 snfcca74 dnvrco74 pitbpa62 nycmny56 nycmny80 sttlwa68 okbril81 chcgcg60 milwwi60 cdknnj63 cdknnj65 hrfrct68 grcyny65 cmbrma79 clmboh69 artnva65 snjsca69 scrmca70 snfcca26 snfcca25 sttlwa82 washdt61 okbril80 cmbrma53 snjsca71 nycmny76 nycmny77 slspmd60 whplny81 ptldor68 ptldor66 sttlwa63 dtrtmi69 clevoh60 grcyny65 snjsca73 cmbrma58 nycmny33 hrfrct60 ptldor63 desmia64 dtrtmi81 cdknnj61 hrbgpa60 nycmny69 cmbrma16 sttlwa67 chcgcg63 okbril63 nycmny31 hrfrct67 artnva67 cmbrma74 cmbrma57 snjsca67 sttlwa60sttlwa61 iplsin65 nycmny72 whplny63 desmia60 chcgil78 chcgil75 desmia67 artnva66 cmbrma73 snfcca28 snjsca70 dtrtmi66 clevoh63 cdknnj82 ptldor69 desmia62 dtrtmi61 pitbpa64 nycmny55 nycmny78 cdknnj68 ptldor13 clmboh67 cmbrma79 snjsca81 snfcca10 snfcca27 sttlwa80 clmboh81 sttlwa65 mplsmn70 chcgcg61 okbril61 nycmny70 pitbpa81 artnva65 cmbrma71 cmbrma52 chcgcg80 hrfrct62artnva62artnva63 cmbrma57 scrmca71 ptldor61 nycmny74 grcyny80 cdknnj70 cmbrma76 iplsin61 clmboh68clmboh66 grcyny60 snjsca72 rlmdil65 hrfrct81 washdc75 cmbrma67 cmbrma59 spknwa62 ptldor70 clevoh62 omahne62 artnva80 cmdnnj61 cmbrma77 sttlwa62sttlwa64 stplmn81 whplny65 desmia61 dtrtmi64 dtrtmi62 mplsmn66 okbril60 clmboh80 nycmny71 clmboh71 artnva66 whplny63 cmbrma55 snjsca66 ptldor14 okbril62 iplsin63 clevoh61clevoh65 hrfrct61artnva64 cmbrma74 cmbrma52 snjsca74 ptldor64 desmia65 clevoh80 nycmny75 slspmd61 cmbrma55 spknwa80 ptldor82 sttlwa73 rlmdil60 cmdnnj60 ptldor12 desmia68 iplsin62 clmboh70 artnva68artnva61whplny62 cmbrma71 ptldor60 sttlwa75 dtrtmi60 pitbpa63 nycmny58 cdknnj64 cdknnj69 cmbrma53 clevoh64 okbril71 mplsmn67 okbril64 clevoh60 nycmny32 nycmny09 clmboh69 artnva63 grcyny60 cmbrma56 ptldor07 desmia66 iplsin60 snjsca67 desmia81 cdknnj62 washdc76 omahne60 iplsin80 cdknnj60 grcyny61 cmbrma83 sttlwa70 okbril65 cmbrma56 omahne61 desmia63 rlmdil61 pitbpa61 cdknnj63 spknwa62 snjsca70 ptldor62 dtrtmi67 pitbpa65 slspmd62 okbril70 mplsmn69 chcgcg64 milwwi65 iplsin65 clevoh63 nycmny33 cdknnj82 hrfrct60 whplny65 cmbrma78 spknwa61 sttlwa66 milwwi60 iplsin64 whplny61whplny64 ptldor65 ptldor80 sttlwa74 sttlwa76 rlmdil63 dtrtmi63 whplny80 ptldor67 ptldor68 nycmny31 cmbrma82 cmbrma79 ptldor06 albyny80 rlmdil64 nycmny70 cdknnj65 washdc74 clmboh62clmboh61 pitbpa70 cmbrma70cmbrma75 cmbrma78cmbrma54 artnva60 grcyny62 okbril69 mplsmn68 chcgcg65 stplmn82 rlmdil81 iplsin61 nycmny72 clmboh67 hrfrct62 whplny62 cmbrma54 bflony80pitbpa67 dtrtmi65 grcyny63 milwwi81 clevoh81 snjsca66 desmia67 clmboh80 pitbpa81 cmbrma57 cmbrma72 spknwa60 okbril63 spknwa61 ptldor13 ptldor66 sttlwa71 sttlwa72 dtrtmi68 nycmny71 hrfrct68 hrfrct66 chcgcg60 milwwi65 ptldor70 sttlwa81 nycmny09 cmbrma52 mplsmn71 mplsmn80 stplmn64 iplsin63 clevoh62 clmboh68 clmboh66 artnva62 cmbrma75 sttlwa69 mplsmn81 chcgcg63 okbril61 clmboh63 rlmdil62 dtrtmi70 nycmny32 clmboh71 clmboh69 milwwi64 clmboh64 cdknnj76 sttlwa63 sttlwa61 hrfrct67 artnva67 grcyny64 sttlwa68 milwwi61 okbril80 whplny60 pitbpa71pitbpa68 hrfrct80 milwwi80 iplsin65 okbril68 mplsmn65 chcgcg62 stplmn62 iplsin60 iplsin62 clevoh61 clmboh70 hrfrct61 artnva64 cmbrma70 okbril60 rlmdil81 spknwa60 sttlwa60 okbril62 chcgcg80 clevoh60 hrfrct60 cmbrma73 milwwi63 ptldor07 ptldor14 ptldor12 nycmny33 artnva65 cmbrma56 stplmn64 cdknnj72 stplmn81 dtrtmi69 clevoh63 nycmny72 clmboh67 grcyny65 sttlwa67 clmboh65 cdknnj73 sttlwa70 cdknnj82 okbril66 mplsmn63 stplmn65 iplsin64 clevoh65 bflony80 pitbpa70 artnva61 chcgcg61okbril64 clmboh60 pitbpa69hrfrct82 sttlwa66 clmboh68 artnva66 cmbrma55 sttlwa65 desmia68 whplny63 okbril81 pitbpa66 okbril65 desmia66 iplsin61 sttlwa62 sttlwa64 clevoh62 hrfrct62 milwwi62 ptldor67 ptldor06 artnva62 artnva63 cmbrma71 okbril67 mplsmn64 stplmn61 milwwi81 clevoh64 pitbpa67 pitbpa68 artnva68 whplny64 milwwi60 dtrtmi66 clevoh65 stplmn60 cdknnj75 clevoh61 clmboh70 clmboh66 cmbrma74 chcgcg64 stplmn62 sttlwa69 clevoh64 hrfrct61 grcyny60 mplsmn70 chcgcg65 cdknnj74 whplny65 cmbrma78 mplsmn61 chcgcg81 milwwi64 clmboh61 pitbpa71 hrfrct82 albyny80 whplny61 mplsmn80 dytnoh80 okbril63 okbril61 iplsin63 iplsin62 bflony80 pitbpa67 stplmn65 sttlwa68 mplsmn81 artnva61 cmbrma54 albyny63 milwwi65 artnva64 artnva68 whplny62 stplmn82 akrnoh80 cdknnj77 okbril81 iplsin60 pitbpa70 mplsmn62 stplmn70 milwwi61 clmboh62 pitbpa69 cdknnj76 hrfrct80 hrfrct03 chcgcg62 stplmn61 dtrtmi82 chcgcg60 clmboh62 clmboh61 albyny80 stplmn63 albyny64 sttlwa67 mplsmn70 chcgcg63 clevoh81 bflony60 hrfrct02hrfrct03 okbril60 iplsin64 pitbpa71 mplsmn66 sttlwa65 mplsmn60 milwwi63 clevoh81 clmboh63 cdknnj72 albyny63 hrfrct02 stplmn64 pitbpa68 cmbrma70 cmbrma75 mplsmn67 albyny60 okbril64 rlmdil81 milwwi64 whplny61 whplny64 mplsmn69 milwwi68 clevoh66 albyny65 milwwi81 hrfrct82 cdknnj76 bflony62 clmboh63 chcgcg71 milwwi62 clmboh64 pitbpa66 cdknnj73 hrfrct65 mplsmn66 mplsmn67 chcgcg61 clmboh64 okbril71 milwwi70milwwi69 bflony61 mplsmn69 pitbpa69 cdknnj72 hrfrct80 okbril71 stplmn62 pitbpa66 milwwi66 albyny62 stplmn60 chcgcg66 stplmn60 clmboh65 cdknnj75 mplsmn68 rlmdil66 clevoh68 syrcny80 albyny61 hrfrct65 chcgcg64 milwwi63 milwwi61 clmboh65 cdknnj73 rlmdil71 milwwi67 bflony65 hrfrct70 hrfrct63hrfrct64 mplsmn68 clmboh60 cdknnj77 chcgcg65 bflony60 cdknnj75 chcgcg68 stplmn63 dtrtmi82 clmboh60 cdknnj77 cdknnj74 mplsmn71 clevoh67 bflony64 okbril70 mplsmn80 stplmn82 stplmn65 cdknnj74 albyny63 okbril70 mplsmn65 rlmdil67 rlmdil70 milwwi71 mplsmn71 chcgcg62 stplmn61 milwwi62 okbril69 bflony63 hrfrct72hrfrct71 okbril69 akrnoh80 bflony62 albyny64 rlmdil66 dytnoh80 akrnoh80 bflony60 albyny64 rlmdil68rlmdil69 hrfrct04 okbril68 dytnoh80 hrfrct70 hrfrct02 hrfrct03 mplsmn63mplsmn64 stplmn63 bflony61 albyny65 hrfrct05 albyny60 rlmdil71 milwwi68 clevoh66 bflony62 albyny65 albyny60 okbril68 chcgcg81 mplsmn65 syrcny80 okbril66 dtrtmi82 okbril66 rlmdil71 rlmdil66 bflony65 stplmn80 hrfrct74hrfrct73 milwwi68 clevoh66 hrfrct71 stplmn70 okbril67 bflony64 hrfrct72 albyny62 hrfrct65 rlmdil67 milwwi69 bflony61 hrfrct70 albyny62 hrfrct63 okbril67 mplsmn61 stplmn71 dtrtmi71dtrtmi75dytnoh65dytnoh60 hrfrct69 mplsmn63 mplsmn64 albyny61 hrfrct63 mplsmn62 chcgcg81 milwwi70 milwwi69 hrfrct64 rlmdil68 milwwi70 milwwi66 bflony65 syrcny80 hrfrct71 albyny61 hrfrct64 rlmdil67 rlmdil70 milwwi66 bflony63 hrfrct74 dtrtmi74dtrtmi79 rlmdil68 clevoh68 hrfrct73 mplsmn60 dytnoh64 mplsmn61 stplmn72 dtrtmi72 akrnoh62 mplsmn62 clevoh67 hrfrct69 hrfrct04 rlmdil70 milwwi67 clevoh68 bflony64 hrfrct72 hrfrct73 hrfrct04 dytnoh62 akrnoh61 stplmn70 rlmdil69 milwwi67 milwwi71 hrfrct05 dytnoh61 stplmn71 stplmn80 rlmdil69 milwwi71 clevoh67 bflony63 hrfrct74 hrfrct05 mplsmn60 dtrtmi76 dytnoh63 akrnoh60 dytnoh65 dytnoh60 akrnoh62 akrnoh61 syrcny61 dtrtmi73 stplmn80 dtrtmi71 dtrtmi75 dytnoh60 akrnoh62 hrfrct69 syrcny61 dtrtmi77 dtrtmi71 dtrtmi75 syrcny60 dtrtmi78 stplmn72 syrcny62 syrcny61 stplmn71 dtrtmi74 dtrtmi79 dytnoh65 akrnoh61 syrcny60 syrcny60 dtrtmi74 dytnoh64 chcgcg71 syrcny62 dtrtmi72 dtrtmi79 stplmn72 dtrtmi72 dytnoh64 akrnoh60 syrcny62 chcgcg70 chcgcg71 dytnoh62 stplmn67 akrnoh60 chcgcg66 stplmn67 dtrtmi73 dytnoh61 chcgcg70 stplmn67 dtrtmi76 dytnoh62 chcgcg70 dtrtmi76 stplmn69 dtrtmi77 chcgcg66 dytnoh63 chcgcg68 chcgcg67 stplmn69 dtrtmi73 dtrtmi77 dytnoh61 chcgcg69chcgcg67stplmn06 stplmn07 dtrtmi78 chcgcg68 stplmn06 stplmn69 stplmn07 stplmn68 chcgcg67 chcgcg69 stplmn06 dtrtmi78 dytnoh63 stplmn09 chcgcg69 stplmn08 stplmn68 stplmn09 stplmn07 stplmn09 stplmn68 stplmn08 stplmn08 badvoro badvoro+PRISM badvoro+VPSC

382528

382566

382436

377908 382572

380526

381775

377924

381710

380448

379972 379341

378362

378666

380298

378656

377971 379169

370706

373300 354546 377980

375134 380475

360672 354771 371942 379968

360144 382827 357430

377908 356116 380526 377380 382409 372568 382528 382528 357769 357538 382566 352010 382436 379972 377924 382566 382572 375557 380448 360839 382436 375039 375024 382572 354757 358866 381775 378666 357793 377908380526 358930 379848 358900 381710 378362 354766 377924 358224 375027 380448 380298 379341 341411 379972 378656 381775 355800 378362379341 375519 381835 374700 378666 377971 377562 354221 378656 380298 354290 353506 379169 374300 381211 374741 382574 358471 377971 379169 371943 370706380475377980 381710 360104 375499 357430 379968 370706 382161 382928 379864 370510 358584 375319 380475 381901 371942 382827 377980 380571 379339 370509 377380 373300 354546 379968 372568 358159 382409 357430 375134 377719 380604 383039 372568 357340 375557 352010 375039 358866 383174 379422 360672 358155 371187 375557 360144 354771 354757 381897 376956 358157 355080 341411 375519 379848 381835 357769 358930 375024 382103 358224 356116 371942382827 358900 358900 354766 356741 354785 377380 382409 358224 381211 374741 374700 377763 341411358866 357538 354221 352010375039 378108 355288 379848 374700 382574 373300 360839 357793 375508 358930375519381835 354546 374300 354290 381211 377220 354878 374741 354290 375024 375134 381901 382928 377562 382928 380604 375499 381901 379864 360672 354221 374886 374300 380604383174379339 357769360144354771354757 383174 371943 375027 357538356116354766 382574 355800 382161381897382103 360839 377562 371943 379339 375027 358471 383039 357793 382161 375319 375507 380571377763375508 379422 358471 355800375499 360104 374886 358584358159360104 381897 382103 375507 357340 379864 358159 358155358157371187355080375319 380571 383039 355080 377719 376956 353506 356741354785 358584 384096 354878355288 371187 378108 377763 375508 357340 370510 377220 379422 359471 377719 374886 353506 370509 358155 384096 358157 354785 375507 380963 376956 359471 370510 379126 356741 354878 355288

359471 380963 370509 384909 384096 379126 378108 377220 379212 384909 380963 380285 380285 379126 379212 384909 379212380285 377801 377801376529 377801 376529 359100 376529 359100 359100 mode mode+PRISM mode+VPSC

Fig. 3. Comparing PRISM and VPSC on two graphs. Original layouts are scaled to have an average edge length that equals 4 times the label size. 214 E.R. Gansner and Y. Hu

Mattheou Karagrigoriou

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Fig. 4. The second largest component from the Mathematics Genealogy Project

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Fig. 5. Close-up view of the center-left part of Fig. 4 drawings with extreme aspect ratios. In fact, for 9 out of 14 test graphs, VPSC produces layouts with extreme aspect ratios. PRISM does not suffer from this problem. We experimented with layouts initially scaled sufﬁciently so that relatively fewer nodes overlap. For example, when initial layouts were scaled to give an average edge length equal to 4 times the average node size, we found that the performance of VPSC was improved. Nevertheless it still suffered from extreme aspect ratio on at least 5 out of the 14 graphs. Figure 3 shows two of these graphs. Efﬁcient Node Overlap Removal Using a Proximity Stress Model 215

Overall, quantitative and visual comparison of the drawings of these 14 graphs, as well as drawings for graphs in the complete Graphviz test suite (a total of 204 graphs in March 2008), shows that PRISM performs very well, and is overall better and faster than VPSC and VORO. The ODNLS algorithm preserves similarity somewhat better than PRISM, but at much higher costs in term of speed and area. As a demonstration of the scalability of PRISM, we consider its application to a large graph. This is a tree from the Mathematics Genealogy Project [24]. Each node is a mathematician, and an edge from node i to node j means that j is the ﬁrst supervisor of i. The graph is disconnected and consists of thousands of components. Here we consider the second largest component with 11766 vertices. This graph took 31 seconds to layout using SFDP, and 15 seconds post-processing using PRISM for overlap removal. Important mathematicians (those with the most offspring) and important edges (those that lead to the largest subtrees) are highlighted with larger nodes and thicker edges. Figure 4 gives the overall layout, which shows that PRISM preserved the tree structure of the layout very well after node overlap removal. Figure 5 gives a close up view of the details of a small area in the center-left part of Fig. 4. Additional drawings of this and other components of the Mathematics Genealogy Project graph, including that of the largest component, are available [15].

5 Conclusions and Future Work

A number of algorithms have been proposed for removing node overlaps in undirected graph drawings. For graphs that are relatively large with nontrivial connectivities, these algorithms often fail to produce satisfactory results, either because the resulting drawing is too large (e.g., scaling, VORO, ODNLS), or the drawing becomes highly skewed (e.g., VPSC). In addition, many of them do not scale well with the size of the graph in terms of computational costs. The main contribution of this paper is a new algorithm for removing overlaps that is both highly effective and efﬁcient. The algorithm is shown to produce layouts that preserve the proximity relations between vertices, and scales well with the size of the graph. It has been applied to graphs of tens of thousands of vertices, 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 are removed in which pass or by gradually removing overlaps using many alternating X 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 to deal with edge node overlaps. We would also like to explore the possibility of using the proximity stress model for packing disconnected components.

Acknowledgments

We would like to thank Tim Dwyer and Wanchun Li for making their implementa- tions of VPSC and ODNLS, respectively, available to us; Yehuda Koren and Stephen 216 E.R. Gansner and Y. Hu

North for helpful discussions; Stephen Kobourov for bringing our attention to the term “Frobenius metric”; and the referees for valuable suggestions and references.

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