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

Efficient, 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 initial layout or as a post-processing step. The core problem in avoiding or removing overlaps is to retain the structural information inherent in a layout while minimizing the additional area required. This paper presents a new node overlap removal algorithm that does well at retaining a graph’s shape while using little additional area and time. As part of the analysis, we consider and evaluate two measures of dissimilarity for two layouts of the same graph.

Submitted: Reviewed: Revised: Accepted: November 2008 March 2009 May 2009 November 2009 Final: Published: November 2009 January 2010 Article type: Communicated by: Regular paper I. G. Tollis and M. Patrignani

E-mail addresses: [email protected] (Emden R. Gansner) [email protected] (Yifan Hu) 54 Gansner and Hu Efficient Node Overlap Removal

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 while preserving the node size until the nodes no longer overlap [30]. 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 ap- proach 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. 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. Such heuristics, however, cannot guarantee all overlaps will be removed, so they typ- ically rely on overly large repulsive forces, or the type of post-processing step considered below. This problem was addressed by Dwyer et al. [5], who showed how 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 using stress majorization [13]. An alternative approach is to remove node 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. In addition to many of the al- gorithms mentioned above that can be adapted for this use, a number of other algorithms have been proposed. For example, the Voronoi cluster busting algo- rithm [15, 29] works by iteratively forming a Voronoi diagram from the current layout and moving each node to the center of its Voronoi cell until no overlaps remain. The idea is that restricting each node to its corresponding Voronoi cell should preserve the relative positions of the nodes. In practice, because of the number of iterations often required and the use of a rectangular bounding box,1 the nodes in the final drawing can be homogeneously distributed within a rectangle, the graph bearing little resemblance to the original layout. Another group of post-processing algorithms is based on setting up pairwise node constraints to remove overlaps and then performing some procedure to

1The latter might be improved by using something like α-shapes [8] to provide a more accurate 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 overlaps while, if desired, minimizing node displacement. All of these techniques rely on solving separate horizontal and vertical problems. They behave differently if the layout is rotated by a degree that is not a multiple of π/2. We have also found that, in practice, this asymmetry often results in a layout with a distorted aspect ratio (e.g., Figure 4, bottom right). One desideratum proposed [31] for overlap removal is the preservation of orthogonal ordering, i.e., the relative ordering of the x and y coordinates of two nodes should be the same in the original layout and the one with overlaps removed. Thus, if a node is above and to the left of another node in the original layout, it is above and to the left in the derived one. The idea is that, in this manner, a user’s “mental map” of the graph is better preserved. Many of the algorithms described above either inherently preserve orthogonal ordering, or have simple variations that do. While preserving orthogonal ordering can be important, it alone cannot ensure that the relative proximity relations [23] between nodes are preserved. At other times, it is too restrictive, causing two highly-related nodes to be moved far apart due to some largely unrelated third node. For these reasons, our approach is to concentrate on proximity relations and not deal with orthogonal ordering. In this paper, we focus on the problem of removing overlaps rather than avoiding them. To this end, we discuss (Section 3) metrics for the similarity between two layouts which we believe better quantify the desired outcome of overlap 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 keeping the relative positions between them as close to those in the original layout as possible. 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 possible node pairs. Because of this sparse stress function, the algorithm is efficient and is able to handle very large graphs. In Section 5, we evaluate our algorithm and others using the proposed similarity measures. Finally, Section 6 presents a 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 and edges, respectively. We let xi represent the current coordinates of vertex i in Euclidean space. For this paper we are interested in 2D layout, therefore 2 xi ∈ R . The aim of graph drawing is to find xi for all i ∈ V so that the resulting 56 Gansner and Hu Efficient Node Overlap Removal drawing gives a good visual representation of the information in the graph. Two popular 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 a minimal energy configuration of a physical system. We shall describe the stress model in more detail since we will use a similar model for the purpose of node overlap removal in Section 4. The stress model assumes that there are springs connecting all nodes of the graph, with the ideal spring length equal to the graph theoretical distance between nodes. The energy of this spring system is

2 wij (kxi − xj k− dij) , (1) i j X=6 where dij is the graph theoretical distance between vertices i and j, and wij is a 2 weight factor, typically 1/dij . The layout that minimizes the above stress en- ergy is an optimal layout of the graph. There are several ways to find a solution of the minimization problem. An iterative approach can be employed. Starting from a random layout, the total spring force on each vertex is calculated, and the 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, until the layout stabilizes. Alternatively, a stress majorization technique can be em- ployed, where the cost function (1) is bounded by a series of quadratic functions from above, and the process of finding an optimum becomes that of solving a series 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 of vertices. There have been attempts (e.g., [2, 13]) to simplify the stress function by 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 efficient approximation scheme for long range repulsive forces [18, 22, 36] are still the most efficient choices to lay out large graphs without 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 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. When total node movement is optimized [6, 24], comparison is reduced to measuring the amount of displacement of the vertices in the new layout from those of the original graph. This measurement does not take into account pos- sible shifts, scalings or rotations, nor the importance of maintaining the relative position among vertices. JGAA, 14(1) 53–74 (2010) 57

As far as we are aware, there is no definitive way to measure similarity of two layouts of the same graph. We adopt two approaches. The first approach is based on measuring changes in lengths of edges. The second approach, which is a modification of the metric of Dwyer et al. [6], is based on measuring the displacement 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 of points in space: points that are “neighbors” to each other in the space form an edge in the proximity graph. There are many ways to create a proximity graph [25]. In this paper, we shall work with the Delaunay triangulation (DT), which is an approximation to the proximity graph, and has the advantage being rigid. Two points are neighbors in DT if and only if there exists a sphere passing through 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 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 [9]. Calculating all pairwise distances is expensive for large graphs, both in CPU time and in the amount of memory. As we want a metric feasible for very large graphs, we instead form a DT of the original graph, then measure the distance between vertices along the edges of the triangulation for the original and new layouts.2 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

kxi − xj k rij = 0 0 , {i, j} ∈ EP , kxi − xj k then define a measure of the dissimilarity as the normalized standard deviation

2 P (rij−r¯) {i,j}∈EP |EP | σ (x0, x)= r , dist r¯ where

1 r¯ = rij |EP | {i,jX}∈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,

2 Another possibility would be to sample O(|V |) out of the |V |(|V |− 1)/2 possible vertex pairs. Some care is needed in making sure that the resulting graph is rigid. 58 Gansner and Hu Efficient Node Overlap Removal

4 3

4 3

1 2 1 2

Figure 1: The edge lengths of these two layouts of a non-rigid graph are exactly the same, but the layouts are clearly different.

but are clearly different (see Figure 1). The rigidity of the triangulation avoids this problem. 0 Notice that σdist(x , x) is not symmetric with regard to which layout comes first. Furthermore, 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 it by defining the dissimilarity between layout x 0 0 0 and x as (σdist(x , x)+ σdist(x, x ))/2. This also resolves the “foldover prob- lem”. The symmetric version may be more appropriate 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 ver- tices of the new layout from the original layout [6]. 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 ro- tation that minimize the displacement. The optimal displacement is then a measure of dissimilarity. We denote by scalars r and θ the scaling and rotation,

cos(θ) sin(θ) T = (2) − sin(θ) cos(θ)   the rotation matrix, p ∈ R2 the translation, and define the displacement dissim- ilarity as 0 0 2 σdisp(x , x) = minp∈R2,θ,r∈R krT xi + p − xi k , (3) i V X∈ This is a known problem in the Procrustes analysis [1, 16] and the solution (the Procrustes statistic) is

0 0 0T T 0 0T 1 2 T σdisp(x , x) = tr(X X ) − (tr((X X X X) 2 ) tr(X X), (4)

0 where tr(A) is the trace of a matrix A, X is a matrix with columns xi − x¯, X 0 0 0 is a matrix with columns xi − x¯ , andx ¯ andx ¯ are the centers of gravity of the new and original layout. JGAA, 14(1) 53–74 (2010) 59

j j

i i

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

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. The quality of an overlap removal algorithm is a combination of how similar the new layout is to the original, and how small an area it occupies. The simplest overlap removal algorithm is that of scaling the layout until all overlaps are removed. This has a dissimilarity of 0, but usually occupies a very large area. The alternative extreme is to pack the nodes as close to each other as possible while ignoring the original layout. This will have the smallest area, but a large dissimilarity. A good solution should be a compromise between these two 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 by maintaining the proximity relations [23] among the nodes. To do this, we first set up a rigid “scaffolding” structure so that while vertices can move around, their relative positions are maintained. This scaffolding is constructed using an approximate 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 node overlaps along that edge. Let wi and hi denote the half width and height of 0 0 the node i, and xi (1) 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 . (5) |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, see Figure 2. Therefore we want to generate a layout such that an edge in the proximity graph has the 60 Gansner and Hu Efficient Node Overlap Removal

0 0 ideal edge length close to tijkxi − xj k. In other words, we want to minimize the following stress function

2 wij (kxi − xj k− dij ) . (6) (i,jX)∈EP 0 0 Here dij = sijkxi − xj k is the ideal distance for the edge {i, j}, sij is a scaling 2 factor related to the overlap factor tij (see (7)), wij = 1/kdijk is a weighting factor, and EP is the set of edges of the proximity graph. We call (6) the proximity stress model in obvious analogy. Because DT is a planar graph, which has no more than 3|V |− 6 edges, the above stress function has no more than 3|V |− 6 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 graph, and they overlap. If we 0 0 try to make the length of the edge equal tijkxi − xj k, 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) (7) and try to remove overlap 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. 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 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. 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 the Delaunay triangulation. Clearly, node 2 and node 4 still overlap. If we plot the Delaunay triangulation (Figure 3(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- JGAA, 14(1) 53–74 (2010) 61

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

Figure 3: (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.

line algorithm [6] 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 (6). This process is repeated until the scan-line algorithm finds no 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) 0 Input: coordinates for each vertex, xi , and bounding box width and height {wi,hi}, i =1, 2,..., |V |. repeat 0 Form a proximity graph GP of x 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 0 Form a proximity graph GP of x by Delaunay triangulation. Find all node overlaps using a scan-line algorithm. Augment GP with edges from 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 in O(l|V |(log |V | + l)) time [6], 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 (6), like the spring model (1), can be solved 62 Gansner and Hu Efficient Node Overlap Removal using the stress majorization technique [13] which is known to be a robust process for finding the minimum of (1). The technique works by bounding (6) with a series of quadratic functions from above, and the process of finding an optimum becomes that of finding the optimum of the series of quadratic functions, which involves solving linear systems with a weighted Laplacian of the proximity graph. We solve each linear system using a preconditioned conjugate gradient algorithm. Because we use DT as our proximity graph and it has no more than 3|V |− 6 edges, each iteration of the conjugate gradient algorithm takes a time of O(|V |). Overall, therefore, Algorithm 1 takes O(t(mk|V | + |V | log |V |)) time, where t is the total number of iterations in the two main loops in Algorithm 1, m is the average number of stress majorization iterations, and k the average number of iterations for the conjugate gradient algorithm. In practice, we found that the majority of CPU time is spent in repeatedly solving 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 the linear system involved in the stress majorization process is less than 0.01. A tighter tolerance is not necessary because the solution of each linear system con- stitutes an intermediate step of the stress majorization. Furthermore, solution of 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 a faster algorithm, and that, in terms of quality, a smaller mmax is often just as good 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 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) [22], a multilevel, 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 time3 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 imple- mentation of the solve VPSC algorithm [6] provided by its authors. We also evaluated the companion algorithm satisfy VPSC. This offered, at best, a 2%

3All timings were derived on a 4 processor, 3.2 GHz Intel Xeon CPU, with 8.16 GB of memory, 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 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

reduction in time, while producing almost identical results concerning displace- ment, dissimilarity and area. For this reason, in the sequel, we only consider the solve VPSC algorithm, hereafter denoted as VPSC.4 We also evaluated the Voronoi cluster busting algorithm [15, 29], denoted by VORO, as well as the ODNLS algorithm of Li et al. [28], which relies on varied edge lengths in a spring embedder. We note, in passing, that we also tested scaling using the algorithm of Marriott et al. [30]. As expected, it outperformed all other algorithms in terms of speed and dissimilarity, but at an unacceptably high cost in area. For example, on the b143 graph, the time and dissimilarity were essentially 0 but the area was 82.2. Overall, scaling produced areas at least 4.5 times larger than PRISM and, more typically, at least an order of magnitude larger. We therefore remove 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 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, fol- lowed 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 ex-

4 Both versions of the VPSC algorithm can be extended to support the preservation of orthogonal ordering. We did not have an opportunity to check these versions. 64 Gansner and Hu Efficient Node Overlap Removal tremely 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 Figure 4 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 differences in aspect ratio. This is evident in drawings of the root graph (Figure 4, second row): VPSC clearly produced a drawing that is overly stretched in the vertical direction, but its σdist is actually smaller! 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 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 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. When starting from a layout that is scaled sufficiently so that relative fewer nodes overlap, VPSC’s performance can be improved. Table 3 compares the four overlap removal algorithms, starting from layouts that are scaled to give an average edge length that equals 4 times the average node size. Here the size of 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 VPSC are now similar, closer to the better performing ODNLS. In terms of drawing area, PRISM is better than VPSC, with VORO and ODNLS much larger. When visually inspected, VPSC again suffers from extreme aspect ratio issue on at least 5 out of the 14 graphs (b100, b143, badvoro, mode, root). Figure 5 shows the layout of badvoro (first row), on which VPSC performed badly based on the two similarity measures. In the same figure, we also show b124 (second row), 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 measures suggest. On the other hand, if we look at b124, VPSC perhaps performed better than PRISM, but not as clearly as the similarity measures suggest. Overall, visual inspection 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. 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. 6 Results shown 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 σdist σdisp area σdist σdisp area b100 0.74 0.38 14.05 0.76 0.72 18.91 b102 0.44 0.25 2.45 0.58 0.8 2.71 b124 0.65 0.37 1.04 0.78 0.73 0.91 b143 0.59 0.35 1.5 0.78 0.83 2.16 badvoro 0.34 0.15 12.58 0.61 0.75 13.85 mode 0.59 0.37 0.79 1.02 0.77 1.29 ngk10 4 0.41 0.16 0.33 0.39 0.3 0.25 NaN 0.4 0.2 0.72 0.54 0.65 0.71 dpd 0.34 0.18 0.25 0.51 0.4 0.18 root 0.71 0.3 16.99 0.6 0.75 17.68 rowe 0.33 0.14 0.22 0.44 0.31 0.19 size 0.37 0.2 0.47 0.77 0.74 0.4 unix 0.39 0.23 0.39 0.51 0.67 0.36 xx 0.42 0.25 3.96 0.57 0.82 3.9 Graph VORO ODNLS σdist σdisp area σdist σdisp area b100 - - - 0.33 0.20 1.02E3 b102 0.8 0.3 31.79 0.30 0.16 53.13 b124 0.86 0.39 13.42 0.33 0.19 14.79 b143 0.99 0.45 22.91 0.49 0.34 23.79 badvoro 2.29 0.65 3.01E3 0.31 0.26 318.66 mode 0.97 0.54 10.84 0.38 0.27 49.45 ngk10 4 0.48 0.26 0.52 0.22 0.13 2.30 NaN 0.56 0.28 5.04 0.26 0.15 5.10 dpd 0.48 0.32 0.45 0.37 0.29 1.30 root 4.09 0.94 6.93E9 0.29 0.22 950.01 rowe 0.49 0.26 0.95 0.27 0.12 2.10 size 0.62 0.35 1.27 0.32 0.20 4.14 unix 0.6 0.35 0.85 0.26 0.13 2.35 xx 0.97 0.34 58.83 0.29 0.14 74.00 66 Gansner and Hu Efficient Node Overlap Removal

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da0d7bbcf30 e3fd0c7b3 52a6c2063bccd83110c32 ff ffd1b1af3b6864078f37e74e1587f3a4d208 30d6138b63eb c72df69b40156a3254 2601085bde1b2450d64509f36 38fa61352f5086d2cb51 04390dec6f1779353c07f5 8df 342E 26fa7360ab81be9d4434a e287d497450664a4c0f4efc338 8cbcf5cb5 9ccd974150a18260b207b6584caa 86633c26be93ada8b e33792703 c360aaeb167487c9578a8f 8c7cd9b93b1cbe48e1 8bdb6a91c6dee925b557c705b3 8066f87a88f4e b9 07 99da1f8a5 31055116421c96b37f72a262bb ff 8f9cdc26798117dd3e9ee4a8770 8df

382E 2d7b7fb6c9ad6821752651f7 275afb2b215b966d9fac51b96b9 8df a2eab7c9fa641e5f

23f94655294d3ff537f2915fa 901b2105a902ee7791 88a4f0337c2189c3fc7b31 0fabab47 a77622f2ff77ffeeb2 8df a77622f2ff77ffeeb2 0a963fef9 444E 4869e993f2bb10f 3cfae 354E 696E 300E 370E 597E 8b06a67a 06eff1db45cdf c0f34a600 bf65cfddeb00ff847feae0c 81dabfaba8 3eecb304bab2136a76deda 08500a6044 fff03efcd 1562abef0d8241 684E b38049e00 9937ccba1469582E 341E ffd1b1af3b6864078f3

ac284d73563 620E 7e74e1587f3a4d208 1d4ea80c7194689d69f9592186616d8a7b630E 8f9cdc26798117dd3e9ee4a8770 797E a2eab7c9fa641e5f 0a963fef9 582E bb92d1da1f38d52f8ff 91ecbe29a71f00ed5a3342E354E 4869e993f2bb10f 1b82200 c5d5be3942 f713a8b311ffa05ce3683ad10 8b06a67a fbdc3ca37406f66635c8b226e 0da9135 fe821bce bf65cfddeb00ff847feae0c 3eecb304bab2136a76deda 422E 234E 52a6c2063bccd83110c32 9937ccba1469 534E 630E 381E 1b13908a9f0763c0ae54af90620808c7cd9b93b1cbe48e1382E620E684E696E300E370E444Ee287d497450664a4c0f4efc338 1d4ea80c7194689d69f9592186 86633c26be93ada8b 616d8a7b

a8e9 503737b64d432c60d6ac557e0e6a78eb40ae56aaa9 a8e9422E234E 3dafe1619016463f521f 91ecbe29a71f00ed5a3

dcc5d5e9d6c4e b62cd1d0a0 342E 354E

444E e64f22a31 81dabfaba8 382E 3cfae07 82d201 bf141dbce 0f167280 8c7cd9b93b1cbe48e1 a8e9 07 369E a8e9

1172dca23 370E 341E6472c2861e0e0dd681bd7aca295ca46e412a3 619E 910b00285f11bb90d0a15641 696E 300E

1b13908a9f0763c0ae54af9062080

dcc5d5e9d6c4ea8e9 95e93c4a1324dfe1a997a3a8173d6c86e9b042865a11d529eb4 a78eb40ae56aaa9 620E 1d4ea80c7194689d69f9592186 d370c12dbc 3f9ddf1ffbc0d38b c96cb3 683E e287d497450664a4c0f4efc338 c72df69b40156a3254

684E 82d201 3f9ddf1ffbc0d38bb62cd1d0a0381E369Ea8e959a8b435ccd5a7a610a8a8f143077ef0f45e70733e1de99da1f8a5e3fd0c7b347b2da3d

234E 422E a0befe6dd1ca7b165786835619E683Ea8e907 c0f34a6000da91350fabab471b82200534E 01 233E 81dabfaba8 503737b64d432c60d6ac557e0e6

a8e9 676bbe7d1c1fb71742df534ce8233E01 2d7b7fb6c9ad6821752651f7bf141dbce30d6138b63ebe64f22a310f167280fe821bce c96cb38cbcf5cb5 d886e4b76537a99bc71b8a9331c94 a0befe6dd1ca7b165786835 07

95e93c4a13 3cfae

c1c30f30d40c4f1f84924622f 6472c2861e0e0dd681 80874aa8 37d80ae421dba4a70730338860 ffccaa9e381dabfaba8 341E 65a1 bd7aca295ca 533E 24dfe1a997a dcc5d5e9d6c4e 46e412a3 fa2afbd869 81dabfaba8

65a1 6472c2861e0e0dd681 a8e9 95e93c4a13 162d8039483d83b1563a23eb945827dbdd8c5d5be3942 37d80ae421dba4a70730338860 3b1563a23eb9

a8e9 86e9b0428

381E 24dfe1a997a

e82f47b8d4949ba4af69b38cbc199f24ba80192c339a64c0f8128d574c356631b8a9 0acc5bb8ca4fbdc3ca37406f66635c8b226ef713a8b311ffa05ce3683ad10eb8eb881dabfaba8 3a8173d6c 1d529eb4 e82f47b8d4949ba4af69b38cbc19

8f143077e e3fd0c7b3

5a7a610a8a 47b2da3d 6448451ac2ceade90715378b5eea496cc301b2a9721b06bbfa920aa95ddac487676a04e4 45827dbdd8eb8eb8eb8 d370c12dbc

23f94655294d3ff537f2915fa 275afb2b215b966d9fac51b96b9 33e1de 99da1f8a5 6aae8d25951 5e9156098c06446f754ea06f070dbc023e571a876 0acc5bb8ca445827dbdd8910b00285f11bb90d0a1564181dabfaba880874aa8 3a8173d6c 3f9ddf1ffbc0d38b 5a7a610a8a 81dabfaba8 3b2f08d52e4bca3f9ca7bbbd6 b62cd1d0a0 f0f45e707 b06bbfa920aa95dd 8f143077e 86e9b0428 81dabfaba8 f0bd1521 369E 59a8b435ccd 07 0fabab47 bf01c8a262 c1c30f30d40c4f1f84924622ffa2afbd86981dabfaba8 533E

a8e9 c0f34a600

619E 1b82200 162d8039483d8 f8128d574c356631b8a9 9f24ba80192c339a64c0 11bb8ed3ae227d3acefc08dade990b22823b2f08d52e4bca3f9ca7bbbd681dabfaba8 0f94064681dabfaba8f0bd1521 683E 0da9135 33e1de 0f940646 a0befe6dd1ca7b165786835 534E fe821bce 676bbe7d1c1fb71742df534ce8 1d163eac141def176461c6c632ad9c42228bb337f78e145a127014eb43345a0c ac487676a04e4 f78e145a127014eb43345a0c 30d6138b63eb 233E a2eab7c9fa641e5f f0f45e707 8cbcf5cb5 84e4f1c582427a98d7bea890ccca2f7c2107351dca32af03698c988b2218115fa32ff1cb99 bf141dbce 5eea496cc301b2a9721 59a8b435ccd fb039d7a2a9fe73b5f468eba9 e64f22a31 cd0d985a366cad7e173fd00917644f0f1f3e30c72a426929600000f53cdc90c57243373efaba65afb039d7a2a9fe73b5f468eba9a27037332d9fa5c43bcfe94c0 2d7b7fb6c9ad6821752651f7 01 c96cb3

0f167280 b38049e00 24431c3eb075102f07cc2c1be75ba8d840070942eb4e737849e3bdbca0e2256fffa8a59018 37d80ae421dba4a70730338860 5e9156098c064 162d8039483d8 ffccaa9e3 81dabfaba8

c5d5be3942 07f8a9e55a16beddb3c9153b04a38abc630c82b0c48dfbf527186276bb1e23f2c7ffcbe82a0 3b1563a23eb9 a78eb40ae56aaa9 bf01c8a262 0acc5bb8ca4 ffccaa9e3 45827dbdd8 9c9e2e0f2da8758e436c a27037332d9fa5c43bcfe94c0 f713a8b311ffa05ce3683ad10 6448451ac2ceade90715378b eb8 9f24ba80192c339a64c0

cd0d985a366cad7e eb8

eb8 45827dbdd8 24431c3eb075102f07cc2c1be e3bdbca0e2256fffa8a59018 eb8 fbdc3ca37406f66635c8b226e

0acc5bb8ca4 81dabfaba8 3cdc90c57243373efaba65a f8128d574c356631b8a9 ffd1b1af3b6864078f3 e82f47b8d4949ba4af69b38cbc19

eb8 676bbe7d1c1fb71742df534ce8 b06bbfa920aa95dd 91ecbe29a71f00ed5a3 eb8 eb8 eb8 6aae8d25951 ac487676a04e4 45827dbdd8 45827dbdd8 9c9e2e0f2da8758e436c 6448451ac2ceade90715378b

eb8 d370c12dbc

0a963fef9 4a38abc630c82b0c48dfbf5271 75ba8d840070942eb4e737849 eb8

5eea496cc301b2a9721 6aae8d25951 45827dbdd8 7e74e1587f3a4d208 46f754ea06f070dbc023e571a876 910b00285f11bb90d0a15641 0acc5bb8ca4 81dabfaba8

45827dbdd8 80874aa8

11bb8ed3ae227d3acefc 5e9156098c064 8b06a67a 1d163eac141def176461c 46f754ea06f070dbc023e571a876 08dade990b2282 0acc5bb8ca4

07f8a9e55a16beddb3c9153b0 86276bb1e23f2c7ffcbe82a0 bf01c8a262 fa2afbd869

1b13908a9f0763c0ae54af9062080 533E

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

Figure 4: 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 nsvltn67 orlnfl71orlnfl75 dnvrco69 okcyok60 stlsmo12 stlsmo63 orlnfl76 tampfl80 jcvlfl64 rlghnc70 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 ftwotx63ftwotx65 brhmal75nworla63nworla62nworla60nworla67nworla66 tampfl81 ojusfl64 wpbhfl64 anhmca69 phnxaz12 tcsnaz64 dllstx77 atlnga56 atlnga70 orlnfl81 atlnga80 chrlnc60 rlghnc61 orlnfl67 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 brhmal60 tampfl60 anhmca65 phnxaz69 stlsmo13 dllstx73 atlnga59 atlnga78 atlnga16 chrlnc61 rlghnc68 stlsmo76hstntx64 brhmal81 tampfl64 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 orlnfl72 hstntx75 brhmal71 tampfl70 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 stlsmo14 dllstx74 ftwotx61 hstntx60 brhmal76 nsvltn81 grdnca60 lsanca72 kscymo68 kscymo65 kscymo61 cncnoh81 phlapa58 washdc80 nybwny82 nybwny74 anhmca75 stlsmo60 atlnga71 atlnga80chrlnc60 stlsmo69 tampfl62 wpbhfl60 wpbhfl65 anhmca65anhmca67 phnxaz67phnxaz12 stlsmo67 atlnga56atlnga64atlnga70 rlghnc62 stlsmo73 brhmal64 brhmal63 brhmal62 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 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scrmca69 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 chcgil36phlapa60phlapa27 rcpknj80 rcpknj61 snfpca70 phnxaz74 lsanca28 bltmmd64 nwrknj74 nwrknj80 rcmdva61 nwrknj20 snjsca73 snjsca81 ptldor81 lsanca33 chcgil77 iplsin67 cncnoh15 pitbpa80 washdc69 washdt60 nycmny28 rcmdva62 shokca67 nwrknj65 cmbrma63 waynpa62 snfpca69 kscymo75 cncnoh71 washdc81 phlapa61 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 scrmca71 ptldor61 snfcca10 omahne62 desmia62 chcgil78 dtrtmi61 pitbpa64 nycmny56 nycmny29 cdknnj81 cdknnj71 cmbrma68 cncnoh67 washdc69 chcgil81 nycmny82 nybwny80 cmbrma61 dnvrco73 cncnoh15 shokca66 slkcut65 snfcca30 chcgil82 cmbrma60 snfcca71 snfcca73 pitbpa80 nwrknj63 okldca60 okldca61 scrmca63 snfcca81 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 cmbrma65 waynpa67 cmdnnj62 cmbrma13 okldca63 lsanca75 scrmca65 snfpca80 cmbrma62 waynpa80 rcpknj62 snfcca28 nycmny56nycmny80 cmbrma68 slkcut60 chcgil61 cncnoh80 whplny69 ptldor70 ptldor80 sttlwa76 rlmdil61 dtrtmi67 dtrtmi63 nycmny55 clmboh81 hrfrct81 grcyny80 cdknnj68 cmbrma58 scrmca68 nycmny69 cdknnj71 whplny66 scrmca62 snfcca10snfcca27 chcgil35 phlapa27 phlapa67 ptldor63 chcgil75 nycmny76 hrbgpa60 snfcca20 lsanca60 lsanca62 nycmny81 bltmmd60 nycmny35 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 lsanca67 snfcca62 chcgil68 washdc73 hrbgpa80 omahne62 rlmdil65 nycmny74 cmdnnj60 whplny68 snfcca83 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 iplsin66 cmbrma68 snjsca72snjsca74spknwa80 sttlwa71sttlwa72 rlmdil62 grcyny64 cmbrma59 snfcca70 omahne80 lsanca33 chcgil77 chcgil73 rlmdil80 washdc83 hrbgpa61 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 scrmca70 snfcca26 snfcca25 sttlwa82 okbril80 cmbrma53 snjsca71 snjsca69 nycmny76 nycmny77 washdt61 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 sttlwa75 sttlwa73 rlmdil60 dtrtmi60 cmdnnj60 ptldor12 desmia68 iplsin62 clmboh70 artnva68artnva61whplny62 cmbrma71 ptldor60 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 dtrtmi67 pitbpa65 okbril70 mplsmn69 chcgcg64 milwwi65 iplsin65 clevoh63 nycmny33 cdknnj82 hrfrct60 whplny65 cmbrma78 spknwa61 sttlwa66 milwwi60 iplsin64 whplny61whplny64 ptldor65 ptldor80 ptldor62 sttlwa74 sttlwa76 rlmdil63 dtrtmi63 whplny80 slspmd62 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 sttlwa60 okbril62 chcgcg80 clevoh60 cmbrma73 milwwi63 spknwa60 ptldor07 ptldor14 ptldor12 nycmny33 hrfrct60 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 pitbpa68 cmbrma70 cmbrma75 mplsmn60 milwwi63 clevoh81 clmboh63 cdknnj72 albyny63 hrfrct02 mplsmn67 albyny60 okbril64 stplmn64 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 mplsmn60 dtrtmi74dtrtmi79 dytnoh64 rlmdil68 clevoh68 hrfrct73 stplmn72 dtrtmi72 akrnoh62 mplsmn61 mplsmn62 clevoh67 hrfrct69 hrfrct04 rlmdil70 milwwi67 clevoh68 bflony64 hrfrct72 hrfrct73 hrfrct04 dytnoh62 akrnoh61 stplmn70 rlmdil69 milwwi67 milwwi71 hrfrct05 stplmn71 rlmdil69 milwwi71 clevoh67 bflony63 hrfrct74 hrfrct05 dytnoh61 stplmn80 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

Red Hat, Inc Red Hat, Inc

Red Hat, Inc

Peter Jones Peter Jones

CERT Coordination Center CERT Coordination Center FreeBSD Security Officer FreeBSD Security Officer Peter Jones

CERT Coordination Center FreeBSD Security Officer

CERT Coordination Center CERT Coordination Center

Linus Torvalds Linus Torvalds CERT Coordination Center Paul S. Traina Paul S. Traina Hunter Matthews (Duke Biology Unix Sysadmin) Hunter Matthews (Duke Biology Unix Sysadmin) Linus Torvalds Paul S. Traina Stephen C. Tweedie (Key signing key)

Stephen C. Tweedie (Key signing key) Hunter Matthews (Duke Biology Unix Sysadmin) Jamie Wilkinson CERT Coordination Center Jamie Wilkinson H. Peter Anvin (hpa) CERT Coordination Center

Daniel Stone (NEC Business Solutions) Stephen C. Tweedie (Key signing key) Daniel Stone (NEC Business Solutions) Linux Kernel Archives Verification Key Andrew G. Malis Andrew G. Malis Jamie Wilkinson CERT Coordination Center H. Peter Anvin (hpa) Eugene H. Spafford AUSCERT Eugene H. Spafford David Elson (Del) Matt Crawford Daniel Stone (NEC Business Solutions) David Elson (Del) Mark Crichton (preferred key) Matt Crawford AUSCERT Linux Kernel Archives Verification Key H. Peter Anvin (hpa) Andrew G. Malis Mark Crichton (preferred key) Theodore Ts’o [SIGNATURE] Eugene H. Spafford Theodore Ts’o [SIGNATURE] Peter Nai Wan David Elson (Del) AUSCERT Andrew S. Nagy Matt Taggart Linux Kernel Archives Verification Key Mark Crichton (preferred key) Matt Crawford Cory F. Cohen Cory F. Cohen Peter Nai Wan Philip R. Zimmermann Peter Nai Wan Andrew S. Nagy Roger So Philip R. Zimmermann Roger So Citizen Gold (MedeMail - Babel Client) Matt Taggart Theodore Ts’o [SIGNATURE] Citizen Gold (MedeMail - Babel Client) Tim O’Shea [tmo] dann frazier Jeffrey I. SchillerLaMont Jones Peter Nai Wan Cory F. Cohen dann frazier Roger So Andrew S. Nagy Matt Taggart Peter Nai Wan Jeffrey I. Schiller LaMont Jones Werner Koch Citizen Gold (MedeMail - Babel Client) Philip R. Zimmermann Douglas F. Calvert (High Security Key Use 0x13300731 For Everyday Communication) Tim O’Shea [tmo] Werner Koch (gnupg sig) Werner Koch (gnupg sig) Theodore Y. Ts’o Werner Koch Tim O’Shea [tmo] Peter Nai Wan M. Drew Streib Theodore Y. Ts’o [email protected] [email protected] dann frazier LaMont Jones DouglasEmily F. ManheimCalvert (High Security Key Use 0x13300731 For Everyday Communication)M. Drew Streib Jeffrey I. Schiller Emily Manheim Stanton McCandlish (POP acct. for file attachments) Werner Koch (gnupg sig) Emily Manheim Werner Koch Stanton McCandlish (POP acct. for file attachments) Christoph Martin Douglas F. Calvert Douglas F. Calvert (High Security Key Use 0x13300731 For Everyday Communication) Theodore Y. Ts’o [email protected] David Walter Brian D. Webb David Walter Douglas F. Calvert WichertChristoph Akkerman Martin M. Drew Streib Dave Del Torto Bradley M.L. Kuhn Sassaman Douglas F. Calvert Dave Del Torto Wichert Akkerman Stanton McCandlish (POP acct. for file attachments) Christoph Martin K. Reid Wightman Brian D. Webb K. Reid Wightman David Walter Wichert Akkerman Douglas F. Calvert Robert van der Meulen Brian D. Webb Robert van der Meulen Dan Nowacki Clint Adams (Debian) Noah Meyerhans Clint Adams (Debian) L. Sassaman Mike Fratto Dave Del Torto Douglas F. Calvert K. Reid Wightman Bradley M. Kuhn L. Sassaman Robert van der Meulen Mike Fratto Douglas F. Calvert Dr. Michael Meskes J. Daniel Backman Dr. Michael Meskes J. Daniel Backman Clint Adams (Debian) Jack M Twilley Jack M Twilley Bradley M. Kuhn Mike Fratto Dan Nowacki Noah Meyerhans Dan Nowacki Charles Gallo at Work Michael StoneNik Clayton Wichert Akkerman Dr. Michael Meskes Michael Stone J. Daniel Backman Nik Clayton Wichert Akkerman Charles Gallo Daniel Jacobowitz Noah Meyerhans Eric S. Raymond Jack M Twilley mauri stott Charles Gallo at Work Michael Stone Nik Clayton Wichert Akkerman mauri stott Charles Gallo at Work Martin Schulze Daniel Jacobowitz Martin Schulze Charles Gallo Daniel Jacobowitz Charles Gallo Gregory L. Pratt Bill Scannell mauri stott Eric S. Raymond Martin Schulze Gregory L. Pratt Bill Scannell

Martin Schulze Eric S. Raymond Martin Schulze Gregory L. Pratt Bill Scannell Daniel Jacobowitz Martin Schulze Daniel Jacobowitz Daniel Jacobowitz

Miro Jurisic Miro Jurisic

Miro Jurisic

Leland Wallace Leland Wallace Leland Wallace

Tim Holmes Tim Holmes Leonard D. Rosenthol Tim Holmes Leonard D. Rosenthol Leonard D. Rosenthol

Vinnie Moscaritolo Vinnie Moscaritolo Vinnie Moscaritolo Darxus Darxus Darxus

Woz Woz Woz b124 b124+PRISM b124+VPSC

Figure 5: Comparing PRISM and VPSC on two graphs. Original layouts are scaled 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. 6 Results shown are σdist, σdisp and area. Area is measured with a unit of 10 square points. Initially the layout is scaled to an average edge length that equals 4 times the average label size.

Graph PRISM VPSC σdist σdisp area σdist σdisp area b100 0.74 0.36 14.59 0.95 0.62 25.33 b102 0.41 0.18 2.62 0.44 0.33 2.77 b124 0.52 0.12 3.39 0.28 0.06 3.04 b143 0.5 0.22 2.59 0.67 0.41 4.26 badvoro 0.35 0.15 11.85 0.65 0.58 11.71 mode 0.57 0.35 0.78 0.84 0.59 1.45 ngk10 4 0.25 0.05 0.54 0.16 0.03 0.53 NaN 0.25 0.06 1.15 0.15 0.04 1.15 dpd 0.14 0.03 0.54 0.1 0.03 0.53 root 0.73 0.28 16.02 0.73 0.7 25.93 rowe 0.16 0.04 0.38 0.11 0.03 0.38 size 0.35 0.11 0.54 0.28 0.08 0.57 unix 0.25 0.07 0.6 0.13 0.04 0.59 xx 0.39 0.18 4.23 0.45 0.34 4.54 Graph VORO ODNLS σdist σdisp area σdist σdisp area b100 - - - 0.49 0.24 1.09E3 b102 1.18 0.47 131.55 0.33 0.15 41.01 b124 0.69 0.36 19.09 0.43 0.26 14.64 b143 0.84 0.39 19.35 0.57 0.36 29.39 badvoro 3.8 0.9 4.45E5 0.34 0.10 166.53 mode 1.09 0.58 20.02 0.44 0.28 45.21 ngk10 4 0.46 0.2 0.79 0.23 0.12 2.21 NaN 0.49 0.22 2.28 0.33 0.18 4.86 dpd 0.43 0.23 0.6 0.38 0.29 1.45 root 4.14 0.94 3.68E8 0.39 0.14 1.52E3 rowe 0.41 0.2 0.68 0.25 0.11 2.01 size 0.62 0.34 2.07 0.28 0.16 3.35 unix 0.55 0.24 0.64 0.20 0.09 2.01 xx 1.13 0.44 185.7 0.30 0.11 45.18 68 Gansner and Hu Efficient Node Overlap Removal

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

sizes # PRISM VPSC VORO ODNLS σdisp area σdisp area σdisp area σdisp area 10–19 1407 0.009 0.16 0.09 0.061 0.14 0.069 0.07 0.34 20–29 839 0.02 0.27 0.11 0.12 0.14 0.15 0.078 0.92 30–39 2036 0.03 0.36 0.13 0.19 0.13 0.26 0.078 1.85 40–49 1800 0.035 0.43 0.14 0.25 0.12 0.36 0.074 2.71 50–59 1045 0.042 0.51 0.15 0.32 0.11 0.53 0.074 4.53 60–69 1172 0.046 0.58 0.17 0.39 0.11 0.70 0.074 6.18 70–79 1008 0.051 0.64 0.17 0.46 0.11 0.94 0.073 8.44 80–89 788 0.05 0.71 0.17 0.52 0.10 1.20 0.07 9.99 90–99 1296 0.054 0.78 0.18 0.59 0.10 1.49 0.069 13.26 100–109 140 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 has a total of 11534 graphs5 of relatively small size. Due to space limitation, we only give the similarity measure σdisp and the area, and we average the results over graphs of similar sizes. Again, PRISM achieves the best compromise between being close to the original drawing, and having a smaller drawing area. We note that while there is no theoretical result guaranteeing that PRISM algorithm converges to an overlap free layout in a finite number of iterations, in practice, out of thousands of graphs tested, some as large as tens of thousands of vertices, PRISM always converges within a few hundred total number of iterations in the two main loops in Algorithm 1. In our implementation we set a limit of 1000 iterations, even though this limit has never been observed to be reached. Table 5 gives the number of iterations taken for the 14 test cases in Tables 2-3. As can be seen, for these graphs, the maximum number of iterations is 122. 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 [32]. 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 lay out using SFDP, and 15 seconds post-processing using PRISM for overlap removal. PRISM converges in 81 itera- tions. Important mathematicians (those with the most offspring) and important edges (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 the layout is scaled to an average edge length that equals 4 times the average label size.

Graph |V | |E| A B b100 1463 5806 122 75 b102 302 611 66 44 b124 79 281 43 19 b143 135 366 46 26 badvoro 1235 1616 53 32 mode 213 269 43 26 ngk10 4 50 100 29 8 NaN 76 121 34 9 dpd 36 108 26 6 root 1054 1083 103 119 rowe 43 68 28 7 size 47 55 29 14 unix 41 49 32 8 xx 302 611 53 46

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

6 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 contributions 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 70 Gansner and Hu Efficient 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 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 imple- mentations of VPSC and ODNLS, respectively, available to us. We would like to thank Yehuda Koren and Stephen North for helpful discussions, and Stephen Kobourov for bringing our attention to the term “Frobenius metric”. Finally, the reviewers’ comments were very helpful, especially in making the exposition clearer and more accurate. JGAA, 14(1) 53–74 (2010) 71

Mattheou Karagrigoriou

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Figure 6: The second largest component from the Mathematics Genealogy Project. Top: overall layout with node overlap removed. Bottom: close up view of a small area of the center-left part of above. 72 Gansner and Hu Efficient Node Overlap Removal

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