Using Spring Algorithms to Remove Node Overlapping

Wanchun Li, Peter Eades and Nikola Nikolov

National ICT Australia Ltd,University of SydneyNSW, 2006, Australia

Email: lpaul@it.usyd.edu.au, {peter.eades,nikola.nikolov}@nicta.com.au

Abstract

Cluttered drawings of graphs cannot effectively con-vey the information of graphs. Two issues mightcause node overlapping when one draws a picture of agraph. The first issue occurs when applying a layoutalgorithm for an abstract graph to a practical appli-cation in which nodes are labeled. The second is thechanging of a node’s size in a dynamic drawing sys-tem. This paper presents two algorithms, DNLS andODNLS, for removing the two kinds of overlapping.The algorithms are based on the well-known springembedder model. The outputs of the algorithmsprovide the features of spring algorithms. Experi-ments are carried out to compare DNLS and ODNLSto the Force Scan(FS) algorithm and its variants.The results demonstrate the advantages of DNLS andODNLS in terms of some aesthetic criteria.

Keywords: graph drawing, node overlapping, springalgorithm

1 Introduction

Graphs are abstract structures that are used to modelrelational information. Good pictures of a graph canconvey the information rapidly and efficiently. Manysuccessful algorithms have been introduced to au-tomatically generate the layout of a graph(Battistaet al. 1998). Most graph layout algorithms treatnodes as points. Such an algorithm is a layout cre-ation algorithm(Eades et al. 1995). A layout gener-ated by a layout creation algorithm might have over-lapping nodes when the layout is applied to practi-cal applications, in which nodes are labeled and aredrawn as boxes or other symbols. Overlapping mightalso occur in a dynamic graph drawing system. Whena user opens some node for more details, the enlargednode might overlap its neighbour nodes.

Cluttered drawings of graphs cannot effectivelyconvey the information of the graphs. Eadeset al. first address the issue of removing nodeoverlapping(Eades et al. 1995). Before giving anyspecific algorithm, they introduce an important con-cept, the mental map. Three mathematical modelsare constructed to define the mental map. One of themodels is orthogonal ordering. Orthogonal orderingmodels the most basic mental map, which preservesup, down, left, and right. Suppose a node v∈V hasthe positions (x′v,y′v) and (xv,yv) in the drawings ofΓ′ and Γ respectively. Then Γ′ preserves the mentalmap of Γ if for ∀u ∈V,

Copyright c©conference year, Australian Computer Society,Inc. This paper appeared at Asia Pacific Symposium on Infor-mation Visualisation (APVIS 2005), Sydney, Australia. Con-ferences in Research and Practice in Information Technology,Vol. 45. Seok-Hee Hong, Ed. Reproduction for academic, not-for profit purposes permitted provided this text is included.

• x′v < x′u ⇔ xv < xu, and

• y′v < y′u ⇔ yv < yu, and

• x′v = x′u ⇔ xv = xu, and

• y′v = y′u ⇔ yv = yu

They then propose three requirements for a layoutadjustment:

1. The adjusted drawing should be compact.

2. Node images should be disjoint.

3. The mental map should be preserved during in-teraction.

One simple solution is to uniformly scale all edges.The overlapping nodes can be disjoint, and the mentalmap is preserved. However, the generated layout isusually not compact.

Eades et al. present the Force Scan(FS) algorithmfor adjusting a cluttered layout (Eades et al. 1995).The algorithm scans in both horizontal and verticaldirection to find overlapping nodes, and move theoverlapping nodes to new positions. The new posi-tions of overlapping nodes are determined by the de-sirable distance. The desirable distance is the shortestdistance to move two overlapping nodes apart alongthe line that connect their centers. The algorithm it-erates the scanning and moving process until there isno further overlapping. They treat all nodes as rec-tangle. If the image of a node is not a rectangle, thebounding box will be considered.

Hayashi et al. prove that it is a NP-hard totransform a given layout of a graph with overlappingrectangular nodes into a minimum-area layout with-out node overlapping which preserves the orthogo-nal order(Hayashi et al. 1998). They also propose aheuristic FS′ that is an improvement of FS. Givena node vi, a set of nodes overlapped {v1, v2, ...,vn}, and the desirable distances {d1, d2, ..., dn}, FSmoves in one iteration all the overlapping nodes bythe distance max(di), 1 ≤ i ≤n, while FS′ movesmin(di). A mathematical proof demonstrates thatFS′ can generate a smaller area of layout than FS.Huang and Lai introduces another version of FS, theForce-Transfer(FT )(Huang, & Lai 2003). Instead ofmoving nodes along the line across the centers of twooverlapping nodes, FT moves the overlapped nodeseither horizontally or vertically. Given a pair of over-lapping nodes, dh is the distance to horizontally movethe nodes to remove the overlapping, and dv is thedistance in the vertical direction. If dv < dh, theoverlapped node is moved vertically, otherwise hori-zontally. Empirical experiments show that the outputof FT has a smaller area than that of FS. All of FS,FS′ and FT can preserve the orthogonal order.

Lyons et al. propose different requirements ofadjustment(Lyons et al. 1998). There are three crite-ria with different priorities. The first is to keep the

adjusted layout in a window with the same size as thewindow of the cluttered layout. The second is moreeven distribution of nodes. Their approach weak-ens the restriction of similarity. Instead of preservingthe exact ordering of two layouts, the approach triesto minimise the difference between them. In otherwords, it tries to make the two layouts “not very dif-ferent”. This is the third criterion, and its priorityis the lowest. They present two algorithms, namely,VDCB and GeoForce. VDCB is based on Voronoidiagram, and GeoForce is an application of the force-directed model. Empirical experiments show thatboth the algorithms satisfy the goals of even distri-bution and similarity in corresponding priorities. Itis not clear how to keep the size of a layout if there isnot enough space for a dense graph or if the labellednode sizes are too big to fit in the fixed area. Gansnerand North propose an algorithm inspired by VDCB(Gansner, & North 1998). The algorithm removes therestriction of the fixed size of the window, which is thefirst goal of VDCB. The algorithm first generates theVoronoi diagram using the node positions in a clut-tered layout as sites. Then it moves each node to thecentroid of its Voronoi cell. It iterates this processuntil there is no longer any overlapping. Experimentsshow that the algorithm has positive results in termsof the area of the drawings.

Marriott et al. view removing overlapping asa constrained optimisation problem(Marriott et al.1995). Four algorithms are designed based onquadratic programming. Experiments show that ittakes the algorithms less time to remove overlappingthan FS needs, but more time to find overlappingthan FS. They state that the layout generated by thealgorithms is better than the output of FS, but nomeasurement for evaluation is mentioned. This al-gorithm does not guarantee to preserve the mentalmap.

All mentioned solutions are post-processing ap-proaches, and are called layout adjustment algo-rithms. Generally, there are three steps to producean uncluttered layout using these approaches. First,an initial layout is generated by a layout creation al-gorithm. Second, labels of nodes are added to thepicture. Finally, the adjustment algorithm cleans thelayout if there is overlapping.

Another general approach is to integrate thelayout adjustment process and the layout creationprocess. Force-directed model is used in such meth-ods. Kamps, Kleinz and Read add to the classicalspring embedder model (Eades 1984) a “node-noderepulsion force” that minimises the overlapping areaof two nodes(Kamps, Kleinz & Read 1995). Wangand Miyamoto increase the repulsive force but can-cels the attractive force if a pair of nodes are over-lapping each other(Wang & Miyamoto 1995). (Harel&Koren 2002) calculates repulsive force using a vari-ant value lb. lb is the distance between the bound-aries between a pair of nodes. If a pair of nodes areoverlapped, lb is a small constant vale ε. The out-put of such algorithms has the advantages of force-directed graph drawing algorithm, eg. display of sym-metry and uniform edge length. A common problemof these algorithms is that they do not guarantee tocompletely separate all overlapping nodes. Note thatthere is no need to preserve the mental map, becausethe process of removing the potential overlapping ismerged into the process of layout creation. In otherwords, there is no “initial layout” at all. However,the lack of a mechanism for preserving the mentalmap makes these algorithms unsuitable for dynamicdrawing systems.

In this paper, we present two algorithms for re-moving overlapping nodes. One is designed for staticdrawings of graph; the other is for dynamic draw-

ings of graphs. The two algorithms are based on thespring embedder model. In the classical spring em-bedder models, the repulsive and attractive force arecalculated by taking two distances into account. Oneis the current distance of two abstract nodes d; theother is the expected balance distance, natural lengthk, between nodes. In such models, k is a constantvalue for all nodes. The proposed algorithms calculatethe forces using a dynamic natural length instead of afixed natural length. The dynamic natural length k′takes into account the sizes of nodes. k′ varies and isdynamically calculated according to the pair of nodesinvolved.

The first algorithm is called Dynamic NaturalLength Spring(DNLS). DNLS is a layout creation al-gorithm which can generate uncluttered layout. It isdesigned for static graph drawing and does not con-sider preserving mental map. The other algorithm isbased on DNLS but has the ability to preserve themental map. We name this algorithm OrthogonalDynamic Natural Length Spring(ODNLS). ODNLS isdesigned for dynamic graph drawing.

DNLS integrates the layout creation process andthe adjustment process, while ODNLS processes alayout generated by another spring algorithm. Theuncluttered layout generated by both algorithms hasthe features of a spring algorithm: display symmetryand uniform edge length.

In this paper we consider nodes with rectangularshape. This is commonly accepted for representationof labeled graph nodes. Our results are also applica-ble for non-rectangular node shapes if we considerthe corresponding rectangles that enclose the nodeshapes.

Empirical experiments are carried out to compareDNLS and ODNLS with FS and its variants, FS′ andFT. The results demonstrate that DNLS/ODNLS im-proves on the FS and its variants in terms of someaesthetic criteria. The measurements of evaluationare such aesthetic criteria as area, L1 metric length,aspect ratio, and edge length ratio. The effectivenessof removing overlapping and running time are alsotested.

The rest of the paper is organised as follows. Insection 2, we introduce some preliminaries. In sec-tion 3, we present our algorithms. In section 4, wedescribe our experiments and give an analysis of ex-periments results. In section 5, we conclude and posesome future work.

2 Graphs and Drawings of Graph

2.1 Graphs

Suppose that G = (V,E) is a graph. V is a finite setof nodes, and E is a finite set of edges. Each edgeis a pair of vertices of G. Nodes represent entities,and edges represent the relationship between entities.If the pair of nodes of an edge is unordered, G is anundirected graph, otherwise a directed graph. A nodevi is said to be adjacent to a vertex vi if vivj is anedge of G. If any two nodes are joined by a path, Gis a connected graph. A tree T is a connected graphwithout cycles.

2.2 Drawings of Graphs

A drawing of a graph is a representation of the graphin the Euclidean space. The drawing of an abstractgraph usually represents a node as a point and anedge as a curve. Sometimes, interaction with a draw-ing of a graph is needed for exploring more details oranalyzing the relationships between entities. The vi-sual elements of the drawing will change in response

to an interaction: some new nodes and edges mightbe displayed, some nodes might change their size andposition, and some nodes and edges might disappear.Such drawings are called dynamic graph drawings.

Some drawings are better than others in conveyingthe information of a graph. A drawing is sometimesmeasured by how well they meet the aesthetic criteria:

• minimisation of crossings

• minimisation of area

• minimisation of L1 metric length(see below fordetails)

• maximisation of symmetry

• minimisation of total edge length, uniform edgelength

• aspect ratio close to specified value

• minimisation of bends

• maximisation of smallest angular resolution

Suppose Γ is a drawing of a graph G in a 2D Euclid-ean space with height h and width w. L1 metriclength is equal to max(h, w). Aspect ratio is equalto max(h, w)/min(h, w). Angular resolution appliesto straight line drawings, and it refers to the small-est angle formed by two edges incident on the samevertex. Note that a drawing of a graph generally cannot simultaneously optimise all the criteria.

2.3 Spring Algorithms

In the drawing of undirected graphs there is a problemof “too much freedom”. The spring algorithm(Eades1984) is a successful layout creation algorithm fordrawing an undirected graph. The basic idea of thespring algorithm is to treat a graph as a mechanicalsystem, in which nodes are replaced by steel ringsand edges by springs connected to the rings. Allthe springs have the same natural length k, and eachspring has a current length d. Given a pair of ringsconnected by a spring, if k>d, then the spring at-tracts the rings; if k<d, then the spring repulses therings; if k=d, then the rings are stable. The springforces will attract or repulse the rings until the sys-tem reaches the minimum energy. This is called thebalanced state.

The strength of the forces is determined by kand d. Eades calculates the forces using the functions:

fa=C1 ∗ log(k/C2)

fr=C3/sqr(d)

where fa is the attractive force, fr is the repulsiveforce, and C1, C2, C3 are coefficients.

There are some variants of the spring algorithmthat adopt different force models. All of these algo-rithms are broadly called force-directed algorithms inmost literature. Eades and Lin review some of thesealgorithms and present a general spring system(Eades&Lin 1999).

The input of a spring algorithm is an undirectedgraph, and the output of the algorithm is a drawingof the embedded graph. The running time of a springalgorithm is O(n2), where n is the number of thenodes. The layout generated by a spring algorithmhas two features: display of symmetry and uniformedge length.

A spring system is not guaranteed to generate agood drawing for all graphs. The forces exerted onsome rings might be too strong or too weak, and

the spring system cannot arrive at the balanced state.One solution is to adjust the coefficients; another isto adopt different force models. Some implementa-tions, however, also set a fixed number of iterations,in order to end the spring forces in a predictable time.

2.4 Cluttered Layouts and Layout Adjust-ment Algorithms

An algorithm for drawing abstract graphs is a layoutcreation algorithm. In such a drawing, the images ofnodes are concrete points. The output of a layout cre-ation algorithm might have overlapping nodes whenlabels are added. Such overlapping is called staticdrawing overlapping in this paper. Another kind ofoverlapping is occurred in dynamic drawing systems.The new nodes or enlarged nodes might overlap theirneighbour nodes. This kind of overlapping is calleddynamic drawing overlapping in this paper.

Cluttered drawings can not convey information ef-fectively. It is essential for the drawings to remove theoverlapping. An algorithm that removes the overlap-ping of a layout is a layout adjustment algorithm.

Note that there are two possible kinds of overlap-ping in a drawing of a graph. One is the image ofa node overlapping the image of another node; an-other is the drawing of an edge crossing the image ofa node. In this paper, we only study the former case:overlapping among images of nodes.

2.5 Notations

In this paper, we denote a graph as G, and a drawingof G as Γ. An edge is referred to as e(v1, v2), wherevi and vj are the two end nodes connected by theedge. dvi,vj is the distance between the center pointsof vi and vj . dx

vi,vj is the distance of projection of thecenter points of vi and vj in the x direction, dy

vi,vj inthe y direction. fa is the attractive spring force in aspring system, and fr the repulsive spring force.

The images of nodes are rectangular shapes, thesides of which are parallel to the coordinate systemaxes. A drawing of a node in Γ is refereed to asvi(xi, yi, wi, hi), where xi and yi are coordinates ofthe node’s center point in the x and y directions, re-spectively, and wi is the width, and hi the height.

3 Algorithms

Most existing spring algorithms use a fixed naturallength k to calculate the spring forces. The gen-erated layout is usually a good drawing for an ab-stract graph. However, there might be overlappingnodes when labels are added. To fix the overlap-ping problem, we proposed variants of the springembedder model, namely, Dynamic Natural LengthSpring(DNLS) and Orthogonal Dynamic NaturalLength Spring (ODNLS).

3.1 Dynamic Natural Length Spring(DNLS)

DNLS is designed for solving the problem of staticdrawing overlapping. DNLS calculates the forces us-ing a dynamic natural length kd that is determined byboth k and the sizes of nodes.

Suppose the drawings of a pair of nodes v1, v2 arerectangles. The images of two nodes will not overlapeach other if dv1,v2 > k′,where

k′=l1/2 + l2/2 + m

l1 is the diagonal of v1l2 is the diagonal of v2

m is the desired margin between v1 and v2

we call k′ the clean distance. we define the dynamicnatural length kd between the nodes v1 and v2 as

kd=max(k′, k)

When the spring mechanism system calculates theforces between a pair of nodes, kd is used to deter-mine the strength of the forces. If there is overlappingbetween a pair of nodes, then dv1,v2 < kd. The springforce will repulse them. If dv1,v2 > kd, the nodesare not overlapping but the edge connecting them islonger than necessary. The spring force will attractthe two nodes. The spring system will not arrive atthe balanced state unless there are no overlappingnodes and the length of every edge is kd.

DNLS is a layout creation algorithm that canavoid overlapping. The idea of dynamic naturallength can be applied in any force model. In ourimplementation, we use the force models below:

fa = d/kd

fr = (kd/d)2

Figure 1 are two sample layouts generated byDNLS.

(a) Drawing of a graph with 9 Nodes

(b) Drawing of a graph with 10 Nodes

Figure 1: Drawings generated by DNLS

3.2 Orthogonal Dynamic Natural LengthSpring(ODNLS)

DNLS is a layout creation algorithm, and does notconsider the mental map. However, it is necessaryto preserve the mental map in dynamic graph draw-ing systems. ODNLS is a post-processing approachfor solving the problem of dynamic drawing overlap-ping. It preserves the mental map measured by the

orthogonal ordering model. The input of ODNLS isa cluttered layout. The output is a clean layout thathas the same orthogonal order as the input layout.

ODNLS uses dynamic natural length kd to calcu-late the energy of a spring system. An extra forcefe is added to the spring system in order to keep theorthogonal ordering the same as that of the input lay-out. fe works as follows.

Suppose v1(x1, y1, w1, y1), v2(x2, y2, w2, h2) aredrawings of a pair of nodes in Γ, x1 < x2, y1 < y2,kd is the dynamic natural length, and m the desiredmargin between the nodes in Γ. Let

p.left=min(x1, x2)

p.right=max(x1, x2)

p.top=min(y1, y2)

p.bottom=max(y1, y2)

If dv1,v2 < kd, fr will repel v1 and v2. Let rxand ry refer to the repulsive distances in the x and ydirections, respectively. For ∀ v(x, y, w, h) ∈ Γ, fesets:

• x := x + rx, if x >= p.right

• x := x− rx, if x <= p.left

• y := y + ry, if y >= p.bottom

• y := y − ry, if y <= p.top

In Figure 2(a), v1 and vs are a pair of overlappingnodes. When fr repulse them, vleft, vtop, vright, andvbottom are pushed by fe left, up, right, and down,respectively. The new drawing is in Figure 2(b).

(a)

(b)

Figure 2: fe repulsive effect

If dv1,v2 > kd, fa will attract v1 and v2. Let axand ay refer to the attractive distances in x and ydirections, respectively. Suppose a set of nodes Vkthat are between v1, v2 in x direction, and vleft is theleftmost node of Vk, and vright the rightmost. Alsosuppose a set of nodes Vj that are between v1, v2 in ydirection, and vtop is the uppermost node of Vj , andvbottom the lowermost. fe will move:

• v1 to where

– dxv1,vleft=w1/2 + wleft/2 + m, and

– dyv1,vtop=h1/2 + htop/2 + m

• v2 to where

– dxv2,vright=w2/2 + wright/2 + m, and

– dyv2,vbottom=h2/2 + hbottom/2 + m

In Figure 3(a), v1 and v2 are attracted by fa, whilefe will not allow the two nodes moving inside thedashed rectangle. Note that the margins between thedashed rectangle and vleft, vtop, vright, and vbottomare the desired margin between nodes in the drawing.Figure 3(b) shows the positions of v1 and v2 deter-mined by fe.

(a)

(b)

Figure 3: fe attractive effect

Now, Let ex1, ey1, ex2 and ey2 refer to the dis-tance between v1 and v2 covered by fe in the x andy directions, respectively. Then

• dx1 = min{0, ex1, ax}• dy1 = min{0, ey1, ay}• dx2 = min{0, ex2, ax}• dy2 = min{0, ey2, ay}

For ∀v(x, y, w, h) ∈ Γ, the spring system sets

• x := x + dx1, if x = x1

• x := x− dx2, if x = x2

• y := y + dy1, if y = y1

• y := y − dy2, if y = y2

By adding fe to the spring system, ODNLS canalso preserve the orthogonal order. Since ODNLS is avariant of spring model as well, the output of ODNLShas the features of the spring algorithms.

Figure 4 is a sample of ODNLS. Figure 4(a) isa cluttered drawing of a graph with 10 nodes. Fig-ure 4(b) is the output of ODNLS.

(a) Cluttered Initial lay-out(INI)

(b) Output of ODNLS

Figure 4: Sample of ODNLS

4 Experiments

4.1 Overview

DNLS and ODNLS are variants of classical springmodels. FS and its variants, FS′ and FT, are actu-ally simplified spring models, which have only repul-sive forces but no attractive forces. Our experimentsevaluate the outputs of the two types of algorithms.

In the experiment, we first generate a layout for adata set by a spring algorithm. The layout becomescluttered after labels are added. We call the clutteredlayout the initial layout(INI). We also generate a lay-out using DNLS. All of the other algorithms, ODNLS,FS, FS′ and FT, are adjustment algorithms, and eachof them produces a clean layout for INI. The experi-ments will evaluate these layouts.

The measurements for evaluation are area, L1 met-ric length, aspect ratio, edge ratio, orthogonal order-ing, running time, and effectiveness of removing over-lapping. Edge ratio refers to the ratio of the longestand the shortest edge length, which is used here toevaluate how even the edges are. We also test thesymmetry presented in the layouts by visual observa-tion. Note that the edge crossing is not considered inthe experiments. The reasons are: 1. the spring algo-rithms have no mechanism to reduce the edge cross-ing, 2. we only study the node overlapping and donot take into account the edges.

All nodes are assigned the same size. There aretwo reasons for this. First, uniform node size is ad-equate for most practice applications. A simple wayof keeping nodes the same size is to adjust font sizeof the labels. Another common way is to compressor truncate long labels. The second reason and moreimportant reason is that if nodes of different sizes areused for experiments, for example, and the sizes arerandomly generated, it might be difficult to analyzeof the results. For example, it is hard to determinewhether the difference of area of different drawingsis caused by the layout algorithms or by the size ofnodes. Therefore, we simplify the experiment by theassumption that all nodes have the same size. In ad-

dition, nodes are drawn as rectangles, and they havea larger width than height. This is the common shapein most practical applications.

Six sets of data are tested in our experiment. Ta-ble 1 gives a description of the data sets. The firstcolumn of the table is the ID of the data set. The sec-ond column is the number of nodes of the data set.The third is the number of overlapping nodes in theinitial layout. The last column is the drawing patternpresented in the initial layout.

Data Nodes Overlapping PatternNo.1 7 16 closed starNo.2 9 12 treeNo.3 15 20 triangleNo.4 21 35 opened starNo.5 25 37 gridNo.6 44 79 tree

Table 1: Experiment data set

4.2 Analysis

Figure 5(a) - 5(e) show the experimental results forarea, L1 metric length, aspect ratio, edge ratio, andrunning time, respectively. In each figure, there are 6histogram sets representing the results of the 6 datasets described in Table 1. The results for area and L1metric length are normalised.

4.2.1 FS and its variants

FS′ and FT are the variants of FS. They can be seenas one algorithm family. FS′ improves the results ofarea and L1 metric length of FS in all cases. It hassimilar results for aspect ratio and edge ratio to FS inmost cases. FS′ moves the nodes in the same directionas FS does, but more efficiently in the cases whereoverlapping occurs between more than two nodes. Soit can improve the area and L1 Metric length of FSbut keep the the aspect ratio and edge ratio.

Results show that FT has the best area in this al-gorithm family. It also has a better L1 metric lengththan that of FS. However, the aspect ratio of FT isworse than those of FS and FT′ in all cases. Theoutput of FT tends to increase more in the verticaldirection than in the horizontal direction. The rea-son is that FT moves nodes in the direction of whichdesirable distance is smaller. Because all the nodeshave the same size and the height of a node is smallerthan the width, there is more possibility of movingin the vertical direction than in horizontal direction.The same reason also causes the large edge ratio insome cases.

The drawings of all the FS family have large edgeratio, that is, the edge length are not uniform. It in-dicates that the advantages of the INI, symmetry anduniform edge length, are destroyed after overlappingremoved.

It takes almost the same time to run the threealgorithms in all cases.

4.2.2 DNLS

DNLS has the smallest area of data sets 4 and 6. Inthe other data sets, DNLS has similar area results toFT, which generates the smallest area drawing in theFS family. DNLS removes overlapping by both push-ing and pulling the overlapped nodes. It also has theability to evenly distribute the nodes. On the otherside, the FS family has the ability to enlarge the draw-ing, but no ability to compress it. When they movethe overlapping nodes, the FS family have to push

(a) Normalized Area

(b) Normalized L1 Metrics Length

(c) Aspect Ratio

(d) Edge Ratio

(e) Running Time

Figure 5: Experiment Results

away some uncluttered nodes in order to preserve theorthogonal ordering. Those uncluttered nodes willnot be pulled closer to the overlapping nodes. So theoutput of DNLS is usually smaller than those of FSfamily.

In all the cases, DNLS has a better L1 metriclength than any member of the FS family. This meansthat the values of width and height of the layouts gen-erated by DNLS are closer than those of the FS family,that is, the aspect ratio should be closer to 1. Theresults of aspect ratio support this conclusion.

Basically, DNLS is a spring algorithm. So it shouldhave the advantages of the spring algorithms: displayof symmetry and uniform edge length. Uniform edgelength is proved by our results for aspect ratio andedge ratio. Figure 5(c) and 5(d) show that DNLShas very similar results to INI, which is generated bya spring algorithm, in terms of aspect ratio and edgeratio in all cases. Visual observation confirms thedisplay of symmetry.

The experimental results show that DNLS has bet-ter performance than the FS family in terms of thetested aesthetic criteria.

The results also indicate an interesting feature ofDNLS. Because node size is constant in the experi-ments, the dynamic natural length k′ of all pairs ofnodes of DNLS is the same. In such cases, DNLSis actually a classical spring algorithm whose naturallength k can prevent nodes overlapping. Thus, wecan say that, in a static graph drawing system thatemployees a classical spring algorithm, it is a bettersolution to produce an uncluttered layout by increas-ing the natural length of the spring system than byadopting the FS family as a post-processing adjust-ment.

It takes DNLS less time to draw a graph in datasets 1 and 2 than the FS family. However, the re-sults also show that DNLS is slightly slower in datasets 4, 5, 6. Note that the running time is the sumof the time of creation and the time of adjustment,if a post-processing adjustment is applied. It seemsstrange that DNLS is slower than the FS family, be-cause DNLS is basically a layout creation spring al-gorithm, while the FS family is a post-processing ofanother layout creation spring algorithm. We believethis is the trade-off of computing the dynamic naturallength k′. DNLS find the spring forces, both attrac-tive and repulsive, by computing k′ for each pair ofnodes in every iteration. Note that the running timeof the spring algorithm is O(n2). When the numberof nodes is small, the computing trade-off is trivial.However, the trade-off will become obvious when thenumber of nodes increases.

4.2.3 ODNLS

ODNLS has similar results for aspect ratio and edgeratio to INI. This means that ODNLS has some fea-tures of a spring algorithm.

ODNLS sometimes costs more space to drawgraphs than DNLS. The results for area and L1 met-ric length are larger in data sets 2, 5, and 6 than thatof DNLS, and similar to DNLS in data sets 1, 3, and4. Compared to the FS family, ODNLS’s performanceof area in its good cases is similar to FT, but neverworse than FS in any case.

Experiments of preserving orthogonal ordering arecarried on for ODNLS and FS family. The resultsshow that they all preserve orthogonal ordering. BothODNLS and the FS family increase the area of thedrawings, but the FS family destroy the aestheticbeauty.

ODNLS is a post-processing adjustment algo-rithm. It is slower than any algorithm tested in the

experiments. The running time of both the FS familyand ODNLS is O(n2). However, in the best case, theFS family runs in time O(n), while ODNLS in timeO(i*(m + O(n2)), where i is the number of iterations,and m is the number of edges. One way to improvethe practical running time of ODNLS is to decrease k.But it might be unsatisfying from the aesthetic pointof view if k is too small. In our implementation, k=20.

4.2.4 Effectiveness of Removing Overlapping

Table 2 shows the result of removing overlapping bythese algorithms. The column 2 to 7 are the resultsof the algorithms when applied to data sets 1 to 6.

Algorithm No1 No2 No3 No4 No5 No6FS 0 0 0 0 0 0FS’ 0 0 0 0 0 0FT 0 0 0 0 0 0

DNLS 0 0 0 5 6 13ODNLS 0 0 0 0 0 0

Table 2: The Results for Removing Overlapping

We can see from Table 2 that all the algorithms ofthe FS family can remove the overlapping completely.

DNLS does not guarantee to remove overlapping.Just as a classical spring algorithm can not find agood drawing for all graphs, DNLS can not remove alloverlapping in drawings of all graphs. The most com-mon solution is to adjust the coefficients, to increaseiteration time, or to change the force model. Butthis would still not guarantee to completely removeoverlapping. An effective method is to use a FS al-gorithm as an additional step after DNLS. The extrapost-processing adjustment will not much change theaesthetic properties. Comparing to Table ??, we cansee that DNLS has significantly reduced the numberof overlapping nodes. Visual observation also showsthe overlapping area of DNLS is much smaller thanthose of FS family. So the number of nodes to bemoved and the distance of the motion are relativelysmall when an algorithm of the FS family cleans theoutput of DNLS. The appendix contains snapshots oflayouts generated for data sets 3 and 4 by all the al-gorithms tested. The output of DNLS of dataset 4is processed by FS′. The drawing still presents moreuniform edges and displays more symmetry than theFS family do.

ODNLS can remove all the overlapping nodes. Webelieve that this is the effect of the extra force fe, al-though no mathematical proof has been given. Ina ODNLS system, when fr repels the overlappingnodes, and fe prevents the repulsed nodes from cre-ating new overlapping. When fa attracts the nodesconnected by an edge, fe will not allow the attractednodes overlapping any nodes in between them. So fepreserves the orthogonal ordering, as well as prevent-ing overlapping.

4.3 Summary of the Experiment

The experiments show that the FS family is guaran-teed to remove overlapping, but they usually destroythe aesthetic beauty of the initial layout. Althoughthe running times of DNLS and ODNLS is a littlelonger than those of the FS family in some cases,the results show that both DNLS and ODNLS havethe one of the features of spring algorithms: uniformedge length. We do not do a quantitative analysis ofsymmetry, but the visual observation confirms thatin most cases DNLS and ODNLS present symmetry.

There is an interesting fact about the static draw-ing overlapping. It is better to adjust the natural

length of a classical spring algorithm to avoid over-lapping nodes than to clean the generated layout bythe algorithms of the FS family.

DNLS is a layout creation algorithm, and doesnot consider the issue of preserving the mental map.ODNLS and the FS family are post-processing adjust-ment algorithms, and they can all preserve the mentalmap according to the orthogonal ordering model.

5 Conclusion and Future Work

5.1 Conclusion

This paper addresses the issue of removing overlap-ping of drawings of graphs. We present two algo-rithms, namely, DNLS and ODNLS. Both algorithmsare based on the spring embedder model. Instead of afixed value of the natural length in the spring system,DNLS and ODNLS use the dynamic natural length kto compute the attractive forces and repulsive forcesexerting on the nodes.

DNLS is designed for static drawing overlapping,and it does not consider the mental map. ODNLS isan algorithm for dynamic drawing overlapping. Itadds an extra force fe to the spring embedder ofDNLS in order to preserve the mental map. Whena pair of nodes are attracted, fe will prevent the pairfrom overlapping the nodes in between them. Whena pair of nodes are repulsed, fe will push in the cor-responding direction all the nodes that are not inbetween the pair. The produced uncluttered layoutkeeps the same orthogonal ordering as the initial lay-out. fe also helps to complete remove overlapping.

Experiments are carried out to compare the out-puts of DNLS and ODNLS to those of the FS family.Six sets of data are tested in the experiments. Re-sults show that the FS family guarantee to removeoverlapping, but they usually destroy the aestheticbeauty of the initial layout. Both DNLS and ODNLShave a better performance than FS family in termsof aesthetic criteria. They have the features of springalgorithms: display symmetry and uniform edge. Theresults of uniform edge are quantitative, while the dis-play of symmetry is visually observed. DNLS does notguarantee to remove all overlapping. But a FS fam-ily algorithm can be employed as a post-processingof DNLS. The output of the pipeline usually still hasthe features spring algorithms.

We also found an interesting fact about removingstatic drawing overlapping. It is better to increasethe natural length of a classical spring algorithm toavoid overlapping nodes than to clean the generatedlayout by the algorithms of the FS family.

5.2 Future works

ODNLS and the FS family can be used in dynamicgraph drawing systems. In this paper, we only eval-uate these algorithms in a static environment. Onefuture work is to evaluate them in a dynamic drawingsystem.

6 Acknowledgement

We would like to thank Damian Merrick and TimDwyer of the University of Sydney for helpful discus-sions about the spring algorithms. We also thank TimDwyer for his help through the writing of the paper.

References

Gansner E. & North S. (1998), Improved Force-Directed Layouts, in ‘Graph Drawing’, LNCS,Montreal, Canada, August 13-15, pp. 364–373.

Hayashi K., Inoue M., Masuzawa T., & Fujiwara H.(1998), A layout Adjustment Problem fo Dis-joint Rectangles Preserving Orthogonal Order,in ‘Graph Drawing’, LNCS, Montreal, Canada,August 13-15, pp. 183–197.

Huang X., & Lai W. (2003), Force-Transfer: A NewApproach to Removing Overlapping Nodes inGraph Layout, in ‘25th Australian ComputerScience Conference’, Adelaida, Australia, 2003.

Kamps T., Kleinz J., & Read J. (1995), Constraint-Based Spring-Model Algorithm for Graph Lay-out, in ‘Graph Drawing’, LNCS, F.J. Branden-burg, Canada, 1996, pp. 349–360.

Generating customized layouts. in ‘Graph Draw-ing’, LNCS, F.J. Brandenburg, Canada, 1996,pp. 504–515.

Di Battista G., Eades P., Tamassia R.,& Tollis I.,(1998), Graph drawing: algorithms for the visu-alization of graphs, Prentice Hall.

Eades P. (1984), ‘A Heuristci Graph Drawing’, Con-gressus Numerantium 42,1984.

Eades P., Lai W., Misue K., & Sugiyama K. (1995),‘Layout Adjustment and the Mental Map’, Jour-nal of Visual Languages and Computing 6, 183–210, 1995.

Eades P., & Lin X. (1999), ‘Spring Algorithms andSymmetry’, Theoretical Computer Science 240-2, 379–405, 2000.

Harel D. &Koren Y. (1999), ‘Drawing Graphs withNon-Uniform Vertices’, in ‘Proceedings of Work-ing Conference on Advanced Visual Interfaces(AVI’02), Italy, 2002, pp. 157–166.

Lyons K., Meijer H.,& Rappaport D., (1998), ‘Al-gorithms for Cluter Busting in Anchored GraphDrawing’, J. Graph Algorithms and Applications2-1, 1–24, 1998.

Marriott K., Stuckey P., Tam V., & He W. (1995),‘Removing Node Overlapping in Graph LayoutUsing Constrained Optimization’, Constraints8, 143–171, 2003.

a) INI b) FS

c) DNLS d) FS’

e) ODNLS f) FT

Appendix 1: Snapshots of data set 3

a) INI b) FS

c) DNLS d) FS’

e) ODNLS f) FT

Appendix 2: Snapshots of data set 4