Automatic Graph Drawing Algorithms

Transcription

1 Autoatic Graph Drawing Algoriths Susan Si Deceber 7, 996. Ebeddings of graphs have been of interest to theoreticians for soe tie, in particular those of planar graphs and graphs that are close to being planar. One definition of a planar graph is one that can be drawn in the plane with no edge crossings. While working on the four-colour proble, Wagner(936) was the first to show that every planar graph has a straight-line ebedding. Tutte (960, 963) showed that every 3-connected planar graph has a convex ebedding. While graph theory was originally an artifact fro atheatics, it has becoe quite prevalent as a eans of solving probles or representing data. With autoatically generated data sets being represented as graphs, cae the need to autoatically generate ebeddings of graphs in a 2-diensional space, such as a coputer terinal or a sheet of paper. A wide variety of fields each with their own requireents utilize autoatic graph drawing algoriths. As a result, any different algoriths have been developed over the last decade. Within the last five years, there have been annual conferences on graph drawing (Di Battista, et al., 993; Taassia and Tollis, 995; Brandenberg, 996), special issues of journals on the topic and this year a onograph was published. (Cruz and Taassia, 994) Three different algoriths fro the field will be presented briefly in this paper: the spring odel algorith, siulated annealing ethod and the Sugiyaa algorith. Applications of Graph Drawing Algoriths Work in this area was done ainly in response to requireents of data visualization techniques and interactive coputer systes. Many fields in coputer science, such as software engineering, electronic circuit design and database design, have found it useful represent data as graphs, with vertices denoting eleents and edges denoting relations between the. These graphs are norally generated by software tools based on

2 inforation in the syste. As the size of a graph generated fro data or constraints grew, so has the sophistication of ebedding algoriths. In software engineering, the architecture of a large software syste can be visualized as a directed graph with vertices representing odules and edges denoting various use relations between the. These systes are often hierarchical in nature and their drawings should reflect this. Furtherore, this inforation can be used to ake the graph drawing task easier. (Müller, et al., 993) Coputer hardware and icrochips are now sufficiently coplex that they are designed using CAD tools. It is then the responsibility of the tool to create a layout of the logical gates and the connections between the on icrochips and circuit boards. This layout should be a grid drawing. An orthogonal drawing is one in which an edge is a chain of horizontal and vertical segents. A grid drawing is an orthogonal drawing in which all the vertices and bends of edges have integer coordinates. (Di Battista, et al., 994) There are any other exaples of applications which use graph drawing algoriths. Entity-relationship diagras in database design can have a visual representation or an algebraic one. One of the requireents of systes analysis and design tools is that a database description need only be entered once in either forat and the alternate forat will be generated. There is a project anageent technique that uses PERT charts (Project Evaluation and Review Technique) to track dependencies aong tasks. These dependencies for a directed graph fro which other inforation can be derived, such as a project critical path. One technique used by the Huan Genoe Project analyses the gene structure by representing raw data as a directed graph (Harley and Bonner, 994). Aesthetics It is not difficult to design a naïve algorith to display a graph. A rando layout places vertices randoly within a finite space. Edges can be drawn as iniu length straight lines between vertices or they ay be polylines, that is, lines with bends in the to avoid

3 2 drawing eleents. A circular layout algoriths places the vertices along the perieter of a circle and edges are drawn across the circle. A siilar strategy places vertices at the intersections of an n n grid along the ain diagonal. (Noik, 996) The circular layout ethod is particularly effective for representing cliques as it ephasizes the regularity of the graph. Unfortunately, ost graphs are not cliques. None of the above ethods pays uch attention to the readability of the resulting graph, either as a collection of lines and dots or the data that they represent. Fro a graph theoretic point of view, isoorphic graphs should look siilar. Also, planar graphs should be drawn without edge crossings so it is easy to visually confir the planarity of the graph. Consequently, the drawings generated by the above strategies ay not be very inforative. The quality or usefulness of a particular ebedding is highly dependent on its application doain. Therefore a graph drawing algorith ust take into account aesthetics: criteria for aking salient characteristics of the graph easily readable. Readability and salient characteristics are highly subjective and dependent on the purpose for which the drawing is generated. Soe aesthetic criteria include: iniize the nuber of edge crossings; draw edges as straight as possible; vertices should be evenly distributed; the ajority directed edges should be drawn pointing in the sae direction; in polyline drawings, iniize bends in the edges; iniize the area of the area drawing; axiize display of syetries; axiize angular resolution. (Di Battista, et al., 994; Eades and Sugiyaa, 990; and Cruz and Taassia, 994) The angular resolution of a line drawing of a graph is the sallest angle fored by two edges incident on the sae vertex. If this angle is too sall, it ay be beyond the resolution of

4 3 graphic display device or even the huan eye, and the two incident edges end up looking like a fuzzy blob. In general, it is not possible to optiize two criteria siultaneously. For exaple, in the two ebeddings of K 4 below, the one on the left iniizes the nuber of edge crossings, whereas the one on the right axiizes the display of syetries. (Cruz and Taassia, 984) Many of these optiization probles are either NP-hard or NP-coplete. Miniizing the nuber of crossings in an ebedding is NP-coplete, even if the graph is hierarchical with only two layers. (Garey and Johnson, 983, in Eades and Sugiyaa, 990; and Di Battista et al., 994) Miniizing the area of a grid drawing is NP-hard. (Kraer and van Leeuwen, 984, in Di Battista et al., 994), as is iniizing the length of the axiu edge length (Miller and Orlin, 984, in Di Battista et al., 994). Polytie graph drawing algoriths tend to either use heuristics to approxiate an NPhard optiization, use innovative techniques to anipulate the layout or soe cobination of the two. The ore coplex algoriths that yield ore aesthetically pleasing graphs are usually able to approxiate optiizations of several criteria. The specific criteria used are often deterined by the specific application doain or graph type for which the algorith was developed. Three such algoriths will be discussed in the reainder of this paper. They are the Spring Model Algorith, Siulated Annealing Algorith, Sugiyaa Algorith. Spring Model Algorith

5 4 There are any different strategies that can be used to draw a general undirected graph. One ethod is to use a planar ebedding algorith because a planar ebedding can be constructed in linear tie. The first step is to test for planarity and if the test is positive construct the ebedding. If the graph is not planar, then it can be planarized by a variety of techniques such as deleting edges, splitting duy vertices or adding duy vertices at edge crossings. Planarity testing can be done in linear tie. Planarization is an NP-hard proble but can be done using heuristics in O(n 2 ) or less. So overall, this ethod is relatively efficient. (Di Battista, et al., 994; Cruz and Taassia, 994) Another ethod is to orient the edges, say using an Eulerian walk and use a directed graph drawing algorith. Finally, there is a faily of force-directed algoriths which involves transforing the vertices and edges into a syste of forces and finding the iniu energy state of the syste. This state is found either by solving differential equations or by running a siulation of the forces. The spring odel is the ost popular algorith in this faily. The spring odel was originally developed by Eades, 984. The version presented here is a descendant of that algorith developed by Kaada (989). Conceptually, the spring ebedder works by replacing edges with springs unit soe natural length. It also adds springs with larger natural lengths between non-adjacent vertices. The vertices are initially placed randoly and a syste of differential equations is solved to find the syste state with iniu energy. By nature, springs attract their endpoints when stretched and repel their endpoints when copressed. Vertices that are adjacent are kept close to each other by shorter springs. While the longer springs between non-adjacent vertices keep the apart, and at the sae tie they liit the overall size of the ebedding. The total energy of the syste is represented by the following suation:

6 5 n ( ) E = k p p l i= n 2 j= i+ ij i j ij 2 p, p 2,, p n are particles on a plane representing vertices v, v 2,, v n l ij is the natural length of the spring between v i and v j and is defined as l = L d, where L is the desired length of the edge in the ebedding and d ij is the shortest length path between v i and v j in the graph. L is soeties chosen as a function of the diaeter of the graph and the available ebedding area. ij ij k ij is the strength of the spring between v i and v j and is defined as k ij = K d 2, where K ij is a constant and d ij is as defined above. Rewriting the energy suation in ters of xy-coordinates, we get: n (( ) ( ) 2 ( ) ( ) ) E = k x x + y y + l l x x + y y i= n 2 j= i+ ij i j i j ij i j i j To find local inia of this suation, we need to take partial derivatives of this and solve to find local inia for each vertex. This results in 2n non-linear non-independent equations. This syste of equations is rather difficult to solve, so the proble is considered one particle at a tie. The particle in the syste with the highest energy is identified and oved to a stable location with low energy. Let this particle be p, with energy function: E = 2 2 E + x y p is oved in a stepwise anner to iniize. This location corresponds to where E x = = 0 is satisfied. This is done by iteratively solving the following two linear E y equations for δx and δ y, where t is the current iteration and adding the to x and y respectively. These steps are repeated until the new energy function stops decreasing.

7 6 2 E 2 x E x y 2 t t E t t E t t ( x, y) δx + ( x, y) δ y = ( x, y) x y x t t E t t E t t ( x, y) δx + ( x y ) y 2, = ( x, y) 2 2 y The algorith is suarized by the following pseudocode.. copute d ij for i j n ; 2. copute l ij for i j n ; 3. copute k ij for i j n ; y 4. initialize p, p 2,, p n 5. while (ax i > ε ) 6. let p be the particle satisfying = ax i ; 7. while ( >ε ) 8. copute δx and δ y ; 9. x x + δx 0. y y + δ y. end while 2. end while When a local iniu has been found, each pair of particles are exchanged and the energy of the syste is tested. If a swap results in a lower energy state, the energy iniization process restarted with the new configuration as a starting state. This exchange and copare process provides a eans of escaping large local inia. Eventually the syste converges to a global iniu. It takes O(n 3 ) tie to find all pairs shortest paths. More efficient algoriths ay be used to lower this bound. O(n) tie is needed to copute each of, δx and δ y during each iteration. With soe bookkeeping, ax i can be found in O() tie. The tie required by the energy iniization process is O(Tn), where T is the total nuber of inner loops. T is difficult to characterize beyond this because it depends on the value of ε, the initial position of the vertices, and the graph itself.

8 7 This algorith reduces the nuber of edge crossings as their presence increases the energy in the syste. Also, drawings produced by this ethod display syetries present in the graph. The sae graph with different initial conditions will converge to the sae drawing. Graphs with siilar structures will also be drawn siilarly. This ethod can be extended to layered hierarchical graphs by assigning vertices in the sae level a fixed y value and allowing only the x position to vary. Siulated Annealing Method In the ost general ters, annealing is the process by which soe substance is heated until it is a liquid and then slowly cooled until it is a solid. The slow cooling allows the olecules to organize theselves into a crystal, a totally ordered for. This process is often used in the anufacturing of products such as steel. Siulations of annealing were developed to analyze the efficiency of these anufacturing processes. Siulated annealing is used as a proble solving technique where the potential solution space is large and a cobinatorial search is infeasible. It has been applied with soe success to probles such as VLSI circuit design, graph partitioning and the traveling salesan proble. Davidson and Harel (996) have applied this technique to drawing undirected graphs with straight-line edges. One big advantage of this algorith is that the relative iportance of different aesthetic criteria can be varied. The basic shape of a siulated annealing algorith is given in the pseudocode below.. Initialize teperature T; 2. Initialize configuration σ; 3. E cost of σ; 4. while(in T not reached) 5. while(terination rule is not satisfied) 6. choose a new configuration σ fro the neighbourhood of σ; 7. E cost of σ ; 8. if ((E < E) OR (rando < 9. σ σ ; 0. E E ;. end if 2. decrease teperature T 3. end while ( E E ) T e )) then

9 8 4. end while A configuration is a proposed ebedding of the input graph. The graph can be entered either as a set of adjacencies or as a hand drawing. If the input is a set of adjacencies then the initial configuration is randoly generated. A neighbourhood of a configuration σ, is the set of configurations that differ fro σ by the location of a single vertex. σ is generated by taking a vertex and placing it on the perieter of a circle drawn around its original location. To siulate the behaviour of olecules in the physical annealing process, the size of this circle is initially large and decreases with T. The cost of a configuration ust be carefully chosen so that it reflects the desired aesthetics of the final graph and is not overly coputationally intensive to copute.. To ensure that vertices are evenly distributed to avoid overcrowding, the ter a ij = λ 2 d is added to the cost function pair of vertices, i, j. d 2 ij is the Euclidean ij straight line distance between i and j. λ is a weighting factor whose value depend in the iportance of this criteria relative to the others in the cost function. 2. A copletely iniized cost function will spread out the vertices indefinitely. To ensure that vertices are drawn on the display area, the following ter is added to the cost function for each node i: i = λ + + +, where r 2 r 2 l 2 t 2 b 2, l, t i i i i i i i and b i are the straight-line distances between the vertex and the right, left, top and botto borders of the display area. Again, λ 2 is a weighting factor. 3. To avoid overly long edges, the following ter is added for each edge k of length d k 2, with weighting factor λ 3 : c k = λ 3 d To reduce the nuber of edge crossings λ 4 is added to the cost function for pair of edges that cross. k

10 9 5. To prevent edges fro being drawn too close together, particularly those that intersect at a vertex, the following ter is added for every vertex i, edge k with distance g ik and weighting factor λ 5 : h ik = λ 5 2 g ik The relative iportance of each of the criteria can be adjusted by varying the weighting factors. The fifth factor is norally used only during an optional fine tuning stage of the algorith. The ain part of the algorith is run with only the first four factors until a terination condition is reached. Then the drawing ay be adjusted further by using the larger cost function. The actual value of the initial teperature depends on how any iterations are desired in the annealing process. At each teperature level, approxiately 30n perturbations should be perfored. The cooling function should be geoetric, i.e. T γ T, with p+ = 0. 6 γ A value of 0.75 gives a relatively rapid cooling with the resulting graphs giving good aesthetic properties. p The terination condition can be a fixed nuber of iterations, say 0, or when the proposed ebedding stops changing significantly over two to three iterations. The siulated annealing algorith runs in O(n 2 ) tie. The nuber of teperature changes required to terinate the outer loop is constant in the sense that it does not depend on the input graph. O(n) perturbations are perfored in the inner loop, say 30n. It takes O(n) tie to update the cost function, which is ore efficient that calculating it fro scratch. Since the location of only one vertex is changed in a new configuration, n vertices need to be updated and for each of these at ost edges will also need to be updated. This technique produces drawings that are coparable to those generated by the spring ethod. This algorith does not produce conventional looking graphs for a several categories of graphs. One such group is a large cycle with no chords. While this is

11 0 norally drawn as a large circle, this algorith tends to draw the cycle curled around itself so there is not a large epty space in the iddle. Siulated annealing is also not very good at bringing out the regularity of a graph such as a clique because it tends to draw all edges with coparable length. Although this has not yet been rigorously shown, this algorith appears to be ore successful with ore coplex graphs than spring algoriths. One useful property of siulated annealing algoriths is that they are easily parallelized. Sugiyaa Algorith Layered graphs are of practical interest in software engineering, PERT chart generating and other fields that require data odeling. The structure of a software progra can be represented as a procedure calling hierarchy. Soeties this structure can be drawn as a tree but ore often it needs to be drawn as a layered graph. Müller, et al. (993), call this an idealized layered software architecture a (k, 2)-partite graph. This notation eans that there are k layers and each layer has edges only to the layers adjacent to it, i.e. above and below. The ost popular algorith for laying out this type of graph was introduced by Sugiyaa, Tagawa, and Toda (98) and is nicknaed after the first author. Soe of its popularity ay be attributed to its early developent. Although the overall structure of this algorith is less coplex than the first two presented, there are any sall anipulations that need to be perfored. Each of these sub-parts can be as coplex as either the spring ethod or the siulated annealing ethod. Consequently, only a high level description of the Sugiyaa algorith will be presented here. The algorith takes an hierarchical directed graph as input and draws it in four stages. In stage I, the graph is transfored into a proper hierarchy, if necessary. In stage II, the vertices at each level are ordered to reduce the nuber of edge crossings. In stage III, the horizontal position of each vertex is anipulated to reduce the length of the edges.

12 Finally, in stage IV the graph is drawn. Stage I is a preparatory step and stages II and III for the ain part of the algorith. A proper hierarchical graph has no cycles, and only has edges between adjacent levels. In stage I Sugiyaa, Tagawa and Toda suggest condensing vertices as a way of eliinating cycles and refers the reader to other sources for ore detailed inforation. Edges that span ore than one level are replaced with placeholder vertices and edges that connect two levels. This hand-waving is interesting, as finding the iniu feedback edge set is an NP-coplete proble. In stage II, an iterated barycentric ethod is used to reduce the nuber of edge crossings. The barycentre of a graph is a vertex with the iniu distance between itself and every other vertex in the graph. A graph ay have ore than one barycentre. Conceptually, these vertices are those closest to the iddle of the graph.(west, 996) The barycentric ethod iniizes the distance between a vertex and its neighbours and as a result reduces the nuber of edge crossings in the drawing. In this stage, and the next, only one level and an adjacent level are considered at a tie, thus siplifying a k-level proble to a 2-level proble, or a bipartite graph. The edges in between the two levels are represented as an adjacency atrix, with the upper level as rows and the lower level as coluns. An adjacency atrix is constructed for each pair of adjacent levels. The barycentre for a the upper level is given as B R ik, where R denotes row and i denotes the current adjacency atrix, i.e. pair of levels, and k the current row in the atrix. Siilarly, B C il is the barycentre for lower level, where C denote colun and I denotes the current adjacency atrix and l the current colun in the atrix. R i i B = l, k =,, V ik q q kl l= l= kl i, the nuber of vertices in the upper level C i i B = k, l =,, V il p p kl k= k= kl i+, the nuber of vertices in the lower level The algorith first calculates the barycentres for each colun in the atrix for the top two levels. It then re-orders the coluns fro sallest to largest barycentre. This change

13 2 is then reflected in the next adjacency atrix. This process is repeated for all adjacency atrices. After the re-ordering the rows of the adjacency atrix between the botto two levels, then the barycentres for the rows are calculated and re-ordered. This change is reflected in the coluns of the next higher atrix. This process is repeated until the first atrix has been re-ordered. In stage III, edge lengths are reduced by adjusting the horizontal positions of vertices within a level using a priority layout ethod. Each vertex in a level is assigned a onotonically increasing priority nuber, say without loss of generality fro left to right. In this way, the vertex order established in stage II. When reducing the spacing of a given vertex, only vertices with a lower priority nuber on the sae level ay be odified. Finally, no two vertices ay lie on top of one another. Levels are adjusted in ultiple passes fro top to botto and back to the top. Specifically, the order of levels adjusted is: 2,..., k, k-,...,, t,..., n, where t is an integer, 2 t n-. Often, three passes are sufficient to ake a nice looking drawing. In the last stage, the graph is displayed on a graphic terinal. Stage II was tested on one hundred randoly generated graphs. The iniu nuber of edge crossings was found through cobinatorial search. The nuber of edge crossing in the final ebedding was copared with the iniu possible and was found to be within 5%. It takes O(n) tie to generate all the adjacency atrices. For each vertex in the graph, its barycentre ust be calculated twice. Each calculation takes O(n) tie, so the total aount of tie spent calculating barycentres is O(n 2 ). In the worst case, each row or colun re-ordering requires the entire adjacency atrix to odified. This can be done in O(n) tie, with a constant coefficient depending in the size of the adjacent levels. So stage II runs in O(n 2 ) tie. One iteration of stage III, in the worst case (i.e. every vertex requires all other vertices with saller priority on the sae level to ove), will take O(n 2 ) tie. In total, this stage will take O(Tn 2 ) tie where T is the nuber of iterations of the up-down process. So while, the algorith is rather coplex it runs relatively efficiently. It takes O(n 2 ) with a large coefficient and large lower order ters.

14 3 Conclusion Since any of the sub-probles in graph drawing are NP-hard or NP-coplete, it is too coputationally expensive to attack the proble directly. As we have seen, innovative odels and heuristics have been developed to the solve the proble. The spring ethod represent a graph ebedding as a physical systes with energy states. By iniizing energy state, an aesthetically pleasing drawing can be generated. Siilarly, the siulated annealing algorith is patterned after a physical process that iproves the ebedding by applying any sall perturbations to interediate drawings. The Sugiyaa algorith uses a statistical ethod to calculate etrics to represent desirable aesthetic properties. These etrics are then used to anipulate the original graph to approxiate the quality of those generated by exhaustive cobinatorial search. The particular algorith used by a coputer scientist depends uch on the proble doain for which it is going to be used. The study of autoatic graph drawing algoriths is a growing field. As ore fields in coputer science start to use graphs as a eans of proble solving or data representation, they will bring to the study their own particular requireents and insights. So far autoatic graph drawing algoriths were of interest only in relation to soe application. But as the field grows, so too will interest in the developent of these algoriths for their own sake.

15 4 References F.J. Brandenberg (editor). Graph Drawing 95, Proceedings of Syposiu on Graph Drawing, Springer Verlag, Berlin, 996. I.F. Cruz and R. Taassia. How to visualize a graph: Specification and Algoriths. Available on the WWW at URL R. Davidson and D. Harel. Drawing Graphs Nicely Using Siulated Annealing. ACM Transactions on Graphics, 5(4):30-33, October, 996. G. Di Battista, P. Eades, R. Taassia, And I. G. Tollis. Algoriths for Drawing Graphs: an Annotated Bibliography. June, 994. Available by anonyous ftp fro ftp.cs.brown.edu:pub/papers/copgeo/. G. Di Battista, H. D Fraysseix, P. Eades, P Rosenstiehl, and R. Taassia (Eds). Graph Drawing 93, Proceedings of the ALCOM International Workshop on Graph Drawing and Topological Graph Algoriths. Available by anonyous ftp fro wila.cs.brown.edu:pub/papers/copgeo/. P. Eades. A Heuristic for Graph Drawing. Congressus Nuerantiu, 42:49-60, 984. P. Eades and K. Sugiyaa. How to Draw a Directed Graph. Journal of Inforation Processing, 3(4): , 990. M.R. Garey and D.S. Johnson. Crossing Nuber is NP-Coplete. SIAM Journal of Algebraic and Discrete Methods, 4(3):32-36, 983. E. Harley and A.J Bonner. A Flexible Approach to Genoe Map Assebly. Proceedings of the International Syposiu on Intelligent Systes for Molecular Biology, pp. 6-69, 994. T. Kaada. Visualizing Abstract Objects and Relations: A Constraint-Based Approach, World Scientific, New Jersey, 989. M.R. Kraer and J. van Leeuwen. The Coplexity of Wire-Routing and Finding Miniu Area Layouts for Arbitrary VLSI Circuits in Advances in Coputing Research, vol. 2, edited by F.P. Preparata, pp , JAI Press, Greenwich, Connecticut, 984. Z. Miller and J.B. Orlin. NP-Copleteness for Miniizing Maxiu Edge Length in Grid Ebeddings. Journal of Algoriths, 6:0-6, 985.

16 5 H.A. Müller, M.A. Orgun, S.R. Tilley and J.S. Uhl. A Reverse-engineering approach to subsyste structure identification. Software Maintenance: Research and Practice, 5:8-204, 993. E.G. Noik. Dynaic Fisheye Views: Cobining Dynaic Queries and Mapping with Database Views. Doctoral thesis, Departent of Coputer Science, University of Toronto, 996. K. Sugiyaa, S. Tagawa, and M. Toda. Methods for Visual Understanding of Hierarchical Syste Structures. IEEE Transactions on Systes, Man and Cybernetics, (2):09-25, 98. R. Taassia and I.G. Tollis (editors). Graph Drawing 94, DIMACS International Workshop, Springer Verlag, Berlin, 995. W.T. Tutte. Convex Representations of Graphs. Proceedings of the London Matheatics Society, 0: , 960. W.T. Tutte. How to Draw an Graph. Proceedings of the London Matheatics Society, 3: , 963. K. Wagner. Bererkungen zu Vierfarbenproble. Jber. Deutsches Matheatiks Verein, 46:26-32, 936. D.B. West. Introduction to Graph Theory. Prentice Hall, Upper Saddle River, New Jersey, 996.