Planarity-Preserving Clustering and Embedding for Large Planar Graphs

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1 Planarity-Presering Clstering and Embedding for Large Planar Graphs Christian A. Dncan, Michael T. Goodrich, and Stephen G. Koboro Center for Geometric Compting The Johns Hopkins Uniersity Baltimore, MD Abstract. In this paper we present a noel approach for clster-based drawing of large planar graphs that maintains planarity. Or techniqe works for arbitrary planar graphs and prodces a clstering which satisfies the conditions for compond-planarity (c-planarity). Using the clstering, we obtain a representation of the graph as a collection of O(log n) layers, where each scceeding layer represents the graph in an increasing leel of detail. At the same time, the difference between two graphs on neighboring layers of the hierarchy is small, ths presering the iewer s mental map. The oerall rnning time of the algorithm is O(n log n), where n is the nmber of ertices of graph G. 1 Introdction The problem of displaying large graphs often arises in the networking and telecommnications areas. While sch application areas typically gie rise to nonplanar graphs, there are neertheless seeral application areas that gie rise to large graphs that are planar. Examples of sch planar graph applications inclde comptations arising in comptational cartography and geographic information systems (GIS). In this paper we are therefore concerned with the isalization of large planar graphs. There are seeral approaches to the isalization of planar graphs, each of which mst address the fact that the resoltion of most display technologies (and possibly een the hman eye) simply cannot display more than a few million pixels. Moreoer, no matter how many pixels a display technology has, these pixels mst display not jst the ertices of a graph of interest, bt also, and more importantly, the edges connecting these ertices. Or approach is based on displaying the graph sing a hierarchical clstering in which the graph is represented by a collection of layers, where each scceeding layer describes the graph in a decreasing leel of detail. That is, together with G one gies a tree T sch that the leaes of T coincide with the ertices of G, and each internal node of T represents the clster defined by the ertices of G associated with the descendent leaes of in T. In this case G can be drawn in a layered manner, Partially spported by NSF grant CCR and ARO grant DAAH Also at Max-Planck-Institt für Informatik. J. Kratochíl (Ed.): GD 99, LNCS 1731, pp , c Springer-Verlag Berlin Heidelberg 1999

2 Planarity-Presering Clstering and Embedding for Large Planar Graphs 187 where we draw each clster on the same layer of T as a region of the plane and connect adjacent clsters by segments. We wold like that each sch layer be drawn planar, with no segments intersecting each other or intersecting the bondary of a non-incident clster region. Ths, the general goal of clstered graph drawing is to presere the global strctre of a graph G by recrsiely clstering smaller sbgraphs of G and drawing these sbgraphs as single nodes or filled-in regions in a rendering of G. By groping ertices together into clsters in this way one can recrsiely diide a gien graph into layers of decreasing detail, which can then be iewed in a top-down fashion. 1.1 Prior Related Work on Clstered Graph Drawing If clsters of a graph are gien as inpt along with the graph itself, then seeral athors gie arios algorithms for displaying these clsters in two or three dimensions [2,3,5,8]. Still, as will often be the case, if clsters of a graph are not gien a priori, then arios heristics can be applied for finding clsters sing properties sch as connectiity, clster size, geometric proximity, or statistical ariation [7,9,11]. If no clsters are gien and no special properties are known in adance, Dncan et al. [1] show how to create a hierarchical decomposition and a 3-dimensional drawing for general graphs. Howeer, for planar graphs, it is possible to introdce edge-region crossings, in which edges cross clster regions they are not part of. Een with no edge-edge crossings, the edge-region crossings are a serios drawback to the readability of a drawing. Eades et al. [3] describe a drawing algorithm that draws a planar graph G, assming that the clsters of G presere certain recrsie conditions, which they collectiely call the c-planarity conditions. They show that if G and its clsters satisfy the c-planarity conditions, then one can prodce a drawing of G sch that each layer of the clster hierarchy is drawn planar, with each ertex drawn as a conex region and each edge drawn as a straight line segment. This approach allows the graph to be represented by a seqence of drawings of increasing detail. As illstrated by Eades and Feng [2], this hierarchical approach to drawing large graphs can be ery effectie. Howeer, we are not aware of any preios work for deterministically prodcing a clstering of an arbitrary planar graph so as to satisfy all the c-planarity conditions. 1.2 Or Reslts In this paper we proide an algorithm for constrcting a clstering of any planar graph so as to satisfy the c-planarity conditions of Eades et al. [3]. Or algorithm rns in O(n log n) time, ses O(n) space, and can be implemented sing simple off-the-shelf data strctres. We also show that the clstering tree T, defined by or algorithm, has the additional property that the nmber of clsters at layer i of T (i.e., the clsters associated with the nodes of T at height i) isa constant fraction fewer than the nmber of clsters at the next higher layer, i + 1. Ths, T has O(log n) height. This in trn implies faster drawing times when T is sed in a clstered graph drawing algorithm.

3 188 C.A. Dncan and M.T. Goodrich, and S.G. Koboro This logarithmic height reslt also implies some nice properties of the clstered drawing itself. For example, had we instead prodced a clstering tree T of depth Θ(n), which is possible if one ses a different clstering algorithm, then we wold hae a hierarchy that takes an extraordinarily long time to traerse for large planar graphs. At the same time, an o(log n) height for T wold imply drastic changes between consectie layers in the hierarchy. In addition to this logarithmic height reslt, or algorithm prodces a clstering sch that the changes between the graphs in consectie layers of the hierarchy T are local. In order to presere the iewer s mental map of the graph when moing from one layer to another, the changes in the graph shold be minimal. Gien the graph in layer i in T, to obtain the graph associated with the next higher layer i+1 in T, we need to grop certain sets of ertices together and replace them by new ertices. In this paper, we consider only changes that affect pairs of ertices, so that the tree T is in fact a binary tree. Ths we restrict or clstering operation so as to allow only the combining of two adjacent clsters, which is an operation typically referred to as an edge contraction. Throgh a seqence of graph contractions, we obtain the layer graphs G 0,G 1,...,G k, where G 0 = G and G k is a singleton graph. If the changes necessary to obtain layer i + 1 from layer i are to be local, then the following three locality conditions for edge contraction mst be met: 1. A ertex can participate in at most one edge contraction. 2. Changes in the drawing of the graph that reslt from the contraction of an edge (, ) shold only affect edges with endpoint or. 3. A contraction of edge (, ) reslts in the creation of ertex w. The placement of w in the drawing shold be close to the edge (, ). Optimally, we wold like that w lie along the line segment defined by (, ). We proide a clstering method that satisfies the aboe locality conditions. One of the main challenges in creating the layers in a clster hierarchy of a planar graph is to define clsters and the drawing algorithm associated with G s clstering in sch a way that no edge crossings are introdced in the drawing of each layer. We proide a drawing algorithm which makes se of or clstering method to prodce a drawing that has neither edge-edge crossings nor edge-region crossings. In addition, we show that one can se or clstering as inpt to the clstered planar graph drawing algorithm of Eades et al. [3]. 2 Hierarchical Embedding of Planar Graphs Let s assme, withot loss of generality, that all the graphs that we are dealing with are maximally planar. If a particlar graph is not maximally planar then we can flly trianglate it. Let G =(V,E) be a maximally planar graph, where V = n. V (G) and E(G) as sal refer to the set of G s ertices and edges, respectiely and the degree of a node in graph G is d G (). Let l G (f i )bethe length of a face f i in G, where by the length of the face we mean the nmber of

4 Planarity-Presering Clstering and Embedding for Large Planar Graphs 189 ertices on that face. Frther, w will refer to the triangle defined by ertices,, w and by the edges (, ), (, w), (w, ). Similar to [2] we define the clstered graph C =(G, T ) to be the graph G and a tree T sch that the ertices of G coincide with the leaes of T. An internal node of T represents a clster, which consists of all the ertices in its sbtree. All the nodes of T at a gien height i represent the clsters of that leel. A iew at leel i, G i =(V (G i ),E(G i )), consists of the nodes of height i in T and a set of representatie edges. The edge (, ) isine(g i ) if there exists an edge between a and b in G, where a is in the sbtree of and b is in the sbtree of. Each node T has an associated region, corresponding to the partition gien by T. We create the graphs G i in a bottom-p fashion, starting with G 0 = G and going all the way p to G k, where k = height(t ). We obtain G i+1 from G i by contracting a careflly chosen set of edges of G i in a certain order. The z- coordinate of a ertex V (G i ) is eqal to i, that is, all the ertices in G i are embedded in the plane gien by z = i. The edges of T are defined by the edge contractions. More precisely, if (, ) E(G i ) is contracted to a ertex w G i+1, then edges (w, ) and (w, ) are added to T. The problem of embedding planar graphs with straight lines and no crossings is well stdied [4,10,12]. Embedding clstered graphs withot crossings poses additional difficlties. To embed the layers, we reerse the seqence of graph contractions: we start with embedding of G k (which has only one ertex). To obtain an embedding for G i 1 from an embedding for G i we consider the set of edges of G i 1 whose contraction reslted in G i. We then reerse the process by careflly expanding and embedding one edge from that set at a time. Throghot this process we maintain the three locality conditions for edge expansions/contractions. 2.1 Edge Contraction and Separating Triangles Contracting an edge is a standard operation on planar graphs, see [6]. We say that an edge e =(, ) ofg is contracted when its endpoints, and, are replaced by a new ertex w sch that all reslting mltiple edges are remoed. Ideally, we wold like to perform edge contractions in a straight-line drawing that can be continosly animated so as to presere planarity. Frthermore, so as to presere the iewer s mental map, we prefer that only the endpoints of the contracted edge moe, reslting in only minimal changes in the drawing. It is well-known that contracting an edge in a planar graph reslts in a planar graph [6]. Note that this does not imply that contracting an edge in a straight line planar drawing of a graph reslts in a straight line planar drawing! More precisely, consider a straight-line planar drawing of a graph and an edge to be contracted. Sppose we are not allowed to moe any other ertices in the drawing except the two inoled. Then there exist drawings in which the contraction of some sch edge introdces a crossing. We show this with an example in Fig. 1. We hae seen one of the problems that occr when an edge in an embedded graph is contracted. Another problem can occr een if we do not hae a fixed embedding. When the contracted edge is a part of a separating triangle, the

5 190 C.A. Dncan and M.T. Goodrich, and S.G. Koboro Fig. 1. A sbgraph of an embedded flly trianglated graph. Edge (, ) cannot be contracted withot introdcing a crossing, if we are to keep all other ertices fixed. reslting graph is not flly trianglated and in fact may hae many different embeddings. We call a triangle in G is a separating triangle if the remoal of its ertices and their adjacent edges disconnects G. Ths, we can diide the edges of G into two categories depending on the effect their contraction has on the reslting graph. We say that an edge is simple if it is not a part of a separating triangle. Edges that are part of separating triangles we call non-simple. Non-simple edges present problems when contracted, so we will be contracting only simple edges, for their contraction can be continosly animated while presering planarity sing straight lines. Moreoer, eliminating the parallel edges after contracting a simple edge in a maximally planar graph reslts in a maximally planar graph. 3 Simple Matching in Maximally Planar Graphs In this section we show that any maximal matching that ses only simple edges contains a constant fraction of all the edges in G, proided G is maximally planar. Next we show how to find a matching that can be sed to contract the graph so that the reslting graph is maximally planar. Frthermore, if the size of that matching is O(n), then after repeating this process O(log n) times we are left with only a constant nmber of ertices. Ths, we need to show that we can constrct a maximal matching with O(n) edges sch that their contraction reslts in a maximally planar graph. Let G be the graph obtained from G by remoing all the non-simple edges. We start by showing that any maximal matching in G contains at least n/12 edges. To proe this claim we constrct a maximal matching in G and consider faces of different lengths. Recall that the length of a face refers to the nmber of ertices on that face. We break the faces of G into three classes, A, B, C, respectiely faces of length 3, faces of length 4, and faces of length 5 or more. We then cont the nmber of nmatched nodes in faces of the different classes. Finally, when we factor in oer-conting we show that any maximal matching mst contain at least n/12 edges.

6 Planarity-Presering Clstering and Embedding for Large Planar Graphs 191 Lemma 1. If f i is a face in A, there is at most one nmatched ertex in f i and this ertex has degree at least 3. If f i is a face in B or C, then there exist at most l(f i )/2 nmatched nodes in f i. For the nmatched ertices on faces in A we can show that they hae degree at least 3 in G. This is not necessarily tre for the nmatched ertices on faces in B or C. We show, howeer, that if a pair of nmatched ertices on a face in B hae degrees 2 in G then G belongs to a special class of graphs H. IfG/ H then for any face f i B, at most one ertex on that face has degree 2. Lemma 2. Let H be the class of maximally planar graphs in which there exist two ertices,, sch that eery other ertex in the graph is adjacent to both and. Then if H H, any maximal matching that ses only simple edges contains n/12 or more edges. Lemma 3. Let G and G be a planar graph and the indced graph on G in which all the non-simple edges hae been remoed. If there exists a face in B with more than one ertex of degree 2, then G H. We hae shown that if G has a face of length for with more than two nodes of degree 2, then G Hand hence any maximal matching in G contains at least n/12 edges (from Lemma 2). Finally we show that the same reslt holds for all maximally planar graphs. Theorem 1. Let G be a maximally planar graph and let M be the set of matched ertices in a maximal matching which ses only simple edges. Then M n/6, where n is the nmber of ertices of G. 4 Algorithm and Analysis Before we can consider a particlar embedding we mst show how to obtain all the graphs in the hierarchy, G 0,G 1,...,G k. Recall that G 0 = G is a flly trianglated planar graph on n ertices. To constrct G i+1 from G i we find a matching E i of G i and perform the graph contraction sing the edges in E i.we repeat this process ntil G i+1 is a singleton graph. Set E i for 0 i<kcontains a maximal matching on the edges of G i with some added constraints. It is important that after the contraction of the edges in E i the reslting graph G i+1 remains flly trianglated. In order to presere the mental map, the three locality conditions mst be maintained. Finally, in order to maintain a small hierarchical height, E i mst be a constant fraction of the edges in G i. Ths, the constraints that we hae on E i are as follows: 1. E i is a matching of simple edges. 2. Using the locality conditions, contracting E i yields a maximally planar G i E i V (G i ) /c, for some constant c>1.

7 192 C.A. Dncan and M.T. Goodrich, and S.G. Koboro w x y x y (a) (b) Fig. 2. Edges (, ) and (x, y) in part (a) are both simple, do not share an endpoint, and can be contracted as a part of a matching. After (, ) is contracted to w in part (b), edge (x, y) becomes a part of a separating triangle and so it shold not be contracted. Note that condition (1) does not imply condition (2), see Fig. 2. Before we proceed we show how to prodce a set E i which satisfies the aboe three conditions. Sppose we hae graph G i and we want to create set E i so that when all the edges in E i are contracted, we get G i+1. We will contract simple edges of G i one at a time. When an edge (, ) is contracted, it is replaced by a node w. The next time an edge is contracted, it cannot hae w as an endpoint. Let W i be the set of ertices that were created as a reslt of contractions in phase i. The edges that we place in E i mst be a matching, and so when a new edge is considered for contraction, it cannot hae an endpoint in W. Finally, let S i be the set of ertices of G i of small degrees. More precisely, let S i = { V (G i ):d Gi () < 39}. In general, G i is transformed into G i+1 one edge contraction at a time sing the edges in E i in the order they were chosen. Call the intermediate graphs from G i to G i+1, G i = G i,0,g i,1,...,g i,j = G i+1, and consider the algorithm on Fig. 3. Lemma 4. Let V (G i ) = n i. Then E i n i /50. Proof Sketch: For G i with more than 3 ertices, E i 1. Then consider the seqence of intermediate graphs G i,0,g i,1,...,g i,j and let G i,j hae no more edges that cold be added to E i. Obsere that we hae contracted exactly j edges of G i and so V (G i,j ) = n i j. Then from Theorem 1 there are (n i j)/12 edges in any maximal matching of G i,j which ses only simple edges. Consider sch a matching M. We are not allowed to add M edges with endpoints in W i. Bt since W i = j, at most j of the edges in M can hae endpoints in W. Also note that if both endpoints of a simple edge in the matching hae degrees greater than or eqal to 39 in G i they cannot be added to M. Note that if there exist at most k nodes of degree greater than or eqal to 39, then there are at most k/2 sch edges. It is easy to show that k<n i /12: Sppose there are k ertices of degrees 39 or more in G i. Since G i is flly trianglated, eery ertex has degree at least 3 and since G i is maximally planar, the sm of the degrees

8 Planarity-Presering Clstering and Embedding for Large Planar Graphs 193 match(g i,e i) j 0 G i,j G i W i while (S i ) Let j S i, S i = S i \{ j} if ( j, j) G i,j, s.t. j / W i and ( j, j) is simple E i E i {( j, j)} Contract ( j, j)tow j to get G i,j+1 W i W i {w j} j j +1 retrn(g i,j 1) Fig. 3. Create G i+1 from G i by contracting a seqence of edges in E i. is twice the sm of the edges. Then 39k +3(n i k) 6n i 12. From this we get that k<n i /12. We stopped selecting good edges from G i when we got to graph G i,j in which we cold not find a simple edge to contract. The only other types of edges that might be aailable in G i,j bt which we cannot take are those that were at some point non-simple, bt later became simple. Also, there can be at most j sch edges. Then (n i j)/12 2j n i /24 0 which implies j n i /50. Ths, if we cannot find another edge to add to the matching, we mst hae E i = j n i /50 which completes the proof sketch. From the reslt aboe it follows that the height of the hierarchy is O(log n). We next arge that one call to match(g i,e i ) takes O(n i log n i ) time, and since n i+1 is a constant fraction of n i, the O(log n) calls to match(g i,e i ) take O(n log n) time oerall ths yielding the desired theorem: Theorem 2. The clstering algorithm rns in O(n log n) time and prodces a seqence of graphs G 0,G 1,...,G k sch that G i is maximally planar for all 0 i k and k = O(log n). 5 Constrcting the Embedding After we obtain the combinatorial graphs G 0,G 1,...,G k we hae to embed them in planes z = 0,z = 1,...,z = k. While constrcting the combinatorial graphs is a bottom p process, constrcting the embedding is a top-down one. The first graph to be embedded is G k, which only has one ertex. We then expand the edges in E k 1 one at a time, in the reerse order of their insertion. We then arge that this can be done in a way which garantees that no crossings are introdced. We need the following lemma.

9 194 C.A. Dncan and M.T. Goodrich, and S.G. Koboro 2 1 x w l q q l x l q q-1 p-1 p y 2 p-1 1 p y 1 2 (a) (b) Fig. 4. Vertex w l and its neighbors in G i,l+1 (a) before expansion (b) after expansion. Lemma 5. Let G be a maximally planar graph embedded in the plane withot crossings. For any V (G), there exists a ball of radis ɛ>0 sch that if is placed anywhere inside that ball, the embedding has no crossings. Proof Sketch: The main idea is to consider the isibility region arond ertex. Any point inside that region can see all the neighbors of. It is not hard to show that this region cannot be empty. This wold imply the existence of ɛ>0 for which the ball of size ɛ fits inside the isibility region. Theorem 3. Gien combinatorial representations of graphs G k,g k 1,...,G 0 we can embed them in the planes z = k, z = k 1,...,z =0so that there are no crossings in any of the drawings. Proof Sketch: We first embed G k in the plane z = k withot crossings sing any straight-line drawing method. Sppose we hae embedded G k,g k 1,...,G i. We will show how to embed G i 1 gien an embedding for G i. Recall that we obtained G i from G i 1 throgh a series of edge contractions from the edge set E i 1 = {( 0, 0 ), ( 1, 1 ),...(j, j)} which prodced graphs G i 1,0,G i 1,1,..., G i 1,j = G i. We now reerse the process and expand G i back to G i 1 throgh the exact opposite seqence of expansions. Since we hae an embedding for G i in the plane z = i, we can embed G i 1,j in the plane z = i 1. Next we expand edge ( j, j ) by replacing ertex w j by the pair j, j. The reslting graph is G i,j 1 and we embed it withot a crossing. We proceed ntil we get to G i,0.we next show how to embed G i,l gien an embedding for G i,l+1, for 0 l<j. Assme we hae an straight-line embedding for G i,l+1 withot crossings on the plane z = i. TogetG i,l we mst expand ertex w l back to edge ( l, l ). Consider the sbgraph on Fig. 4. Let x and y be the neighbors in common for l and l. We then consider the ball of maximal radis arond w l which sees all neighbors (we know it is of radis ɛ>0 from Lemma 5). Consider a diameter in this ball which is perpendiclar to the line connecting x and y. Place l and l on the two ends of the diagonal. We define the drawing of a clstered graph C =(G, T ) as in [3]. Graph G is drawn as sal, while for eery node T the clster is drawn as a simple closed region R sch that:

10 Planarity-Presering Clstering and Embedding for Large Planar Graphs 195 all sb-clster regions of R are completely contained in the interior of R. all other clster regions are completely contained in the exterior of R. if there is an edge e between two ertices contained in a clster ν, then the drawing of e is completely contained in R. Following the definitions of Eades et al., the drawing of edge e and region R hae an edge crossing if the drawing of e crosses the bondary of R more than once. A drawing of a clstered graph is c-planar if there are no edge crossings or edge-region crossings. Graphs with c-planar drawings are c-planar. Theorem 4. The clstered graph C =(G, T ) prodced by or algorithm is c- planar and a c-planar embedding can be obtained in O(n 2 ) time. Proof Sketch: It sffices to show that there exists a drawing of C which has no edge crossings and no edge-region crossings. Let s embed G sing any planar embedding algorithm. Define the region of a clster, ν to be the simple closed cre arond the sbgraph of G indced by the clster, G(ν). By the definition of the clstering in or algorithm, the sbgraph G(ν) is connected. If is a ertex not in clster ν, then cannot be contained inside the region R. Assme that is contained in R. If we contract the edges of ν in the order defined by or algorithm, eentally will be inside a trianglar face. Bt then none of the edges on that face can be contracted. This is a contradiction since ν is eentally contracted to one ertex. Finally, since G is embedded in the plane withot crossings and the regions are connected there can be neither edge crossings nor edge-region crossings. Therefore C is c-planar and from [3] it follows that the c-planar embedding can be prodced in O(n 2 ) time. References 1. C. A. Dncan, M. T. Goodrich, and S. G. Koboro. Balanced aspect ratio trees and their se for drawing ery large graphs. Proc. of 6th Symposim on Graph Drawing (GD 98), LNCS 1190:1 112, P. Eades and Q. W. Feng. Mltileel isalization of clstered graphs. Proc. of 4th Symposim on Graph Drawing (GD 96), LNCS 1190:1 112, P. Eades, Q. W. Feng, and X. Lin. Straight-line drawing algorithms for hierarchical graphs and clstered graphs. Proc. of the 4th Symposim on Graph Drawing (GD 96), LNCS 1190: , I. Fary. On straight lines representation of planar graphs. Acta Sci. Math. Szeged, 11: , Q.-W. Feng, R. F. Cohen, and P. Eades. Planarity for clstered graphs. ESA 95, LNCS 979: , R. J. Lipton and R. E. Tarjan. A separator theorem for planar graphs. SIAM J. Appl. Math., 36: , F. J. Newbery. Edge concentration: A method for clstering directed graphs. In Proceedings of the 2nd International Workshop on Software Configration Management, pages 76 85, Princeton, New Jersey, October 1989.

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