Automatic Layout of PERT Diagrams with X-PERT*

Transcription

1 Automatic Layout of PERT Diagrams with X-PERT* G. Di Battistaq E. Pietrosanti v, R. TamassiaO and I.G. Tollis O Dipartimento di Informatica e Sistemistica - University of Rome Via Buonmti, Rome, Italy Department of Computer Science - Brown University Box Providence, RI 029 Department of Computer Science - The University of Texas at Dallas P.O. Box , Mp Richardson, TX ABSTRACT We describe techniques for visualizing PERT diagrams. These techniques are used in the graphic tool X -PERT for computer-aided development and analysis of PERT diagrams, which provides an integrated environment to construct, display, and analyze PERT diagrams. Several graphic standards are supported, allowing for various styles of vertices (circles, boxes, etc.) and edges (straight-lines, polygonal lines, horizontal and vertical segments, etc.). INTRODUCTION The problem of visualization of data is very important and has received much attention recently. In this paper we describe techniques for visualizing PERT (Program Evaluution and Review Technique) diagrams. These techniques are used in the graphic tool X-PERT for computer-aided development and analysis of PERT diagrams, which provides an integrated environment to construct, display, and analyze PERT diagrams. Several graphic standards are sup ported, allowing for various styles of vertices (circles, boxes, etc.) and edges (straight-lines, polygonal lines, horizontal and vertical segments, etc.). X-PERT can be used as a production tool to assist project managers in planning and decision-making activities and as an instructional tool for teaching network management techniques. PERT diagrams, which represent scheduling constraints among the individual tasks of a project, are very important tools in the management and scheduling of large, complicated projects. Thus PERT diagrams are one of the fundamental techniques of modem management science and have been successfully used in many application areas [4,]. A PERT diagram is a drawing of a digraph G (called PERT digraph) whose edges are associated with the individual tasks of the project, and whose vertices are associated with designated events in the evolution of the project, i.e., Research supported in part by Cadre Technologies Inc. (USA), DIgital Equipment S.p.A. (Italy), and the Texas Advanced Research Program under grant no the start and completion of the various tasks. Each edge has a length, which represents the (expected) duration of the task. The tasks are partially ordered due to technical constraints. Correspondingly, in the PERT digraph all tasks associated with edges outgoing from a vertex v can start only if all the tasks associated with the edges incoming in v are completed. A PERT digraph G has the following fundamental properties: (1) G is acyclic, i.e., it has no directed cycles; (2) G has two designated vertices, denoted s (source) and t (sink), such that s has no incoming edges, and t has no outgoing edges. Vertices s and t are associated with the start and end of the project, respectively. (3) Every vertex of G distinct from s and t is on some directed simple path from s to t. An planar st-graph is a planar PERT digraph embedded in the plane with s and t on the boundary of the external face. An important and innovative feature of X-PERTis an automatic layout capability that takes into account several aesthetic criteria for better readability, such as the reduction of crossings, the left-to-right orientation of the edges, and the visualization of symmetries and isomorphic subgraphs. The automatic layout algorithm also supports a variety of constraints imposed by the user, such as the assignment of specific levels (x-coordinates) to the vertices. Note that existing software packages for PERT diagrams either do not have automatic layout capabilities or such capabilities are rather limited. The construction of a diagram can be done either through a graphic editor or by textual input of the list of events (vertices) and tasks (edges). In the latter case, the layout of the diagram is automatically generated. The diagram can be hierarchically specified so that it can be displayed at various levels of detail. In the analysis of a PERT diagram, several project parameters can be computed, such as earliest/latest start and finish times for each individual task. This can be done by computing for each vertex v the lengths of longest paths TH0277-4/89/0000/0171$ IEEE 171

2 from s to v and from v to t. The earliest completion time for the project corresponds to the length of a longest path from s to t, and any such longest sr -path is called a critical path, since an increase in duration of any task (edge) in a critical path causes a delay in the completion of the project. The algorithms for performing the above computations can be graphically animated for instructional purposes. Also. an option can be specified to modify the layout so that the edges of the critical path are arranged on a straight line as shown in Fig. 1. This feature of X-PERT is especially valuable for decision support, since the diagram so generated portrays very effectively the scheduling constraints of the project. Further analyses can be performed by introducing a probabilistic framework, and by studying cosdtime tradeoffs when the durations of tasks can be speeded up by allocating more resources ( crashing ). For a detailed treatment of PERT diagrams and their analysis, the reader is referred to [4,7,1. OVERVIEW OF X-PERT X-PERT consists of the following modules: (1) Graphic Interface: The graphic interface of X-PERT is a multi-window environment, where each window displays a diagram. Commands are selected from menus or entered through the keyboard. A macro capability allows experi; enced users to design their own commands. Several graphic standards for the diagrams are supported. Vertices can be drawn with different shapes, such as circles, ovals, rectangles, or diamonds. The user can also design new shapes, and add them to the existing selection. Edges can be drawn as straight-lines, polygonal lines, or splines. The lines can be chosen in different styles (simple, double, dashed, dotted) and the arrow can be put in various places along the edge. (2) Edit Module: The construction of a diagram can be done either in graphic mode or in textual mode. In graphic mode, the user places the vertices and draws the edges using a graphic editor. To assist in drawing edges, the system can determine routings that avoid or minimize crossings. This can be done efficiently with a variation of the graph embedding techniques described in [9]. Several checks are performed before executing a command to verify that the pmperties of a legal PERT digraph are satisfied. In textual mode, the user enters the list of vertices, which are temporarily displayed on a circle, and defines the edges by clicking with the mouse. The layout is then automatically generated according to some user-selected graphic standard. An additional primitive for educational use is a random PERT-diagram generator, which provides a virtually unlimited series of sample diagrams for testing algorithms. (3) Display Module: The visualization of the diagram on the screen is controlled by geometric zoom (enlarg4shriwtranslate the window on the diagram) and hierarchical zoom (change the detail level for an edgehubgraph). In other words, the diagrams are hierarchically represented to allow for the visualization of the tasks at different levels of detail. A general top-down primitive allows to specify the replacement of an edge (U,v) with a subgraph GUY, which is itself a PERT digraph with source U and sink v. The replacement can be performed by copying G, from another diagram, or by a sequence of operations from a library of predefined elementary expansions. A symmetric bottom-up primitive allows to replace a (PERT) subgraph of G with an edge. Further, hiding and highlight mechanisms are provided to help the user in focusing in specific aspects of the diagram. The above primitives are also available in a software library, so that the user can conveniently animate his/her own algorithms. (4) Analysis Module: When the diagram is completed, the analysis can be performed using either the traditional Critical Path Method, which assumes the exact durations of the tasks to be known, or more sophisticated techniques, where the duration of each task is a random variable with a given probability distribution. In an instructional environment, the algorithms for computing the various timing parameters (earliest and latest start and finish times for the tasks) are animated. Namely, the depth-fist search computations are visualized by appropriately highlighting the vertices and edges currently being examined. (5) Layout Module: X-PERT features a powerful automatic layout capability that supports all the commonly used graphic standards: (1) polyline diagrams: the edges are drawn as polygonal lines; (2) spline diagrams: the edges are drawn as splines; (3) orthogonal diagrams: the edges are drawn as sequences of horizontal and vertical segments. In all cases, the diagrams generated have the following important monotonicity property: all the edges are curves monotonically increasing in the x - (or y -) direction. This allows to effectively visualize the temporal dimension of the project along the horizontal direction. The diagram is implicitly embedded in a grid, so that each vertex is placed at a point with integer coordinates. The following aesthetic criteria are taken into account to produce layouts that are clear and easy to understand: (1) minimization of crossings between edges; (2) minimization of bends along the edges; (3) minimization of the total edge length; (4) minimization of the area; (5) display of symmetries; (6) display of isomorphic subgraphs. The input to this module is a PERT digraph G. It produces as output a corresponding diagram D, taking into account the above aesthetic criteria. The user specifies the graphic standard to be used (polyline, spline, orthogonal), and a set I72

3 of options and construints, to be discussed in the following sections. The layout strategy considers first the crossing minimization aesthetic. If the digraph is not planar (note that the monotonicity requirement must also be verified), it is planarized by introducing fictitious vertices at crossings. The resulting planar embedding, which describes the topology of the layout, is then processed to obtain the final drawing. This second phase is performed using different algorithms depending on the graphic standard adopted and on the relative importance attributed by the user to the remaining aesthetics. A distinctive feature of the layout algorithms is that the user can specify several constraints on the layout. For example, it is possible to have all the edges of a given path to be drawn on the same horizontal line. This provides a very effective way of visualizing critical paths. In addition, the user can specify the x coordinates of the vertices, provided they are consistent with the monotonicity requirement. A different layout algorithm can detect and display symmetries and isomorphic subgraphs. These layout algorithms have a solid theoretical foundation [2,3,8], and are computationally efficient both asymptotically and in practice. For a survey on graph drawing algorithms, see [5,10]. (6) Database Module: For each diagram, the system stores the names of vertices (events) and edges (tasks), the duration of each task, the connectivity information (vertices connected by each edge), the hierarchical structure of the graph, and the geometric information on the layout. If the analysis has been performed, the various timing parameters are also recorded. The diagrams can be grouped into projects, which in turn are stored in a database. Each project has a unique name and a set of keywords, which include the names of the tasks. Additional data, such as administrative and financial information, can also be stored. Searching in the database is performed either by project name or by keywords. The revision history of each project is recorded in a tree, which allows to restore previous versions and explore concurrently various alternatives. The database module also handles the copying of subdiagrams, which are selected by the user either graphically, or by a selection query. Finally, a centrally maintained library of task definitions is provided, which allows the enforcement of consistent project management within the same organization. TOPOLOGICAL EMBEDDING As discussed earlier, the layout algorithm consists essentially of two phases. Phase 1 computes a topological embedding for the digraph G, while Phase 2 determines the final geometric embedding. Phase 1 is the same for all the graphic standards, while Phase 2 depends on the graphic standard and on the set of options and constraints specified by the user. First, we test the planarity of G, subject to the monotonicity constraint on the edges. This can be done by adding to G the edge (s,t), and testing the resulting graph for (normal) planarity. If G is planar, we compute for it a topological embedding, i.e., the circular sequences of edges incident upon each vertex. If G is not planar, we extract from it a planar subgraph G, compute a topological embedding for G, and add the remaining edges one at a time. Graph G is incrementally constructed using a depthfirst-search technique on G, starting from the source s. The reintroduction of each nonplanar edge is performed so that each crossing is replaced by a fictitious vertex. The algorithm for this step introduces fictitious vertices (crossings), subject to the constraint that the resulting graph is planar and acyclic. This can be done by means of a shortest path computation on a subgraph of the dual graph of G. The example of Fig. 2, shows two ways of introducing the edge (U,v). The solution which introduces only one crossing is not feasible because it creates a directed cycle. After all the nonplanar edges have been added, we have a planarized version of G, which is a planar graph where the fictitious vertices represent crossings. The heuristic method adopted for this phase is justified by the fact that the crossing minimization problem is NP-hard. The running time for this phase is 0 (6 (n +c)), where n is the number of vertices, c is the number of crossings, and K is the number of nonplanar edges. It is interesting to observe that most PERT diagrams used in real-life applications are planar, and when they are not planar, the number of crossings is usually very small (see for example the diagrams for the construction industry given in [l]). This implies that our algorithm works very well in practice. GEOMETRIC EMBEDDING Since the previous phase has planarized the graph, we can assume with insignificant loss of generality that the input of this phase is a planar st -graph G = (V,A ) with a set of faces F. We visualize a planar st -graph G as drawn in the plane from left to right. F contains two representatives for the external face: the upper external face SI, which is incident with the edges on the upper boundary of G, and the lower external face t*, which is incident with the edges on the lower boundary of G. Each face f of G consists of two directed paths with common origin, called origv)), and common destination, called dest(f). For each edge U we define orig (a ) and dest (a ) as the tail and head vertices of a. Also, we define below(a) (above(a)) to be the face below (above) a. The incoming edges for each vertex v appear consecutively around v, and so do the outgoing edges. The face separating the incoming from the outgoing edges in the clockwise direction is called ubove(v) and the other separating face is called below (v), see Fig. 3. Let G =(V,A) be a digraph, where I V I =n and I E I = m. Given positive weights w (a ) on the edges of G, a I73

4 weighted topological numbering of G is a function that maps every vertex v of G to a number t(v) such that,(v)-$(u)2w(a), for every edge (u,v). A digraph G admits a weighted topological numbering if and only if it is acyclic. A weighted topological numbering is optimal if the quantity max I,(v) -,(U) 1 is minimized. u.v Lemma1 Let G be an acyclic digraph with positive weights on the edges. An optimal weighted topological numbering for G can be computed in 0 (n +m ) time. Two distinct algorithms can be used for Phase 2. The first one supports all the three graphic standards. provides a variety of layout options, and is able to satisfy user defined constraints. The second algorithm supports only the polyline standard, but has the advantage of recognizing and displaying symmetries and isomorphisms. Both algorithms have optimal 0 (n) time complexity. The first algorithm is based on the construction of a visibility representation for the graph G [8]. A visibility representation Y for a planar st-graph G maps each vertex v into a vertical segment Y(v), and each edge a into a horizontal segment "(a) such that: (1) segments Y(u) and Y(v) are disjoint for distinct vertices U and v : (2) segment Y(a) has its left endpoint on Y(orig(a)), its right endpoint on Y(dest(a)) and does not intersect any other segment. An interesting feature of Visibility representations is the possibility of aligning edges. Two paths x1 and x2 of G are said to be nonintersecting if they are edge-disjoint and they do not cross at common vertices. Let TI be a collection of nonintersecting paths of G. It is possible to construct a visibility representation of G such that the edges of every path in TI are horizontally aligned. An example of a visibility representation with two horizontally aligned paths from the source to the sink is given in Fig. 4. Algorithm Constrained-Visibility Input: planar st-graph G; set TI of nonintersecting paths covering the edges of G : Output: visibility representation Y of G such that y ("(a ')) = y (Y(a ")) for any two edges a ' and a " in the Same path x of n. (1) Construct the graph Gn with vertex set Fun and edge set ((f,x) If = above(a) for some edge a of path x) U (((x,g) Ig =below (a) for some edge a of path x) * Graph Gn is a planar st-graph. (2) (3) Compute an optimal weighted topological numbering X of G with unit weights. Compute an optimal weighted topological numbering Y of Gn with unit weights. (4) foreach vertex v do let Y(v) be the vertical segment with (5) x (Y(v)>=X (v); y I(WV 1) = Y (above(v)); y2(y(v)) = Y(below(v)). foreach path x in TI do foreach edge a in x do let Y(a) be the horizontal segment with y(y(a))=y(x); xdwa))=x(orig(a)); x 2(Y(a)) = X (dest(a )). Theorem 1 Let G be a planar st-graph with n vertices, and TI a set of nonintersecting paths covering the edges of G. Algorithm Constrained-Visibility computes in 0 (n ) time a visibility representation with integer coordinates and 0 (n 2, area, such that the edges of every path n in II are horizontally aligned. From the visibility representation Y, we can derive an orthogonal diagram in linear time [ 111. If a polyline diagram is instead requested, it can be constructed from Y with the following algorithm provided the paths in TI are vertex disjoint (see Fig. 5): for each vertex v do replace the vertex-segment Y(v) with a point P(v)=(x(v),y(v)) on Y(v) as follows: if v belongs to an aligned path x then x(v)=x(v); y(v)=y(x); else any point on Y(v) endfor for each edge (U,v) do ifx(v)-x(u)= 1 then ( short edge ) replace the edge-segment Y(u,v) with the segment P (u)+p (v) else begin ( long edge ) replace the edge-segment Y(u,v) with the polygonal line: P (U) -+ (x (u)+l,y ("(U,v))) (x (v >-I Y (W,v 1)) -3 p (v ); end endfor Any choice of P(v) along Y(v) guarantees the correctness of the algorithm and a small number of bends (at most two per edge). Theorem 2 Let G be a planar st-graph with n vertices, and TI a set of nonintersecting vertex-disjoint paths of G. A polyline planar diagram r for G with the following properties can be computed in 0 (n) time: (1) all 'edges of r increase monotonically in the x- direction (2) vertices and bends have integers coordinates, (3) r has 0 (n2) area, (4) r has at most 4n -10 bends, and (5) the edges of every path x in I'I are horizontally aligned. I14

5 Point (5) of this theorem allows one to effectively visualize specific paths, e.g. critical paths. The user has several options to specify the placement of P(v) (e.g., lowest endpoint, highest endpoint, or median intersection point of Y(v) with the edge-segments incident upon it). Each such option gives raise to diagrams with different aesthetic flavor. Specific choices of P (v) can be shown to produce no more that 3.3n - 10 bends 1, and total edge length within a 36 factor of the optimal [61. Depending on the user s choice, the resulting polyline digram can be converted into a spline diagram by means of a smoothing process. In this case the edges cannot be aligned. The above algorithm can be extended to draw graphs with prespecified x-coordinates for a set of vertices. This is very useful in constructing timed PERT diagrams, where the x coordinate of each vertex denotes the completion time of the corresponding activity. With respect to the second algorithm for Phase 2, it assigns the coordinates of the vertices using a procedure that detects and displays the symmetries of the graph and the isomorphisms of its subgraphs [3] (see Fig. 6). Another interesting feature of this algorithm is that the existence of directed paths between vertices is graphically visualized by the geometric dominance relation between the points of the plane where the vertices are located. Namely, there is a directed path from vertex U to vertex v if and only if x(u) Ix(v) and y (u)iy(v). WORKING WITH X-PERT Now we will see how a user can utilize X-PERT. Suppose that a user is interested in an application represented by the diagram of Fig. 1. Further, suppose the hdshe wants to insert the edge (v,u) with length 45 as shown in Fig. 7. The new edge will clearly introduce a crossing with edge (d,h). X-PERT notifies the user that there is a different drawing of the graph that preserves planarity. If the user prefers, he/she can have the planar drawing instead. The new drawing is shown in Fig. 8. It is interesting to note that the introduction of the new edge has changed the critical path. X-PERT realizes this and asks the user if he/she prefers to see a drawing of the diagram where all the edges in the new critical path lie on a straight line. If the answer is positive X-PERT draws the diagram of Fig. 9. If the newly introduced edge (v,u) corresponds to a subgraph which is stored in the database module, the user can invoke the hierarchical zoom capability of the display module, in order to see what is represented by the single edge (v,u). X-PERT can draw the new diagram with the edges of the new critical path on a straight line, see Fig. 10. X-PERT is currently being implemented. A prototype version has been developed and it runs on IBM PC and Macintosh. REFERENCES J.M. Antill and R.W. Woodhead, Critical Path Methods in Construction Practice, John Wiley & Sons, G. Di Battista and R. Tamassia, Algorithms for Plane Representations of Acyclic Digraphs, Theoretical Computer Science, vol. 61, no. 3, pp , G. Di Battista, R. Tamassia, and I.G. Tollis, Area Requirement and Symmetry Display in Drawing Graphs, Proc. Fifth ACM Symposium on Computational Geometry, Saarbruecken, West Germany, pp ,1989. C.B. Chapman, D.F. Cooper, and MJ. Page, Management for Engineers, John Wiley & Sons, P. Eades and R. Tamassia, Algorithms for Automatic Graph Drawing: An Annotated Bibliography, Technical Report CS-89-09, Dept. of Computer Science, Brown Univ., E. Pietrosanti, Algorithmi di Layout per Diagrammi PERT, Thesis, Dipartimento di Informatica e Sistemistica, University of Rome, L.R. Shaffer, J.B. Ritter, and W.L. Meyer, The Critical-Path Method, McGraw-Hill, R. Tamassia and I.G. Tollis, A Uniiied Approach to Visibility Representations of Planar Graphs, Discrete (e Computational Geometry, vol. 1, no. 4, pp , R. Tamassia, A Dynamic Data Structure for Planar Graph Embedding, Automata, Languages and Programming (F roc. 15th ICALP, Tampere, Finland, 1988), T. Lepisto and A. Salomaa (Eds.), Lecture Notes in Computer Science, vol. 317, pp , Springer-Verlag, R. Tamassia, C. Batini, and G. Di Battista, Automatic Graph Drawing and Readability of Diagrams, IEEE Transactions on Systems, Man and Cybernerics, vol. SMC-18, no. 1, pp , R. Tamassia and I.G. Tollis, Planar Grid Embedding in Linear Time, IEEE Transactions on Circuits and Systems, vol. 36, 1989 (to appear). N.L. Wu and J.A. Wu, Introduction to Management Science: A Contemporary Approach, Rand McNally, I75

6 d U Figure 1: A PERT diagram. Figure 6. A symmetric drawing of a graph. d Y v Figure 2. Introduction of a new edge (u.v). Figure 7. Addition of edge (v,u). g=above (e) below (v) Figure 3. Properties of a planar st-graph. Figure 8. The graph of Figure 7 drawn in a planar way. t d \ \ Figure 4. A visibility representation. b 20 f Figure 9. The graph of Figure 8 drawn so that the new critical path is a straight line. \Fe h b 20 f 17 t Figure 5. A polyline diagram. Figure 10. The graph of Figure 9 with the edge (v.u) drawn as a subgraph.