Algoritmo de Sugiyama--básico

GRAFOSArtigo: “Methods for visual understanding ofhierarchical system structures”K. Sugiyama, S. Tagawa, M. Toda. Methods for visual undestanding ofhierarchical system structures. IEEE Trans. Syst. Man Cybern., pp. 109-125,1981.MARCO ANTONIO GARCIA DE CARVALHOMarço de 2010Grafos e Aplicações

MotivationThis paper is intended to present methods to generate a visually understandable drawingof a hierarchy automatically by a computer.MARCO ANTONIO GARCIA DE CARVALHOMarço de 2010Grafos e Aplicações

MotivationThe rows and columns of a graphmatrix representation can bepermuted, allowing different viewsof the data set.(# of rows)! x (# of columns) ) !The user has a chance to detectpatterns in the new presentation andto gain insight into the data.E. Makinen, H. Siirtola. Reordering the reoderable matrix as an algorithmicproblem. In: Proc. Of the First International Conference on Theory andApplication of Diagrams, pp. 453-467, 2000.MARCO ANTONIO GARCIA DE CARVALHOMarço de 2010Grafos e Aplicações

Schematic diagramsMARCO ANTONIO GARCIA DE CARVALHOMarço de 2010Grafos e Aplicações

(P1) Definition of a Hierarchy: A hierarchy is defined by terms of graph theory.(P2) Identification of Readability Elements: In the case of hierarchies, the former, regularlayout of vertices, is identfied as the following readability element:Element A: “Hierarchical” layout of vertices;Element B: “Less-Crossings” of lines (edges);Element C: “Straightness” of lines;Element D: “Close” layout of vertices connected to each other.Element E: “Balanced” layout of lines coming into or going from a vertex.(P3) Specification of Basic Rules of Drawing: In order to draw a hierarchy, we shoulddetermine a layout of vertices and how to draw edges. We specify the basic rules regardingthese aspects as follows:Rule a) Vertices are placed on horizontal lines in each level of the hierarchywithout overlapping;Rule b) Each edge is drawn with a straight line.MARCO ANTONIO GARCIA DE CARVALHOMarço de 2010Grafos e Aplicações

(P4) Formulation as a Multstage Multiobjective Problem: The whole algorithmdeveloped includes four step.Step I: A “proper” hierarchy is formed from a given set of directed pairwise relationsamong elements of a system.Step II: The number of crossings of edges in the proper hierarchy is reduced bypermuting orders of vertices in each level.Step III: Horizontal positions of vertices are determined. The orders of the vertices inStep II is given as constraints to preserve the reduced number of crossings.Step IV: A picture of the hierarchy is automatically drawn.(P5) Theorical and Heuristcs Aproaches: We have focus on heuristhic methods.Therefore, we developed to Step II a method called Barycentric (BC) and to Step III wehave been developed the priority (PR) layout method where the computing cost isMARCO ANTONIO GARCIA DE CARVALHOMarçosignificatlyde 2010lower.Grafos e Aplicações

Basic definitionsMARCO ANTONIO GARCIA DE CARVALHOMarço de 2010Grafos e Aplicações

Basic definitionsMARCO ANTONIO GARCIA DE CARVALHOMarço de 2010Grafos e Aplicações

Numerical exampleOriginal graphMatrix realizationBarycentric MethodWe consider a heuristic method for the reordering of the row order σ 1 to reduce thenumber of crossings under the fixed column order σ 2 in M(σ 1 , σ 2 ).The essential idea of this method is to reorder σ 1 according to the row barycenterscalculated by (9)We can reduce the number of crossings by repeating the barycentric ordering of rowsand column in turn.MARCO ANTONIO GARCIA DE CARVALHOMarço de 2010Grafos e Aplicações