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Drawing Graphs with Nonuniform Nodes Using Potential Fields

Drawing Graphs with Nonuniform Nodes Using Potential Fields. Jen-Hui Chuang 1 , Chun-Cheng Lin 2 , Hsu-Chun Yen 2 1 Dept. of Computer and Information Science, National Chiao-Tung University, Taiwan 2 Dept. of Electrical Engineering, National Taiwan University, Taiwan. Outline.

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Drawing Graphs with Nonuniform Nodes Using Potential Fields

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  1. Drawing Graphs with Nonuniform Nodes Using Potential Fields Jen-Hui Chuang1, Chun-Cheng Lin2, Hsu-Chun Yen2 1 Dept. of Computer and Information Science, National Chiao-Tung University, Taiwan 2 Dept. of Electrical Engineering, National Taiwan University, Taiwan

  2. Outline • Introduction • Force-directed Method using Potential Fields • Experimental Results • Conclusion

  3. Definition • Graph • G = ( V, E ) • V : the set of nodes • E : the set of edges • Graph with nonuniform nodes • G = ( P, E ) • P : the set of nonuniform nodes • 2D: polygon • 3D: polyhedron

  4. Motivation • In practice, entities (nodes) may not be zero-sized. • Harel and Koren, 2002 • Propose two methods to draw this kind of graphs • Elliptic spring method • Modified spring method • Not considering degree of inclination of each nonuniform node

  5. Nodes → charges → repulsive force Edges → springs → attractive force let it go Force-directed method( a.k.a. Spring algorithm )

  6. Extended force-directed method • Nonuniform nodes → uniformly charged → repulsive force & torque • Edges → springs → attractive force & torque

  7. Our Model • 3 formulas in our model • Attractive force ( spring force ) • fa( d ) = C1 × log ( d / C2 ) • Repulsive force • Torque

  8. + + + + + + + + + + + + + + + ++++ + ++ ++ + +++++ Potential Field Method • Motion planning or Path planning ( Chuang and Ahuja, 1998) G + + S

  9. The attractive force on B due to A The attractive torque on B due to A 2-D force model(Chuang and Ahuja, 1998 ) a2 A A B B a3 a1 b1 b2 a4 b3 The repulsive force on border line b1 due to A The repulsive force on each border line bi of B due to A The repulsive force between two line segments The repulsive force on B due to A The potential at a point due to a point charge The potential at a point due to a line segment charge = Σ ( The repulsive force on each border line of B due to A ) = Σ Σ ( The repulsive force on each border line of B due to each border line of A ) The repulsive force on B due to A The repulsive torque on B due to A

  10. The force at a point due to the polyhedron A is formulated as The force at a point due to a surface is formulated as Assume that the potential is inversely proportional to the distance of the third order. The potential at a point due to a surface is expressed as Those functions are analytically tractable. The repulsive force on B due to A is formulated as 3-D force model(Chuang, 1998 ) A B The repulsive force on B due to A = Σ ( The repulsive force on each sampling point of B due to A ) = Σ Σ ( The repulsive force on each sampling point of B due to each surface of A )

  11. 2-D Mesh structure Initial drawing Final drawing

  12. 3-D Cases (A) Mesh. (B) Cube. (C) Flower. (D) Hypercube.

  13. Application to Clustered Graphs

  14. Application to Clustered Graphs (cont) • Advantage of our approach • Suppose new nodes are added to or deleted from a clustered graph. Instead of running the drawing algorithm on the new graph all over again, our approach allows us to keep the internal drawings of those unaffected clusters intact, while the redrawing only need to be applied to a much smaller graph, giving rise to a much better performance

  15. Conclusion • A potential-based approach, coupled with a force-directed method, has been proposed and implemented for drawing graphs with nodes of different sizes and shapes • The formulas are analytically tractable, making our algorithm computationally efficient • An application to clustered graphs has been proposed

  16. The End Thank you~

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