2. 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

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

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

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

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

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

8. Our Model 3 formulas in our model 3 formulas in our model –Attractive force ( spring force ) f a ( d ) = C1 × log ( d / C2 ) f a ( d ) = C1 × log ( d / C2 ) –Repulsive force –Torque

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

10. The repulsive force on each border line b i of B due to A A B 2-D force model ( Chuang and Ahuja, 1998 ) A B a1a1 a2a2 a3a3 a4a4 b1b1 The potential at a point due to a point charge The potential at a point due to a line segment charge The repulsive force between two line segments b2b2 b3b3 The repulsive force on border line b 1 due to A The repulsive force on B due to A The repulsive torque on B due to A The attractive force on B due to A The attractive torque on B due to A The repulsive force on B due to A = 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 )

11. 3-D force model (Chuang, 1998 ) 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 The force at a point due to a surface is formulated as Those functions are analytically tractable. A B The force at a point due to the polyhedron A is formulated as The repulsive force on B due to A is formulated as 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 )

12. 2-D Mesh structure Initial drawingFinal drawing

13. 3-D Cases (A) Mesh.(B) Cube. (D) Hypercube. (C) Flower.

14. Application to Clustered Graphs

15. Application to Clustered Graphs (cont) Advantage of our approach 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

16. 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 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 The formulas are analytically tractable, making our algorithm computationally efficient An application to clustered graphs has been proposed An application to clustered graphs has been proposed

17. The End Thank you~