1. A Note on the Self-Similarity of some Orthogonal Drawings Maurizio “Titto” Patrignani Roma Tre University, Italy GD2004 NYC 28 Sept – 2 Oct 2004

2. Orthogonal drawings

3. Are orthogonal drawings self-similar?

12. Purpose of this note  Prove that orthogonal drawings with a reduced number of bends are actually self-similar  How? Explore the implications of self-similarity Find some measurable property of self-similar objects Perform measures on a suitable number of orthogonal drawings obtained with different approaches and different types of graphs

13. Self-similarity and dimension number of copies scaling factor 22 44 number of copies scaling factor 42 164 number of copies scaling factor 82 644 = dimension 1 = 1 = 2 = 2 = 3 = 3

14. Recursively defined self-similar objects  Koch curve: recursively replace each segment with four segments whose length is 1/3 of the original

15. Recursively defined self-similar objects  Koch curve: recursively replace each segment with four segments whose length is 1/3 of the original

16. Dimension of the Koch curve number of copiesscaling factor 43 169 = dimension d = d log(4) = d log(3) log(4) log(3) d = = 1.26 4 = 3 d log(4) = log(3 ) d

17. Strategy  Self-similarity implies fractal dimension  To prove that orthogonal drawings are self- similar it suffices to show that they have a fractal dimension  We may choose between a number of “fractal dimensions”: Similarity dimension Hausdorf dimension Box-counting dimension Correlation dimension …

18. Box-counting fractal dimension log(box side length) log(#non empty boxes) N  l -d slope -d 15 non empty boxes98 non empty boxes

19. Box-counting fractal dimension box-side length l = 1 non empty boxes = N 0 box-side length l = 1/ non empty boxes = N = cN 0 scaling factor =  number of copies = c c =  d Hp similarity dimension d given by N = cN 0 NdNd c = N/N 0 N/N 0 = d N  l -d

20. Box-counting fractal dimension  PROS Easy to compute Also accounts for “statistical” self-similarity  CONS Defined for finite geometric objects only Defined for plane geometric objects only

21. Graph drawing and box-counting  We used FracDim Package [L. Wu and C. Faloutsos]

22. Graph drawing and box-counting Doubling the size of the boxes the number of non-empty boxes doesn’t change N  l 0 A B C D

23. Graph drawing and box-counting Doubling the size of the boxes the number of non-empty boxes is divided by two N  l -1 A B C D

24. Graph drawing and box-counting Doubling the size of the boxes the number of non-empty boxes is divided by four N  l -2 A B C D

25. Graph drawing and box-counting Doubling the size of the boxes the number of non-empty boxes doesn’t change N  l 0 A B C D

26. Graph drawing and box-counting If this segment exists then the geometrical object is a fractal A B C D

27. A test-suite of planar graphs  Using P.I.G.A.L.E. [H. de Fraysseix, P. Ossona de Mendez], we generated three test suites of random graphs planar connected, planar biconnected and planar triconnected ranging from 500 to 3,000 edges, increasing each time by 500 edges 10 graphs for each type  After the generation we removed multiple edges and self-loops

28. Three Orthogonal drawing approaches Orthogonal From Visibility approach (OFV) Construct a visibility representation of a biconnected graph Transform it into an orthogonal drawing [Di Battista et al. 99] Relative Coordinates Scenario (RCS) We used the “simple algorithm” described in [Papakostas & Tollis 2000] for biconnected graphs Topology-Shape-Metrics approach (TSM) Planarization: we used [Boyer & Myrvold 99] Orthogonalization: [Tamassia 87], [Fossmeier & Kaufmann 96] Compaction: rectangularization of the faces [Tamassia 87]

29. The Fractal Dimension of Orthogonal Drawings (OFV = Orth. From Visibility, RCS = Rel. Coord. Scenario, TSM = Topology-Shape-Metrics)

30. Conclusions and open problems  We assessed a fractal dimension (box-counting) of about 1.7 for orthogonal drawings with a reduced number of bends  Open problems: Do other graph drawing standards also produce self- similar drawings of large graphs? Can alternative measures of fractal dimension, like the correlation dimension, help deepening our understanding of this phenomenon? Can we lose self-similarity without adding too many bends to the drawings?

31. Biconnected graph with 1500 vert. drawn with OFV approach

32. Biconnected graph with 1500 vert. drawn with RCS approach

33. Maximal Planar (LEDA) 5000 vert. drawn with TSM approach

34. A test-suite of planar graphs