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Northridge, Los Angeles September 26, 2015 Graph Drawing Pixel & Voxel Representations of Graphs Md. Jawaherul Alam Tiomas Blsius Ignaz Rutuer Torsuen Ueckerdt Alexander Wolff . Motivation Build contact representation of graphs

  1. Northridge, Los Angeles – September 26, 2015 Graph Drawing Pixel & Voxel Representations of Graphs Md. Jawaherul Alam Tiomas Bläsius Ignaz Rutuer Torsuen Ueckerdt Alexander Wolff

  2. . Motivation Build contact representation of graphs Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  3. . Motivation Build contact representation of graphs Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  4. . Motivation Build contact representation of graphs Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  5. . Motivation Build contact representation of graphs Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  6. . Contact Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize polygonal complexity GD 2015 9 9 8 6 5 8 7 7 4 6 2 5 1 9 4 3 2 3 1 8 2 7 3 5 9 9 4 6 1 8 8 4 5 7 7 6 6 6 5 5 3 4 7 4 1 2 3 3 2 8 2 1 1 9 ■ Vertices ⇒ Geometric objects (polygons, arcs, polyhedra) ■ Edges ⇒ Contacts

  7. . Contact Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize polygonal complexity GD 2015 9 9 8 6 5 8 7 7 4 6 2 5 1 9 4 3 2 3 1 8 2 7 3 5 9 9 4 6 1 8 8 4 5 7 7 6 6 6 5 5 3 4 7 4 1 2 3 3 2 8 2 1 1 9 ■ Vertices ⇒ Geometric objects (polygons, arcs, polyhedra) ■ Edges ⇒ Contacts

  8. . Motivation Build contact representation of graphs Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  9. . Motivation Build contact representation of graphs from unit blocks How many unit blocks are required? Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  10. . Motivation Build contact representation of graphs from unit blocks How many unit blocks are required? Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  11. . Pixel and Voxel Representations Pixel in 2D, Voxel in 3D Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Building contact representation from unit blocks

  12. . Pixel and Voxel Representations Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Building contact representation from unit blocks ■ Pixel in 2D, Voxel in 3D

  13. . Pixel and Voxel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels/voxels GD 2015 1 1 5 3 2 4 3 5 5 4 4 2 3 1 2 ■ Vertices ⇒ Blobs (connected sets of pixels/voxels) ■ Edges ⇒ Adjacent (face-to-face) pixels/voxels in two blobs

  14. . Pixel and Voxel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels/voxels GD 2015 1 1 5 3 2 4 3 5 5 4 4 2 3 1 2 ■ Vertices ⇒ Blobs (connected sets of pixels/voxels) ■ Edges ⇒ Adjacent (face-to-face) pixels/voxels in two blobs

  15. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 5 2 4 3 5 4 2 3 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  16. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 1 5 3 5 2 4 3 4 5 4 2 3 2 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  17. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 1 5 3 5 2 4 3 4 5 4 2 3 2 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  18. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 1 5 3 5 33 23 2 4 3 4 5 4 2 3 2 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  19. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 1 1 5 5 3 12 3 5 33 23 4 2 4 3 4 5 2 4 2 3 2 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  20. . Pixel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of pixels GD 2015 1 1 1 1 5 5 3 12 3 5 33 23 4 2 4 3 4 5 2 4 2 3 2 ■ Vertices ⇒ Blobs (connected sets of pixels) ■ Edges ⇒ Adjacent (face-to-face) pixels in two blobs

  21. . Voxel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of voxels GD 2015 1 46 3 8 4 5 3 1 5 5 4 4 2 2 3 1 2 ■ Vertices ⇒ Blobs (connected sets of voxel) ■ Edges ⇒ Adjacent (face-to-face) voxels in two blobs

  22. . Voxel Representations Md. Jawaherul Alam Pixel & Voxel Representations Goal: minimize total number of voxels GD 2015 1 46 3 8 4 5 3 1 5 5 4 4 2 2 3 1 2 ■ Vertices ⇒ Blobs (connected sets of voxel) ■ Edges ⇒ Adjacent (face-to-face) voxels in two blobs

  23. . Related Work Contact Representations Point-contact with circles [Koebe, 1936] Point-contact with triangles [De Fraysseix et al., 1994] Side-contact with hexagons [Gansner et al., 2010], [Bonichon et al., 2010] Pixel & Voxel Representations Md. Jawaherul Alam GD 2015

  24. . Related Work Contact Representations [Gansner et al., 2010], [Bonichon et al., 2010] Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Point-contact with circles [Koebe, 1936] ■ Point-contact with triangles [De Fraysseix et al., 1994] ■ Side-contact with hexagons

  25. . Related Work Contact Representations with Rectilinear Polygons [Yeap and Sarrafzadeh, 1993], [He, 1999], [Liao et al., 2003] Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Contact with 8-sided rectilinear polygons:

  26. . Contact Representation of nonplanar graphs Md. Jawaherul Alam Pixel & Voxel Representations Related Work GD 2015 [Felsner and Francis, 2011] [Thomassen, 1986], [Bremner et al., 2012] Contact Representations in 3D ■ Contact representation of planar graphs with cuboids ■ Improper contact representation of planar graphs with cubes 2 3 4 1 5 6 7 8 9

  27. . [Felsner and Francis, 2011] Md. Jawaherul Alam Pixel & Voxel Representations Related Work GD 2015 Contact Representations in 3D [Thomassen, 1986], [Bremner et al., 2012] ■ Contact representation of planar graphs with cuboids ■ Improper contact representation of planar graphs with cubes ■ Contact Representation of nonplanar graphs 2 3 4 1 5 6 7 8 9

  28. . Related Work Vertex Contact Graphs of Paths on a Grid (VCPG) Mosaic Drawing Contact of square or hexagonal tilies [Cano et al., 2015] Same representation, different objective! Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Contact graphs of grid paths [Aerts and Felsner, 2014]

  29. . Related Work Vertex Contact Graphs of Paths on a Grid (VCPG) Mosaic Drawing Same representation, different objective! Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Contact graphs of grid paths [Aerts and Felsner, 2014] ■ Contact of square or hexagonal tilies [Cano et al., 2015]

  30. . Related Work Vertex Contact Graphs of Paths on a Grid (VCPG) Mosaic Drawing Same representation, different objective! Pixel & Voxel Representations Md. Jawaherul Alam GD 2015 ■ Contact graphs of grid paths [Aerts and Felsner, 2014] ■ Contact of square or hexagonal tilies [Cano et al., 2015]

  31. . Min-Pixel-Representation Md. Jawaherul Alam Pixel & Voxel Representations Our Result Min-Voxel-Representation GD 2015 Find a grid drawing with unit edge lengths Input: a planar max-degree-4 graph Reduction from: NP-complete in both 2D and 3D Computational Complexity ■ Finding minimum-size representation is 1 5 2 4 3 1 2 3 4 5 6 7 1 2 3 4 5 6 7

  32. . Min-Pixel-Representation Md. Jawaherul Alam Pixel & Voxel Representations Our Result Min-Voxel-Representation GD 2015 Find a grid drawing with unit edge lengths Computational Complexity NP-complete in both 2D and 3D ■ Finding minimum-size representation is 1 5 Reduction from: P 2 4 3 Input: a planar max-degree-4 graph G P ⇓ 1 2 3 4 ⇓ 5 6 7 1 2 3 4 5 6 7

  33. . , Md. Jawaherul Alam Pixel & Voxel Representations Our Result voxels are sufficient , For a graph with genus voxels are necessary and sufficient GD 2015 For a graph with treewidth voxels are sufficient Voxel Representation Computational Complexity Pixel Representation NP-complete in both 2D and 3D ■ Finding minimum-size representation is 1 5 2 3 4 ■ For a k -outerplanar graph, Θ( kn ) pixels are necessary and sufficient 1 2 3 4 5 6 7 1 2 3 4 5 6 7

  34. . , Md. Jawaherul Alam Pixel & Voxel Representations Our Result voxels are sufficient , For a graph with genus voxels are necessary and sufficient GD 2015 For a graph with treewidth Voxel Representation Computational Complexity Pixel Representation NP-complete in both 2D and 3D ■ Finding minimum-size representation is 1 5 2 3 4 ■ For a k -outerplanar graph, Θ( kn ) pixels are necessary and sufficient ■ O ( n 2 ) voxels are sufficient 1 2 3 4 5 6 7 1 2 3 4 5 6 7

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