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Morphing and Visiting Drawings of Graphs Vincenzo Roselli XXVI Ciclo Advisors: Prof. Giuseppe Di Battista Prof. Maurizio Patrignani Dipartimento Di Ingegneria June 9, 2014 ROMA TRE UNIVERSIT DEGLI STUDI Graphs a set V of objects,

  1. Morphing and Visiting Drawings of Graphs Vincenzo Roselli XXVI Ciclo Advisors: Prof. Giuseppe Di Battista Prof. Maurizio Patrignani Dipartimento Di Ingegneria June 9, 2014 ROMA TRE UNIVERSITÀ DEGLI STUDI

  2. Graphs a set V of objects, called vertices ; and a (multi)set E of relationships, called edges , between pairs of vertices . . . . . preliminaries morphing visiting publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation A graph G is a pair ( V , E ) of: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  3. Graphs . Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries . . . . . … … transportation geographical maps … biological social computer networks knowledge representation . . Areas of Application Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  4. Graphs . Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries . . . . . … … transportation geographical maps … biological social computer networks knowledge representation . . Areas of Application Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  5. Graphs . Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries . . . . . … … transportation geographical maps … biological social computer networks knowledge representation . . Areas of Application Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  6. Drawings of Graphs . Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing Ph.D. Dissertation preliminaries . . . . . Drawing a graph is a very natural way to analyze it . . Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  7. Drawings of Graphs . Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries . . . . . Obtain nice and readable graphical representations of graphs . . Goal . Ph.D. Dissertation . Research field dealing with the visualization of graphs Graph I n Theory V f o i r s m u a a l t i i z o a n t Graph i o n Drawing Computational Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  8. Conventions & Æsthetic Criteria angular resolution Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries . . . . . … number of crossings . minimum distance . . Criteria . … convex, orthogonal, upward curve, poly-/straight-line planarity . . Conventions Ph.D. Dissertation b t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  9. Equivalent Drawings A drawing of a graph defines: Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries . . . . . rotation schemes + crossing sequences + external face Two drawings are equivalent if they induce the same embedding : Ph.D. Dissertation a circular ordering ( rotation scheme ) of the edges incident to each vertex a sequence of crossings along each edge a partition of the plane into regions called faces b f b e f e a c a c a c f e d d b d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  10. Morphing and Visiting losing the “correspondences” between two drawings is Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries . . . . . long-standing open, challenging, ticklish problem morphing helps the user in switching from a drawing to another Motivations confusing, time-wasting, bad! even if equivalent, drawings can be very different Morphing: namely, the user wants to look from different perspectives and hence in several drawings a single user is typically interested in several properties different drawings emphasize different aspects of the structure Visiting: . . People (unawares) do look at drawings of graphs . Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  11. Morphing Drawings of Graphs Metamorphosis I – M. C. Escher . . . . . preliminaries morphing visiting publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  12. Morphing Planar Graph Drawings Transformation of a planar drawing of a graph into another preserving planarity moving vertices at constant speed along straight-lines . . . . . preliminaries morphing visiting publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  13. Morphing Planar Graph Drawings Transformation of a planar drawing of a graph into another preserving planarity moving vertices at constant speed along straight-lines . . . . . preliminaries morphing visiting publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  14. Planar Linear Morphing Steps For some pairs of drawings, the morph requires several steps The complexity of a morphing algorithm is defined as the number of (planar linear morphing) steps it performs . . . . . preliminaries morphing visiting publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  15. Planar Linear Morphing Steps For some pairs of drawings, the morph requires several steps The complexity of a morphing algorithm is defined as the number of (planar linear morphing) steps it performs . . . . . preliminaries morphing visiting publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  16. One Hundred Years of Morphs preliminaries morphing visiting publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  17. Tietze, 1914: Quadrilaterals Rendiconti del Circolo Matematico di Palermo , 38(1):247-304,1914 . . . . . preliminaries morphing visiting publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  18. Polygons Smith, 1917 Veblen, 1917 Alexander, 1923 . . . . . preliminaries morphing visiting publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  19. Cairns, 1944: Triangulations First algorithmic proof . . . . . preliminaries morphing visiting publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  20. Cairns’s Idea O Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries . n T n = O T n T n . . . Number of Steps Ph.D. Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  21. Cairns’s Idea = Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries n T n O O T n T n . . . Number of Steps . Ph.D. Dissertation contract v on O (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  22. Cairns’s Idea O Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries n O = T n T n T n . . . Number of Steps . Ph.D. Dissertation contract v on contract v on O (1) O (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  23. Cairns’s Idea . Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries n O T n = O T n T n . . . Ph.D. Dissertation Number of Steps contract v on contract v on Convex drawing of G \ { } ⇒ ⇒ O (1) O (1) ? ? O (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  24. Cairns’s Idea . Department of Engineering - Roma Tre Vincenzo Roselli publications visiting morphing preliminaries . . . . . . . Number of Steps Ph.D. Dissertation contract v on contract v on Convex drawing of G \ { } ⇒ ⇒ O (1) O (1) O (1) recursion recursion T ( n − 1) T ( n − 1) T ( n ) = 2 T ( n − 1) + O (1) = ⇒ T ( n ) ∈ O (2 n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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