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COMP4048 Planar and Orthogonal Graph Drawing Algorithms

COMP4048 Planar and Orthogonal Graph Drawing Algorithms. Richard Webber National ICT Australia. Lecture Overview. Planarity Testing Planarity Tessellation Drawings Visibility Drawings Polyline Drawings Orthogonal Drawings via Visibility Drawings Orthogonal Drawings via Network Flow

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COMP4048 Planar and Orthogonal Graph Drawing Algorithms

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  1. COMP4048Planar and Orthogonal Graph Drawing Algorithms Richard WebberNational ICT Australia

  2. Lecture Overview • Planarity • Testing Planarity • Tessellation Drawings • Visibility Drawings • Polyline Drawings • Orthogonal Drawings via Visibility Drawings • Orthogonal Drawings via Network Flow • Degree > 4 • Bend Stretching • Degree > 4 again

  3. Planarity • A graph is planar if it can be drawn such that no edges cross • A drawing is planar if it is drawn with no edges crossing

  4. Planarity • Straight-line drawing with no edge crossings = Fáry Drawing (Fáry 1948) • 2D Fáry Drawing = Planar (Graph) Drawing •  Planar Drawing   Planar Embedding  Planar Graph • Plane Graph (drawing) = 2d

  5. Planarity • Simple Planar Embedding n + f = m + 2 m = O(n) and f = O(n) m  3n-6 Euler (http://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.html)

  6. Testing Planarity • (Di Battista et al. 1999, Goldstein 1963) • Trees and SP Digraphs = planar • Graph = planar  connect components = planar • Connect components = planar  biconnected components = planar • Biconnected  two vertex-disjoint paths

  7. Testing Planarity • Find a cycle C in G (biconnected  cycle must exist) • Decompose remaining edges into piecesPi • Connected without passing vertices of C • Incident vertices in C are attachments of Pi • If C  2+ pieces then C is separating • If C  1 piece then C is non-separating • C non-separating and P1  a path   separating C

  8. Testing Planarity

  9. Testing Planarity • Each piece must lie entirely inside or outsideC • Two pieces interlace if they cannot both be inside (outside) C without breaking planarity • Interlacement graphI of Gwith respect toC • Vertices = pieces of G • Edges between interlacing pieces

  10. Testing Planarity • Biconnected G with cycle C is planar iff • For each piece P, P’ = P  C is planar; and • Interlacement graph Iis bipartite • Planarity of P’ determined recursively

  11. Testing Planarity • Compute piece of G with respect to C • For each non-path piece P • P’ = P  C • C’ = cycle of P’by replacing C between consecutive attachments with a path through P • Recursively test P’ with C’ – return if “non-planar” • Compute interlacement graph I • Return “non-planar” if I not bipartite • Return “planar”

  12. Testing Planarity • Computing pieces and finding C’: O(n) • Computing I and testing bipartite: O(n2) • Each invocation = O(n2), O(n) invocations  O(n3) running time • Can be improved to O(n) (Hopcroft+Tarjan 1974) • Can construct planar embedding • use bipartite interlacement graph to alternate inside/outside pieces • path-pieces trivially inserted • non-path-pieces constructed recursively

  13. Planar st-Graphs • (Di Battista et al. 1999, Lempel et al. 1967) • Digraphs only • s = source, t = sink – only one of each • Add dummies if needed • Topological numbering – number(v) for v  V such that (u, v)  E  number(v) > number(u) • Topological sorting – numbering  [0..n-1] • For weighted edges number(v)  number(u) + weight(u, v) • number(s) = 0; number(v) by max over BFS • optimal in O(n + m) time

  14. Planar st-Graphs

  15. Planar st-Graphs • F = faces of planar st-graph G such that external face split: left s* and right t* • orig(e), dest(e), left(e), right(e) • left(v), right(v), orig( f ), dest( f ) • orig(v) = dest(v) = v; left( f ) = right( f ) = f • G* = ( F, { ( left(e), right(e) ) | e E } ) • G* is also planar st-graph

  16. Planar st-Graphs

  17. Planar st-Graphs

  18. Tessellation Drawings • (Di Battista et al. 1999, Tamassia+Tollis 1989) • Vertices / Edges / Faces = Objects • Object o drawn as a rectangle (o) • Possibly degenerate • (o1)  (o2) =  • Union over allo  V  E  F = rectangle • (o)s horizontally adjacent  os left/right • (o)s vertically adjacent  os orig/dest

  19. Tessellation Drawings • G* from G • Topological numbering Y of G • Topological numbering X of G* • For each o  V  E  F • xL(o) = X(left(o)) • xR(o) = X(right(o)) • yB(o) = Y(orig(o)) • yT(o) = Y(dest(o)) • O(n) time and O(n2) area

  20. Tessellation Drawings

  21. Visibility Drawings • (Di Battista et al. 1999, Tamassia+Tollis 1986) • Vertices = Horizontal lines • Edges = Vertical lines • Intersections only where edges meet end-points • Tessellation Drawing  Visibility Drawing • degenerate vertices, non-degenerate faces

  22. Visibility Drawings • G* from G • weight(e) = 1 – Optimal topological numbering Y of G • weight(e*) = 1 – Optimal topological numbering X of G* • For each v  V • y(v) = Y(v); xL(v) = X(left(v)); xR(v) = X(right(v))-1 • For each e  E • x(e) = X(left(e)); yB(e) = Y(orig(e)); yT(e) = Y(dest(e)) • O(n) time and O(n2) area

  23. Visibility Drawings

  24. Constrained Visibility • (Di Battista et al. 1999, Di Battista et al. 1992) • Identify non-intersecting pathsiin G • No common edges • No “crossings” • Can “touch” at vertices

  25. Constrained Visibility • Set of paths  covers G–Add single-edge paths • Duplicate each path, adding faces to G* gives G • weight(e) = 1, Y(s) = 0 – Optimal topological numbering Y of G • weight(e*) = 0.5, X(s*) = -0.5 – Optimal topological numbering X of G

  26. Constrained Visibility • For each   : for eache   • x(e) = X() • yB(e) = Y(orig(e)) • yT(e) = Y(dest(e)) • For each v  V • y(v) = Y(v) • xL(v) = minv X() • xR(o) = maxv X() • O(n) time and O(n2) area

  27. Constrained Visibility

  28. Polyline Drawings • (Di Battista et al. 1999, Di Battista et al. 1992) • Construct a visibility drawing • Place vertex vi at an arbitrarypi on its line segment • Draw short edge (vi, vj) as line pipj • Draw long edge (vi, vj) as polyline pi (x(u, v), yu+1)  (x(u, v), yv-1) pj

  29. Polyline Drawing • Place vertex at mid-point of its line segment • O(n) time and O(n2) area •  6n-12 bends (2 per edge)

  30. Polyline Drawing • Place vertex above long edges if they exist • O(n) time and O(n2) area •  (10n-31)/3 bends

  31. Polyline Drawing • Use constrained visibility • Place vertex on path • O(n) time and O(n2) area •  4n-10 bends

  32. Orthogonal via Visibility • (Di Battista et al. 1999) • Input = planar st-graph • Create subpaths v for v  {s,t} • 2 incoming edges  leftmost-inrightmost-out • 1 or 3 incoming edges  median-inmedian-out

  33. Orthogonal via Visibility

  34. Orthogonal via Visibility • Unify subpaths with common edges to give  • Apply Constrained-Visibilityalgorithm

  35. Orthogonal via Visibility • Create orthogonal drawing • Place vertex v  {s,t} on path v • Place s(t) on path of median of out (in) edges • Routes general edges via paths • Route s(t) edges as …  

  36. Orthogonal via Visibility • O(n) time, O(n2) area,  2n+4 bends

  37. Orthogonal via Network Flow • (Di Battista et al. 1999, Tamassia 1987) • Visibility guarantees O(1) bends per edge • Want to minimise total bends for embedding • minimising over all embeddings in NP-hard • Represent angles as a commodity • Produced by vertices, consumed by faces, transferred by bends • Apply a cost to each bend • Minimising bends = minimising cost of flow!

  38. Orthogonal via Network Flow • Replace each (undirected) edge (u, v) with two darts(u, v) and (v, u) • dart = counterclockwise for ff is on left • (u, v)·/2 = angle from dart (u, v) to next dart counterclockwise about u • (u, v) = number of “left” bends in (u, v) • Orthogonal representation = all (, ) • Same representation  same number bends

  39. Orthogonal via Network Flow

  40. Orthogonal via Network Flow • Network Nsuch that… • Source (sink) v produces (consumes) (v) • Arc (u, v) has • Lower bound (u, v) • Capacity (u, v) • Cost (u, v) • Flow (u, v) such that (u, v)  (u, v)  (u, v) • Sum  into v  {s,t} = sum  out • Cost of flow  in N = sum all (u, v)·(u, v)

  41. Orthogonal via Network Flow • Embed Graph G into Network N by… • Nodes of N = vertices and faces of G • Vertex-node v produces (v) = 4 • Internal face-node f consumes (f) = 2a(f)-4 • External face-node f consumes (f) = 2a(f)+4 • a(f) = number vertex-angles in face f

  42. Orthogonal via Network Flow • Dart (u, v) with left (right) face f (g) • arc (u, f): (u, f) = 1, (u, f) = 4, (u, f) = 0  (u, v) • arc (f, g): (f, g) = 0, (f, g) = , (f, g) = 1  (u, v)

  43. Orthogonal via Network Flow • Construct N from G – O(n) time • Compute minimum cost flow for N – O(n2 log n) (Ahuja et al. 1993) or O(n7/4 log n) (Garg+Tamassia 1997) time • Map Nto orthogonal representation for G – O(n) time

  44. Orthogonal via Network Flow • To map orthogonal representation to drawing… • Divide the faces into rectangles • e  corner(e)  next(e) – counterclockwise • turn(e) = +1 (left), 0 (straight), –1 (right) • front(e) = 1st next(e’) s.t. sum e..e’ = +1 • If turn(e) = –1 then insert • Vertex project(e) in front(e) • Edge extend(e) = (corner(e), project(e))

  45. Orthogonal via Network Flow • External face by enclosing in a rectangle • Total O(n+b) time – b = number of bends

  46. Orthogonal via Network Flow • Assign edge lengths • Minimising lengths/area – compaction • Interior rectangles: (u, v)  2, (u, v) = 0 • Exterior rectangle: (u, v)  2, (u, v) = 0 • Use horizontal and vertical flow networks, Nhorand Nver

  47. Orthogonal via Network Flow • Horizontal Flow Network Nhor • Nodes = interior faces of G plus lower s and upper t outer face • Arcs (f, g)  face f shares horizontal edge with face g – f below g • (f, g) = 1, (f, g) = , (f, g) = 1 • (f, g) = length of horizontal edge • Nveris analogous

  48. Orthogonal via Network Flow

  49. Orthogonal via Network Flow • Run-time dominated by network flow • O(n2 log n) or O(n7/4 log n) • Guarantees minimal width/height/length/area • Alternative Method • Place dummy vertices in external corners • Treat vertical (horizontal) paths as vertices • Calculate topological ordering X (Y) • Edge length = X(v)-X(u) (Y(v)-Y(u)) • O(n) time, but no guarantee of minimal total edge length

  50. Degree > 4 • (Di Battista et al. 1999, Fößmeier+ Kaufmann 1996) • Replace vertex v of degree d > 4 with a cycle v1, …, vd – eachviincident to one edge incident to v • Solve using Network Flow such that cycle edges have no bends… • For edge (u, v) separating faces f and g, (f, g) = (g, f) = 0 • By planarity, still O(n) vertices

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