COMP4048 Planar and Orthogonal Graph Drawing Algorithms

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COMP4048Planar and Orthogonal Graph Drawing
Algorithms

• Richard WebberNational ICT Australia

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Lecture Overview

• Planarity
• Testing Planarity
• Tessellation Drawings
• Visibility Drawings
• Polyline Drawings
• Orthogonal Drawings via Visibility Drawings
• Orthogonal Drawings via Network Flow
• Degree gt 4
• Bend Stretching
• Degree gt 4 again

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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

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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

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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/Bio
graphies/Euler.html)
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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

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Testing Planarity

• Find a cycle C in G (biconnected ? cycle must
  exist)
• Decompose remaining edges into pieces Pi
• 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

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Testing Planarity
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Testing Planarity

• Each piece must lie entirely inside or outside C
• Two pieces interlace if they cannot both be
  inside (outside) C without breaking planarity
• Interlacement graph I of G with respect to C
• Vertices pieces of G
• Edges between interlacing
• pieces

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Testing Planarity

• Biconnected G with cycle C is planar iff
• For each piece P, P P ? C is planar and
• Interlacement graph I is bipartite
• Planarity of P
• determined
• recursively

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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

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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) (HopcroftTarjan 1974)
• Can construct planar embedding
• use bipartite interlacement graph to alternate
  inside/outside pieces
• path-pieces trivially inserted
• non-path-pieces constructed recursively

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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) gt 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

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Planar st-Graphs
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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

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Planar st-Graphs
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Planar st-Graphs
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Tessellation Drawings

• (Di Battista et al. 1999, TamassiaTollis 1989)
• Vertices / Edges / Faces Objects
• Object o drawn as a rectangle ?(o)
• Possibly degenerate
• ?(o1) ? ?(o2) ?
• Union over all o ? V ? E ? F rectangle
• ?(o)s horizontally adjacent ? os left/right
• ?(o)s vertically adjacent ? os orig/dest

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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

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Tessellation Drawings
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Visibility Drawings

• (Di Battista et al. 1999, TamassiaTollis 1986)
• Vertices Horizontal lines
• Edges Vertical lines
• Intersections only where edges meet end-points
• Tessellation Drawing ? Visibility Drawing
• degenerate vertices, non-degenerate faces

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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

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Visibility Drawings
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Constrained Visibility

• (Di Battista et al. 1999, Di Battista et al.
  1992)
• Identify non-intersecting paths ?i in G
• No common edges
• No crossings
• Can touch at vertices

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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?

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Constrained Visibility

• For each ? ? ? for each e ? ?
• 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

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Constrained Visibility
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Polyline Drawings

• (Di Battista et al. 1999, Di Battista et al.
  1992)
• Construct a visibility drawing
• Place vertex vi at an arbitrary pi on its line
  segment
• Draw short edge (vi, vj) as line pi ? pj
• Draw long edge (vi, vj) as polyline pi ? (x(u,
  v), yu1) ? (x(u, v), yv-1) ? pj

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Polyline Drawing

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

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Polyline Drawing

• Place vertex above long edges if they exist
• O(n) time and O(n2) area
• ? (10n-31)/3 bends

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Polyline Drawing

• Use constrained visibility
• Place vertex on path
• O(n) time and O(n2) area
• ? 4n-10 bends

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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

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Orthogonal via Visibility
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Orthogonal via Visibility

• Unify subpaths with common edges to give ?
• Apply Constrained-Visibility algorithm

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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

?
?
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Orthogonal via Visibility

• O(n) time, O(n2) area, ? 2n4 bends

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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!

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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

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Orthogonal via Network Flow
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Orthogonal via Network Flow

• Network N such 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)

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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

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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)

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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)
  (GargTamassia 1997) time
• Map N to orthogonal representation for G O(n)
  time

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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))

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Orthogonal via Network Flow

• External face by enclosing in a rectangle
• Total O(nb) time b number of bends

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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, Nhor
  and Nver

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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
• Nver is analogous

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Orthogonal via Network Flow
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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

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Degree gt 4

• (Di Battista et al. 1999, Fößmeier Kaufmann
  1996)
• Replace vertex v of degree d gt 4 with a cycle v1,
  , vd each vi incident 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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Degree gt 4
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Bend Stretching

• (Di Battista et al. 1999, TamassiaTollis 1989)
• Take any planar orthogonal drawing
• Identify configurations that can be transformed
  to reduce bends
• Iterate
• General case requires O(n2)
• Identifying special cases requires O(n)

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Bend Stretching
?
?
?
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Bend Stretching
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Bend Stretching
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Bend Stretching
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Bend Stretching
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Alternative Degree gt 4

• (Di Battista et al. 1999, PapakostasTollis 1997)
• Vertices are rectangular boxes
• width max(1, out-degree-1)
• height max(1, ?in-degree/2?-1)
• Place vertices in order by st-numbering
• Place above previous vertex
• Place between median in-coming edges
• Route in-coming edges to left and right sides
• Route out-going edges from top

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Degree gt 4
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References

• I. Fáry (1948) On Straight Lines
  Representations of Planar Graphs in Acta
  Scientiarum Mathematicarum, 11229-233
• G. Di Battista, P. Eades, R. Tamassia, I. G.
  Tollis (1999) Graph Drawing Algorithms for the
  Visualization of Graphs, Prentice-Hall
• A. J. Goldstein (1963) An Efficient and
  Constructive Algorithm for Testing Whether a
  Graph can be Embedded in the Plane, Proc. Graph
  and Combinatorics Conf.
• J. Hopcroft, R. E. Tarjan (1974) Efficient
  Planarity Testing, J. ACM, 21(4)549-568
• A. Lempel, S. Even, I. Celerbaum (1967) An
  Algorithm for Planarity Testing of Graphs, Proc.
  Int. Symp. Theory of Graphs (1966), pp. 215-232,
  Gordon and Breach

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References

• R. Tamassia, I. G. Tollis (1989) Tessellations
  Representations of Planar Graphs, Proc. 27th
  Allerton Conf. Communication, Control and
  Computing
• R. Tamassia, I. G. Tollis (1986) A Unified
  Approach to Visibility Representations of Planar
  Graphs, Discrete and Computational Geometry,
  1(4)321-341
• G. Di Battista, R. Tamassia, I. G. Tollis (1992)
  Constrained Visibility Representations of
  Graphs, Information Processing Letters, 411-7
• R. Tamassia (1987) On Embedding a Graph in the
  Grid with the Minimum Number of Bends, SIAM J.
  Computing, 16(3)421-444
• R. K. Ahuja, T. L. Magnanti, J. B. Orlin (1993)
  Network Flows Theory, Algorithms and
  Applications, Prentice-Hall

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References

• A. Garg, R. Tamassia (1997) A New Minimum Cost
  Flow Algorithm with Applications to Graph
  Drawing, Proc. Graph Drawing (1996),
  Springer-Verlag, LNCS 1190193-200
• U. Fößmeier, M. Kaufmann (1996) Drawing High
  Degree Graphs with Low Bend Numbers, Proc. Graph
  Drawing (1995), Springer-Verlag, LNCS
  1027254-266
• R. Tamassia, I. G. Tollis (1989) Planar Grid
  Embedding in Linear Time, IEEE Trans. Circuits
  and Systems, 36(9)1230-1234
• A. Papakostas, I. G. Tollis (1997) Orthogonal
  Drawing of High Degree Graphs with Small Area and
  Few Bends, Proc. 5th Work. Algorithms and Data
  Structures, Springer-Verlag, LNCS, 1272354-367