comp4048 planar and orthogonal graph drawing algorithms

COMP4048Planar and Orthogonal Graph

Drawing Algorithms

Richard WebberNational ICT Australia

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

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

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

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)

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

Testing Planarity

1. Find a cycle C in G (biconnected cycle must exist)

2. 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

Testing Planarity

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

Testing Planarity

• Biconnected G with cycle C is planar iff1. For each piece P, P’ = P C is planar;

and2. Interlacement graph I is bipartite

• Planarity of P’

determinedrecursively

Testing Planarity

1. Compute piece of G with respect to C2. For each non-path piece P

1. P’ = P C

2. C’ = cycle of P’ by replacing C between consecutive attachments with a path through P

3. Recursively test P’ with C’ – return if “non-planar”

3. Compute interlacement graph I4. Return “non-planar” if I not bipartite5. Return “planar”

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

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

Planar st-Graphs

Planar st-Graphs

• F = faces of planar st-graph G such thatexternal 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

Planar st-Graphs

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 all o V E F = rectangle (o)s horizontally adjacent os left/right (o)s vertically adjacent os orig/dest


1. G* from G2. Topological numbering Y of G3. Topological numbering X of G*

4. 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

Visibility Drawings


• Vertices = Horizontal lines• Edges = Vertical lines• Intersections only where edges meet

end-points• Tessellation Drawing Visibility

Drawing– degenerate vertices, non-degenerate faces

Visibility Drawings

1. G* from G2. weight(e) = 1 – Optimal topological

numbering Y of G3. weight(e*) = 1 – Optimal topological

numbering X of G*4. For each v V

– y (v) = Y(v); xL(v) = X(left(v)); xR(v) = X(right(v))-1

5. For each e E– x (e) = X(left(e)); yB(e) = Y(orig(e)); yT(e) = Y(dest(e))


Visibility Drawings

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


• Set of paths covers G – Add single-edge paths

1. Duplicate each path, adding faces to G* gives G

2. weight(e) = 1, Y(s) = 0 – Optimal topological numbering Y of G

3. weight(e*) = 0.5, X(s*) = -0.5 – Optimal topological numbering X of G


4. For each : for each e – x (e) = X()– yB(e) = Y(orig(e))– yT(e) = Y(dest(e))

5. For each v V– y(v) = Y(v)– xL(v) = minv X()– xR(o) = maxv X()


Polyline Drawings

• (Di Battista et al. 1999, Di Battista et al. 1992)

1. Construct a visibility drawing2. Place vertex vi at an arbitrary pi on

its line segment3. Draw short edge (vi, vj) as line pi pj

4. Draw long edge (vi, vj) as polyline pi (x(u, v), yu+1) (x(u, v), yv-1) pj

Polyline Drawing

• Place vertex at mid-point of its line segment


6n-12 bends (2 per edge)

Polyline Drawing

• Place vertex above long edges if they exist


(10n-31)/3 bends

Polyline Drawing

• Use constrained visibility

• Place vertex on path


4n-10 bends

Orthogonal via Visibility

• (Di Battista et al. 1999)• Input = planar st-graph1.Create subpaths v for v {s,t}

– 2 incoming edges leftmost-inrightmost-out

– 1 or 3 incoming edges median-inmedian-out


2. Unify subpaths with common edges to give

3. Apply Constrained-Visibility algorithm


4. 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 …


• O(n) time, O(n2) area, 2n+4 bends

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!


• 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


• 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)


• 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


• 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)


1. Construct N from G – O(n) time2. Compute minimum cost flow for N –

O(n2 log n) (Ahuja et al. 1993) or O(n7/4 log n) (Garg+Tamassia 1997) time

3. Map N to orthogonal representation for G – O(n) time


• To map orthogonal representation to drawing…

1. 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))


• External face by enclosing in a rectangle• Total O(n+b) time – b = number of bends


2. 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


• 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


• 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

Degree > 4

• (Di Battista et al. 1999, Fößmeier+ Kaufmann 1996)

• Replace vertex v of degree d > 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

Degree > 4

Bend Stretching


1. Take any planar orthogonal drawing2. Identify configurations that can be

transformed to reduce bends3. Iterate• General case requires O(n2)

• Identifying special cases requires O(n)

Bend Stretching

Alternative Degree > 4

• (Di Battista et al. 1999, Papakostas+Tollis 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

Degree > 4

References

• I. Fáry (1948): “On Straight Lines Representations of Planar Graphs” in Acta Scientiarum Mathematicarum, 11:229-233

• 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

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, 41:1-7

• 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

References

• A. Garg, R. Tamassia (1997): “A New Minimum Cost Flow Algorithm with Applications to Graph Drawing”, Proc. Graph Drawing (1996), Springer-Verlag, LNCS 1190:193-200

• U. Fößmeier, M. Kaufmann (1996): “Drawing High Degree Graphs with Low Bend Numbers”, Proc. Graph Drawing (1995), Springer-Verlag, LNCS 1027:254-266

• 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, 1272:354-367

comp4048 planar and orthogonal graph drawing algorithms

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