Download presentation
Presentation is loading. Please wait.
1
COMP4048 Planar and Orthogonal Graph Drawing Algorithms Richard Webber National 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)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 1.Find a cycle C in G (biconnected cycle must exist) 2.Decompose remaining edges into pieces P i –Connected without passing vertices of C –Incident vertices in C are attachments of P i –If C 2+ pieces then C is separating –If C 1 piece then C is non-separating –C non-separating and P 1 a path separating C
8
Testing Planarity
9
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
10
Testing Planarity Biconnected G with cycle C is planar iff 1.For each piece P, P’ = P C is planar; and 2.Interlacement graph I is bipartite Planarity of P’ determined recursively
11
Testing Planarity 1.Compute piece of G with respect to C 2.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 I 4.Return “non-planar” if I not bipartite 5.Return “planar”
12
Testing Planarity Computing pieces and finding C’ : O(n) Computing I and testing bipartite: O(n 2 ) Each invocation = O(n 2 ), O(n) invocations O(n 3 ) 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
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
18
Tessellation Drawings (Di Battista et al. 1999, Tamassia+Tollis 1989) Vertices / Edges / Faces = Objects Object o drawn as a rectangle (o) –Possibly degenerate (o 1 ) (o 2 ) = Union over all o V E F = rectangle (o) s horizontally adjacent o s left/right (o) s vertically adjacent o s orig/dest
19
Tessellation Drawings 1. G* from G 2.Topological numbering Y of G 3.Topological numbering X of G* 4.For each o V E F –x L (o) = X(left(o)) –x R (o) = X(right(o)) –y B (o) = Y(orig(o)) –y T (o) = Y(dest(o)) O(n) time and O(n 2 ) 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 1.G* from G 2.weight(e) = 1 – Optimal topological numbering Y of G 3.weight(e*) = 1 – Optimal topological numbering X of G* 4.For each v V –y (v) = Y(v) ; x L (v) = X(left(v)) ; x R (v) = X(right(v))-1 5.For each e E –x (e) = X(left(e)) ; y B (e) = Y(orig(e)) ; y T (e) = Y(dest(e)) O(n) time and O(n 2 ) area
23
Visibility Drawings
24
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
25
Constrained Visibility 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
26
Constrained Visibility 4.For each : for each e –x (e) = X( ) –y B (e) = Y(orig(e)) –y T (e) = Y(dest(e)) 5.For each v V –y(v) = Y(v) –x L (v) = min v X( ) –x R (o) = max v X( ) O(n) time and O(n 2 ) area
27
Constrained Visibility
28
Polyline Drawings (Di Battista et al. 1999, Di Battista et al. 1992) 1.Construct a visibility drawing 2.Place vertex v i at an arbitrary p i on its line segment 3.Draw short edge (v i, v j ) as line p i p j 4.Draw long edge (v i, v j ) as polyline p i (x (u, v), y u +1) (x (u, v), y v -1) p j
29
Polyline Drawing Place vertex at mid-point of its line segment O(n) time and O(n 2 ) area 6n-12 bends (2 per edge)
30
Polyline Drawing Place vertex above long edges if they exist O(n) time and O(n 2 ) area (10n-31)/3 bends
31
Polyline Drawing Use constrained visibility Place vertex on path O(n) time and O(n 2 ) area 4n-10 bends
32
Orthogonal via Visibility (Di Battista et al. 1999) Input = planar st -graph 1.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
2.Unify subpaths with common edges to give 3.Apply Constrained-Visibility algorithm
35
Orthogonal via Visibility 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 …
36
Orthogonal via Visibility O(n) time, O(n 2 ) 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
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)
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 1.Construct N from G – O(n) time 2.Compute minimum cost flow for N – O (n 2 log n) (Ahuja et al. 1993) or O(n 7/4 log n) (Garg+Tamassia 1997) time 3.Map N to orthogonal representation for G – O(n) time
44
Orthogonal via Network Flow 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))
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 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, N hor and N ver
47
Orthogonal via Network Flow Horizontal Flow Network N hor –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 N ver is analogous
48
Orthogonal via Network Flow
49
Run-time dominated by network flow –O(n 2 log n) or O(n 7/4 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 v 1, …, v d – each v i 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
51
Degree > 4
52
Bend Stretching (Di Battista et al. 1999, Tamassia+Tollis 1989) 1.Take any planar orthogonal drawing 2.Identify configurations that can be transformed to reduce bends 3.Iterate General case requires O(n 2 ) Identifying special cases requires O(n)
53
Bend Stretching
58
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
59
Degree > 4
60
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
61
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
62
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.