Graph Drawing Computational Geometry

Transkript

1 1 / 46 Computational Geometry Universität zu Lübeck Institute for Theoretical Computer Science February 7, 2012

2 2 / 46 Outline

3 3 / 46 What is? How to draw nice graphs?

4 4 / 46 What is a nice graph? Visualization of the graph should have nice look No precisely definition. Criteria on nice drawings Symmetries Less crossing as possible Less bends as possible Drawing should represents special properties Topological sorting

5 5 / 46 (a) (b) (c) (d) Figure : : a polyline (a), straight-line (b) and (c) an orthogonal drawing of K 3,3 ; (d) upward drawing

6 Motivation Is it useful for me? Dreeblöcken Kowitzberg Scharbeutz Timmendorfer Strand Niendorf 40 Pariner-/ Groß Parin 35 Kirschblüten- Linden Hagelstein Brodten Bad Schwartau HowingsTravemünde weg Gneversdorf 1 brook Lembke Distelkrog Am NordmeerRensefeld Hoppenhof 1 Steen- LeegerRensefeld Friedhof Mittschiffs wall kamp Eutin NorwegenTravemünde Kirche MittelRantzauallee Bollbrüch Strandbahnhof Labradorweg Am DänemarkAm Hoch- Kurpark 6007 Linien-Taxi 18 Rönkweg 35 Riesebusch kamp [18] Orkneyweg Stockelsdorf FehlingAm 35 Ludwig-Jahn-Str. Rose 33 Schleswig-Holstein-Tarif Außenallee Fahrenberg Gewerbegebiet Markt Raiffeisen Kirchen Eutiner Strand 5510 Altenheim Bohnrader Weg Gneversdorfer Weg Trelleborgallee NordlandBahnhof Ring Rodenkathen Stockelsdorfer Weg ring Travemünde Clever Segeberger Str. Mittlerer Nikolaus Teutendorf Am Lotsenberg 35 (17) Land Albert7 9 Wassergang Dorf Schwartauer Str. ZOB Teich Brodtener Kirchsteig Hafen Weidenweg Königsberger Kurgarten Schweitzer-Str. Cleverhofer Bad Schwartau Tannenberg31 2 Bäcker38 Gneversdorfer Weg Str. Bollbrügg Langer Hamburger Str. Weg AugustBerg Ravensbusch Garten (17) 9 Paul-Brümmer-Str. 9 gang Tor 30 Bogen Mollwo Teutenbrink Kaltenhöfer Schul- Sereetz Schulweg Hafenbahnhof LinienHindenburgKolberger MarienHohlweg Str Personen 1 35 Taxi Ring 7 10 burg Siebenberg 10 Str. MaxTravemünde fähre (Mai - Sept.) Teutendorfer Weg Rönnauer Ring DrögenMorier Hamerich-Str. Rathaus5520 Leibniz-Gymnasium Lübeck-Travemünder Priwallfähre Margarethenmarkt Dänischburg Mühleneck 2 Str. Sereetzer Weg Rönnauer Weg Bruhn 2 (17) Verkehrsgesellschaft 3 Cleverbrücker Str Vorwerker Friedhof Priwall Priwallfähre 6007 Ostsee PommernBahnhof Am Autofähre Groß Sereetzer Tannen KonradMorier- Bei den zentrum Tremskamp Steinrader WellenMecklenburger Steinrade hof Tannen Skandinavienkai 2 3 Ratekauer Behnckenhof Reinsbeker 1 Damm Grüner Jäger Dänischburger schlag Eingang Teich Land 10 Am Weg Bogen Terminal (17) Str. 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Utechter Bleeken AhornAm Ruschweg Fachhochschule Staunsfeld Weg 2 Buslinie, Haltestelle Grönauer Baum weg Kamp Stephenson 16 UniVorrade Grace-Hopper-Str. Nienhüsen Moorweg Auskunft kliniken 2 (zeitweilige Bedienung) 7 (17) Neptun Englisch Bahn 1 UniHerrnburg Alexander-Fleming-Str. Sporthuus Moor9 Hagen LibellenMaria- Bessemer VerwalSchule Schärenweg Herrnburg Str. Goeppert7 garten Stadtverkehr Lübeck Schwerin tung Halt nur in einer Richtung 9020 St.Hubertus weg 5 Wulfsdorf Beidendorf Str. Bahnhof 1 Universitäts(0451) Uhr Sundweg Liniennetz Region Lübeck I state-machines I UML class diagrams I electrical circuits / chip-design I traffic-nets (e. g. bus map) Graph Node R1 Edge R2 Faces el ün gr z n it inien rote L in PlanarGraph 6 / 46

7 7 / 46 Used for straight line drawings Model the graph-vertices and edges as physical objects with forces Distance of adjacent vertices (v, w) should be nearly the same distance d(v, w) No overlapping vertices

8 8 / Generate random representation of G with random positions p(v) for v V 2. Calculate forces between each pair of vertices v i, v j : f (vi, v j ) 3. Calculate resulting force f (v) for each node v V : f (v) w V \{v} f (v, w) 4. Update positions p(v) = p(v) + f (v)

9 9 / 46 Remember: we want a unit distance d(v, w) if v, w are adjacent Model edges as springs with a unit length unit length no force attracting force (adjacent vertices): f a = c a log d repelling force (non-adjacent vertices): f r = c r d 2 stepwise simulatable { f a (v, w), (v, w) E f (v) = ( p(w) p(v)) f r (v, w), (v, w) E w V \{v} p(v) := p(v) + f (v)

10 10 / 46 Sample

11 11 / 46 Advantages & Disadvantages easy implementable usefull for interactive implementation time-complexity: O(n V 2 ) slow with huge graphs non-predictable result

12 12 / 46 Goal: optimize drawing σ to selected criterias No forces, but physical model Inspired by cool-down process in liquids Define an energy-function E : σ R Low value E(σ) σ fits many criteria High temperature T high propabilty P finding a better configuration σ Low energy value E(σ) low propability P finding a better configuration σ

13 13 / 46 Algorithm 1: input : A graph G = (V, E), an initial temperature T output: A straight-line drawing of G begin Create initial drawing/configuration σ with random pos. for each vertex v V ; while do Choose a new configuration σ from the neighborhood of σ; Let E and E be the energy values of σ and σ ; if E < E or with a probability of e (E E)/T then σ σ ; Decrease T

14 14 / 46 Energy function should depend on the criteria low time complexity Example (by R. Davidson and D. Harel.): f = λ 1 1 d(n i, n j ) i j + λ 2 d bord (n) n V + λ 3 d(e) e E distribution border dist. edge length + λ 4 crossings edge crossings + λ 5 d(n, e) edge-distance e E,n V

15 15 / 46 Neighborhood & energy function How can we define the neighborhood? change the position of a single node in σ

16 R. Davidson, D. Harel Drawing Graphs Nicely Using, / 46 Simulated annealing example

17 17 / 46 Goal: reduce number of crossings in a drawing of a non-planar graph G We know: planar graphs can be drawn without crossings Approximate Graph as a planar graph Idea: convert non-planar graphs into planar graphs and replace non-removable crossings as virtual vertices Planarizataion : create a planar embedding of the new graph

18 18 / Find maximum planar subgraph and embedding, without crossing 2. Create embedding of the graph 3. Insert the non-planar edges with a minimum number of crossing, use crossings as virtual vertices

19 18 / Find maximum planar subgraph and embedding, without crossing 2. Create embedding of the graph 3. Insert the non-planar edges with a minimum number of crossing, use crossings as virtual vertices Problem Finding a maximum-planar subgraph is N P-hard. trade off find a maximal planar subgraph

20 19 / 46 Maximal planar subgraph Algorithm 2: Maximal planar subgraph input : a graph G = (V, E) output: a maximal planar subgraph G = (V, E ) of G begin E = ; while there is an edge e E \ E that G = (V, E {e}) is planar do E E {e}; return G ;

21 20 / 46 Definition A dual graph D G = (F, E) of a planar graph G = (V, E) is a (multi-)graph where F represents the faces in a planar embedding of G. Each pair f 1, f 2 F of adjacent faces is splitted by an edge in G. E(F) E(G)

22 21 / 46 Insert non-planar edge with min. crossings Insert non-planar edge e = (v, w) E 1. Find shortest path p in the dual-graph D G 2. Insert e over path p generate new virtual nodes and split faces

23 22 / 46 Insert non-planar edge with min. crossings

24 23 / 46 Insert non-planar edge with min. crossings

25 24 / 46 Shortest Path in the Dual Graph k e k e k+1 k+1

26 25 / 46 Improve with parallel substructures Problem: for some edges the graph can easily be modified to reduce number of crossings if the 2 vertices of the edge are in different blocks of a parallel substructure switch the order to decrease number of crossings

27 26 / 46 s e t (a) trivial case e (b) parallel case e (c) series case e (d) rigid case Figure : the four different types of nodes in a

28 27 / 46 a d c b e g f h k l i m j n o p

29 28 / 46 a f d b c k h e l i j m g n k o m l h p f m h d h h h j h a d c d e e e e e e h n j j j j i b e n n j g n n o p

30 29 / 46 s a k m l h f d c b d d e e g e e h e e n h e n h n h j n n h j j j o j j m h i R p S P S R P S P S P S P R

31 30 / 46 s b a c e d f h l k i j m g n o p R P S P R S P S P S S P R

32 31 / 46 s e b c d a f h i k l j m g n o p R P S P R S P S P S S P R

33 Γ(E) for each edge e E is a set of orthogonal lines Γ(v) for each vertex v V is a point in R 2 Definition We call Γ an orthogonal planar drawing of G if 1. v, w V : Γ(v) = Γ(w V ) w = v (invective) 2. e = (v, w) E : Γ(v), Γ(w) are endpoints of Γ(e) 3. e 1, e 2 E : e 1 e 2 Γ(e 1 ), Γ(e 2 ) only intersect at a point Γ(v) of a common vertex v (if exists) 32 / 46

34 33 / 46 two different planar orthogonal representations Γ, Γ of the same Graph G, but with a different number of edges and the same faces

35 34 / 46 Bend as If we have a planar representation Γ of a planar Graph, keeping the the same faces we can minimize the number of bends with solving a minimum cost flow problem.

36 35 / 46 Bend as Definition A min-cost flow network N = (D, u, l, b, c) consists of directed (multi-)graph D = (W, A) capacities u : A N 0 { }, l : A Z node demands b : W Z costs c : A N 0. f : A N 0 is a flow on N if l(a) f (a) u(a) b(v) = (v,w) A f (v, w) (w,v) A ( v W (b(v) = 0)) f (w, v) f is a min-cost-flow if a A c(a)f (a) is minimal.

37 36 / 46 Define the flow network for the problem node demands & capacities v 4 v 5 v 1 v 2 v 3 α(v 5, v 3 ) = 2 τ(v 3, v 5 ) = 2; τ(v 5, v 3 ) = 1; α(v 3, v 5 ) = 1 define labels α(v, w), τ(v, w) for each edge (v, w) in G: τ : (v, w) E #of L-bends in Γ((v, w)) α : (v, w i ) E betw. Γ((v, w i ))andγ((v, w i+1 ))

38 37 / 46 Define the flow network for the problem node demands & capacities Remember: We want to minimize the total number of bends on edges in the drawing Γ(G). The total number is (v,w) E τ(v, w) + τ(w, v). So the goal is: minimize (v,w) E τ(v, w) + τ(w, v)

39 Theorem The sum of angles(at vertices or bends) around an inner/outer face f F is: π (deg(f ) + #bends 2) This implies: (α(v, w) + τ(w, v) τ(v, w)) = 2 deg(f ) 4 (v,w) E(f ) 38 / 46

40 39 / 46 (α(v, w) + τ(w, v) τ(v, w)) = 2 deg(f ) 4 (v,w) E(f ) Let the face-to-face edges (producers of bends on an edge) get edges in the flow networks with costs 1

41 40 / 46 Define the flow network for the problem The directed (multi-) Graph D G D G = (W, A) with W = V F consists D F as the face-induced subgraph of F by Γ(G) demands { in the flow-network N : 2 deg(x) ± 4, x F (inner / outer face) b(x) = : 4, x V capacities: e A : u(e) = ; e A F : l(e) = 0; e A F : l(e) = 1 costs: e F A F : c(e F ) = 1; e A F : c(e) = 0

42 41 / 46 Assign flow Given a flow f : E N f (v, f v,w ): angle(α) at node v in face f v,w left of (v, w) f ( (f 1, f 2 ) e ) : #LeftBends (τ) on the edge (f1, f 2 ) e between the faces f 1 and f 2 A flow f is a valid model for a valid drawing if: for all nodes v: b(v) = (v,w) E f (v, w) = 4 for all faces f : (α (v, w) + τ(w, v) τ(v, w)) = 2 deg(f ) 4 (v,w) E(f ) inflow outflow A valid flow implies a valid graph representation Γ A cost-minimal flow implies a bend-minimal graph representation Γ

43 41 / 46 Assign flow Given a flow f : E N f (v, f v,w ): angle(α) at node v in face f v,w left of (v, w) f ( ) (f 1, f 2 ) e : #LeftBends (τ) on the edge (f1, f 2 ) e between the faces f 1 and f 2 A flow f is a valid model for a valid drawing if: for all nodes v: b(v) = (v,w) E f (v, w) = 4 for all faces f : (α (v, w) + τ(w, v) τ(v, w)) = 2 deg(f ) 4 (v,w) E(f ) inflow outflow A valid flow implies a valid graph representation Γ A cost-minimal flow implies a bend-minimal graph representation Γ

44 41 / 46 Assign flow Given a flow f : E N f (v, f v,w ): angle(α) at node v in face f v,w left of (v, w) f ( ) (f 1, f 2 ) e : #LeftBends (τ) on the edge (f1, f 2 ) e between the faces f 1 and f 2 A flow f is a valid model for a valid drawing if: for all nodes v: b(v) = (v,w) E f (v, w) = 4 for all faces f : (α (v, w) + τ(w, v) τ(v, w)) = 2 deg(f ) 4 (v,w) E(f ) inflow outflow A valid flow implies a valid graph representation Γ A cost-minimal flow implies a bend-minimal graph representation Γ

45 42 / 46 Existence of a valid flow: Inflow=Outflow to proof: v W b(v) =! 0 b(v) = b(v) + b(f ) v W v V f F = V 4 + ( 2 deg f + 4) 2 deg f o 4 f F\{f o} = V 4 + ( F 2)4 2 f ) f F(deg = V 4 + ( F 2)4 4 E = 4( V + F E 2) = 0

46 43 / 46 Bend example

47 44 / 46 graph drawing is a wide area with many problems most problems are N P hard problems depend on the type of drawings and the slected criterias good approximation is possible for many problems combination of algorithms may be possible there exists many tools for graph drawing 1. yed 2. GraphViz 3. Gephi 4. Tulip

48 45 / 46 GraphViz example circo-layout: twopi-layout: 1graph G{ 2 node[label=""]; 3 a1 -- a2; a1 -- a3; 4 a1 -- a4; 5 a1 -- a5; a1 -- a6; 6 a2 -- a3; a2 -- a4; 7 a2 -- a5; a2 -- a6; 8 a3 -- a4; a3 -- a5; 9 a3 -- a6; 10 a4 -- a5; a4 -- a6; 11 a5 -- a6; 12}

49 46 / 46 yed