Algorithms for Graph Visualization Layered Layout

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1 Algorithms for Graph Visualization Layered Layout INSTITUT FÜR THEORETISCHE INFORMATIK FAKULTÄT FÜR INFORMATIK Tamara Mchedlidze Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

2 Example 2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

3 Layered Layout Given: directed graph D = (V, A) Find: drawing of D that emphasized the hierarchy 3-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

4 Layered Layout Given: directed graph D = (V, A) Find: drawing of D that emphasized the hierarchy many edges pointing to the same direction nodes lie on (few) horizontal lines 3-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

5 Layered Layout Given: directed graph D = (V, A) Find: drawing of D that emphasized the hierarchy edges as straight as possible and short few edge crossings nodes distributed evenly 3-3 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

6 Application: Java Profiler yed Gallery: Java profiler JProfiler using yfiles 4 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

7 Application: Storylines Source: Design Considerations for Optimizing Storyline Visualizations Tanahashi et al. 5 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

8 Application: Storylines Source: ABC news, Australia 6 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

9 Application: Text-Variant graphs Source: Improving the Layout for Text Variant Graphs Jänicke et al. 7 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

10 Application: Mythological Creatures and Gods Source: Visualization that won the Graph Drawing contest Klawitter&Mchedlidze 8 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

11 Sugiyama Framework (Sugiyama, Tagawa, Toda 1981) given resolve cycles layer assignement 9-1 crossing minimization node positioning edge drawing Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

12 Sugiyama Framework (Sugiyama, Tagawa, Toda 1981) given resolve cycles layer assignement 9-2 crossing minimization node positioning edge drawing Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

13 Sugiyama Framework (Sugiyama, Tagawa, Toda 1981) given resolve cycles layer assignement 9-3 crossing minimization node positioning edge drawing Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

14 Sugiyama Framework (Sugiyama, Tagawa, Toda 1981) given resolve cycles layer assignement 9-4 crossing minimization node positioning edge drawing Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

15 Sugiyama Framework (Sugiyama, Tagawa, Toda 1981) given resolve cycles layer assignement 9-5 crossing minimization node positioning edge drawing Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

16 Sugiyama Framework (Sugiyama, Tagawa, Toda 1981) given resolve cycles layer assignement 9-6 crossing minimization node positioning edge drawing Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

17 Sugiyama Framework (Sugiyama, Tagawa, Toda 1981) paper cited more than 1500 times (300 in the past three years) implemented in yed graphviz/dot tulip... given resolve cycles layer assignement 9-7 crossing minimization node positioning edge drawing Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

18 Sugiyama Framework (Sugiyama, Tagawa, Toda 1981) given resolve cycles layer assignement 9-8 crossing minimization node positioning edge drawing Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

19 Sugiyama Framework (Sugiyama, Tagawa, Toda 1981) given resolve cycles layer assignement 9-9 crossing minimization node positioning edge drawing Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

20 Step 1: Resolve Cycles 10 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

21 Feedback Arc Set Idea: find maximum acyclic subgraph inverce the directions of the other edges 11-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

22 Feedback Arc Set Idea: find maximum acyclic subgraph inverce the directions of the other edges Maximum Acyclic Subgraph Given: directed graph D = (V, A) Find: acyclic subgraph D = (V, A ) with maximum A 11-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

23 Feedback Arc Set Idea: find maximum acyclic subgraph inverce the directions of the other edges Maximum Acyclic Subgraph Given: directed graph D = (V, A) Find: acyclic subgraph D = (V, A ) with maximum A Minimum Feedback Arc Set (FAS) Given: directed graph D = (V, A) Find: A f A, with D f = (V, A \ A f ) acyclic with minimum A f 11-3 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

24 Feedback Arc Set Idea: find maximum acyclic subgraph inverce the directions of the other edges Maximum Acyclic Subgraph Given: directed graph D = (V, A) Find: acyclic subgraph D = (V, A ) with maximum A Minimum Feedback Arc Set (FAS) Given: directed graph D = (V, A) Find: A f A, with D f = (V, A \ A f ) acyclic with minimum A f Minimum Feedback Set (FS) Given: directed graph D = (V, A) Find: A f A, with D f = (V, A \ A f rev(a f )) acyclic with minimum A f 11-4 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

25 Feedback Arc Set Idea: find maximum acyclic subgraph inverce the directions of the other edges Maximum Acyclic Subgraph Given: directed graph D = (V, A) Find: acyclic subgraph D = (V, A ) with maximum A Minimum Feedback Arc Set (FAS) Given: directed graph D = (V, A) Find: A f A, with D f = (V, A \ A f ) acyclic with minimum A f Minimum Feedback Set (FS) Given: directed graph D = (V, A) Find: A f A, with D f = (V, A \ A f rev(a f )) acyclic with minimum A f All three problems are NP-hard! 11-5 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

26 Heuristric 1 (Berger, Shor 1990) A := ; foreach v V do if N (v) N (v) then A := A N (v); else A := A N (v); remove v and N(v) from D. return (V, A ) N (v) := {(v, u) : (v, u) A} N (v) := {(u, v) : (u, v) A} N(v) := N (v) N (v) 12-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

27 Heuristric 1 (Berger, Shor 1990) A := ; foreach v V do if N (v) N (v) then A := A N (v); else A := A N (v); remove v and N(v) from D. return (V, A ) N (v) := {(v, u) : (v, u) A} N (v) := {(u, v) : (u, v) A} N(v) := N (v) N (v) D = (V, A ) is a DAG A \ A is a feedback arc set 12-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

28 Heuristric 1 (Berger, Shor 1990) A := ; foreach v V do if N (v) N (v) then A := A N (v); else A := A N (v); remove v and N(v) from D. return (V, A ) N (v) := {(v, u) : (v, u) A} N (v) := {(u, v) : (u, v) A} N(v) := N (v) N (v) D = (V, A ) is a DAG A \ A is a feedback arc set Work with your neighbour(s) and then share Why D does not contain cycles? What one can say about A in terms of V? 7 min 12-3 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

29 Heuristric 1 (Berger, Shor 1990) A := ; foreach v V do if N (v) N (v) then A := A N (v); else A := A N (v); remove v and N(v) from D. return (V, A ) N (v) := {(v, u) : (v, u) A} N (v) := {(u, v) : (u, v) A} N(v) := N (v) N (v) Task togo: Is D = (V, A rev(a \ A )) acyclic? What is the running time? D = (V, A ) is a DAG A \ A is a feedback arc set Work with your neighbour(s) and then share Why D does not contain cycles? What one can say about A in terms of V? 7 min 12-4 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

30 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

31 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

32 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-3 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

33 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-4 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

34 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-5 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

35 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-6 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

36 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-7 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

37 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-8 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

38 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-9 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

39 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-10Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

40 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-11Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

41 Heuristic 2 (Eades, Lin, Smyth 1993) 1 A := ; 2 while V do 3 while in V exists a sink v do 4 A A N (v) 5 remove v and N (v): {V, n, m} sink 6 Remove all isolated node from V : {V, n, m} iso 7 while in V exists a source v do 8 A A N (v) 9 remove v and N (v): {V, n, m} source 10 if V then 11 let v V such that N (v) N (v) maximal; 12 A A N (v) 13 remove v and N(v): {V, n, m} {=,<} 13-12Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

42 Heuristic 2 Analysis Theorem 1: Let D = (V, A) be a connected, directed graph without 2-cycles. Heuristic 2 computes a set of edges A with A A /2 + V /6. The running time is O( A ) Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

43 Heuristic 2 Analysis Theorem 1: Let D = (V, A) be a connected, directed graph without 2-cycles. Heuristic 2 computes a set of edges A with A A /2 + V /6. The running time is O( A ). Exact Solution: integer linear programming, using branch-and-cut technique (Grötschel et al. 1985) 14-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

44 Example 15-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

45 Example 15-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

46 Step 2: Layer Assignement dummy node 16-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

47 Step 2: Layer Assignement dummy node Think for a minute and then share What could we optimize when doing the layer assignment? 1 min 16-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

48 Step 2: Layer Assignement Given.: directed acyclic graph (DAG) D = (V, A) Find: Partition the vertex set V into disjoint subsets (layers) L 1,..., L h s.t. (u, v) A, u L i, v L j i < j Def: y-coordinate y(u) = i u L i 17-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

49 Step 2: Layer Assignement Given.: directed acyclic graph (DAG) D = (V, A) Find: Partition the vertex set V into disjoint subsets (layers) L 1,..., L h s.t. (u, v) A, u L i, v L j i < j Def: y-coordinate y(u) = i u L i Criteria that we discuss minimize the number of layers h (= height of the layouts) minimize the total length of edges ( number of dummy nodes) minimize width, e.g. max{ L i 1 i h} 17-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

50 Height Optimization Idea: assign each node v to the layer L i, where i is the length of the longest simple path from a source to v all incomming neighbours lie below v the resulting height h is minimized 18-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

51 Height Optimization Idea: assign each node v to the layer L i, where i is the length of the longest simple path from a source to v all incomming neighbours lie below v the resulting height h is minimized Algorithm L 1 the set of sources in D set y(u) max v N (u){y(v)} Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

52 Example 19-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

53 Example 19-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

54 Total Edge Length Can be formulated as an integer linear program: min (u,v) A(y(v) y(u)) subject to y(v) y(u) 1 (u, v) A y(v) 1 v V y(v) Z v V 20-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

55 Total Edge Length Can be formulated as an integer linear program: min (u,v) A(y(v) y(u)) subject to y(v) y(u) 1 (u, v) A y(v) 1 v V y(v) Z v V One can show that: Constraint-Matrix is totally unimodular Solution of the relaxed linear program is integer The total edge length can be minimized in a polynomial time 20-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

56 Width of the Layout 21-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

57 Width of the Layout bound the width! 21-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

58 Layer Assignement with Fixed Width Fixed-Width Layer Assignment Given: directed acyclic graph D = (V, A), width B Find: layer assignement L of minumum height with at most B nodes per layer B = Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

59 Layer Assignement with Fixed Width Fixed-Width Layer Assignment Given: directed acyclic graph D = (V, A), width B Find: layer assignement L of minumum height with at most B nodes per layer M 1 M 2 M 3 B = 4 equivalent to the following scheduling problem: Minimum Precedence Constrained Scheduling (MPCS) Given: n Jobs J 1,..., J n with identical unit processing time, precedence constraints J i < J k, and B identical machines Find: Schedule of minimum length, that satisfies all the precendence constraints 22-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout M 4

60 Fixed Width: Complexity Theorem 2: It is NP-hard to decide, whether for n jobs J 1,..., J n of identical length, given partial ordering constraints, and number of machinces B, there exists a schedule of height at most T, even if T = Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

61 Fixed Width: Complexity Theorem 2: It is NP-hard to decide, whether for n jobs J 1,..., J n of identical length, given partial ordering constraints, and number of machinces B, there exists a schedule of height at most T, even if T = 3. Corollary: If P N P, there is no polynomial algorithm for MPCS with approximation factor < 4/ Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

62 Fixed Width: Complexity Theorem 2: It is NP-hard to decide, whether for n jobs J 1,..., J n of identical length, given partial ordering constraints, and number of machinces B, there exists a schedule of height at most T, even if T = 3. Corollary: If P N P, there is no polynomial algorithm for MPCS with approximation factor < 4/3. Work with your neighbour(s) and then share Why the corollary holds? 5 min 23-3 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

63 Fixed Width: Complexity Theorem 2: It is NP-hard to decide, whether for n jobs J 1,..., J n of identical length, given partial ordering constraints, and number of machinces B, there exists a schedule of height at most T, even if T = 3. Corollary: If P N P, there is no polynomial algorithm for MPCS with approximation factor < 4/3. Work with your neighbour(s) and then share Why the corollary holds? 5 min 23-4 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

64 Fixed Width: Algorithm Theorem 3: There exist an approximation algorithm for MPCS with factor 2 1 B Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

65 Fixed Width: Algorithm Theorem 3: There exist an approximation algorithm for MPCS with factor 2 1 B. List-Scheduling-Algorithm: order jobs arbitrarily as a list L when a machine is free, select an allowed job from L; Machine is idle of there is no such job 24-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

66 Summary given resolve cycles layer assignement crossing minimization node positioning edge drawing 25-1 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

67 Summary given resolve cycles layer assignement Resolve cycles equivalent to minimum feedback set problem, and is NP-hard Heuristic with A A /2 Heuristic with A A /2 + V /6 crossing minimization node positioning edge drawing 25-2 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

68 Summary given resolve cycles layer assignement Layer assignement Height optimization: topological numbering Total edge length: polynomial alg. through integer linear program Height optimization with fixed width: equivalent to MPCS. NP-hard for 3 levels. Approximation algorithm with factor 2 1 B. crossing minimization node positioning edge drawing 25-3 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout

69 Summary given resolve cycles layer assignement crossing minimization node positioning edge drawing 25-4 Dr. Tamara Mchedlidze Algorithmen zur Visualisierung von Graphen Layered Layout