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Compact Layered Drawings of General Directed Graphs

Adalat Jabrayilov1, Sven Mallach2, Petra Mutzel1, Ulf R¨ uegg3, and Reinhard von Hanxleden3

1TU Dortmund University, Germany 2University of Cologne, Germany 3University of Kiel, Germany

24th International Symposium on Graph Drawing & Network Visualization 19-21 September 2016

Adalat Jabrayilov 1/18

Compact Graph Layering for Digraphs

Classic Layering on A4 Compact Layering on A4

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 2/18

Layering for Digraphs: Notation

u v w 1 2 3 Layers Regular Vertex Dummy Vertex Forward Arc Reverse Arc Width Height

Adjacent vertices on different layers total arc lengths = #arcs + #dummy vertices

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 3/18

Compact vs Classic Layering for Digraphs

Classic Layering [Sugiyama et. al, 1981] Preprocessing: cycle breaking → DAG

minimize: number of reverse arcs Rev

Layering of the DAG

minimize: total arc lengths Len

Compact Layering integrates cycle breaking & layering

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 4/18

Compact Layering Models for Digraphs

Generalised Layering (GLP) [R¨ uegg et. al, 2016]

minimize: weighted sum of Rev, Len for given Height H

Compact Generalised Layering (CGLP)[this talk]

GLP + Width (including dummy vertices) minimize: weighted sum of Rev, Len, Width for given Height H

Min+Max Length Layering (MMLP)[this talk]

slight modification of CGLP

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 5/18

Classic

CGLP

MMLP

Compact Generalised Layering Problem (CGLP)

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 6/18

a 1 b 2 c Partial Ordering

Mixed Integer Linear Program CGL

models CGLP as Partial Ordering Problem (POP) Let ℓ(v) the position of vertex v in ordering POP variables for each pair u = v ∈ V : yu,v = 1 ℓ(u) < ℓ(v)

• therwise

for each pair u = v ∈ V : yu,v + yv,u ≤ 1

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 7/18

u v w 1 2 H

POP Variables yu,v =

    

1 ℓ(u) < ℓ(v) 0 otherwise

Mixed Integer Linear Program CGL

Consider layers as help vertices {1, 2, · · · , H} with ℓ(k) = k for each k ∈ {1, 2, · · · , H} Variables for Layering: for each v ∈ V , k ∈ {1, 2, · · · , H}

POP variable yv,k, yk,v

Reverse arcs: for each arc (u, v)

POP variable yu,v, yv,u

Dummy vertices: for each (u, v) ∈ A, k ∈ {1, 2, · · · , H}

zuv,k = 1 arc (u, v) causes dummy on layer k

• therwise

Width:

W ∈ R

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 8/18

u v w 1 2 H

Mixed Integer Linear Program CGL

min ωrev

• (u,v)∈A

yv,u + ωlen

• (u,v)∈A

H

• k=1

zuv,k + ωwid W s.t. yv,1 = 0 ∀v ∈ V yH,v = 0 ∀v ∈ V yk,v + yv,k+1 = 1 ∀v ∈ V , 1 ≤ k ≤ H − 1 yk,v − yk+1,v ≥ 0 ∀v ∈ V , 1 ≤ k ≤ H − 1 yu,v + yv,u = 1 ∀(u, v) ∈ A yv,k + yk,u − yv,u ≥ 0 ∀(u, v) ∈ A, 1 ≤ k ≤ H yu,k + yk,v − yu,v ≥ 0 ∀(u, v) ∈ A, 1 ≤ k ≤ H yk,u + yv,k − zuv,k ≤ 1 ∀(u, v) ∈ A, 1 ≤ k ≤ H yk,v + yu,k − zuv,k ≤ 1 ∀(u, v) ∈ A, 1 ≤ k ≤ H

• v∈V

(1 − yv,k − yk,v) +

• (u,v)∈A

zuv,k ≤ W 1 ≤ k ≤ H y ∈ {0, 1}, z ∈ [0, 1], W ∈ R

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 9/18

Mixed Integer Linear Program EXT

extends DAG Layering Model [Healy and Nikolov, 2002] models CGLP as Assignment Problem (AP) uses so called assignment variables for layering of vertices much slower than CGL

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Classic CGLP

MMLP

Min+Max Length Problem (MMLP)

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 11/18

Mixed Integer Linear Program MML

slight modification of CGL without variables and constraints corresponding to dummies MML objective vs CGL objective

CGL min ωrev

• (u,v)∈A

yv,u + ωlen

• (u,v)∈A

H

• k=1

zuv,k + ωwid W MML min ωrev

• (u,v)∈A

yv,u + ωlen

• (u,v)∈A

H

• k=1

(yk,v − yk,u) + ωwid Wr

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Evaluation

Used Hard/Software System: Intel Core i7-4790, 3.6 GHz with 32 GB RAM, Linux Solver: MIP Solver Gurobi 6.5 Number of used threads: 1 Parameters H = ⌈ √

|V | 0.6 ⌉

ωrev = |A| · H ωlen = 1 ωwid = 1

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Evaluation

Benchmark Sets: ATTar: extraction of 146 acyclic AT&T graphs with final drawings aspect ratio with classic layout:

final drawings Width final drawings Height ≤ 0.5

Vertices: 20-99 Arcs: 20-168 On average: |A|

|V | = 1.5

Random: 340 Random (non acyclic) directected graphs Vertices: 17-100 Arcs: 30-158 On average: |A|

|V | = 1.5

Compact Layered Drawings of General Directed Graphs Adalat Jabrayilov 14/18

Final drawings aspect ratio

Classic vs CGLP vs MMLP (ATTar Graphs)

Classic CGLP MMLP 0.0 0.2 0.4 0.6 0.8

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Time [s] |V |

CGL vs MML (ATTar Graphs)

[15, 30) [30, 45) [45, 60) 2 4 6 8 [60, 75) [75, 90) [90, 105) 5 15 25

EXT: 29 timeouts (time limit 1h)

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Time [s] |V |

CGL vs MML (Random Graphs)

[15, 30) [30, 45) [45, 60) 5 10 15 [60, 75) [75, 90) [90, 105) 50 150 250

EXT: 143 timeouts (time limit 1h)

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Conclusion

We introduced: Two Compact Layout Problems: CGLP and MMLP Two new ILP models based on POP variables: CGL and MML Our experiments showed: Both models can improve aspect ratio Both ILP formulations can be solved for each ATTar instance within 30 seconds.

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