SLIDE 1
Track Layout is Hard
Michael J. Bannister1 William E. Devanny2 Vida Dujmovi´ c3 David Eppstein2 David R. Wood4
1Santa Clara University 2University of California, Irvine 3University of Ottawa 4Monash University
Track Layout is Hard Michael J. Bannister 1 William E. Devanny 2 c 3 - - PowerPoint PPT Presentation
Track Layout is Hard Michael J. Bannister 1 William E. Devanny 2 c 3 - - PowerPoint PPT Presentation
Track Layout is Hard Michael J. Bannister 1 William E. Devanny 2 c 3 Vida Dujmovi David Eppstein 2 David R. Wood 4 1 Santa Clara University 2 University of California, Irvine 3 University of Ottawa 4 Monash University Track Layout Track Layout
SLIDE 2
SLIDE 3
Track Layout
SLIDE 4
Track Layout
SLIDE 5
Track Layout
SLIDE 6
Track Layout
SLIDE 7
Track Layout
SLIDE 8
Track Layout
SLIDE 9
Track Layout
SLIDE 10
Track Layout
SLIDE 11
Track Layout
SLIDE 12
Track Layout
Track layouts find applications in minimizing the volume of 3D drawings
[Di Giacomo, 2004; Di Giacomo and Meijer, 2004; Dujmovi´ c and Wood, 2004; Felsner et al, 2003; Hasunuma, 2004]
SLIDE 13
Track Layout
Given a graph G can we compute the track number for G? A graph is 2-track if and only if it is a forest of caterpillars Track layouts find applications in minimizing the volume of 3D drawings
[Di Giacomo, 2004; Di Giacomo and Meijer, 2004; Dujmovi´ c and Wood, 2004; Felsner et al, 2003; Hasunuma, 2004]
SLIDE 14
Track Layout
Given a graph G can we compute the track number for G? A graph is 2-track if and only if it is a forest of caterpillars Track layouts find applications in minimizing the volume of 3D drawings
[Di Giacomo, 2004; Di Giacomo and Meijer, 2004; Dujmovi´ c and Wood, 2004; Felsner et al, 2003; Hasunuma, 2004]
3-track and higher?
SLIDE 15
Leveled planarity
Perfect Sugiyama-style layered drawing No edge crossing No dummy vertices Recognition is NP-complete
[Heath and Rosenberg, 1992]
SLIDE 16
Leveled planarity
Perfect Sugiyama-style layered drawing No edge crossing No dummy vertices Recognition is NP-complete
[Heath and Rosenberg, 1992]
Will show leveled planar graphs are the same as bipartite 3-track graphs
SLIDE 17
Lemma: Every leveled planar graph has a 3-track layout.
SLIDE 18
Lemma: Every leveled planar graph has a 3-track layout.
SLIDE 19
Lemma: Every leveled planar graph has a 3-track layout. 1 2 3 1 2 3
SLIDE 20
Lemma: Every leveled planar graph has a 3-track layout. 1 2 3 1 2 3 1 2 3
SLIDE 21
Lemma: Every leveled planar graph has a 3-track layout. 1 2 3 1 2 3 1 2 3
SLIDE 22
Lemma: Every leveled planar graph has a 3-track layout. 1 2 3 1 2 3 1 2 3
SLIDE 23
+1 +1 +1 +1 −1 Label counterclockwise edges −1 Label clockwise edges +1
SLIDE 24
+1 +1 +1 +1 −1 Label counterclockwise edges −1 Label clockwise edges +1
- e∈Clabel(e) =
0 if |C| is even ±3 if |C| is odd +1 + 1 + 1 − 1 + 1 = +3 Lemma: Proof:
SLIDE 25
Label counterclockwise edges −1 Label clockwise edges +1
- e∈Clabel(e) =
0 if |C| is even ±3 if |C| is odd +1 + 1 + 1 − 1 + 1 = +3 Lemma: Proof: The cycle forms a polygon and has at least two ears
SLIDE 26
Label counterclockwise edges −1 Label clockwise edges +1
- e∈Clabel(e) =
0 if |C| is even ±3 if |C| is odd +1 + 1 + 1 − 1 + 1 = +3 Lemma: Proof: The cycle forms a polygon and has at least two ears If the edges of an ear are +1 and −1 remove it
SLIDE 27
Label counterclockwise edges −1 Label clockwise edges +1
- e∈Clabel(e) =
0 if |C| is even ±3 if |C| is odd +1 + 1 + 1 − 1 + 1 = +3 Lemma: Proof: The cycle forms a polygon and has at least two ears If the edges of an ear are +1 and −1 remove it If the edges of every ear have the same sign than the polygon is a triangle
SLIDE 28
Label counterclockwise edges −1 Label clockwise edges +1
- e∈Clabel(e) =
0 if |C| is even ±3 if |C| is odd +1 + 1 + 1 − 1 + 1 = +3 Lemma: Proof: The cycle forms a polygon and has at least two ears If the edges of an ear are +1 and −1 remove it If the edges of every ear have the same sign than the polygon is a triangle The winding number of a cycle around the central point is 0 if the cycle has even length and is 1 if the cycle has odd length
SLIDE 29
Lemma: Every bipartite 3-track graph is leveled planar.
SLIDE 30
Lemma: Every bipartite 3-track graph is leveled planar.
SLIDE 31
Lemma: Every bipartite 3-track graph is leveled planar.
SLIDE 32
Lemma: Every bipartite 3-track graph is leveled planar.
SLIDE 33
Lemma: Every bipartite 3-track graph is leveled planar.
SLIDE 34
Lemma: Every bipartite 3-track graph is leveled planar. Non-tree edges form cycles in the tree ⇒ Edges connect consecutive levels
SLIDE 35
Order vertices by their track order ⇒ No crossings Lemma: Every bipartite 3-track graph is leveled planar. Non-tree edges form cycles in the tree ⇒ Edges connect consecutive levels
SLIDE 36
Order vertices by their track order ⇒ No crossings Lemma: Every bipartite 3-track graph is leveled planar. Non-tree edges form cycles in the tree ⇒ Edges connect consecutive levels Thm: Recognizing 3-track graphs is NP-complete
SLIDE 37
Bipartite Outerplanar Graphs
Which 3-track graphs can we recognize and layout 3-track?
SLIDE 38
Bipartite Outerplanar Graphs
Which 3-track graphs can we recognize and layout 3-track? Pick a vertex and perform a breadth first layering
SLIDE 39
Bipartite Outerplanar Graphs
Which 3-track graphs can we recognize and layout 3-track? Pick a vertex and perform a breadth first layering How to layout vertices with a level? Each face cycle has a closest vertex to the chosen vertex Otherwise there is a K2.3 subdivision
SLIDE 40
Bipartite Outerplanar Graphs
Which 3-track graphs can we recognize and layout 3-track? Pick a vertex and perform a breadth first layering How to layout vertices with a level? Each face cycle has a closest vertex to the chosen vertex Order vertices of each face cycle independently
SLIDE 41
Bipartite Outerplanar Graphs
Which 3-track graphs can we recognize and layout 3-track? Pick a vertex and perform a breadth first layering How to layout vertices with a level? Each face cycle has a closest vertex to the chosen vertex Order vertices of each face cycle independently
SLIDE 42
Bipartite Outerplanar Graphs
Which 3-track graphs can we recognize and layout 3-track? Pick a vertex and perform a breadth first layering How to layout vertices with a level? Each face cycle has a closest vertex to the chosen vertex Order vertices of each face cycle independently
SLIDE 43
Bipartite Outerplanar Graphs
Which 3-track graphs can we recognize and layout 3-track? Pick a vertex and perform a breadth first layering How to layout vertices with a level? Each face cycle has a closest vertex to the chosen vertex Order vertices of each face cycle independently
SLIDE 44
Bipartite Outerplanar Graphs
Which 3-track graphs can we recognize and layout 3-track? Pick a vertex and perform a breadth first layering How to layout vertices with a level? Each face cycle has a closest vertex to the chosen vertex Order vertices of each face cycle independently
SLIDE 45
Bipartite Outerplanar Graphs
Which 3-track graphs can we recognize and layout 3-track? Pick a vertex and perform a breadth first layering How to layout vertices with a level? Each face cycle has a closest vertex to the chosen vertex Order vertices of each face cycle independently Because every face cycle has a valid layout, the entire graph has a valid layout
[Abel et al, 2014]
SLIDE 46
Squaregraphs
Every bounded face is a 4-cycle and every interior vertex has at least 4 neighbors
SLIDE 47
Squaregraphs
Every bounded face is a 4-cycle and every interior vertex has at least 4 neighbors Again pick a vertex and perform a breadth first layering
SLIDE 48
Squaregraphs
Every bounded face is a 4-cycle and every interior vertex has at least 4 neighbors Again pick a vertex and perform a breadth first layering Use [Bandelt et al, 2010] for ordering layers
SLIDE 49
Squaregraphs
Every bounded face is a 4-cycle and every interior vertex has at least 4 neighbors Again pick a vertex and perform a breadth first layering Use [Bandelt et al, 2010] for ordering layers
SLIDE 50
Squaregraphs
Every bounded face is a 4-cycle and every interior vertex has at least 4 neighbors Again pick a vertex and perform a breadth first layering Use [Bandelt et al, 2010] for ordering layers
SLIDE 51
Squaregraphs
Every bounded face is a 4-cycle and every interior vertex has at least 4 neighbors Again pick a vertex and perform a breadth first layering Use [Bandelt et al, 2010] for ordering layers
SLIDE 52
Squaregraphs
Every bounded face is a 4-cycle and every interior vertex has at least 4 neighbors Again pick a vertex and perform a breadth first layering Use [Bandelt et al, 2010] for ordering layers
SLIDE 53
Squaregraphs
Every bounded face is a 4-cycle and every interior vertex has at least 4 neighbors Again pick a vertex and perform a breadth first layering Use [Bandelt et al, 2010] for ordering layers
SLIDE 54
Dual graphs of x-monotone curves
where each curves projection covers the entire x-axis
SLIDE 55
Dual graphs of x-monotone curves
where each curves projection covers the entire x-axis
SLIDE 56
Dual graphs of x-monotone curves
where each curves projection covers the entire x-axis
SLIDE 57
Dual graphs of x-monotone curves
where each curves projection covers the entire x-axis
SLIDE 58
Dual graphs of x-monotone curves
where each curves projection covers the entire x-axis
SLIDE 59
Computing the track number for a graph is NP-hard Recognizing 3-track graphs is NP-complete
- holds for larger track number
3-track layouts for special graph classes
- Bipartite outerplanar graphs
- Squaregraphs
- Dual graph of x-monotone curves that cover the x-axis
Some common FPT approaches do not work
Summary Open questions
Is computing track number FPT or W[1]-hard?
SLIDE 60
Thank you!
SLIDE 61
FPT Results Summary
- 1. Courcelle’s Theorem cannot be applied
Find a family of graphs that are not finite state
- 2. 2-core kernelization for k-almost-tree will not work
Found pairs of graphs with the same 2-core and different track number
- 3. Track number is non-uniformly FPT in tree depth