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Ordering Metro Lines by Block Crossings
Martin Fink Lehrstuhl f¨ ur Informatik I Universit¨ at W¨ urzburg
Joint work with Sergey Pupyrev
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Ordering Metro Lines by Block Crossings Martin Fink Lehrstuhl f - - PowerPoint PPT Presentation
Ordering Metro Lines by Block Crossings Martin Fink Lehrstuhl f - - PowerPoint PPT Presentation
Ordering Metro Lines by Block Crossings Martin Fink Lehrstuhl f ur Informatik I Universit at W urzburg Joint work with Sergey Pupyrev 1 /18 Metro Maps Vienna 2 /18 Metro Maps Paris 3 /18 Metro Maps Metro Lines 4
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SLIDE 3
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Metro Maps – Paris
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Metro Maps – Metro Lines
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Metro Maps – Metro Lines
Previous work, e.g. [N¨
- llenburg and Wolff, 2011]
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Metro Maps – Metro Lines
Previous work, e.g. [N¨
- llenburg and Wolff, 2011]
Focus on drawing underlying graph
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Metro Line Crossing Minimization
Insert all lines L into embedded graph G = (V , E) such that ...
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Metro Line Crossing Minimization
Insert all lines L into embedded graph G = (V , E) such that ...
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Metro Line Crossing Minimization
Insert all lines L into embedded graph G = (V , E) such that ... the number of crossings is minimized.
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Metro Line Crossing Minimization
Insert all lines L into embedded graph G = (V , E) such that ... the number of crossings is minimized. crossings are placed on edges if possible.
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Metro Line Crossing Minimization
Insert all lines L into embedded graph G = (V , E) such that ... the number of crossings is minimized. crossings are placed on edges if possible.
✗
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Metro Line Crossing Minimization
Insert all lines L into embedded graph G = (V , E) such that ... the number of crossings is minimized. crossings are placed on edges if possible.
✗
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Metro Line Crossing Minimization
Insert all lines L into embedded graph G = (V , E) such that ... the number of crossings is minimized. crossings are placed on edges if possible.
✗
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Metro Line Crossing Minimization
Insert all lines L into embedded graph G = (V , E) such that ... the number of crossings is minimized. crossings are placed on edges if possible. NP-hard [Bekos et al., 2007] & [Fink, Pupyrev, 2013]
✗
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Path Terminal Property
Line terminals are leaves of the graph.
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Path Terminal Property
Line terminals are leaves of the graph. Follow two lines:
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Path Terminal Property
Line terminals are leaves of the graph. Follow two lines:
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Path Terminal Property
Line terminals are leaves of the graph. Follow two lines:
- crossing
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Path Terminal Property
Line terminals are leaves of the graph. Follow two lines:
- crossing
Crossing Minimization: Solvable in linear time [Pupyrev et al., 2012]
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Path Terminal Property
Line terminals are leaves of the graph. Follow two lines:
- crossing
Crossing Minimization: Solvable in linear time [Pupyrev et al., 2012] additional restriction: two lines intersect in a path
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New Model: Block Crossings
4 single crossings
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New Model: Block Crossings
4 single crossings 1 block crossing
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New Model: Block Crossings
4 single crossings 1 block crossing
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New Model: Block Crossings
4 single crossings 1 block crossing 12 single crossings
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New Model: Block Crossings
4 single crossings 1 block crossing 12 single crossings 3 block crossings
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Block Crossing Minimization
New problem variants: BCM: Minimize the number of block crossings
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Block Crossing Minimization
New problem variants: BCM: Minimize the number of block crossings MBCM: Minimize the number of monotone block crossings
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Block Crossing Minimization
New problem variants: BCM: Minimize the number of block crossings MBCM: Minimize the number of monotone block crossings no double crossings!
✗
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Single Edge
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Single Edge
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Single Edge
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Single Edge
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Single Edge
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Single Edge
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Single Edge
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Single Edge
1 2 3 4 5 1 2 3 4 5
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Single Edge
1 2 3 4 5 1 2 3 4 5
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Single Edge
1 2 3 4 5 1 2 3 4 5 Equivalent to Sorting by Transpositions for permutations
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Single Edge
1 2 3 4 5 1 2 3 4 5 Equivalent to Sorting by Transpositions for permutations NP-hard [Bulteau et al., 2012]
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Single Edge
1 2 3 4 5 1 2 3 4 5 Equivalent to Sorting by Transpositions for permutations NP-hard [Bulteau et al., 2012] simple 3-approximation [Bafna, Pevzner, 1998]
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs no good pair for simple permutation
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs no good pair for simple permutation
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs no good pair for simple permutation
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs no good pair for simple permutation
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs no good pair for simple permutation
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs no good pair for simple permutation
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs no good pair for simple permutation ≤ |L| − 1 crossings
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs no good pair for simple permutation ≤ |L| − 1 crossings
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs no good pair for simple permutation ≤ |L| − 1 crossings
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs no good pair for simple permutation ≤ |L| − 1 crossings ≤ 3 good pairs can be created per crossing 3-approx.
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Single Edge – 3-approximation
1 2 3 4 5 1 2 3 4 5 |L| + 1 good pairs no good pair for simple permutation ≤ |L| − 1 crossings ≤ 3 good pairs can be created per crossing 3-approx. solution uses monotone block crossings!
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Path
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Path
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Path
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Path
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Path
lines end
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Path
lines end
- lines start
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Path
lines end
- lines start
Redefine good pairs:
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Path
lines end
- lines start
Redefine good pairs: – lines end together
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Path
lines end
- lines start
Redefine good pairs: – lines end together – inheritance b a1 a2
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Path
lines end
- lines start
Redefine good pairs: – lines end together – inheritance b a1 a2 ≤ 3 good pairs created per crossing
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Path – 3-approximation
e treat edges from left to right
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Path – 3-approximation
e treat edges from left to right
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Path – 3-approximation
e treat edges from left to right identify good pairs
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Path – 3-approximation
e treat edges from left to right
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Path – 3-approximation
e treat edges from left to right
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Path – 3-approximation
e treat edges from left to right
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Path – 3-approximation
e treat edges from left to right bring ending lines to top/bottom keeping good pairs together
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Path – 3-approximation
e treat edges from left to right bring ending lines to top/bottom keeping good pairs together create 1 good pair per block crossing crossing
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Path – 3-approximation
e treat edges from left to right bring ending lines to top/bottom keeping good pairs together create 1 good pair per block crossing crossing
- ptimum creates up to 3
3-approximation
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Path – 3-approximation
e treat edges from left to right bring ending lines to top/bottom keeping good pairs together create 1 good pair per block crossing crossing
- ptimum creates up to 3
3-approximation Special Case: Destroying good pairs e l
a c b a
- ptimal strategy available:
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Path – 3-approximation
e treat edges from left to right bring ending lines to top/bottom keeping good pairs together create 1 good pair per block crossing crossing
- ptimum creates up to 3
3-approximation Special Case: Destroying good pairs e l
a c b a
- ptimal strategy available:
– preferably destroy non-initial good pair – if none: destroy longest-living good pair
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Path – 3-approximation
e treat edges from left to right bring ending lines to top/bottom keeping good pairs together create 1 good pair per block crossing crossing
- ptimum creates up to 3
3-approximation algorithm can be adjusted for monotone block crossings
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Path – 3-approximation
e treat edges from left to right bring ending lines to top/bottom keeping good pairs together create 1 good pair per block crossing crossing
- ptimum creates up to 3
3-approximation algorithm can be adjusted for monotone block crossings adjust inheritance of good pairs: c b a1 a2
✗
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Path – 3-approximation
e treat edges from left to right bring ending lines to top/bottom keeping good pairs together create 1 good pair per block crossing crossing
- ptimum creates up to 3
3-approximation algorithm can be adjusted for monotone block crossings adjust inheritance of good pairs: c b a1 a2
✗
new strategy when destroying good pairs: ensure monotonicity
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Trees - an upper bound
root at some leave
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Trees - an upper bound
root at some leave after treating edge recursively order subtrees
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Trees - an upper bound
root at some leave after treating edge recursively order subtrees
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Trees - an upper bound
root at some leave after treating edge recursively order subtrees
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Trees - an upper bound
root at some leave after treating edge recursively order subtrees insert lines between subtrees
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Trees - an upper bound
root at some leave after treating edge recursively order subtrees insert lines between subtrees ≤ 2 crossings per line
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Trees - an upper bound
root at some leave after treating edge recursively order subtrees insert lines between subtrees ≤ 2 crossings per line right insertion order needed for: – avoiding vertex crossings – avoiding double crossings
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Trees - an upper bound
root at some leave after treating edge recursively order subtrees insert lines between subtrees ≤ 2 crossings per line right insertion order needed for: – avoiding vertex crossings – avoiding double crossings worst-case instances: 2|L| − 3 crossings necessary
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Upward Trees
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Upward Trees
simplification use tree algorithm 6-approximation for monotone block crossings
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General Graphs
Process edges in arbitrary order
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General Graphs
Process edges in arbitrary order Completely sort lines on an edge
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General Graphs
Process edges in arbitrary order Completely sort lines on an edge
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General Graphs
Process edges in arbitrary order Completely sort lines on an edge lines l, l′ seen together on edge will never cross (again) l l′
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General Graphs
Process edges in arbitrary order Completely sort lines on an edge lines l, l′ seen together on edge will never cross (again) l l′ lines l, l′ seen together for the first time information gain
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General Graphs – Sorting an Edge
e
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General Graphs – Sorting an Edge
e follow lines
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General Graphs – Sorting an Edge
e follow lines
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General Graphs – Sorting an Edge
e follow lines
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General Graphs – Sorting an Edge
e follow lines find cut edges
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General Graphs – Sorting an Edge
e follow lines find cut edges
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General Graphs – Sorting an Edge
e follow lines find cut edges
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General Graphs – Sorting an Edge
e follow lines find cut edges identify groups of lines
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General Graphs – Sorting an Edge
e follow lines find cut edges identify groups of lines group stays parallel
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General Graphs – Sorting an Edge
e follow lines find cut edges identify groups of lines group stays parallel consider pairs of groups
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General Graphs – Sorting an Edge
e follow lines find cut edges identify groups of lines group stays parallel consider pairs of groups
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General Graphs – Sorting an Edge
e follow lines find cut edges identify groups of lines group stays parallel consider pairs of groups
✗
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General Graphs – Sorting an Edge
e follow lines find cut edges identify groups of lines group stays parallel consider pairs of groups merge lines if possible
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General Graphs – Sorting an Edge
e follow lines find cut edges identify groups of lines group stays parallel consider pairs of groups merge lines if possible sort by insertion into largest group
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General Graphs – Sorting an Edge
e follow lines find cut edges identify groups of lines group stays parallel consider pairs of groups merge lines if possible sort by insertion into largest group undo merging
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General Graphs – Analysis
e finally: all edges
- rdered
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General Graphs – Analysis
e finally: all edges
- rdered
pairs of lines cross at most once monotone block crossings
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General Graphs – Analysis
e finally: all edges
- rdered
pairs of lines cross at most once monotone block crossings (bc(e))2 ≤ I(e) information gain block crossings on edge
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General Graphs – Analysis
e finally: all edges
- rdered
pairs of lines cross at most once monotone block crossings (bc(e))2 ≤ I(e) information gain block crossings on edge
- e∈E(bc(e))2 ≤ |L|2
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General Graphs – Analysis
e finally: all edges
- rdered
pairs of lines cross at most once monotone block crossings (bc(e))2 ≤ I(e) information gain block crossings on edge
- e∈E(bc(e))2 ≤ |L|2
Using Cauchy-Schwarz:
- e∈E bc(e) ≤ |L|
- |E ′|
SLIDE 116
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General Graphs – Analysis
e finally: all edges
- rdered
pairs of lines cross at most once monotone block crossings (bc(e))2 ≤ I(e) information gain block crossings on edge
- e∈E(bc(e))2 ≤ |L|2
Using Cauchy-Schwarz:
- e∈E bc(e) ≤ |L|
- |E ′|
worst-case instances: Ω(|L|
- |E ′|) block
crossings necessary
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Conclusion
new model for counting crossings approximations for paths and upward trees tight upper bound for trees algorithm for general graphs upper bound asymptotically tight
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Conclusion
new model for counting crossings approximations for paths and upward trees tight upper bound for trees algorithm for general graphs upper bound asymptotically tight Open Questions: Complexity of monotone block crossings on a single edge? Approximations for trees / general graphs?
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Conclusion
new model for counting crossings approximations for paths and upward trees tight upper bound for trees algorithm for general graphs upper bound asymptotically tight Open Questions: Complexity of monotone block crossings on a single edge? Approximations for trees / general graphs?