Ordering Metro Lines by Block Crossings Martin Fink Lehrstuhl f - PowerPoint PPT Presentation

Path – 3-approximation e treat edges from left to create 1 good pair per right block crossing crossing bring ending lines to optimum creates up to 3 top/bottom keeping 3-approximation good pairs together algorithm can be adjusted for monotone block crossings 12 /18

Path – 3-approximation adjust inheritance of good pairs: a 2 c a 1 b e ✗ treat edges from left to create 1 good pair per right block crossing crossing bring ending lines to optimum creates up to 3 top/bottom keeping 3-approximation good pairs together algorithm can be adjusted for monotone block crossings 12 /18

Path – 3-approximation adjust inheritance of good pairs: a 2 c a 1 b e ✗ treat edges from left to create 1 good pair per right block crossing crossing new strategy when destroying good pairs: bring ending lines to optimum creates up to 3 ensure monotonicity top/bottom keeping 3-approximation good pairs together algorithm can be adjusted for monotone block crossings 12 /18

Trees - an upper bound root at some leave 13 /18

Trees - an upper bound root at some leave after treating edge recursively order subtrees 13 /18

Trees - an upper bound root at some leave after treating edge recursively order subtrees 13 /18

Trees - an upper bound root at some leave after treating edge recursively order subtrees 13 /18

Trees - an upper bound root at some leave after treating edge recursively order subtrees insert lines between subtrees 13 /18

Trees - an upper bound root at some leave after treating edge recursively order subtrees insert lines between subtrees ≤ 2 crossings per line 13 /18

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 13 /18

Trees - an upper bound root at some leave after treating edge recursively order subtrees worst-case instances: insert lines between 2 | L | − 3 crossings subtrees necessary ≤ 2 crossings per line right insertion order needed for: – avoiding vertex crossings – avoiding double crossings 13 /18

Upward Trees 14 /18

Upward Trees simplification use tree algorithm 6-approximation for monotone block crossings 14 /18

General Graphs Process edges in arbitrary order 15 /18

General Graphs Process edges in arbitrary order Completely sort lines on an edge 15 /18

General Graphs Process edges in arbitrary order Completely sort lines on an edge 15 /18

General Graphs Process edges in arbitrary order Completely sort lines on an edge l l ′ lines l , l ′ seen together on edge will never cross (again) 15 /18

General Graphs Process edges in arbitrary order Completely sort lines on an edge l l ′ lines l , l ′ seen together on edge will never cross (again) lines l , l ′ seen together for the first time information gain 15 /18

General Graphs – Sorting an Edge e 16 /18

General Graphs – Sorting an Edge e follow lines 16 /18

General Graphs – Sorting an Edge e follow lines 16 /18

General Graphs – Sorting an Edge e follow lines 16 /18

General Graphs – Sorting an Edge e follow lines find cut edges 16 /18