Abstract Order Type Extension and New Results on the Rectilinear Crossing Number

1. Abstract Order Type Extension and New Results on the Rectilinear Crossing Number Oswin Aichholzer Institute for Softwaretechnology Graz University of Technology Graz, Austria Hannes Krasser Institute for Theoretical Computer Science Graz University of Technology Graz, Austria European Workshop on Computational Geometry, Eindhoven, The Netherlands, 2005.

2. Point Sets - finite point sets in the real plane R2 - in general position - with different crossing properties

3. Crossing Properties 4 points: no crossing crossing

4. b b c c a a Order Type order type of point set: mapping that assigns to each ordered triple of points its orientation Goodman, Pollack, 1983 orientation: left/positive right/negative

5.  Order Type Point sets of same order type there exists a bijection s.t. eitherall (or none) corresponding triples are of equal orientation Point sets of same order type

6. Enumerating Order Types Task: Enumerate all order types of point sets in the plane (for small, fixed size and in general position) Order type data base for n≤10 pointsAichholzer, Aurenhammer, Krasser, Enumerating order types for small point sets with applications. 2001 Our work: extension to n=11 points,same approach with improved methods

7. Order Type Data Base Extended order type data base 16-bit integer coordinates, >100 GB

8. Order Type Extension Extension to n=12, 13, … ? - approx. 750 billion order types for n=12 - too many for complete data base - partial extension of data base - obtain results on „suitable applications“ for 12 and beyond…

9. Subset Property „suitable applications“: subset property Property valid for Sn and there exists Sn-1s.t. similar property holds for Sn-1 Sn .. order type of n points Sn-1.. subset of Sn of n-1 points

10. Order Type Extension • Order type extension with subset property: • order type data base  result set of order types for n=11 • - enumerate all order types of 12 points that contain one of these 11-point order types as a subset • - filter 12-point order types according to subset property

11. Order Type Extension Order type extension algorithm: - extending point set realizations of order types with one additional point is not applicable  extension of abstract order types

12. Order Type Extension • Abstract order type extension: • duality: point sets  line arrangements order type  intersection sequences • abstract order type  pseudoline arrangement

13. Order Type Extension line arrangement

14. Order Type Extension pseudoline arrangement

15. Order Type Extension • Abstract order type extension: • duality: point sets  line arrangements order type  intersection sequences • abstract order type  pseudoline arrangement • extend pseudoline arrangement with an additional pseudoline in all combinatorial different ways • decide realizability of extended abstract order type (optional)

16. Order Type Extension Problem: Order types of size 12 may contain multiple start order types of size 11  some order types are generated in multiple Avoiding multiple generation of order types - Order type extension graph: nodes .. order types in extension algorithm edges .. for each generated order type of size n+1 (son) define a unique sub-order type of size n (father)

17. Order Type Extension • - Extension only along edges of order type extension graph  each order type is generated exactly once • distributed computing can be applied to abstract order type extension: independent calculation for each starting 11-point order type

18. Rectilinear Crossing Number Application: Rectilinear crossing number of complete graph Kn minimum number of crossings attained by a straight-line drawing of the complete graph Kn in the plane

19. Rectilinear Crossing Number What numbers are known so far? cr(Kn) .. rectilinear crossing number of Kn dn .. number of combinatorially different drawings Aichholzer, Aurenhammer, Krasser, On the crossing number of complete graphs. 2002

20. Rectilinear Crossing Number Subset property of rectilinear crossing number of Kn: Drawing of Kn on Sn has c crossings  at least one drawing of Kn-1on Sn-1 has at most c·n/(n-4) crossings Parity property: n odd  c  ( ) (mod 2) Extension graph: point causing most crossings n 4

21. Rectilinear Crossing Number Not known: cr(K13)=229 ? K13 .. 227 crossings  K12 .. 157crossings K12 .. 157 crossings  K11 .. 104crossings Not known: d13= ? K13 .. 229 crossings  K12 .. 158crossings K12 .. 158 crossings  K11 .. 104crossings

22. Rectilinear Crossing Number

23. Rectilinear Crossing Number Extension of the complete data base: 2 334 512 907 order types for n=11 Extension for rectilinear crossing number:

24. Rectilinear Crossing Number New results on the rectilinear crossing number: cr(Kn) .. rectilinear crossing number of Kn dn .. number of combinatorially different drawings

25. best known lower bound: Balogh, Salazar, On k-sets, convex quadrilaterals, and the rectilinear crossing number of Kn. Rectilinear Crossing Constant Problem: rectilinear crossing constant,asymptotics of rectilinear crossing number

26. Rectilinear Crossing Constant - best known upper bound: large point set with few crossings, lens substitution - improved upper bound: set of 54 points with 115 999 crossings, lens substitution Aichholzer, Aurenhammer, Krasser, On the crossing number of complete graphs. 2002

27. Further Applications • „Happy End Problem“: • What is the minimum number g(k) s.t. each point set with at least g(k) points contains a convex k-gon? • No tight bounds are known for k6. • Conjecture: Erdös, Szekeres, A combinatorial problem in geometry. 1935

28. Further Applications Subset property: Sn contains a convex k-gon  each subset Sn-1contains a convex k-gon Future goal: Solve the case of 6-gons by a distributed computing approach.

29. Further Applications • Counting the number of triangulations: • exact values for n≤11 • best asymptotic lower bound is based on these result Aichholzer, Hurtado, Noy, A lower bound on the number of triangulations of planar point sets. 2004 • subset property: adding an interior point increases the number of triangulations by a constant factor 1.806 • calculations: to be done…

30. Abstract Order Type… Thank you!