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 R 2 - in general position - with different crossing properties

3. Crossing Properties no crossing 4 points: crossing

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

5. Order Type Point sets of same order type  there exists a bijection s.t. either all (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 points Aichholzer, 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. Enumerating Order Types How to create an order type data base: 1. Generate a candidate list of abstract order types 2. Group abstract order types into projective classes, decide realizability 3. Realize all realizable order types by point sets with „nice“ representation

8. Enumerating Order Types 1. Generate a candidate list of abstract order types - duality: point sets  line arrangements order type  intersection sequences - abstract order types  pseudoline arrangements - purely combinatorial - 2 343 203 071 abstract order types for n=11

9. Enumerating Order Types 2. Group abstract order types into projective classes, decide realizability - equivalent order types in the projective plane - heuristics for deciding realizability - realizability proof: point set coordinates by geometrical extension, simulated annealing - non-realizability proof: system of linear inequalities from Grassmann-Plücker relations Bokowski, Richter, On the finding of final polynomials. 1990

10. Enumerating Order Types 3. Realize all realizable order types by point sets with „nice“ representation - high reliability for applications - 16-bit integer coordinates - 2 334 512 907 order types for n=11

11. Order Type Data Base number of points34567891011 projective abstract o.t.1114111354 38231235641 848 591 - thereof non-realizable1242155 214 = project. order types1114111354 381312 11441 693 377 abstract order types123161353 315158 83014 320 1822 343 203 071 - thereof non-realizable1310 6358 690 164 = order types123161353 315158 81714 309 5472 334 512 907 Extended order type data base 16-bit integer coordinates, >100 GB

12. 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…

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

14. 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

15. 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

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

17. Order Type Extension line arrangement

18. Order Type Extension pseudoline arrangement

19. 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)

20. 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)

21. 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

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

23. Rectilinear Crossing Number n3456789101112 cr(K n )00139193662102153 dndn 111132102374 cr(K n ).. rectilinear crossing number of K n d n.. number of combinatorially different drawings Aichholzer, Aurenhammer, Krasser, On the crossing number of complete graphs. 2002 What numbers are known so far?

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

25. Not known: cr(K 13 )=229 ? K 13.. 227 crossings  K 12.. 157 crossings K 12.. 157 crossings  K 11.. 104 crossings Not known: d 13 = ? K 13.. 229 crossings  K 12.. 158 crossings K 12.. 158 crossings  K 11.. 104 crossings

26. Rectilinear Crossing Number n11121314151617 12 a≤100≤152 12 b≤102≤153 13 a≤104≤157≤227 13 b≤158≤229 14 a≤323 14 b≤106≤159≤231≤324 15 a≤326≤445 15 b≤161≤233≤327≤447 16 a≤108≤162≤235≤330≤451≤602 16 b≤603 17 a≤164≤237≤333≤455≤608≤796 17 b≤110≤165≤239≤335≤457≤610≤798

27. Rectilinear Crossing Number crossings102104106108110 order types3743 98417 89647 471102 925 Extension of the complete data base: 2 334 512 907 order types for n=11 Extension for rectilinear crossing number:

28. Rectilinear Crossing Number n34567891011121314151617 cr(K n )00139193662102153229324447603798 dndn 11113210237414534201600136  37269 cr(K n ).. rectilinear crossing number of K n d n.. number of combinatorially different drawings New results on the rectilinear crossing number:

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

30. - 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 Rectilinear Crossing Constant Aichholzer, Aurenhammer, Krasser, On the crossing number of complete graphs. 2002

31. 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

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

33. 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…

34. Abstract Order Type… Thank you!