1. Order Types of Point Sets in the Plane Hannes Krasser Institute for Theoretical Computer Science Graz University of Technology Graz, Austria supported by FWF

2. Point Sets How many different point sets exist? - point sets in the real plane 2 - finite point sets of fixed size - point sets in general position - point sets with different crossing properties

3. Crossing Properties point set complete straight-line graph K n crossingno crossing

4. Crossing Properties 3 points: no crossing

5. Crossing Properties no crossing 4 points: crossing

6. 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

7. Crossing Determination a b c d b a d c line segments ab, cd crossing different orientations abc, abd and different orientations cda, cdb line segments ab, cd

8. Crossing Determination point quadruple abcd crossing number of positively oriented triples abc, abd, acd, bcd is even a b c d

9. Enumerating Order Types Task: Enumerate all different order types of point sets in the plane (in general position)

10. Enumerating Order Types 3 points: 1 order type triangle

11. Enumerating Order Types no crossing 4 points: 2 order types crossing

12. arrangement of lines cells Enumerating Order Types geometrical insertion

13. Enumerating Order Types geometrical insertion: - for each order type of n points consider the underlying line arrangement - insert a point in each cell of each line arrangement order types of n+1 points

14. Enumerating Order Types 5 points: 3 order types

15. Enumerating Order Types geometrical insertion: no complete data base of order types line arrangement not unique

16. Enumerating Order Types point-line duality: p T(p) a b c T(a) T(b) T(c) bc ac ab

17. Enumerating Order Types point-line duality: p T(p) a b c T(a) T(b) T(c) ab ac bc

18. Enumerating Order Types order type local intersection sequence (point set) (line arrangement) point-line duality: p T(p)

19. Enumerating Order Types line arrangement

20. Enumerating Order Types pseudoline arrangement

21. Enumerating Order Types wiring diagram

22. Enumerating Order Types creating order type data base: - enumerate all different local intersection sequences abstract order types - decide realizability of abstract order types order types easy hard

23. Enumerating Order Types realizability of abstract order types stretchability of pseudoline arrangements

24. Realizability Pappuss theorem

25. Realizability non-Pappus arrangement is not stretchable

26. Realizability Deciding stretchability is NP-hard. [Mnëv, 1985] Every arrangement of at most 8 pseudolines in P 2 is stretchable. [Goodman, Pollack, 1980] Every simple arrangement of at most 9 pseudo- lines in P 2 is stretchable except the simple non-Pappus arrangement. [Richter, 1988]

27. Realizability heuristics for proving realizability: - geometrical insertion - simulated annealing heuristics for proving non-realizability: - linear system of inequations derived from Grassmann-Plücker equations

28. Order Type Data Base main result: complete and reliable data base of all different order types of size up to 11 in nice integer coordinate representation

29. Order Type Data Base number of points 34567891011 abstract order types 123161353 315158 83014 320 1822 343 203 071 - thereof non- realizable 1310 6358 690 164 = order types123161353 315158 81714 309 5472 334 512 907 8-bit16-bit 24-bit

30. Order Type Data Base number of points 34567891011 abstract order types 123161353 315158 83014 320 1822 343 203 071 - thereof non- realizable 1310 6358 690 164 = order types123161353 315158 81714 309 5472 334 512 907 550 MB

31. Order Type Data Base number of points 34567891011 abstract order types 123161353 315158 83014 320 1822 343 203 071 - thereof non- realizable 1310 6358 690 164 = order types123161353 315158 81714 309 5472 334 512 907 140 GB

32. Order Type Data Base number of points 34567891011 projective abstract o.t. 1114111354 38231235641 848 591 - thereof non- realizable 1242155 214 = projective order types 1114111354 381312 11441 693 377 abstract order types 123161353 315158 83014 320 1822 343 203 071 - thereof non- realizable 1310 6358 690 164 = order types123161353 315158 81714 309 5472 334 512 907 1.7 GB

33. Applications problems relying on crossing properties: - crossing families - rectilinear crossing number - polygonalizations - triangulations - pseudo-triangulations and many more...

34. Applications how to apply the data base: - complete calculation for point sets of small size (up to 11) - order type extension

35. Applications motivation for applying the data base: - find counterexamples - computational proofs - new conjectures - more insight

36. Applications Problem: What is the minimum number n of points such that any point set of size at least n admits a crossing family of size 3? crossing family: set of pairwise intersecting line segments

37. Applications Problem: What is the minimum number n of points such that any point set of size at least n admits a crossing family of size 3? Previous work: n37 [Tóth, Valtr, 1998] New result: n10, tight bound

38. Applications Problem: (rectilinear crossing number) What is the minimum number cr(K n ) of crossings that any straight-line drawing of K n in the plane must attain? Previous work: n9 [Erdös, Guy, 1973] Our work: n16

39. Applications

40. n345678910111213141516 cr(K n )00139193662102153229324447603 dndn 11113210237414534201600136 data base order type extension cr(K n )... rectilinear crossing number of K n d n... number of combinatorially different drawings

41. Applications Problem: (rectilinear crossing constant)

42. Previous work: [Brodsky, Durocher, Gethner, 2001] Our work: Latest work: [Lovász, Vesztergombi, Wagner, Welzl, 2003] Applications

43. Problem: (Sylvesters Four Point Problem) What is the probability q(R) that any four points chosen at random from a planar region R are in convex position? [Sylvester, 1865] choose independently uniformly at random from a set R of finite area, q * = inf q(R) q * = [Scheinerman, Wilf, 1994]

44. Applications Problem: Give bounds on the number of crossing-free Hamiltonian cycles (polygonalizations) of an n-point set. crossing-free Hamiltonian cycle of S: planar polygon whose vertex set is exactly S

45. Applications Conjecture: [Hayward, 1987] Does some straight-line drawing of K n with minimum number of edge crossings necessarily produce the maximal number of crossing-free Hamiltonian cycles? NO! Counterexample with 9 points.

46. Applications Problem: What is the minimum number of triangulations any n-point set must have? New conjecture: double circle point sets Observation: true for n11

47. Applications Problem: What is the minimum number of pointed pseudo-triangulations any n-point set must have? New conjecture: convex sets theorem [Aichholzer, Aurenhammer, Krasser, Speckmann, 2002]

48. Applications Problem: (compatible triangulations) Can any two point sets be triangulated in the same manner?

49. Applications Conjecture: true for point sets S 1, S 2 with |S 1 |=|S 2 |, |CH(S 1 )|=|CH(S 2 )|, and S 1, S 2 in general position. [Aichholzer, Aurenhammer, Hurtado, Krasser, 2000] Observation: holds for n9 Note: complete tests for all pairs with n=10,11 points take too much time

50. Order Types... Thank you!