Order Types of Point Sets in the Plane

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 no crossing crossing complete straight-line graph Kn

4. Crossing Properties 3 points: no crossing

5. Crossing Properties 4 points: no crossing crossing

6. 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

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

8. Enumerating Order Types Task: Enumerate all differentorder types of point sets in the plane(in general position)

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

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

11. Enumerating Order Types arrangement of lines geometrical insertion  cells

12. 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

13. Enumerating Order Types 5 points: 3 order types

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

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

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

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

18. Enumerating Order Types line arrangement

19. Enumerating Order Types pseudoline arrangement

20. easy hard 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

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

22. Realizability Pappus‘s theorem

23. Realizability non-Pappus arrangement is not stretchable

24. Realizability Deciding stretchability is NP-hard. [Mnëv, 1985] Every arrangement of at most 8 pseudolines in P2 is stretchable. [Goodman, Pollack, 1980] Every simple arrangement of at most 9 pseudo-lines in P2 is stretchable except the simplenon-Pappus arrangement. [Richter, 1988]

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

26. 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

27. 8-bit 16-bit 24-bit Order Type Data Base

28. Order Type Data Base  550 MB

29. Order Type Data Base  140 GB

30.  1.7 GB Order Type Data Base

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

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

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

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

35. Applications Problem: What is the minimum number n of points such that any point set of size at least nadmits a crossing family of size 3? Previous work:n≥37 [Tóth, Valtr, 1998] New result:n≥10, tight bound

36. Applications Problem: (rectilinear crossing number) What is the minimum number cr(Kn) of crossings that any straight-line drawing of Kn in the plane must attain? Previous work: n≤9 [Erdös, Guy, 1973] Our work: n≤16

37. Applications

38. data base order type extension Applications cr(Kn) ... rectilinear crossing number of Kn dn ... number of combinatorially different drawings

39. Applications Problem: (rectilinear crossing constant)

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

41. Applications Problem: (“Sylvester‘s 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]

42. 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

43. Applications Conjecture: [Hayward, 1987] Does some straight-line drawing of Kn with minimum number of edge crossingsnecessarily produce the maximal numberof crossing-free Hamiltonian cycles? NO! Counterexample with 9 points.

44. Applications Problem: What is the minimum number of triangulations any n-point set must have? New conjecture: double circle point sets Observation: true for n≤11

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

46. Applications Problem: (compatible triangulations) “Can any two point sets be triangulatedin the same manner?“

47. Applications Conjecture: true for point sets S1, S2 with |S1|=|S2|, |CH(S1)|=|CH(S2)|, and S1, S2 in general position. [Aichholzer, Aurenhammer, Hurtado, Krasser, 2000] Observation: holds for n≤9 Note: complete tests for all pairs with n=10,11 points take too much time

48. Order Types... Thank you!