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 ACM Symposium on Computational Geometry (SoCG), Pisa, Italy, 2005

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

3. Crossing Properties point set no crossing crossing complete straight-line graph Kn

4. Crossing Properties 4 points: no crossing crossing

5. 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

6. 2 3 4 3  5 1 5 1 4 2 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

7. Order Type • How to decide whether 2 point sets are of the same order type? • encoding order types: λ-matrixGoodman, Pollack, Multidimensional Sorting. 1983 • S={p1, ..,pn} .. labelled point setλ(i,j) .. number of points of S on the left of the oriented line through pi and pj • - Theorem: order type λ-matrixGoodman, Pollack, Multidimensional Sorting. 1983

8. Order Type

9. Order Type • natural λ-matrix: p1 on the convex hull, p2, ..,pn sorted clockwise around p1

10. Order Type • natural λ-matrix: p1 on the convex hull, p2, ..,pn sorted clockwise around p1 • lexicographically minimal λ-matrix: unique „fingerprint“ for an order type • same order type  identicallexicographically minimal λ-matrices

11. Order Type Extension • complete order type extension: • input: order type Sn of n points • output: all different order types Sn+1 of n+1 points that contain Sn as a sub-order type

12. Order Type Extension arrangement of lines  cells

13. Order Type Extension extending point set realizations of order types with one additional point is not a complete order type extension line arrangement not unique

14. b T(a) c a T(b) T(c) bc ac ab Order Type Extension point-line duality: pT(p)

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

16. Order Type Extension point-line duality: pT(p) order type  local intersection sequence (point set) (line arrangement)

17. Order Type Extension line arrangement

18. Order Type Extension pseudoline arrangement

19. Order Type Extension point-line duality: pT(p) order type  local intersection sequence (point set) (line arrangement) abstract  local intersection sequence order type (pseudoline arrangement)

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

21. Enumerating Order Types Task: Enumerate all order types of point sets in the plane (for small, fixed size and in general position)

22. Order Type Data Base 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 16-bit integer coordinates, >100 GB

23. Order Type Extension Extension to n=12, 13, … ? -  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…

24. 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

25. 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

26. 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

27. 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

28. Rectilinear Crossing Number Order type extension (rectilinear crossing number of Kn): Enumerate order types with „few“ crossings Subset property: Drawing of Kn on Sn with „few“ crossings contains at least one drawing of Kn-1on Sn-1 with „few“ crossings

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

30. 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

31. Rectilinear Crossing Number

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

33. 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)

34. 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

35. Rectilinear Crossing Number • Extension graph (rectilinear crossing number): • point causing most crossings • largest index in the lexicographically minimal λ-matrix representation

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

37. Rectilinear Crossing Constant Problem: rectilinear crossing constant,asymptotics of rectilinear crossing number

38. - best known lower bound: Balogh, Salazar, On k-sets, convex quadrilaterals, and the rectilinear crossing number of Kn. Rectilinear Crossing Constant - lower bound: Lovász, Vesztergombi, Wagner, Welzl, Convex quadrilaterals and k-sets. 2003

39. 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

40. Rectilinear Crossing Constant • - further improvement: set of 45 points with 54 213 crossings, recursive substitution • possible further improvement: abstract set of 96 points with 1 238 508 crossings realizable ??

41. 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 exact values g(k) are known for k6. • Conjecture: Erdös, Szekeres, A combinatorial problem in geometry. 1935

42. Further Applications Order type extension (6-gon problem): Enumerate all order types that do not contain a convex 6-gon Subset property: Sn contains no convex 6-gon  each subset Sn-1contains no convex 6-gon

43. Further Applications Start: n=11 ... 235 987 328 order types n=12 ... 14 048 972 314 (abstract) o.t. n=13 ...  800 109 order types Future goal: Solve the case of convex 6-gons by a distributed computing approach

44. 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 a point increases the number of triangulations by a constant factor • calculations: to be done…

45. Abstract Order Type… Thank you!