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.

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

Crossing Properties no crossing 4 points: crossing

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

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 

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 Our work: extension to n=11 points, same approach with improved methods

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

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 abstract order types for n=11

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

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

Order Type Data Base number of points projective abstract o.t thereof non-realizable = project. order types abstract order types thereof non-realizable = order types Extended order type data base 16-bit integer coordinates, >100 GB

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…

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

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

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

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

Order Type Extension line arrangement

Order Type Extension pseudoline arrangement

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)

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)

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

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

Rectilinear Crossing Number n cr(K n ) dndn 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 What numbers are known so far?

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

Not known: cr(K 13 )=229 ? K crossings  K crossings K crossings  K crossings Not known: d 13 = ? K crossings  K crossings K crossings  K crossings

Rectilinear Crossing Number n a≤100≤ b≤102≤ a≤104≤157≤ b≤158≤ a≤ b≤106≤159≤231≤ a≤326≤ b≤161≤233≤327≤ a≤108≤162≤235≤330≤451≤ b≤ a≤164≤237≤333≤455≤608≤ b≤110≤165≤239≤335≤457≤610≤798

Rectilinear Crossing Number crossings order types Extension of the complete data base: order types for n=11 Extension for rectilinear crossing number:

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

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.

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

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

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.

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 subset property: adding an interior point increases the number of triangulations by a constant factor calculations: to be done…

Abstract Order Type… Thank you!