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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

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

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

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Crossing Properties no crossing 4 points: crossing

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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

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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 5 5

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Order Type How to decide whether 2 point sets are of the same order type? - encoding order types: λ-matrix Goodman, Pollack, Multidimensional Sorting S={p 1,..,p n }.. labelled point set λ(i,j).. number of points of S on the left of the oriented line through p i and p j - Theorem: order type λ-matrix Goodman, Pollack, Multidimensional Sorting. 1983

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Order Type

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- natural λ-matrix: p 1 on the convex hull, p 2,..,p n sorted clockwise around p 1

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Order Type - natural λ-matrix: p 1 on the convex hull, p 2,..,p n sorted clockwise around p 1 - lexicographically minimal λ-matrix: unique „fingerprint“ for an order type - same order type identical lexicographically minimal λ-matrices

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

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arrangement of lines cells Order Type Extension

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extending point set realizations of order types with one additional point is not a complete order type extension line arrangement not unique

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

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

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

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Order Type Extension line arrangement

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Order Type Extension pseudoline arrangement

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

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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)

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

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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 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 16-bit integer coordinates, >100 GB

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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…

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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

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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

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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

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Rectilinear Crossing Number n cr(K n ) dndn 11 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?

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

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Subset property: 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-4)/n crossings Parity property: n odd c ( ) (mod 2) Rectilinear Crossing Number n 4

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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

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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

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

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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)

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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

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

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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:

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

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

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- 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

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- further improvement: set of 45 points with crossings, recursive substitution - possible further improvement: abstract set of 96 points with crossings Rectilinear Crossing Constant realizable ??

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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

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

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Start: n= order types n= (abstract) o.t. n=13... order types Future goal: Solve the case of convex 6-gons by a distributed computing approach

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

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Abstract Order Type… Thank you!

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