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

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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

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

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Crossing Properties 3 points: no 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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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

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

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

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Enumerating Order Types 3 points: 1 order type triangle

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

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arrangement of lines cells Enumerating Order Types geometrical insertion

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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

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Enumerating Order Types 5 points: 3 order types

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

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

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

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

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Enumerating Order Types line arrangement

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Enumerating Order Types pseudoline arrangement

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Enumerating Order Types wiring diagram

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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

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

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Realizability Pappuss theorem

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Realizability non-Pappus arrangement is not stretchable

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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]

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

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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

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Order Type Data Base number of points abstract order types thereof non- realizable = order types bit16-bit 24-bit

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Order Type Data Base number of points abstract order types thereof non- realizable = order types MB

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Order Type Data Base number of points abstract order types thereof non- realizable = order types GB

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Order Type Data Base number of points projective abstract o.t thereof non- realizable = projective order types abstract order types thereof non- realizable = order types GB

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

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

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

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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

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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

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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

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Applications

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

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Applications Problem: (rectilinear crossing constant)

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

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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]

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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

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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.

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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

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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]

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

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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

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Order Types... Thank you!

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