order types of point sets in the plane hannes krasser institute for theoretical computer science...
TRANSCRIPT
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Order Types of Point Sets in the Plane
Hannes KrasserInstitute for Theoretical Computer
ScienceGraz 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 Kn
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/positive right/negative
a
bc
a
bc
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Crossing Determination
a
b
c
d
ba
d
c
line segments ab, cd crossing different orientations abc, abd anddifferent orientations cda, cdb
line segments ab, cd
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Enumerating Order Types
Task: Enumerate all differentorder 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
cT(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
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
Pappus‘s 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 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]
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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
3 4 5 6 7 8 9 10 11
abstract order types
1 2 3 16 135 3 315 158 830
14 320 182
2 343 203 071
- thereof non- realizable
13 10 635 8 690 164
= order types 1 2 3 16 135 3 315 158 817
14 309 547
2 334 512 907
8-bit 16-bit 24-bit
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Order Type Data Base
number of points
3 4 5 6 7 8 9 10 11
abstract order types
1 2 3 16 135 3 315 158 830
14 320 182
2 343 203 071
- thereof non- realizable
13 10 635 8 690 164
= order types 1 2 3 16 135 3 315 158 817
14 309 547
2 334 512 907
550 MB
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Order Type Data Base
number of points
3 4 5 6 7 8 9 10 11
abstract order types
1 2 3 16 135 3 315 158 830
14 320 182
2 343 203 071
- thereof non- realizable
13 10 635 8 690 164
= order types 1 2 3 16 135 3 315 158 817
14 309 547
2 334 512 907
140 GB
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Order Type Data Base
number of points
3 4 5 6 7 8 9 10 11
projective abstract o.t.
1 1 1 4 11 135 4 382 312356 41 848 591
- thereof non-
realizable
1 242 155 214
= projective order types
1 1 1 4 11 135 4 381 312 114 41 693 377
abstract order types
1 2 3 16 135 3 315 158 830
14 320 182
2 343 203 071
- thereof non- realizable
13 10 635 8 690 164
= order types 1 2 3 16 135 3 315 158 817
14 309 547
2 334 512 907 1.7 GB
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Applications
problems relying on crossing properties:- crossing families- rectilinear crossing number- polygonalizations- triangulations- pseudo-triangulationsand 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: n≥37 [Tóth, Valtr, 1998]New result: n≥10, tight bound
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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
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Applications
153)( 12 Kcr
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Applications
n 3 4 5 6 7 8 9 10 11 12 13 14 15 16
cr(Kn) 0 0 1 3 9 19 36 62 102
153
229
324
447 603
dn1 1 1 1 3 2 10 2 374 1 453
420 1600
136
data base
order type extension
cr(Kn) ... rectilinear crossing number of Kn
dn ... number of combinatorially different drawings
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Applications
Problem: (rectilinear crossing constant)
)(lim
4/)()(
* n
nKcrn
n
n
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Previous work: [Brodsky, Durocher, Gethner, 2001]
Our work:
Latest work:[Lovász, Vesztergombi, Wagner, Welzl, 2003]
Applications
3838.03001.0 *
3808.03328.0 * 5* 10for 375.0
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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]
*
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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 Kn
with minimum number of edge crossingsnecessarily produce the maximal numberof 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 n≤11
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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 triangulatedin the same manner?“
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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≤9Note: complete tests for all pairs with n=10,11 points take too much time
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Order Types...
Thank you!