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Order Types of Point Sets in the Plane Hannes Krasser Institute for Theoretical Computer Science...
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Transcript of Order Types of Point Sets in the Plane Hannes Krasser Institute for Theoretical Computer Science...
Order Types of Point Sets in the Plane
Hannes KrasserInstitute for Theoretical Computer
ScienceGraz University of Technology
Graz, Austria
supported by FWF
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
Crossing Properties
point set
complete straight-line graph Kn
crossingno crossing
Crossing Properties
3 points:
no crossing
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/positive right/negative
a
bc
a
bc
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
Enumerating Order Types
Task: Enumerate all differentorder types of point sets in the plane(in general position)
Enumerating Order Types
3 points: 1 order type
triangle
Enumerating Order Types
no crossing
4 points: 2 order types
crossing
arrangement of lines cells
Enumerating Order Types
geometrical insertion
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
Enumerating Order Types
5 points: 3 order types
Enumerating Order Types
geometrical insertion:no complete data base of order types
line arrangement not unique
Enumerating Order Types
point-line duality: p T(p)
a
b
cT(a)
T(b)
T(c)
bc ac ab
Enumerating Order Types
point-line duality: p T(p)
a
b
c
T(a)
T(b)
T(c)
ab ac bc
Enumerating Order Types
order type local intersection sequence (point set) (line arrangement)
point-line duality: p T(p)
Enumerating Order Types
line arrangement
Enumerating Order Types
pseudoline arrangement
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
Enumerating Order Types
realizability of abstract order types stretchability of pseudoline arrangements
Realizability
Pappus‘s theorem
Realizability
non-Pappus arrangement is not stretchable
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]
Realizability
heuristics for proving realizability:- geometrical insertion- simulated annealing
heuristics for proving non-realizability:- linear system of inequations derived from Grassmann-Plücker equations
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
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
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
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
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
Applications
problems relying on crossing properties:- crossing families- rectilinear crossing number- polygonalizations- triangulations- pseudo-triangulationsand many more ...
Applications
how to apply the data base:- complete calculation for point sets of small size (up to 11)- order type extension
Applications
motivation for applying the data base:- find counterexamples - computational proofs- new conjectures- more insight
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
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
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
Applications
153)( 12 Kcr
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
Applications
Problem: (rectilinear crossing constant)
)(lim
4/)()(
* n
nKcrn
n
n
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
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]
*
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
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.
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
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]
Applications
Problem: (compatible triangulations)“Can any two point sets be triangulatedin the same manner?“
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
Order Types...
Thank you!