Order Types - Graz University of Technology · – Cn≥1=...
Transcript of Order Types - Graz University of Technology · – Cn≥1=...
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22nd European Workshop on Computational Geometry
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Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG
Order Types (from geometry to combinatorics)
DCG
22nd European Workshop on Computational Geometry
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Triangulations
Triangulation of a point set S:
– Maximal plane straight-line
graph with vertex set S
– Decomposition of the convex hull of S into non-overlapping
triangles
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Triangulations
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Triangulations
Delaunay Triangulation
– Dual to Voronoi diagram
Greedy Triangulation
Minimum Weight Triangulation
– One of 3 unsolved of 14 open Problems in
Garey and Johnson’s book on NP-Completeness [Computers and Intractability,1979]
– Solved: Minimum Weight Triangulation is NP-hard [W. Mulzer, G. Rote, SoCG June 2006]
How many triangulations do exist?
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How Many Triangulations?
Number of triangulations of a convex n -gon
Number of complete binary trees with fixed root and n-1 leaves
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5.1
1
0
1
42
1
1
nn
n
nCCC
nn
k
knkn
B(m) = k=1..m-1B(k) B(m-k); m=n-1
T(n) = B(m) =…= Cm-1 = Cn-2
Catalan numbers:
– Cn≥1 = 1,2,5,14,42,132,429,1430,4862,16796,58786,…(C0 =1)
– belong to the ‘top five’ numbers in Combinatorics:
• Complete Binary trees with n leaves Cn-1
• Completely paranthesizing a product of n letters Cn-1
• Two (robot) soccer teams play n:n. How many goal-sequences exist, such that team A is never behind team B? Cn
• Triangulations Cn-2
k m-k
How Many Triangulations?
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So for convex n-gons there
are 4n triangulations
“And of course there are
much more triangulations
in the general case!”
Lower Bound?
? F.A.
O.A.
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Of Course There Are More …
Well, not really.
Maybe asymptotically ... n = 3
n = 4
n = 5
1
2 1
2 3 5
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How Many Point Sets?
How many ways to draw n points in the plane?
n =3
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How Many Point Sets?
How many ways to draw n points in the plane?
n =4
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Why Crossing Properties?
Pla
ne
Spannin
g T
rees
Pla
ne
Perf
ect
Matc
hin
gs
Pla
ne
Tri
ang
ula
tions
Pla
ne
Ham
ilto
nia
n
Cyc
les
Co
un
t fo
r
n=
3,4
,5
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Crossing Properties
complete straight-line graph Kn
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
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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order type of point set:
mapping that assigns to each ordered triple of points
i ts orientation [Goodman, Pollack, 1983]
orientation:
Order Type
left/positive right/negative
a
b
c
a
b
c
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Order Type
n =4: 2 order types n =3: 1 order type
“Generate all point sets”
Enumerate all dif ferent order types
of point sets in the plane
1 2
3 4 3 3
4
1 1 2 2
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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
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4 1
3 3
1
2
4
5
5
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Order Type
How to decide whether two point sets are of the same order type?
encoding order types: λ-matrix [Goodman, Pollack, Multidimensional Sorting. 1983]
S ={p1, ..,pn} .. labelled point set
λ(i,j) .. number of points of S on the left
of the oriented line through pi and pj
Theorem: order type λ-matrix [Goodman, Pollack, Multidimensional Sorting. 1983]
λ(i,j)=2
i
j
Think: Why true? 16 22nd European Workshop on Computational Geometry
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Order Type
Find the typo!
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Order Type
natural λ-matrix: p1 on the convex hull, p2, ..,pn sorted clockwise around p1
lexicographically minimal λ-matrix:
unique „fingerprint“ for an order type
same order type identical lexicographically minimal λ-matrices
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Order Type
+ symmetric sets
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Order Type
Properties of the natural λ-matrix:
λ(i,j )=0 pi and pj on the convex hull
λ(i,j )+λ(j,i ) = n – 2
p1 on the convex hull
p2, ..,pn sorted clockwise around p1
lexicographically minimal matrix
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Enumerating Order Types
Generating order types, iterative approach:
complete order type extension:
- input: order type Sn of n points
- output: all different order types Sn+1
of n +1 points that contain Sn as a sub-order type
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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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arrangement of lines cells
Enumerating Order Types
geometrical insertion n=4? n=5?
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Enumerating Order Types
5 points: 3 order types
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Enumerating Order Types
Is geometrical iterative insertion complete?
Line arrangement not unique for fixed order type!
NO!
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Enumerating Order Types
Same order type, but different arrangements!!
?
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Enumerating Order Types
point-line duality: p T(p)
T : p p tl =1 , l =(x,y )t
p
T (p) p
T (p)
p
T (p)
(Unit Circ le Duality)
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point-line duality: p T(p)
T : p p tl =1 , l =(x,y )t
Properties:
- p l T-1(l ) T(p) incidence preservation
- p l+ T-1(l ) T(p)+ order preservation
Enumerating Order Types
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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
a
b
c
T(a)
T(b)
T(c)
ab ac bc
point-line duality: p T(p)
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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
again: (line) arrangement
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Enumerating Order Types
pseudoline arrangement
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Enumerating Order Types
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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Enumerating Order Types
abstract duality for natural λ-matrices
Find the error!
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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)
Enumerating Order Types
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Enumerating Order Types
1,5,3,2,4,6
1,6,5,3,4,2
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Enumerating Order Types
wiring diagram
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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) hard
easy
Enumerating Order Types
abstract order types
order types
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Realizability
Realizability of abstract order types
stretchability of pseudoline arrangements
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 simple
non-Pappus arrangement. [Richter, 1988]
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Realizability
Pappus‘s theorem
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Realizability
non-Pappus arrangement
is not stretchable
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Realizability
Group abstract order types into projective classes
– combinatorial simulation of the rotation (of -matrices)
without (knowledge of) realization
– either all or none elements of a projective class are realizeable
heuristics for proving realizability:
– geometrical insertion + local optimization (simulated
annealing)
heuristics for proving non-realizability:
– l inear system of inequations derived from Grassmann-
Plücker equations [Bokowski, Richter, 1990] 43
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Order Type Data Base
Data Base:
Complete and reliable data base of all
different order types of size up to 11 in nice 16-bit integer coordinate representation
(Using natural labeling and lexicographical
minimal lambda matrices)
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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
projective
abstract o.t.
1 1 1 4 11 135 4 382 312 356 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
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Applications
Motivation for using the data base:
find counterexamples
computational proofs
new conjectures
structural insight
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Applications
Problems investigated include:
recti l inear crossing number
polygonalizations (Hamilton circles)
crossing families
triangulations
pseudo-triangulations
spanning trees
(empty) convex polygons
decomposition, partition, covering
perfect matching
reflexivity
and more ... 47
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Crossing Families
What is the minimum number n (k ) of points such that any point set of size at least n (k ) admits a crossing family of size k ?
crossing family: set of k pairwise intersecting line segments
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Crossing Families
crossing family for k = 2: n (k ) = 5
Think!
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Crossing Families
k =3?
– Any set of n ≥ 10 points contains at
least 3 pair w ise crossing edges.
The bound on n is tight.
– Best previous bound on w as n ≥ 37
[Tóth, Valtr, 1998]
New 2015: k =4: n ≥ 15 points contain at
least 4 pair wise crossing edges
Why interesting? Proofs using divide & conquer
Lower bound for triangulations …
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Ramsey Type Results
Further Ramsey-type results observed from the database:
Every set of 8 points contains
either an
empty convex pentagon or two disjoint
empty convex quadrilaterals
Provided ‘human readable’ proof afterwards. [Aichholzer, Huemer, Kappes, Speckmann, Toth, 2005/06]
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Ramsey Type Results
Further Ramsey-type results observed from the database:
Every set of 11 points contains
either an
empty convex hexagon or an
empty convex pentagon and a disjoint
empty convex quadrilateral
Provided ‘human readable’ proof afterwards. [A, Huemer, Kappes, Speckmann, Toth, 2005/06]
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Decomposition
Convex decomposition Pseudo-Convex decomposition
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Constant Asymptotics
We ‘win’ 3 faces per 9 points …
Ramsey-type results for constant size objects might
lead to improvements for arbitrary size:
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Decomposition
Convex decomposition Pseudo-Convex decomposition
7
18
7
102
11
12 nDn C
nDn PC10
7
10
6
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How to generate order types for n ≥12: Complete extension intractable
- ≈750 billion order types for n =12
- Disk space > 30 TB - Realizability problem
- > 200 years of computation Partial Extension on “suitable” sets
Abstract Extension
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Abstract Extension
Sub-Set property: For a set of n points with a required
property there exists at least one set of n -1 points with
an according property. Generate all sets with desired property for n =11
Only extend these sets to n =12,13,… No need to realize intermediate sets: only realize
counterexamples or special sets Choose unique predecessor for each set (paternity
test), proceed tree-like (generate each set only once!)
Use distributed computing: each set of the starting base can be extended independently
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Happy End Problem
„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 7. Conjecture: g(k ) =2k-2+1 (g(6)=17 solved 2006)
[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: Sn contains no convex 6-gon each
subset Sn-1 contains no convex 6-gon
Start:
n =11 ... 235 987 328 order types
n =12 ... 14 048 972 314 (abstract) order types
n =13 ... 800 109 (abstract) order types
Use improved algorithms and havy distributed computing
Happy End Problem
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cr (Kn) … recti l inear crossing number of Kn
minimum number of edge crossings attained
by a drawing of Kn with straight l ines
Crossing Number
Think: n=6?
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Crossing Number
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Crossing Number: History
Turán’s brick factory problem, cr(Km,n ), 1944
Bring bricks on small
vehicles from ovens to
stores, each oven was connected with each
store by rail.
Troubles occurred at crossings: cars
jumped out, bricks fell
down
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Crossing Number: History
n 4 5 6 7 8 9 10 11
cr(Kn ) 0 1 3 9 19 36 62 102
dn 1 1 1 3 2 10 2 374
[Erdös, Guy, 1960-1973]
[Brodsky, Durocher, Gether, 2001] [AAK, 2001]
[AAK, 2004]
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Subset property:
– Each n -1 point subgraph of Kn contains at least cr (Kn-1) crossings, each being counted n-4 times:
– A drawing of Kn must contain a drawing of Kn -1 with
drawings.
– Parity property for n odd:
Crossing Number
)(
4)( 1nn Kcr
n
nKcr
nn
Kcr n mod4
)(
)(
4nKcr
n
n
2
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Crossing Number
draw ing w ith 153 crossings
cr(K12)=153
1531028
12)( 12
Kcr
Every 11-sub graph is
drawn optimally
Unique optimal drawing
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cr(K13)=229 ? K13 .. 227 crossings K12 .. 157 crossings K12 .. 157 crossings K11 .. 104 crossings
d13= ?
K13 .. 229 crossings K12 .. 158 crossings K12 .. 158 crossings K11 .. 104 crossings
Crossing Number
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Crossing Number n 11 12 13 14 15 16 17
12 a ≤100 ≤152
12 b ≤102 ≤153
13 a ≤104 ≤157 ≤227
13 b ≤158 ≤229
14 a ≤323
14 b ≤106 ≤159 ≤231 ≤324
15 a ≤326 ≤445
15 b ≤161 ≤233 ≤327 ≤447
16 a ≤108 ≤162 ≤235 ≤330 ≤451 ≤602
16 b ≤603
17 a ≤164 ≤237 ≤333 ≤455 ≤608 ≤796
17 b ≤110 ≤165 ≤239 ≤335 ≤457 ≤610 ≤798
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crossings 102 104 106 108 110
order types 374 3 984 17 896 47 471 102 925
Range for optimal cr(n)
12 14 15 17 19
Sets to extend
374 (0.00001%)
4 358 (0.0002 %)
22 254 (0.0010%)
69 725 (0.0029%)
172 650 (0.0073%)
Extension of the complete data base: 2 334 512 907 order types for n=11
Extension for rectilinear crossing number:
Crossing Number
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n 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
cr (Kn) 0 0 1 3 9 19 36 62 102 153 229 324 447 603 798
dn 1 1 1 1 3 2 10 2 374 1 4534 20 16001 36 37269
data base order type extension
cr (Kn) ... rectilinear crossing number of Kn
dn ... number of combinatorially different drawings
Crossing Number
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cr(K18)=1029 (New: unique optimal example)
Crossing Number
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Structural results:
Sets minimizing the rectilinear crossing
number have a triangular convex hull
… and nested inner triangular layers
There alw ays exist sets maximizing the number
of halving-edges w ith a triangular convex hull
[A, García, Orden, Ramos, 2006]
Conjecture: For any n there always exists a optimal set
which contains an optimal (n -1)-set:
- n =18 provides a counterexample
Crossing Number
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Structural results:
Sets minimizing the rectilinear crossing
number have a triangular convex hull
… and nested inner triangular layers
There alw ays exist sets maximizing the number
of halving-edges w ith a triangular convex hull
[A, García, Orden, Ramos, 2006]
Conjecture: For any n there always exists a optimal set
which contains an optimal (n -1)-set:
- n =18 provides a counterexample
Crossing Number
Meanwhile obtained results:
cr(18)=1029; RCN Dec. 2006 (new 2014/15: unique)
cr(19),cr(21) using triangular structure [AGOR, 2006] cr(20),cr(22)..cr(27) and cr(30) using best sets and
pseudo lines / circular sequences [AMLS, 04/2007] cr(28) ϵ {7233,7234}, cr(29) ϵ {8421,8423}
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rectilinear crossing constant:
)(lim
4/)()(
* n
nKcrn
n
n
Crossing Number
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Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG
Lower Bounds on *
> 0.3001 [Brodsky, Durocher, Gethner, 2001]
> 0.3115 [Ai, Aurenhammer, Krasser, 2002]
> 0.3288 [Wagner, 2003]
> 0.3328 [Ai, Aurenhammer, Krasser, 2003]
≥ 0.3750 [Ábrego,Férnandez-M erchant, 2003]
(general crossing number 3/8)
> 0.3750+ > 3/8 [Lovász, Vesztergombi, Wagner, Welzl, 2003]
> 0.37533 [Balogh, Salazar, 2004]
> 0,37968 [Ai, García, Orden, Ramos, 2006][A,B,F-M,L,S]
> 0,379972 [Ábrego,Férnandez-M erchant, Leaños, Salazar, 2008]
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Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG
380473.0379972.0 *
Upper Bounds on *
< 0.38888 [Jensen, 1974]
< 0.38460 [Singer, 1971, Manuscript]
< 0.38380 [Brodsky, Durocher, Gethner, 2001]
< 0.38074 n=36, cr=21191 [A, Aurenhammer Krasser, 2002]
< 0.38058 n=54, cr=115999 [A, Krasser,2005]
< 0.38055 n=30, cr=9726 [Ábrego,Férnandez-Merchant, 2006]
< 0.38054 n=90, cr=951534 [A, Kornberger 2006/07]
< 0.380473 n=75, cr=450 492 [R.Fabila-Monroy, J.Lopez, 2014]
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22nd European Workshop on Computational Geometry
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Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG
What is the probability q(R) that any four points chosen at random from a planar regio R are in convex position? [Sylvester, 1865]
– choose independently uniformly at random from a set R of finite area, q* = inf q(R)
Convex sets: 2/3 ≤q(R)≤ 1-35/(122) 0.7
Infimum for open sets: q* =* [Scheinerman,Wilf,1994]
Sylvester‘s Four Point Problem
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Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG
Crossing Number
How to construct asymptotically good drawings:
– take “best” known small drawing D
– recursively replace each vertex of D by a copy of D
[Singer 1971]
78
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Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG
Crossing Number
79 22nd European Workshop on Computational Geometry
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Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG
Crossing Number
80
22nd European Workshop on Computational Geometry
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Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG
Halving property:
the replacement
halves the remaining set into groups of
sizes n/2 and n/2-1 (n even)
Crossing Number
81 22nd European Workshop on Computational Geometry
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Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG
Questions & Future Work
Are crossing-optimal abstract order types always realizable?
– For n 18: yes. Open for general n ≥19 (Example??)
Do crossing-optimal drawings always contain optimal sub-drawings?
– For n 17: yes. No for n =18 n =18: unique optimal example has NO optimal n=17 sub-set
Do optimal drawings have 3 extreme vertices?
– Yes! [A, García, Orden, Ramos, 2006]
True constant in asymptotics?
– Best bounds: 0,379972 … 0.380473 (gap reduced from 1/10 to 1/2000)
Value of cr (Kn ) for larger n ...
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22nd European Workshop on Computational Geometry
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Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG
The database is used for …
• rectil inear crossing number
• crossing families
• Happy End Problem • convex k -gons
• empty convex k -gons • convex covering
• convex partitioning
• convex decomposition • number of triangulations
• fl ip distances
• isomorphic triangulations
• sequential triangulations
• minimum pseudo-triangulations • Hamiltonian cycles
• min. reflex polygonalizations • spanning trees
• crossing-free matchings
• k –sets • Hayward-Conjecture
Further applications, subset properties, distributed computing …
86 22nd European Workshop on Computational Geometry
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Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG
Thanks!
Thanks for your attention …
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