Order Types - Graz University of Technology · – Cn≥1= 1,2,5,14,42,132,429,1430,4862,16796,58786,…(C 0 =1) – belong to the ‘top five’ numbers in Combinatorics: ... duality

1

22nd European Workshop on Computational Geometry

Institute for Software Technology

Points, from geometry to combinatorics: order types Oswin Aichholzer TU-Graz DCG

Order Types (from geometry to combinatorics)

DCG




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

1




Triangulations

2 22nd European Workshop on Computational Geometry



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?

3




How Many Triangulations?

Number of triangulations of a convex n -gon

Number of complete binary trees with fixed root and n-1 leaves




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?

5

2




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.




Of Course There Are More …

Well, not really.

Maybe asymptotically ... n = 3

n = 4

n = 5

1

2 1

2 3 5

7




How Many Point Sets?

How many ways to draw n points in the plane?

n =3




How Many Point Sets?

How many ways to draw n points in the plane?

n =4

9




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




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

11

3




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




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

13




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




Order Type

Point sets of same order type

there exists a bijection s.t. either all (or none)

corresponding triples are of equal orientation

2

4 1

3 3

1

2

4

5

5

15




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



Order Type

Find the typo!

17

4




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




Order Type

+ symmetric sets

19




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




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

21





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




arrangement of lines cells


geometrical insertion n=4? n=5?

23

5





5 points: 3 order types





Is geometrical iterative insertion complete?

Line arrangement not unique for fixed order type!

NO!

25





Same order type, but different arrangements!!

?





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)

27





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







a

b

c T(a)

T(b)

T(c)

bc ac ab

29

6





a

b

c

T(a)

T(b)

T(c)

ab ac bc






order type local intersection sequence

(point set) (line arrangement)


31





again: (line) arrangement





pseudoline arrangement

33





order type local intersection sequence

(point set) (line arrangement)


abstract local intersection sequence

order type (pseudoline arrangement)





abstract duality for natural λ-matrices

Find the error!

35

7




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)






1,5,3,2,4,6

1,6,5,3,4,2

7

37





wiring diagram




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


abstract order types

order types

39




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]




Realizability

Pappus‘s theorem

41

8




Realizability

non-Pappus arrangement

is not stretchable




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




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)




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

45




Applications

Motivation for using the data base:

find counterexamples

computational proofs

new conjectures

structural insight




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

9




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




Crossing Families

crossing family for k = 2: n (k ) = 5

Think!

49




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 …




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]

51




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]




Decomposition

Convex decomposition Pseudo-Convex decomposition

53

10




Constant Asymptotics

We ‘win’ 3 faces per 9 points …

Ramsey-type results for constant size objects might

lead to improvements for arbitrary size:




Decomposition

Convex decomposition Pseudo-Convex decomposition

7

18

7

102

11

12 nDn C

nDn PC10

7

10

6

55




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




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

57




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]




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

59

11




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?




Crossing Number

61




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




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]

63




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




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

65

12




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




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

67




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




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

69




cr(K18)=1029 (New: unique optimal example)

Crossing Number




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

71

13




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}




rectilinear crossing constant:

)(lim

4/)()(

* n

nKcrn

n

n

Crossing Number

73




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]




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]

75




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




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

14




Crossing Number




Crossing Number

80




Halving property:

the replacement

halves the remaining set into groups of

sizes n/2 and n/2-1 (n even)

Crossing Number




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 ...

85




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 …




Thanks!

Thanks for your attention …

87

Documents

Order Types - Graz University of Technology · – Cn≥1= 1,2,5,14,42,132,429,1430,4862,16796,58786,…(C 0 =1) – belong to the ‘top five’ numbers in Combinatorics: ... duality