Abstract Order Type Extension and New Results on the Rectilinear Crossing

Number

Oswin Aichholzer

Institute for SoftwaretechnologyGraz University of Technology

Graz, Austria

Hannes KrasserInstitute for Theoretical Computer

ScienceGraz University of Technology

Graz, Austria

ACM Symposium on Computational Geometry (SoCG), Pisa, Italy, 2005

Point Sets

- finite point sets in the real plane 2

- in general position- with different crossing properties

Crossing Properties

point set

complete straight-line graph Kn

crossingno 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

Order Type

Point sets of same order type there exists a bijection s.t. eitherall (or none) corresponding triples are of equal orientation

Point sets of same order type

2

41

33

1

2

4

5

5

Order Type

How to decide whether 2 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

Order Type

Order Type

- natural λ-matrix: p1 on the convex hull, p2, ..,pn sorted clockwise around p1

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 Extension

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

arrangement of lines cells

Order Type Extension

Order Type Extension

extending point set realizations of order types with one additional point is not a complete order type extension

line arrangement not unique

Order Type Extension

point-line duality: p T(p)

a

b

cT(a)

T(b)

T(c)

bc ac ab

Order Type Extension

point-line duality: p T(p)

a

b

c

T(a)

T(b)

T(c)

ab ac bc

Order Type Extension

order type local intersection sequence (point set) (line arrangement)

point-line duality: p T(p)

Order Type Extension

line arrangement

Order Type Extension

pseudoline arrangement

Order Type Extension

order type local intersection sequence (point set) (line arrangement)

point-line duality: p T(p)

abstract local intersection sequence order type (pseudoline arrangement)

Order Type Extension

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)

Enumerating Order Types

Task: Enumerate all order types of point sets in the plane (for small, fixed size and in general position)

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

= project. 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

Order type data base for n≤10 pointsAichholzer, Aurenhammer, Krasser, Enumerating order types for small point sets with applications. 2001

Our work: extension to n=11 points 16-bit integer coordinates, >100 GB

Order Type Extension

Extension to n=12, 13, … ?- 750 billion order types for n=12- too many for complete data base - partial extension of data base- obtain results on „suitable

applications“ for 12 and beyond…

Subset Property

„suitable applications“: subset property

Property valid for Sn and there exists Sn-1 s.t. similar property holds for Sn-1

Sn .. order type of n pointsSn-1 .. subset of Sn of n-1 points

Order Type Extension

Order type extension with subset property:- order type data base result set of order types for n=11 - enumerate all order types of 12 points that contain one of these 11-point order types as a subset- filter 12-point order types according to subset property

Rectilinear Crossing Number

Application: Rectilinear crossing number of complete graph Kn

minimum number of crossings attained by a straight-line drawing of the complete graph Kn in the plane

Rectilinear Crossing Number

n 3 4 5 6 7 8 9 10 11 12

cr(Kn

)0 0 1 3 9 19 36 62 10

215

3

dn1 1 1 1 3 2 10 2 374 1

cr(Kn) .. rectilinear crossing number of Kn

dn .. number of combinatorially different drawings

Aichholzer, Aurenhammer, Krasser, On the crossing number of complete graphs. 2002

What numbers are known so far?

Order type extension (rectilinear crossing number of Kn): Enumerate order types with „few“ crossings

Subset property: Drawing of Kn on Sn with „few“ crossings contains at least one drawing of Kn-1 on Sn-1

with „few“ crossings

Rectilinear Crossing Number

Subset property: Drawing of Kn on Sn has c crossings at least one drawing of Kn-1 on Sn-1

has at most c·(n-4)/n crossings

Parity property: n odd c ( ) (mod 2)

Rectilinear Crossing Number

n

4

Rectilinear Crossing Number

Not known: cr(K13)=229 ? K13 .. 227 crossings K12 .. 157 crossings K12 .. 157 crossings K11 .. 104 crossings

Not known: d13= ? K13 .. 229 crossings K12 .. 158 crossings K12 .. 158 crossings K11 .. 104 crossings

Rectilinear 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

Rectilinear Crossing Number

crossings 102 104 106 108 110

order types 374 3 984 17 896 47 471 102 925

Extension of the complete data base: 2 334 512 907 order types for n=11

Extension for rectilinear crossing number:

Order Type Extension

Problem: Order types of size 12 may contain multiple start order types of size 11 some order types are generated in multiple

Avoiding multiple generation of order types- Order type extension graph: nodes .. order types in extension algorithm edges .. for each generated order type of size n+1 (son) define a unique sub-order type of size n (father)

Order Type Extension

- Extension only along edges of order type extension graph each order type is generated exactly once- distributed computing can be applied to abstract order type extension: independent calculation for each starting 11-point order type

Extension graph (rectilinear crossing number): - point causing most crossings- largest index in the lexicographically minimal λ-matrix representation

Rectilinear Crossing Number

Rectilinear 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 10

215

322

932

4447 60

3798

dn1 1 1 1 3 2 10 2 374 1 453

420 1600

136 3726

9

cr(Kn) .. rectilinear crossing number of Kn

dn .. number of combinatorially different drawings

New results on the rectilinear crossing number:

Rectilinear Crossing Constant

Problem: rectilinear crossing constant,asymptotics of rectilinear crossing number

)(lim

4/)()(

* n

nKcrn

n

n

Rectilinear Crossing Constant

- best known lower bound: Balogh, Salazar, On k-sets, convex quadrilaterals, and the rectilinear crossing number of Kn.

37533.0*

- lower bound: Lovász, Vesztergombi, Wagner, Welzl, Convex quadrilaterals and k-sets. 2003

5* 10for 375.0

- best known upper bound: large point set with few crossings, lens substitution

- improved upper bound: set of 54 points with 115 999 crossings, lens substitution

38058.0*

Rectilinear Crossing Constant

38074.0* Aichholzer, Aurenhammer, Krasser, On the crossing number of complete graphs. 2002

- further improvement: set of 45 points with 54 213 crossings, recursive substitution

- possible further improvement: abstract set of 96 points with 1 238 508 crossings

38046.0*

Rectilinear Crossing Constant

38056.0*

realizable ??

Further Applications

„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 k6.- Conjecture:

Erdös, Szekeres, A combinatorial problem in geometry. 1935

12g(k) 2k

17g(6)

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

Further Applications

Further Applications

Start: n=11 ... 235 987 328 order types n=12 ... 14 048 972 314 (abstract) o.t. n=13 ... 800 109 order types

Future goal: Solve the case of convex 6-gons by a distributed computing approach

Further Applications

Counting the number of triangulations:- exact values for n≤11- best asymptotic lower bound is based on these result Aichholzer, Hurtado, Noy, A lower bound on the number of triangulations of planar point sets. 2004

- subset property: adding a point increases the number of triangulations by a constant factor- calculations: to be done…

Abstract Order Type…

Thank you!