AlgoPerm2012 - 10 Xavier Goaoc

1-1

Geometric permutations

Xavier Goaoc

2-1

Geometric structures

To compute with geometric objects we often consider combinatorial structures they induce.

Point set P

2-2



Point set P

Convex hull

2-3



Point set P

Convex hull Delaunay triangulation

2-4



Point set P

Convex hull Delaunay triangulation

2-5



Point set P

Convex hull

Minimum spanning tree

Delaunay triangulation

2-6



Point set P

Convex hull

Minimum spanning tree

Delaunay triangulation

Arrangement of thelines spanned by P

3-1

Complexity of geometric structures

Predicting the size of a geometric structure is important (eg for complexity analysis).

3-2



? expressed as a function of the number n of elementary geometric primitives,

? ignoring the bit-complexity of coordinates (they are arbitrary real numbers),

? considering the worst-case over all families of size n.

3-3





Sometimes a direct counting argument is enough...

E.g.: ”the convex hull of n points in Rd has Θ(nbd/2c

)faces in the worst-case”.


3-4





Sometimes a direct counting argument is enough...

E.g.: ”the convex hull of n points in Rd has Θ(nbd/2c

)faces in the worst-case”.

Sometimes intermediate combinatorial objects are useful...


4-1

Example: lower enveloppe of segments in R2

What is the worst-case complexity of the lower enveloppe of n segments in R2?

4-2


a

bc

d

a b c da b


4-3


a

bc

d

a b c da b


An alternation ab corresponds to an endpoint or a crossing of segments a and b.

ab

4-4


a

bc

d

a b c da b



ab

⇒ maximum length of a word on n letters with no sub-word of the form ababa?

4-5


a

bc

d

a b c da b



ab

⇒ maximum length of a word on n letters with no sub-word of the form ababa?

Davenport-Schinzel sequence λ3(n) = Θ(nα(n)).

Tight bound for this geometric problem!

5-1

Introduction

Line transversals and geometric permutations

More Davenport-Schinzel sequences

Excluded patterns

Extrapolation methods: VC dimension and shatter functions

6-1

Line transversals

F = C1, . . . , Cn

Disjoint compact convex sets in Rd

6-2

Line transversals

F = C1, . . . , Cn


T (F )

Set of line transversals to F

6-3

Line transversals

F = C1, . . . , Cn


T (F )


T (F ) ⊂ RG2,d, the (2d− 2)-dimensional manifold of lines in RPd.

6-4

Line transversals

F = C1, . . . , Cn


T (F )


Question: What is the complexity of T (F )?


6-5

Line transversals

F = C1, . . . , Cn


T (F )




Motivation: T (F ) underlies algorithmic questions such as”smallest enclosing cylinder computation”.

6-6

Line transversals

F = C1, . . . , Cn


T (F )





6-7

Line transversals

F = C1, . . . , Cn


T (F )





7-1

Which complexity?

Topologist says: compute the Betti numbers of T (F ).

Polytopist says: restrict F to polytopes and count the faces of T (F ).

7-2

Which complexity?



A simpler approach: compute the number of geometric permutations.

7-3

Which complexity?



A simpler approach: compute the number of geometric permutations.

S

W E

N

SWEN

SEWNOriented line transversal to disjoint convex sets

' permutation of these sets

Unoriented lines

' pair of (reverse) permutations

= geometric permutation.

NWES

NEWS

8-1

A hard nut

Question: What is the asymptotic behavior of g(d, n)?

g(d, n) = maxF family of n disjoint convex sets in Rd #geom. perm. of F

8-2

A hard nut



A few tight bounds:

? g(2, n) = 2n− 2

? at most 4 for disjoint translates of a planar convex set

? at most 2 for n ≥ 9 disjoint unit balls in Rd

General case open for ∼20 years:

? g(d, n) is O(n2d−3 log n

)and Ω

(nd−1

).

8-3

A hard nut



A few tight bounds:

? g(2, n) = 2n− 2

? at most 4 for disjoint translates of a planar convex set

? at most 2 for n ≥ 9 disjoint unit balls in Rd

General case open for ∼20 years:

? g(d, n) is O(n2d−3 log n

)and Ω

(nd−1

).

Davenport-Schinzel sequences

Excluded patterns

9-1

Introduction


Excluded patterns



10-1

Construction showing that g(2, n) ≥ 2n− 2.

Lower bound in the plane

10-2



10-3



10-4



11-1

Upper bound in the plane

Charge every geometric permutation

to a LL bitangent line transversal.

11-2




11-3




L(~u) = left-most object in direction ~u.(first object entirely on the left when sweeping from left to right)

11-4


Divide S1 in intervals with same L(·).




11-5



Get a circular word w = L(−→u1)L(−→u2) . . . L(−→uk)

# LL bitangent line transversals ≤ |w|




11-6



Get a circular word w = L(−→u1)L(−→u2) . . . L(−→uk)

# LL bitangent line transversals ≤ |w|

w has no abab subword ⇒ |w| ≤ 2n− 2.




12-1

Introduction


Excluded patterns



13-1

Another approach...

Observation. If A,B,C,D has a line transversal in the order ABCD thenit cannot have a line transversal in the order BADC.

Let A, B, C and D be disjoint convex sets in the plane.

13-2

Another approach...


Let A, B, C and D be disjoint convex sets in the plane. A

B

CD

B

ADC

13-3

Another approach...



B

CD

B

ADC

13-4

Another approach...



B

CD

B

ADC

13-5

Another approach...



B

CD

B

ADC

Constraints on restrictions of geometric permutations.

cannot be two geom. perm. of the same disjoint convex planar sets.1234567

1432756

13-6

Another approach...



B

CD

B

ADC


cannot be two geom. perm. of the same disjoint convex planar sets.1234567

14327561432756

1234567

13-7

Another approach...



B

CD

B

ADC


cannot be two geom. perm. of the same disjoint convex planar sets.

(ABCD,BADC) is an excluded pattern for disjoint planar convex sets.

1234567

14327561432756

1234567

14-1

Excluded patterns: definition

σ ∈ Sn contains τ ∈ Sk if there exists 1 ≤ i1 < i2 < . . . < ik ≤ n such that

∀1 ≤ a, b ≤ k, σ−1(ia) < σ−1(ib) ⇔ τ−1(a) < τ−1(b)

If σ does not contain τ then σ avoids τ .

Classical permutation patterns:

14-2


Patterns in geometric permutations:

(σ1, σ2) ∈ (Sn)2 contains (τ1, τ2) ∈ (Sk)2 if there exists 1 ≤ i1 < i2 < . . . < ik ≤ n such that

for x = 1, 2 and 1 ≤ a, b ≤ k, σ−1x (ia) < σ−1

x (ib) ⇔ τ−1x (a) < τ−1

x (b)

If (σ1, σ2) does not contain (τ1, τ2) then (σ1, σ2) avoids (τ1, τ2).


∀1 ≤ a, b ≤ k, σ−1(ia) < σ−1(ib) ⇔ τ−1(a) < τ−1(b)



14-3


Patterns in geometric permutations:

(σ1, σ2) ∈ (Sn)2 contains (τ1, τ2) ∈ (Sk)2 if there exists 1 ≤ i1 < i2 < . . . < ik ≤ n such that

for x = 1, 2 and 1 ≤ a, b ≤ k, σ−1x (ia) < σ−1

x (ib) ⇔ τ−1x (a) < τ−1

x (b)

If (σ1, σ2) does not contain (τ1, τ2) then (σ1, σ2) avoids (τ1, τ2).

Previous example:

If F is a family of disjoint convex sets in R2,

Any pair of permutations of F induced by oriented line transversals avoids (1234, 2143).


∀1 ≤ a, b ≤ k, σ−1(ia) < σ−1(ib) ⇔ τ−1(a) < τ−1(b)



15-1

Excluded patterns in the plane

(1234, 2143) is an excluded pattern for convex sets.

(1234, 3214) is an excluded pattern for translates of a convex set.

⇒ disjoint translates of a convex set have at most 3 geometric permutations.

(1234, 1342) and (1234, 3142) are excluded pattern for unit disks.

⇒ n ≥ 4 disjoint unit disks have at most 2 geometric permutations.

Application: Helly-type theorems for sets of line transversals.

16-1

Excluded patterns in higher dimension

All pairs of patterns are realizable!

Pick two non-coplanar lines.

Place points labelled from 1 to n in the desired orders.

16-2





Connect pairs of points with the same labels.

Non-coplanarity of lines ⇒ disjointedness of segments.

16-3







There exist excluded triples...

(123456, 456123, 246135) is excluded for convex sets in R3.

16-4







There exist excluded triples...

(123456, 456123, 246135) is excluded for convex sets in R3.

There are excluded pairs in restricted settings...

(1234, 4123), (1234, 1432), (1234, 3412) and (1234, 3142) are excluded for unit balls in Rd.

Contrary to the planar case, it is open whether (1234, 1342) is excluded...

17-1

Introduction


Excluded patterns



18-1

A detour via hypergraphs

Consider a hypergraph H ⊆ 2V with vertex set V .

Associate to H the shatter function fH : N∗ → N defined by:

fH(k) = maxX∈(Vk) #e ∩X | e ∈ H

”fH(k) is the size of the largest trace of H on a k element subset of V ”

18-2






Sauer’s Lemma. If fH(k) < 2k then fH(n) = O(nk−1

).

18-3






Sauer’s Lemma. If fH(k) < 2k then fH(n) = O(nk−1

).

The largest k such that fH(k) = 2k is the Vapnik-Chervonenkis (VC) dimension of H.

Applications in computational learning theory, approximation algorithms...

19-1

VC-dimension of families of permutations

Consider a family of permutations F ⊆ Sn.

Associate to F the shatter function φF : N∗ → N defined by:

fH(k) = maxX∈(Vk) #σ|X | σ ∈ F

where if X = i1, . . . , ik then ∀1 ≤ a, b ≤ k, σ−1|X (a) < σ−1

|X (b) ⇔ σ−1(ia) < σ−1(ib)

Define the VC dimension of F as the largest k such that φF (k) = k!.

19-2






|X (b) ⇔ σ−1(ia) < σ−1(ib)


Theorem (Raz). There is a constant C such that any family F ⊆ Sn with VC-dimensionat most 2 has size O (Cn).

19-3






|X (b) ⇔ σ−1(ia) < σ−1(ib)



Raz conjectured that bounded VC-dimension ⇒ at most exponential size.

Generalizes excluded patterns and the Stanley-Wilf conjecture discussed in the next talk

19-4






|X (b) ⇔ σ−1(ia) < σ−1(ib)



Raz conjectured that bounded VC-dimension ⇒ at most exponential size.

Recently disproved by Cibulka-Kyncl: the right bound is between α(n)n and (log∗ n)n.

Generalizes excluded patterns and the Stanley-Wilf conjecture discussed in the next talk

20-1

Introduction


Excluded patterns


... a few open problems (come see me for more :) )


21-1

Some bounds on Davenport-Schinzel sequences remain with gap.

g(3, n) is only known to be Ω(n2) and O(n3 log n)... The gap widens in higher dimension.

Can we refine the ”bootstrapping” mechanism of the VC-dimension?

What does fH(k) = m guarantee in terms of asymptotic estimates when m < 2k?

Same question for families of permutations...

Is there some reasonable shattering condition that would imply g(3, n) = O(n2)?

How to find excluded patterns in dimension 3 and higher ?

Incompatibility of (1234, 1342) remains open (would close gaps and improve Helly numbers).

How hard is it to test if a d-tuple of permutations is excluded for convex sets in Rd?

22-1

A few pointers...

Tight bounds on the maximum size of a set of permutations with bounded VC-dimension

Jan Kyncl and Josef Cibulka, arXiv:1104.5007v2 (SODA 2012)

VC-Dimension of Sets of Permutations

Ran Raz, Combinatorica 20: 1-15 (2000)

ε-nets and simplex range queries

David Haussler and Emo Welzl, Discrete & Computational Geometry 2:127-151 (1987)

Geometric permutations of disjoint unit spheresOtfried Cheong, X. G. and Hyeon-Suk Na

Comp. Geom. Theor. and Appl. 30: 253–270 (2005).

Geometric permutations in the plane and in Euclidean spaces of higher dimension

Andrei Asinowski, PhD thesis (2005)

Improved bounds for geometric permutations

Nathan Rubin, Haim Kaplan and Micha Sharir, to appear in SICOMP (FOCS 2010)

Davenport-Schinzel sequences and their geometric applications

Micha Sharir and Pankaj Agarwal, Cambridge Univ. Press

AlgoPerm2012 - 10 Xavier Goaoc

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