Post on 20-Jun-2015
description
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
Geometric structures
To compute with geometric objects we often consider combinatorial structures they induce.
Point set P
Convex hull
2-3
Geometric structures
To compute with geometric objects we often consider combinatorial structures they induce.
Point set P
Convex hull Delaunay triangulation
2-4
Geometric structures
To compute with geometric objects we often consider combinatorial structures they induce.
Point set P
Convex hull Delaunay triangulation
2-5
Geometric structures
To compute with geometric objects we often consider combinatorial structures they induce.
Point set P
Convex hull
Minimum spanning tree
Delaunay triangulation
2-6
Geometric structures
To compute with geometric objects we often consider combinatorial structures they induce.
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
Complexity of geometric structures
Predicting the size of a geometric structure is important (eg for complexity analysis).
? 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
Complexity of geometric structures
Predicting the size of a geometric structure is important (eg for complexity analysis).
? expressed as a function of the number n of elementary geometric primitives,
? ignoring the bit-complexity of coordinates (they are arbitrary real numbers),
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”.
? considering the worst-case over all families of size n.
3-4
Complexity of geometric structures
Predicting the size of a geometric structure is important (eg for complexity analysis).
? expressed as a function of the number n of elementary geometric primitives,
? ignoring the bit-complexity of coordinates (they are arbitrary real numbers),
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...
? considering the worst-case over all families of size n.
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
Example: lower enveloppe of segments in R2
a
bc
d
a b c da b
What is the worst-case complexity of the lower enveloppe of n segments in R2?
4-3
Example: lower enveloppe of segments in R2
a
bc
d
a b c da b
What is the worst-case complexity of the lower enveloppe of n segments in R2?
An alternation ab corresponds to an endpoint or a crossing of segments a and b.
ab
4-4
Example: lower enveloppe of segments in R2
a
bc
d
a b c da b
What is the worst-case complexity of the lower enveloppe of n segments in R2?
An alternation ab corresponds to an endpoint or a crossing of segments a and b.
ab
⇒ maximum length of a word on n letters with no sub-word of the form ababa?
4-5
Example: lower enveloppe of segments in R2
a
bc
d
a b c da b
What is the worst-case complexity of the lower enveloppe of n segments in R2?
An alternation ab corresponds to an endpoint or a crossing of segments a and 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
Disjoint compact convex sets in Rd
T (F )
Set of line transversals to F
6-3
Line transversals
F = C1, . . . , Cn
Disjoint compact convex sets in Rd
T (F )
Set of line transversals to F
T (F ) ⊂ RG2,d, the (2d− 2)-dimensional manifold of lines in RPd.
6-4
Line transversals
F = C1, . . . , Cn
Disjoint compact convex sets in Rd
T (F )
Set of line transversals to F
Question: What is the complexity of T (F )?
T (F ) ⊂ RG2,d, the (2d− 2)-dimensional manifold of lines in RPd.
6-5
Line transversals
F = C1, . . . , Cn
Disjoint compact convex sets in Rd
T (F )
Set of line transversals to F
Question: What is the complexity of T (F )?
T (F ) ⊂ RG2,d, the (2d− 2)-dimensional manifold of lines in RPd.
Motivation: T (F ) underlies algorithmic questions such as”smallest enclosing cylinder computation”.
6-6
Line transversals
F = C1, . . . , Cn
Disjoint compact convex sets in Rd
T (F )
Set of line transversals to F
Question: What is the complexity of T (F )?
T (F ) ⊂ RG2,d, the (2d− 2)-dimensional manifold of lines in RPd.
Motivation: T (F ) underlies algorithmic questions such as”smallest enclosing cylinder computation”.
6-7
Line transversals
F = C1, . . . , Cn
Disjoint compact convex sets in Rd
T (F )
Set of line transversals to F
Question: What is the complexity of T (F )?
T (F ) ⊂ RG2,d, the (2d− 2)-dimensional manifold of lines in RPd.
Motivation: T (F ) underlies algorithmic questions such as”smallest enclosing cylinder computation”.
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?
Topologist says: compute the Betti numbers of T (F ).
Polytopist says: restrict F to polytopes and count the faces of T (F ).
A simpler approach: compute the number of geometric permutations.
7-3
Which complexity?
Topologist says: compute the Betti numbers of T (F ).
Polytopist says: restrict F to polytopes and count the faces of T (F ).
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
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
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
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
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
Line transversals and geometric permutations
Excluded patterns
Extrapolation methods: VC dimension and shatter functions
More Davenport-Schinzel sequences
10-1
Construction showing that g(2, n) ≥ 2n− 2.
Lower bound in the plane
10-2
Construction showing that g(2, n) ≥ 2n− 2.
Lower bound in the plane
10-3
Construction showing that g(2, n) ≥ 2n− 2.
Lower bound in the plane
10-4
Construction showing that g(2, n) ≥ 2n− 2.
Lower bound in the plane
11-1
Upper bound in the plane
Charge every geometric permutation
to a LL bitangent line transversal.
11-2
Upper bound in the plane
Charge every geometric permutation
to a LL bitangent line transversal.
11-3
Upper bound in the plane
Charge every geometric permutation
to a LL bitangent line transversal.
L(~u) = left-most object in direction ~u.(first object entirely on the left when sweeping from left to right)
11-4
Upper bound in the plane
Divide S1 in intervals with same L(·).
Charge every geometric permutation
to a LL bitangent line transversal.
L(~u) = left-most object in direction ~u.(first object entirely on the left when sweeping from left to right)
11-5
Upper bound in the plane
Divide S1 in intervals with same L(·).
Get a circular word w = L(−→u1)L(−→u2) . . . L(−→uk)
# LL bitangent line transversals ≤ |w|
Charge every geometric permutation
to a LL bitangent line transversal.
L(~u) = left-most object in direction ~u.(first object entirely on the left when sweeping from left to right)
11-6
Upper bound in the plane
Divide S1 in intervals with same L(·).
Get a circular word w = L(−→u1)L(−→u2) . . . L(−→uk)
# LL bitangent line transversals ≤ |w|
w has no abab subword ⇒ |w| ≤ 2n− 2.
Charge every geometric permutation
to a LL bitangent line transversal.
L(~u) = left-most object in direction ~u.(first object entirely on the left when sweeping from left to right)
12-1
Introduction
Line transversals and geometric permutations
Excluded patterns
Extrapolation methods: VC dimension and shatter functions
More Davenport-Schinzel sequences
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...
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. A
B
CD
B
ADC
13-3
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. A
B
CD
B
ADC
13-4
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. A
B
CD
B
ADC
13-5
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. A
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...
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. A
B
CD
B
ADC
Constraints on restrictions of geometric permutations.
cannot be two geom. perm. of the same disjoint convex planar sets.1234567
14327561432756
1234567
13-7
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. A
B
CD
B
ADC
Constraints on restrictions of geometric permutations.
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
Excluded patterns: definition
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).
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:
14-3
Excluded patterns: definition
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).
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:
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
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.
Connect pairs of points with the same labels.
Non-coplanarity of lines ⇒ disjointedness of segments.
16-3
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.
Connect pairs of points with the same labels.
Non-coplanarity of lines ⇒ disjointedness of segments.
There exist excluded triples...
(123456, 456123, 246135) is excluded for convex sets in R3.
16-4
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.
Connect pairs of points with the same labels.
Non-coplanarity of lines ⇒ disjointedness of segments.
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
Line transversals and geometric permutations
Excluded patterns
Extrapolation methods: VC dimension and shatter functions
More Davenport-Schinzel sequences
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
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 ”
Sauer’s Lemma. If fH(k) < 2k then fH(n) = O(nk−1
).
18-3
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 ”
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
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!.
Theorem (Raz). There is a constant C such that any family F ⊆ Sn with VC-dimensionat most 2 has size O (Cn).
19-3
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!.
Theorem (Raz). There is a constant C such that any family F ⊆ Sn with VC-dimensionat most 2 has size O (Cn).
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
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!.
Theorem (Raz). There is a constant C such that any family F ⊆ Sn with VC-dimensionat most 2 has size O (Cn).
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
Line transversals and geometric permutations
Excluded patterns
Extrapolation methods: VC dimension and shatter functions
... a few open problems (come see me for more :) )
More Davenport-Schinzel sequences
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