A presentation on a volumetric contact dynamics model to the SYDE 652 class, March 27, 2012. Includes model framework, experiments, and implementation in MapleSim with Graph Theoretic Modelling
A presentation on a volumetric contact dynamics model to the SYDE 652 class, March 27, 2012. Includes model framework, experiments, and implementation in MapleSim with Graph Theoretic Modelling
Characterizing the Distortion of Some Simple Euclidean EmbeddingsDon Sheehy
This talk addresses some upper and lower bounds techniques for bounding the distortion between mappings between Euclidean metric spaces including circles, spheres, pairs of lines, triples of planes, and the union of a hyperplane and a point.
Abstract: Algebraic structures in which the property of commutativity is substituted by the me- diality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or e-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with (n-1) associators of the arity (2n-1) satisfying a (n^2-1)-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.
Parallel Evaluation of Multi-Semi-JoinsJonny Daenen
Presentation given on VLDB 2016: 42nd International Conference on Very Large Data Bases.
Paper: http://dx.doi.org/10.14778/2977797.2977800
ArXiv: https://arxiv.org/abs/1605.05219
Poster: https://zenodo.org/record/61653 (doi 10.5281/zenodo.61653)
Gumbo Software: https://github.com/JonnyDaenen/Gumbo
Abstract
While services such as Amazon AWS make computing power abundantly available, adding more computing nodes can incur high costs in, for instance, pay-as-you-go plans while not always significantly improving the net running time (aka wall-clock time) of queries. In this work, we provide algorithms for parallel evaluation of SGF queries in MapReduce that optimize total time, while retaining low net time. Not only can SGF queries specify all semi-join reducers, but also more expressive queries involving disjunction and negation. Since SGF queries can be seen as Boolean combinations of (potentially nested) semi-joins, we introduce a novel multi-semi-join (MSJ) MapReduce operator that enables the evaluation of a set of semi-joins in one job. We use this operator to obtain parallel query plans for SGF queries that outvalue sequential plans w.r.t. net time and provide additional optimizations aimed at minimizing total time without severely affecting net time. Even though the latter optimizations are NP-hard, we present effective greedy algorithms. Our experiments, conducted using our own implementation Gumbo on top of Hadoop, confirm the usefulness of parallel query plans, and the effectiveness and scalability of our optimizations, all with a significant improvement over Pig and Hive.
Jean Cardinal (Computer Science Department, Université Libre de Bruxelles)
Sorting and a Tale of Two Polytopes
Algorithms & Permutations 2012, Paris.
http://igm.univ-mlv.fr/AlgoB/algoperm2012/
Mireille Bousquet-Mélou (LABRI, CNRS)
The Number of Inversions After n Adjacent Transpositions
Algorithms & Permutations 2012, Paris. http://igm.univ-mlv.fr/AlgoB/algoperm2012/
Characterizing the Distortion of Some Simple Euclidean EmbeddingsDon Sheehy
This talk addresses some upper and lower bounds techniques for bounding the distortion between mappings between Euclidean metric spaces including circles, spheres, pairs of lines, triples of planes, and the union of a hyperplane and a point.
Abstract: Algebraic structures in which the property of commutativity is substituted by the me- diality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or e-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with (n-1) associators of the arity (2n-1) satisfying a (n^2-1)-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.
Parallel Evaluation of Multi-Semi-JoinsJonny Daenen
Presentation given on VLDB 2016: 42nd International Conference on Very Large Data Bases.
Paper: http://dx.doi.org/10.14778/2977797.2977800
ArXiv: https://arxiv.org/abs/1605.05219
Poster: https://zenodo.org/record/61653 (doi 10.5281/zenodo.61653)
Gumbo Software: https://github.com/JonnyDaenen/Gumbo
Abstract
While services such as Amazon AWS make computing power abundantly available, adding more computing nodes can incur high costs in, for instance, pay-as-you-go plans while not always significantly improving the net running time (aka wall-clock time) of queries. In this work, we provide algorithms for parallel evaluation of SGF queries in MapReduce that optimize total time, while retaining low net time. Not only can SGF queries specify all semi-join reducers, but also more expressive queries involving disjunction and negation. Since SGF queries can be seen as Boolean combinations of (potentially nested) semi-joins, we introduce a novel multi-semi-join (MSJ) MapReduce operator that enables the evaluation of a set of semi-joins in one job. We use this operator to obtain parallel query plans for SGF queries that outvalue sequential plans w.r.t. net time and provide additional optimizations aimed at minimizing total time without severely affecting net time. Even though the latter optimizations are NP-hard, we present effective greedy algorithms. Our experiments, conducted using our own implementation Gumbo on top of Hadoop, confirm the usefulness of parallel query plans, and the effectiveness and scalability of our optimizations, all with a significant improvement over Pig and Hive.
Jean Cardinal (Computer Science Department, Université Libre de Bruxelles)
Sorting and a Tale of Two Polytopes
Algorithms & Permutations 2012, Paris.
http://igm.univ-mlv.fr/AlgoB/algoperm2012/
Mireille Bousquet-Mélou (LABRI, CNRS)
The Number of Inversions After n Adjacent Transpositions
Algorithms & Permutations 2012, Paris. http://igm.univ-mlv.fr/AlgoB/algoperm2012/
Olivier Hudry (INFRES-MIC2 Télécom ParisTech)
A Branch and Bound Algorithm to Compute a Median Permutation
Algorithms & Permutations 2012, Paris.
http://igm.univ-mlv.fr/AlgoB/algoperm2012/
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
2. Geometric structures
To compute with geometric objects we often consider combinatorial structures they induce.
Point set P
2-1
3. Geometric structures
To compute with geometric objects we often consider combinatorial structures they induce.
Convex hull
Point set P
2-2
4. Geometric structures
To compute with geometric objects we often consider combinatorial structures they induce.
Convex hull Delaunay triangulation
Point set P
2-3
5. Geometric structures
To compute with geometric objects we often consider combinatorial structures they induce.
Convex hull Delaunay triangulation
Point set P
2-4
6. Geometric structures
To compute with geometric objects we often consider combinatorial structures they induce.
Convex hull Delaunay triangulation
Point set P
Minimum spanning tree
2-5
7. Geometric structures
To compute with geometric objects we often consider combinatorial structures they induce.
Convex hull Delaunay triangulation
Point set P
Minimum spanning tree Arrangement of the
lines spanned by P
2-6
8. Complexity of geometric structures
Predicting the size of a geometric structure is important (eg for complexity analysis).
3-1
9. 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.
3-2
10. 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.
Sometimes a direct counting argument is enough...
E.g.: ”the convex hull of n points in Rd has Θ n d/2
faces in the worst-case”.
3-3
11. 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.
Sometimes a direct counting argument is enough...
E.g.: ”the convex hull of n points in Rd has Θ n d/2
faces in the worst-case”.
Sometimes intermediate combinatorial objects are useful...
3-4
12. Example: lower enveloppe of segments in R2
What is the worst-case complexity of the lower enveloppe of n segments in R2 ?
4-1
13. Example: lower enveloppe of segments in R2
a
b
c
d
a b c a b d
What is the worst-case complexity of the lower enveloppe of n segments in R2 ?
4-2
14. Example: lower enveloppe of segments in R2
ab
a
b
c
d
a b c a b d
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.
4-3
15. Example: lower enveloppe of segments in R2
ab
a
b
c
d
a b c a b d
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.
⇒ maximum length of a word on n letters with no sub-word of the form ababa?
4-4
16. Example: lower enveloppe of segments in R2
ab
a
b
c
d
a b c a b d
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.
⇒ 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!
4-5
17. Introduction
Line transversals and geometric permutations
More Davenport-Schinzel sequences
Excluded patterns
Extrapolation methods: VC dimension and shatter functions
5-1
18. Line transversals
F = {C1 , . . . , Cn }
Disjoint compact convex sets in Rd
6-1
19. Line transversals
F = {C1 , . . . , Cn } T (F )
Disjoint compact convex sets in Rd Set of line transversals to F
6-2
20. Line transversals
F = {C1 , . . . , Cn } T (F )
Disjoint compact convex sets in Rd Set of line transversals to F
T (F ) ⊂ RG2,d , the (2d − 2)-dimensional manifold of lines in RPd .
6-3
21. Line transversals
F = {C1 , . . . , Cn } T (F )
Disjoint compact convex sets in Rd Set of line transversals to F
T (F ) ⊂ RG2,d , the (2d − 2)-dimensional manifold of lines in RPd .
Question: What is the complexity of T (F )?
6-4
22. Line transversals
F = {C1 , . . . , Cn } T (F )
Disjoint compact convex sets in Rd Set of line transversals to F
T (F ) ⊂ RG2,d , the (2d − 2)-dimensional manifold of lines in RPd .
Question: What is the complexity of T (F )?
Motivation: T (F ) underlies algorithmic questions such as
”smallest enclosing cylinder computation”.
6-5
23. Line transversals
F = {C1 , . . . , Cn } T (F )
Disjoint compact convex sets in Rd Set of line transversals to F
T (F ) ⊂ RG2,d , the (2d − 2)-dimensional manifold of lines in RPd .
Question: What is the complexity of T (F )?
Motivation: T (F ) underlies algorithmic questions such as
”smallest enclosing cylinder computation”.
6-6
24. Line transversals
F = {C1 , . . . , Cn } T (F )
Disjoint compact convex sets in Rd Set of line transversals to F
T (F ) ⊂ RG2,d , the (2d − 2)-dimensional manifold of lines in RPd .
Question: What is the complexity of T (F )?
Motivation: T (F ) underlies algorithmic questions such as
”smallest enclosing cylinder computation”.
6-7
25. Which complexity?
Topologist says: compute the Betti numbers of T (F ).
Polytopist says: restrict F to polytopes and count the faces of T (F ).
7-1
26. 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.
7-2
27. 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.
SEW N
Oriented line transversal to disjoint convex sets
SW EN
permutation of these sets
N
W E
Unoriented lines
pair of (reverse) permutations S
= geometric permutation. N EW S
N W ES
7-3
28. A hard nut
g(d, n) = maxF family of n disjoint convex sets in Rd
#geom. perm. of F
Question: What is the asymptotic behavior of g(d, n)?
8-1
29. A hard nut
g(d, n) = maxF family of n disjoint convex sets in Rd
#geom. perm. of F
Question: What is the asymptotic behavior of g(d, n)?
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 .
8-2
30. A hard nut
g(d, n) = maxF family of n disjoint convex sets in Rd
#geom. perm. of F
Question: What is the asymptotic behavior of g(d, n)?
A few tight bounds:
Davenport-Schinzel sequences
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 R d }
Excluded patterns
General case open for ∼20 years:
g(d, n) is O n2d−3 log n and Ω nd−1 .
8-3
31. Introduction
Line transversals and geometric permutations
More Davenport-Schinzel sequences
Excluded patterns
Extrapolation methods: VC dimension and shatter functions
9-1
32. Lower bound in the plane
Construction showing that g(2, n) ≥ 2n − 2.
10-1
33. Lower bound in the plane
Construction showing that g(2, n) ≥ 2n − 2.
10-2
34. Lower bound in the plane
Construction showing that g(2, n) ≥ 2n − 2.
10-3
35. Lower bound in the plane
Construction showing that g(2, n) ≥ 2n − 2.
10-4
36. Upper bound in the plane
Charge every geometric permutation
to a LL bitangent line transversal.
11-1
37. Upper bound in the plane
Charge every geometric permutation
to a LL bitangent line transversal.
11-2
38. 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)
11-3
39. 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)
Divide S1 in intervals with same L(·).
11-4
40. 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)
Divide S1 in intervals with same L(·).
Get a circular word w = L(− )L(− ) . . . L(− )
→ u
u1 →
2
→
uk
# LL bitangent line transversals ≤ |w|
11-5
41. 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)
Divide S1 in intervals with same L(·).
Get a circular word w = L(− )L(− ) . . . L(− )
→ u
u1 →
2
→
uk
# LL bitangent line transversals ≤ |w|
w has no abab subword ⇒ |w| ≤ 2n − 2.
11-6
42. Introduction
Line transversals and geometric permutations
More Davenport-Schinzel sequences
Excluded patterns
Extrapolation methods: VC dimension and shatter functions
12-1
43. Another approach...
Let A, B, C and D be disjoint convex sets in the plane.
Observation. If {A, B, C, D} has a line transversal in the order ABCD then
it cannot have a line transversal in the order BADC.
13-1
44. Another approach...
C
D
Let A, B, C and D be disjoint convex sets in the plane. A A
B
C
D
B
Observation. If {A, B, C, D} has a line transversal in the order ABCD then
it cannot have a line transversal in the order BADC.
13-2
45. Another approach...
C
D
Let A, B, C and D be disjoint convex sets in the plane. A A
B
C
D
B
Observation. If {A, B, C, D} has a line transversal in the order ABCD then
it cannot have a line transversal in the order BADC.
13-3
46. Another approach...
C
D
Let A, B, C and D be disjoint convex sets in the plane. A A
B
C
D
B
Observation. If {A, B, C, D} has a line transversal in the order ABCD then
it cannot have a line transversal in the order BADC.
13-4
47. Another approach...
C
D
Let A, B, C and D be disjoint convex sets in the plane. A A
B
C
D
B
Observation. If {A, B, C, D} has a line transversal in the order ABCD then
it cannot have a line transversal in the order BADC.
Constraints on restrictions of geometric permutations.
1234567
cannot be two geom. perm. of the same disjoint convex planar sets.
1432756
13-5
48. Another approach...
C
D
Let A, B, C and D be disjoint convex sets in the plane. A A
B
C
D
B
Observation. If {A, B, C, D} has a line transversal in the order ABCD then
it cannot have a line transversal in the order BADC.
Constraints on restrictions of geometric permutations.
1234567
cannot be two geom. perm. of the same disjoint convex planar sets.
1432756
13-6
49. Another approach...
C
D
Let A, B, C and D be disjoint convex sets in the plane. A A
B
C
D
B
Observation. If {A, B, C, D} has a line transversal in the order ABCD then
it cannot have a line transversal in the order BADC.
Constraints on restrictions of geometric permutations.
1234567
cannot be two geom. perm. of the same disjoint convex planar sets.
1432756
(ABCD, BADC) is an excluded pattern for disjoint planar convex sets.
13-7
50. Excluded patterns: definition
Classical permutation patterns:
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 .
14-1
51. Excluded patterns: definition
Classical permutation patterns:
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 .
Patterns in geometric permutations:
(σ1 , σ2 ) ∈ (Sn )2 contains (τ1 , τ2 ) ∈ (Sk )2 if there exists 1 ≤ i1 < i2 < . . . < ik ≤ n such that
−1 −1 −1 −1
for x = 1, 2 and 1 ≤ a, b ≤ k, σx (ia ) < σx (ib ) ⇔ τx (a) < τx (b)
If (σ1 , σ2 ) does not contain (τ1 , τ2 ) then (σ1 , σ2 ) avoids (τ1 , τ2 ).
14-2
52. Excluded patterns: definition
Classical permutation patterns:
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 .
Patterns in geometric permutations:
(σ1 , σ2 ) ∈ (Sn )2 contains (τ1 , τ2 ) ∈ (Sk )2 if there exists 1 ≤ i1 < i2 < . . . < ik ≤ n such that
−1 −1 −1 −1
for x = 1, 2 and 1 ≤ a, b ≤ k, σx (ia ) < σx (ib ) ⇔ τx (a) < τx (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).
14-3
53. 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.
15-1
54. 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.
16-1
55. 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.
16-2
56. 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 .
16-3
57. 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...
16-4
58. Introduction
Line transversals and geometric permutations
More Davenport-Schinzel sequences
Excluded patterns
Extrapolation methods: VC dimension and shatter functions
17-1
59. 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∈(V ) #{e ∩ X | e ∈ H}
k
”fH (k) is the size of the largest trace of H on a k element subset of V ”
18-1
60. 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∈(V ) #{e ∩ X | e ∈ H}
k
”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 .
18-2
61. 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∈(V ) #{e ∩ X | e ∈ H}
k
”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...
18-3
62. 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∈(V ) #{σ|X | σ ∈ F }
k
−1 −1
where if X = {i1 , . . . , ik } then ∀1 ≤ a, b ≤ k, σ|X (a) < σ|X (b) ⇔ σ −1 (ia ) < σ −1 (ib )
Define the VC dimension of F as the largest k such that φF (k) = k!.
19-1
63. 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∈(V ) #{σ|X | σ ∈ F }
k
−1 −1
where if X = {i1 , . . . , ik } then ∀1 ≤ a, b ≤ k, σ|X (a) < σ|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-dimension
at most 2 has size O (C n ).
19-2
64. 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∈(V ) #{σ|X | σ ∈ F }
k
−1 −1
where if X = {i1 , . . . , ik } then ∀1 ≤ a, b ≤ k, σ|X (a) < σ|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-dimension
at most 2 has size O (C n ).
Raz conjectured that bounded VC-dimension ⇒ at most exponential size.
Generalizes excluded patterns and the Stanley-Wilf conjecture discussed in the next talk
19-3
65. 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∈(V ) #{σ|X | σ ∈ F }
k
−1 −1
where if X = {i1 , . . . , ik } then ∀1 ≤ a, b ≤ k, σ|X (a) < σ|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-dimension
at most 2 has size O (C n ).
Raz conjectured that bounded VC-dimension ⇒ at most exponential size.
Generalizes excluded patterns and the Stanley-Wilf conjecture discussed in the next talk
Recently disproved by Cibulka-Kyncl: the right bound is between α(n)n and (log∗ n)n .
19-4
66. Introduction
Line transversals and geometric permutations
More Davenport-Schinzel sequences
Excluded patterns
Extrapolation methods: VC dimension and shatter functions
... a few open problems (come see me for more :) )
20-1
67. 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.
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 ?
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 )?
21-1
68. A few pointers...
Davenport-Schinzel sequences and their geometric applications
Micha Sharir and Pankaj Agarwal, Cambridge Univ. Press
Improved bounds for geometric permutations
Nathan Rubin, Haim Kaplan and Micha Sharir, to appear in SICOMP (FOCS 2010)
Geometric permutations in the plane and in Euclidean spaces of higher dimension
Andrei Asinowski, PhD thesis (2005)
Geometric permutations of disjoint unit spheres
Otfried Cheong, X. G. and Hyeon-Suk Na
Comp. Geom. Theor. and Appl. 30: 253–270 (2005).
-nets and simplex range queries
David Haussler and Emo Welzl, Discrete & Computational Geometry 2:127-151 (1987)
VC-Dimension of Sets of Permutations
Ran Raz, Combinatorica 20: 1-15 (2000)
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)
22-1