combinatorial macbeath regions for semi-algebraic set …combinatorial macbeath regions for...
TRANSCRIPT
![Page 1: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/1.jpg)
Combinatorial Macbeath Regions forSemi-Algebraic Set Systems
Arijit Ghosh1
1Indian Statistical InstituteKolkata, India
College de France 2017
Ghosh Combinatorial Macbeath Regions
![Page 2: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/2.jpg)
Coauthors
Kunal Dutta (DataShape, INRIA)
Bruno Jartoux (Universite Paris-Est, LIGM)
Nabil H. Mustafa (Universite Paris-Est, LIGM)
Ghosh Combinatorial Macbeath Regions
![Page 3: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/3.jpg)
Geometric set systems
Point-disk incidences: an example of geometric set system.
Ghosh Combinatorial Macbeath Regions
![Page 4: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/4.jpg)
Geometric set systems
Typical applications: range searching, point set queries.
Map
Ghosh Combinatorial Macbeath Regions
![Page 5: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/5.jpg)
Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε contains one of them.
K
h
≥ ε
Ghosh Combinatorial Macbeath Regions
![Page 6: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/6.jpg)
Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε contains one of them.
K
h
≥ ε
Ghosh Combinatorial Macbeath Regions
![Page 7: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/7.jpg)
Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε contains one of them.
K
h
≥ ε
Ghosh Combinatorial Macbeath Regions
![Page 8: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/8.jpg)
Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε contains one of them.
K
h
≥ ε
Ghosh Combinatorial Macbeath Regions
![Page 9: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/9.jpg)
Mnets, or combinatorial Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε includes one of them.
Mnets – for halfplanes
For a set K of n points and ε > 0, an Mnet is a collection ofsubsets of Θ(εn) points such that any halfplane h with|h ∩ K | ≥ εn includes one of them.
Goal: discrete analogue of Macbeath’s tool.
Ghosh Combinatorial Macbeath Regions
![Page 10: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/10.jpg)
Mnets, or combinatorial Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε includes one of them.
Mnets – for halfplanes
For a set K of n points and ε > 0, an Mnet is a collection ofsubsets of Θ(εn) points such that any halfplane h with|h ∩ K | ≥ εn includes one of them.
Goal: discrete analogue of Macbeath’s tool.
Ghosh Combinatorial Macbeath Regions
![Page 11: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/11.jpg)
Mnets, or combinatorial Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε includes one of them.
Mnets – for disks
For a set K of n points and ε > 0, an Mnet is a collection ofsubsets of Θ(εn) points such that any disk h with |h ∩ K | ≥ εnincludes one of them.
Goal: discrete analogue of Macbeath’s tool.
Ghosh Combinatorial Macbeath Regions
![Page 12: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/12.jpg)
Mnets, or combinatorial Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε includes one of them.
Mnets – for [shapes]
For a set K of n points and ε > 0, an Mnet is a collection ofsubsets of Θ(εn) points such that any [shape] h with |h ∩ K | ≥ εnincludes one of them.
Goal: discrete analogue of Macbeath’s tool.
Ghosh Combinatorial Macbeath Regions
![Page 13: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/13.jpg)
Mnets, or combinatorial Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε includes one of them.
Mnets – for [shapes]
For a set K of n points and ε > 0, an Mnet is a collection ofsubsets of Θ(εn) points such that any [shape] h with |h ∩ K | ≥ εnincludes one of them.
Goal: discrete analogue of Macbeath’s tool.
Ghosh Combinatorial Macbeath Regions
![Page 14: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/14.jpg)
Bounds on Mnets
Question
What is the minimum size of an Mnet?
Theorem (D.–G.–J.–M. ’17)
Semialgebraic set systems with VC-dim. d <∞ and shallowcell complexity ϕ have an ε-Mnet of size
O
(d
ε· ϕ(d
ε, d
)).
X Disks
X Rectangles
X Lines
X ‘Fat’ objects
× General convex sets
Theorem (D.–G.–J.–M. ’17)
This is tight for hyperplanes.
Ghosh Combinatorial Macbeath Regions
![Page 15: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/15.jpg)
Bounds on Mnets
Question
What is the minimum size of an Mnet?
Theorem (D.–G.–J.–M. ’17)
Semialgebraic set systems with VC-dim. d <∞ and shallowcell complexity ϕ have an ε-Mnet of size
O
(d
ε· ϕ(d
ε, d
)).
X Disks
X Rectangles
X Lines
X ‘Fat’ objects
× General convex sets
Theorem (D.–G.–J.–M. ’17)
This is tight for hyperplanes.
Ghosh Combinatorial Macbeath Regions
![Page 16: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/16.jpg)
Bounds on Mnets
Question
What is the minimum size of an Mnet?
Theorem (D.–G.–J.–M. ’17)
Semialgebraic set systems with VC-dim. d <∞ and shallowcell complexity ϕ have an ε-Mnet of size
O
(d
ε· ϕ(d
ε, d
)).
X Disks
X Rectangles
X Lines
X ‘Fat’ objects
× General convex sets
Theorem (D.–G.–J.–M. ’17)
This is tight for hyperplanes.
Ghosh Combinatorial Macbeath Regions
![Page 17: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/17.jpg)
Bounds on Mnets
Question
What is the minimum size of an Mnet?
Theorem (D.–G.–J.–M. ’17)
Semialgebraic set systems with VC-dim. d <∞ and shallowcell complexity ϕ have an ε-Mnet of size
O
(d
ε· ϕ(d
ε, d
)).
X Disks
X Rectangles
X Lines
X ‘Fat’ objects
× General convex sets
Theorem (D.–G.–J.–M. ’17)
This is tight for hyperplanes.
Ghosh Combinatorial Macbeath Regions
![Page 18: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/18.jpg)
Abstract set systems
X := arbitrary n-point setΣ := collection of subsets of X , i.e., Σ ⊆ 2X
The pair (X ,Σ) is called a set system
Set systems (X ,Σ) are also referred to as hypergraphs,range spaces
Ghosh Combinatorial Macbeath Regions
![Page 19: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/19.jpg)
Abstract set systems
Projection:For Y ⊆ X ,
ΣY := {S ∩ Y : S ∈ Σ}
andΣkY := {S ∩ Y : S ∈ Σ and |S ∩ Y | ≤ k}
Ghosh Combinatorial Macbeath Regions
![Page 20: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/20.jpg)
Primal shatter dimension and shallow cell complexity
Primal Shatter function Given (X ,Σ), primal shatter function isdefined as
πΣ(m) := maxY⊆X , |Y |=m
|ΣY |
VC dimension: d0 := max {m |πΣ(m) = 2m}Primal Shatter dimension: A set system has primal shatterdimension d if for all m ≤ n, πΣ(m) ≤ O(md).
Shallow cell complexity ϕ(·, ·) If ∀Y ⊆ X ,∣∣∣ΣkY
∣∣∣ ≤ |Y | × ϕ(|Y |, k).
Ghosh Combinatorial Macbeath Regions
![Page 21: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/21.jpg)
Primal shatter dimension and shallow cell complexity
Primal Shatter function Given (X ,Σ), primal shatter function isdefined as
πΣ(m) := maxY⊆X , |Y |=m
|ΣY |
VC dimension: d0 := max {m |πΣ(m) = 2m}
Primal Shatter dimension: A set system has primal shatterdimension d if for all m ≤ n, πΣ(m) ≤ O(md).
Shallow cell complexity ϕ(·, ·) If ∀Y ⊆ X ,∣∣∣ΣkY
∣∣∣ ≤ |Y | × ϕ(|Y |, k).
Ghosh Combinatorial Macbeath Regions
![Page 22: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/22.jpg)
Primal shatter dimension and shallow cell complexity
Primal Shatter function Given (X ,Σ), primal shatter function isdefined as
πΣ(m) := maxY⊆X , |Y |=m
|ΣY |
VC dimension: d0 := max {m |πΣ(m) = 2m}Primal Shatter dimension: A set system has primal shatterdimension d if for all m ≤ n, πΣ(m) ≤ O(md).
Shallow cell complexity ϕ(·, ·) If ∀Y ⊆ X ,∣∣∣ΣkY
∣∣∣ ≤ |Y | × ϕ(|Y |, k).
Ghosh Combinatorial Macbeath Regions
![Page 23: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/23.jpg)
Primal shatter dimension and shallow cell complexity
Primal Shatter function Given (X ,Σ), primal shatter function isdefined as
πΣ(m) := maxY⊆X , |Y |=m
|ΣY |
VC dimension: d0 := max {m |πΣ(m) = 2m}Primal Shatter dimension: A set system has primal shatterdimension d if for all m ≤ n, πΣ(m) ≤ O(md).
Shallow cell complexity ϕ(·, ·) If ∀Y ⊆ X ,∣∣∣ΣkY
∣∣∣ ≤ |Y | × ϕ(|Y |, k).
Ghosh Combinatorial Macbeath Regions
![Page 24: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/24.jpg)
Shallow cell complexity of some geometric set systems
1. Points and half-spaces O(|Y |bd/2c−1kdd/2e)or orthants in Rd
2. Points and balls O(|Y |b(d+1)/2c−1kd(d+1)/2e)in Rd
3. (d − 1)-variate polynomial |Y |d−2+εk1−ε
function of constant degreeand points in Rd
Ghosh Combinatorial Macbeath Regions
![Page 25: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/25.jpg)
δ-packing number
Parameter: Let δ > 0 be a integer parameter
δ-separated: A set system (X ,Σ) is δ-separated if for all S1, S2
in Σ, if the size of the symmetric difference (Hamming distance)S1∆S2 is greater than δ, i.e. |S1∆S2| > δ.
δ-packing number: The cardinality of the largest δ-separatedsubcollection of Σ is called the δ-packing number of Σ.
Ghosh Combinatorial Macbeath Regions
![Page 26: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/26.jpg)
δ-packing number
Parameter: Let δ > 0 be a integer parameter
δ-separated: A set system (X ,Σ) is δ-separated if for all S1, S2
in Σ, if the size of the symmetric difference (Hamming distance)S1∆S2 is greater than δ, i.e. |S1∆S2| > δ.
δ-packing number: The cardinality of the largest δ-separatedsubcollection of Σ is called the δ-packing number of Σ.
Ghosh Combinatorial Macbeath Regions
![Page 27: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/27.jpg)
Shallow packing result
Theorem (Dutta-Ezra-G.’15 and Mustafa’16)
Let (X ,Σ) be a set system with VC-dim d and shallow cellcomplexity ϕ(·) on a n-point set X . Let δ ≥ 1 and k ≤ n be twointeger parameters such that:
1. ∀S ∈∑
, |S | ≤ k , and
2.∑
is δ-packed.
Then
|Σ| ≤ dn
δϕ
(d n
δ,d k
δ
)
We can show that the above bound is tight.
Ghosh Combinatorial Macbeath Regions
![Page 28: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/28.jpg)
Shallow packing result
Theorem (Dutta-Ezra-G.’15 and Mustafa’16)
Let (X ,Σ) be a set system with VC-dim d and shallow cellcomplexity ϕ(·) on a n-point set X . Let δ ≥ 1 and k ≤ n be twointeger parameters such that:
1. ∀S ∈∑
, |S | ≤ k , and
2.∑
is δ-packed.
Then
|Σ| ≤ dn
δϕ
(d n
δ,d k
δ
)We can show that the above bound is tight.
Ghosh Combinatorial Macbeath Regions
![Page 29: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/29.jpg)
Polynomial partitioning lemma (incorrect version)
Theorem (Matousek-Patakova 15)
Given a set of n-points P ⊂ Rm. Then there exists a polynomialf (X1, . . . , Xm) of degree at most r1/m such that
1. Rm \ Z (f ) has at most r maximally connected components,i.e, Rm \ Z (f ) = ω1 t · · · t ωt where ωi are maximallyconnected components and t ≤ r .
2. |ωi ∩ P| ≤ nr and |Z (f ) ∩ P| = 0
3. Any semialgebraic set O crosses at most r1− 1m connected
components of Rm \ Z (f ).
(Def. of “Crossing”) We say a set A crosses a set B ifA ∩ B 6∈ {∅,B}.
Ghosh Combinatorial Macbeath Regions
![Page 30: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/30.jpg)
Proof of Mnets bound(Sketching the) Upper bound construction
1 Build a maximal packing.
7 / 11
Assume all the large sets are of size exactly εn
Build a maximal packing with k = εn and δ = εn/2. Numberof sets in the packing is ≤ d
εϕ(dε , d)
(via Shallow packinglemma)
Any set of size εn in the set system is either in the packing orhas a large intersection with a set in the packing (size ofintersection ≥ εn/2)
Ghosh Combinatorial Macbeath Regions
![Page 31: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/31.jpg)
Proof of Mnets bound(Sketching the) Upper bound construction
1 Build a maximal packing.
7 / 11
Assume all the large sets are of size exactly εn
Build a maximal packing with k = εn and δ = εn/2. Numberof sets in the packing is ≤ d
εϕ(dε , d)
(via Shallow packinglemma)
Any set of size εn in the set system is either in the packing orhas a large intersection with a set in the packing (size ofintersection ≥ εn/2)
Ghosh Combinatorial Macbeath Regions
![Page 32: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/32.jpg)
Proof of Mnets bound(Sketching the) Upper bound construction
1 Build a maximal packing.
7 / 11
Assume all the large sets are of size exactly εn
Build a maximal packing with k = εn and δ = εn/2. Numberof sets in the packing is ≤ d
εϕ(dε , d)
(via Shallow packinglemma)
Any set of size εn in the set system is either in the packing orhas a large intersection with a set in the packing (size ofintersection ≥ εn/2)
Ghosh Combinatorial Macbeath Regions
![Page 33: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/33.jpg)
Proof of Mnets bound(Sketching the) Upper bound construction
1 Build a maximal packing.
7 / 11
Assume all the large sets are of size exactly εn
Build a maximal packing with k = εn and δ = εn/2. Numberof sets in the packing is ≤ d
εϕ(dε , d)
(via Shallow packinglemma)
Any set of size εn in the set system is either in the packing orhas a large intersection with a set in the packing (size ofintersection ≥ εn/2)
Ghosh Combinatorial Macbeath Regions
![Page 34: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/34.jpg)
Proof: How to get a candidate Mnet set?
Partition each set in the maximal packing with ar -partitioning polynomial (satisfying those magical properties),where r is a large constant to be fixed later.
Include in your Mnets set all the subsets ωi ∩ P where|ωi ∩ P| ≥ εn
r2 .
Do this for all the sets in the maximal packing and you willget an Mnet satisfying the bound in the theorem.
Why is this a valid Mnet?
Ghosh Combinatorial Macbeath Regions
![Page 35: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/35.jpg)
Proof: How to get a candidate Mnet set?
Partition each set in the maximal packing with ar -partitioning polynomial (satisfying those magical properties),where r is a large constant to be fixed later.
Include in your Mnets set all the subsets ωi ∩ P where|ωi ∩ P| ≥ εn
r2 .
Do this for all the sets in the maximal packing and you willget an Mnet satisfying the bound in the theorem.
Why is this a valid Mnet?
Ghosh Combinatorial Macbeath Regions
![Page 36: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/36.jpg)
Proof: How to get a candidate Mnet set?
Partition each set in the maximal packing with ar -partitioning polynomial (satisfying those magical properties),where r is a large constant to be fixed later.
Include in your Mnets set all the subsets ωi ∩ P where|ωi ∩ P| ≥ εn
r2 .
Do this for all the sets in the maximal packing and you willget an Mnet satisfying the bound in the theorem.
Why is this a valid Mnet?
Ghosh Combinatorial Macbeath Regions
![Page 37: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/37.jpg)
Proof: How to get a candidate Mnet set?
Partition each set in the maximal packing with ar -partitioning polynomial (satisfying those magical properties),where r is a large constant to be fixed later.
Include in your Mnets set all the subsets ωi ∩ P where|ωi ∩ P| ≥ εn
r2 .
Do this for all the sets in the maximal packing and you willget an Mnet satisfying the bound in the theorem.
Why is this a valid Mnet?
Ghosh Combinatorial Macbeath Regions
![Page 38: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/38.jpg)
Proof: Why the construction is a valid Mnet?
Let S ∈ Σ with |S | = εn.
Since the set system is a semialgebraic set system, there existsa semialg. object O such that O ∩ P = S .
Since we have a maxi. packing, ∃Si from the packing suchthat |S ∩ Si | ≥ εn/2.
Maximum contribution to |S ∩ Si | from connected regionscrossed by O and the small set is at most
εn
r× r1− 1
m +εn
r2× r = εn
(1
r1/m+
1
r
)� εn
2
Ghosh Combinatorial Macbeath Regions
![Page 39: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/39.jpg)
Proof: Why the construction is a valid Mnet?
Let S ∈ Σ with |S | = εn.
Since the set system is a semialgebraic set system, there existsa semialg. object O such that O ∩ P = S .
Since we have a maxi. packing, ∃Si from the packing suchthat |S ∩ Si | ≥ εn/2.
Maximum contribution to |S ∩ Si | from connected regionscrossed by O and the small set is at most
εn
r× r1− 1
m +εn
r2× r = εn
(1
r1/m+
1
r
)� εn
2
Ghosh Combinatorial Macbeath Regions
![Page 40: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/40.jpg)
Proof: Why the construction is a valid Mnet?
Let S ∈ Σ with |S | = εn.
Since the set system is a semialgebraic set system, there existsa semialg. object O such that O ∩ P = S .
Since we have a maxi. packing, ∃Si from the packing suchthat |S ∩ Si | ≥ εn/2.
Maximum contribution to |S ∩ Si | from connected regionscrossed by O and the small set is at most
εn
r× r1− 1
m +εn
r2× r = εn
(1
r1/m+
1
r
)� εn
2
Ghosh Combinatorial Macbeath Regions
![Page 41: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/41.jpg)
Proof: Why the construction is a valid Mnet?
Let S ∈ Σ with |S | = εn.
Since the set system is a semialgebraic set system, there existsa semialg. object O such that O ∩ P = S .
Since we have a maxi. packing, ∃Si from the packing suchthat |S ∩ Si | ≥ εn/2.
Maximum contribution to |S ∩ Si | from connected regionscrossed by O and the small set is at most
εn
r× r1− 1
m +εn
r2× r = εn
(1
r1/m+
1
r
)� εn
2
Ghosh Combinatorial Macbeath Regions
![Page 42: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/42.jpg)
From Mnets to ε-nets
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
This gives ε-nets of size dε logϕ
(dε , d)for semialgebraic set
systems.
Yields best known bounds on ε-nets for geometric set systemswith bounded VC-dim.
Table: Upper bounds on Mnets and ε-nets
Mnet ε-net
Disks ε−1 ε−1
Rectangles 1ε log 1
ε1ε log log 1
ε
Halfspaces (Rd) O(ε−bd/2c) d
ε log 1ε
Ghosh Combinatorial Macbeath Regions
![Page 43: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/43.jpg)
From Mnets to ε-nets
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
This gives ε-nets of size dε logϕ
(dε , d)for semialgebraic set
systems.
Yields best known bounds on ε-nets for geometric set systemswith bounded VC-dim.
Table: Upper bounds on Mnets and ε-nets
Mnet ε-net
Disks ε−1 ε−1
Rectangles 1ε log 1
ε1ε log log 1
ε
Halfspaces (Rd) O(ε−bd/2c) d
ε log 1ε
Ghosh Combinatorial Macbeath Regions
![Page 44: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/44.jpg)
From Mnets to ε-nets
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
This gives ε-nets of size dε logϕ
(dε , d)for semialgebraic set
systems.
Yields best known bounds on ε-nets for geometric set systemswith bounded VC-dim.
Table: Upper bounds on Mnets and ε-nets
Mnet ε-net
Disks ε−1 ε−1
Rectangles 1ε log 1
ε1ε log log 1
ε
Halfspaces (Rd) O(ε−bd/2c) d
ε log 1ε
Ghosh Combinatorial Macbeath Regions
![Page 45: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/45.jpg)
Probabilistic proof
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
Proof.
1 M is such an Mnet. Let p = 1τεn log(εM).
2 Pick every point into a sample S with probability p.
3 ∀m ∈M, Pr[S ∩m = ∅] = (1− p)|m| ≤ e−p|m| ≤ 1εM
4 In expectation, |S |+ |m ∈M : S ∩m = ∅| ≤ np + 1ε .
5 so there is an ε-net of size ≤ np + 1ε (why?).
Ghosh Combinatorial Macbeath Regions
![Page 46: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/46.jpg)
Probabilistic proof
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
Proof.
1 M is such an Mnet. Let p = 1τεn log(εM).
2 Pick every point into a sample S with probability p.
3 ∀m ∈M, Pr[S ∩m = ∅] = (1− p)|m| ≤ e−p|m| ≤ 1εM
4 In expectation, |S |+ |m ∈M : S ∩m = ∅| ≤ np + 1ε .
5 so there is an ε-net of size ≤ np + 1ε (why?).
Ghosh Combinatorial Macbeath Regions
![Page 47: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/47.jpg)
Probabilistic proof
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
Proof.
1 M is such an Mnet. Let p = 1τεn log(εM).
2 Pick every point into a sample S with probability p.
3 ∀m ∈M, Pr[S ∩m = ∅] = (1− p)|m| ≤ e−p|m| ≤ 1εM
4 In expectation, |S |+ |m ∈M : S ∩m = ∅| ≤ np + 1ε .
5 so there is an ε-net of size ≤ np + 1ε (why?).
Ghosh Combinatorial Macbeath Regions
![Page 48: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/48.jpg)
Probabilistic proof
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
Proof.
1 M is such an Mnet. Let p = 1τεn log(εM).
2 Pick every point into a sample S with probability p.
3 ∀m ∈M, Pr[S ∩m = ∅] = (1− p)|m| ≤ e−p|m| ≤ 1εM
4 In expectation, |S |+ |m ∈M : S ∩m = ∅| ≤ np + 1ε .
5 so there is an ε-net of size ≤ np + 1ε (why?).
Ghosh Combinatorial Macbeath Regions
![Page 49: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/49.jpg)
Probabilistic proof
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
Proof.
1 M is such an Mnet. Let p = 1τεn log(εM).
2 Pick every point into a sample S with probability p.
3 ∀m ∈M, Pr[S ∩m = ∅] = (1− p)|m| ≤ e−p|m| ≤ 1εM
4 In expectation, |S |+ |m ∈M : S ∩m = ∅| ≤ np + 1ε .
5 so there is an ε-net of size ≤ np + 1ε (why?).
Ghosh Combinatorial Macbeath Regions
![Page 50: Combinatorial Macbeath Regions for Semi-Algebraic Set …Combinatorial Macbeath Regions for Semi-Algebraic Set Systems Arijit Ghosh1 1Indian Statistical Institute Kolkata, India College](https://reader033.vdocuments.net/reader033/viewer/2022042021/5e78a56996fe153ef75f5131/html5/thumbnails/50.jpg)
Conclusion
Ideally we want a combinatorial proof of the Mnets bound forset systems.
Improve the current lower bound.
Find more applications/connections of Mnets in combinatorialgeometry.
Ghosh Combinatorial Macbeath Regions