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Bi- Lipschitz Bijection between the Boolean Cube and the Hamming Ball. Gil Cohen. Joint work with Itai Benjamini and Igor Shinkar. Cube vs. Ball. Majority. Dictator. n. Strings with Hamming weight k. k. 0. The Problem. - PowerPoint PPT Presentation
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Bi-Lipschitz Bijection between the Boolean
Cube and the Hamming BallGil Cohen
Joint work with Itai Benjamini and Igor Shinkar
Cube vs. Ball
{0 ,1 }π π΅π={π₯β {0 ,1 }π+1:|π₯|>π /2}π : β
{π₯β {0 ,1 }π+1:π₯π+1=1}
Dictator
Majority
Strings with Hamming weight
k
n
0
k
The Problem
Open Problem [LovettViola11]. Prove that for any bijection ,
.
π (π₯ )={ π₯β0|π₯|>π/2ππππ (π₯ )β1ππ‘ hπππ€ππ π
Average stretch
The βNaΓ―veβ Upper Bound.
Motivation. Related to proving lower bounds for sampling a natural distribution by circuits.
Main Theorem
Theorem. such that
1)
of
π΅πof
Main Theorem
Theorem. such that
1)
2) is computable in DLOGTIME-uniform
(Majority is -reducible to )
3) is very local
15β€πππ π‘ (π (π₯ ) ,π (π¦ ) )
πππ π‘ (π₯ , π¦ )β€4
β πβ [π ]Prπ₯
[π (π₯ )πβ π₯π ]β€π ( 1βπ )
The BTK Partition [DeBruijnTengbergenKruyswijk51]
1 0 0 1 1 0 0 1
Partition of to symmetric chains.
A symmetric chain is a path .
1 0 0 1 1 0 0 1
^^
1 0 0 1 1 0 0 1
^^^^
1 0 0 1 1 0 0 1
^^^^^^1 0 _ 1 1 0 0 _
0 0
1 0 _ 1 1 0 0 _
1 0 _ 1 1 0 0 _
11
1 0 _ 1 1 0 0 _
10
The BTK Partition
The Metric Properties
xy
πΆ π₯
πΆ π¦ 1)
The Metric Properties
xy
πΆ π₯πΆ π¦ 1)
2)
The Metric Properties
xy
πΆ π₯πΆ π¦ 1)
2)
The Metric Properties β Proof
π₯=π 0 π‘ π¦=π 1π‘
.. ..
0π1π0π+11π 0π1π+10π 1π
π>π
0π1πβπβ11π+10π+11π^ ^
πππππ‘ h=π+πβπ+π0π1π+1βπ 1π0π1π^^
πππππ‘ h=π+πβπ+π+2
π π
ππ : {0 ,1 }πβπ΅π= {π₯β {0 ,1 }π+1
:|π₯|>π /2}
0
1
1
0
1
0
1
π₯
πΆ π₯πΆ π¦
π¦
π (π1 )=π1β1π (π2)=π1β0π (π3 )=π2β1
π (π4 )=π2β0π (π5 )=π3β1π (π6 )=π3β0
π (π7 )=π4β1
π1π2
π3π4
π5π6
π7
The Hamming Ball isbi-Lipschitz Transitive
Defintion. A metric space M is called k bi-Lipschitz transitive if for any there is a bijection such that , and
Example. is 1 bi-Lipschitz transitive.
π (π’)=π’+π₯+π¦
The Hamming Ball isbi-Lipschitz Transitive
Corollary. is 20 bi-Lipschitz transitive.
π (π’)=π (πβ1 (π’)+πβ1 (π₯ )+πβ1 ( π¦ ) ) is convex in
Open Problems
The Constants
Are the 4,5 optimal ?
We know how to improve 4 to 3 at the expense of unbounded inverse.
Does the 20 in the corollary optimal ?
General Balanced Halfspaces
Switching to notation
π : {π₯ :π₯π+1>0 }β {π₯ :π₯1+β―+π₯π+1>0}
Does the result hold for general balanced halfspaces ?ππ : {π₯ :π₯π+1>0 }β {π₯ :π1 π₯1+β―+ππ+1π₯π+1>0 }π1 ,β¦ ,ππ+1β R
One possible approach: generalize BTK chains.Applications to FPTAS for counting solutions to 0-1 knapsack problem [MorrisSinclair04].
Lower Bounds for Average Stretch
Exhibit a density half subset such that any bijection has super constant average stretch.
Conjecture: monotone noise-sensitive functions like Recursive-Majority-of-Three (highly fractal) should work.
We believe a random subset of density half has a constant average stretch.
Even average stretch 2.001 ! (for 2 take XOR).
Goldreichβs Question
Is it true that for any with density, say, half there exist and , both with density half, with bi-Lipschitz bijection between them ?
Thank youfor your attention!
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