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Guy Kindler. Microsoft Research. On the Error Parameter in Dispersers. Ronen Gradwohl, Omer Reingold, and Amnon Ta-Shma. Joint work with. Guy Kindler. Microsoft Research. On the Error Parameter in Dispersers. Ronen Gradwohl, Omer Reingold, and Amnon Ta-Shma. Joint work with. this talk: - PowerPoint PPT Presentation
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On the Error Parameter in On the Error Parameter in
DispersersDispersersOn the Error Parameter in On the Error Parameter in
DispersersDispersers
Ronen Gradwohl, Omer Ronen Gradwohl, Omer
Reingold, and Amnon Ta-Shma.Reingold, and Amnon Ta-Shma.
Joint work with
Guy KindlerGuy KindlerMicrosoft ResearchMicrosoft Research
On the Error Parameter in On the Error Parameter in
DispersersDispersersOn the Error Parameter in On the Error Parameter in
DispersersDispersers
Ronen Gradwohl, Omer Ronen Gradwohl, Omer
Reingold, and Amnon Ta-Shma.Reingold, and Amnon Ta-Shma.
Joint work with
Guy KindlerGuy KindlerMicrosoft ResearchMicrosoft Research
this talk:this talk:
Goal: better explicit Goal: better explicit bipartite Ramseybipartite Ramsey constructions constructions
We have: some seeded dispersers and extractors.We have: some seeded dispersers and extractors.
Observe: Observe: bipartite Ramseybipartite Ramsey µµ strong seeded strong seeded
dispersers.dispersers.
Draw a path from where we are to where we want to Draw a path from where we are to where we want to
go.go.
Make some steps on that path.Make some steps on that path.
Entropy (not really…)Entropy (not really…)
Define:Define: The entropy of a set The entropy of a set XX by by H(X)=logH(X)=log22((||XX||))
X
(n-bit strings)
Ram
0/1
Bipartite Ramsey GraphsBipartite Ramsey Graphs
A function A function Ram:Ram:{{0,10,1}}nnxx{{0,10,1}}nn!!{{0,10,1}} is is ((kk,,kk)) bipartite bipartite
Ramsey if Ramsey if 88 X,YX,Yµµ{{0,10,1}}nn, , H(X),H(Y)H(X),H(Y)>>k, k,
Ram(X,Y)=Ram(X,Y)={{0,10,1}}..
(n-bit strings)
|X|¸ 2k |Y|¸ 2k x
y
Bipartite Ramsey GraphsBipartite Ramsey Graphs
A function A function Ram:Ram:{{0,10,1}}nnxx{{0,10,1}}nn!!{{0,10,1}} is is ((kk,,kk)) bipartite bipartite
Ramsey if Ramsey if 88 X,YX,Yµµ{{0,10,1}}nn, , H(X),H(Y)H(X),H(Y)>>k, k,
Ram(X,Y)=Ram(X,Y)={{0,10,1}}..
Known to exists for Known to exists for k=O(log n)k=O(log n)..
[GV 88][GV 88] k=n/2k=n/2
[?][?] (O(log n),n/2 )-(O(log n),n/2 )-bipartite Ramsey graph.bipartite Ramsey graph.
[BKSSW 05][BKSSW 05] k=k=nn
[BRSSW 06][BRSSW 06] k=nk=n
Seeded DispersersSeeded Dispersers
DD ::{{0,10,1}}nnxx{{0,10,1}}ss!!{{0,10,1}}mm is a is a (k,(k,)) disperser if for disperser if for
H(X)H(X)>>kk,,
(m-bit strings)
D
x|X|¸ 2k
(s-bit string)
r
If If ss>>mm, take , take D(x,r)=rD(x,r)=r[m][m]
Interesting only when Interesting only when
mm>>ss! !
Strong DispersersStrong Dispersers
DD ::{{0,10,1}}nnxx{{0,10,1}}ss!!{{0,10,1}}mm is a is a (k,(k,)) strong disperser if for strong disperser if for
H(X)H(X)>>kk,,
D
x|X|¸ 2k
(s-bit string)
r
(m-bits) (s-bits)
Strong DispersersStrong Dispersers
DD ::{{0,10,1}}nnxx{{0,10,1}}ss!!{{0,10,1}}mm is a is a (k,(k,)) strong disperser if for strong disperser if for
H(X)H(X)>>kk,,
D
x|X|¸ 2k
(s-bit string)
r
(m-bits) (s-bits)
..
For all but For all but fraction of fraction of rr’s, ’s,
0/1
Strong DispersersStrong Dispersers
DD ::{{0,10,1}}nnxx{{0,10,1}}ss!!{{0,10,1}}mm is a is a (k,(k,)) strong disperser if for strong disperser if for
H(X)H(X)>>kk,,
D
x|X|¸ 2k
(s-bit string)
r
(s-bits)
. .
For all but For all but fraction of fraction of
r’s, r’s,
DD ::{{0,10,1}}nnxx{{0,10,1}}ss!!{{0,10,1}}mm is a is a (k,(k,)) strong disperser if for strong disperser if for
H(X)H(X)>>kk,, For all but For all but fraction of fraction of
rr’s,’s,
||YY|>|>¢¢22ss, , !! 99 rr22YY s.t. s.t.
0/1
Strong DispersersStrong Dispersers
D
x|X|¸ 2k
(s-bit string)
(s-bits)
r
0/1
Strong DispersersStrong Dispersers
DD ::{{0,10,1}}nnxx{{0,10,1}}ss!!{{0,10,1}}mm is a is a (k,(k,)) strong disperser if for strong disperser if for
H(X)H(X)>>kk,,
D
x|X|¸ 2k
(s-bit string) r
For all but For all but fraction of fraction of
rr’s,’s,
||YY|>|>¢¢22ss, , !! 99 rr22YY s.t. s.t.
DD is is (k,s-log(1/(k,s-log(1/))))
Ramsey! Ramsey!
DD ::{{0,10,1}}nnxx{{0,10,1}}ss!!{{0,10,1}} is a is a (k,(k,)) strong disperser. strong disperser.
DD is is (k,s-log(1/(k,s-log(1/)))) Ramsey. Ramsey.
kk¸̧ (log n)(log n)
s=O(log n)+log(1/s=O(log n)+log(1/))
In that case In that case DD is is (k,O(log n))(k,O(log n))-Ramsey!-Ramsey!
For extractors:For extractors: ss¸̧ O(log n)+ O(log n)+ 22¢¢log(1/log(1/))
If If s=log(n)+s=log(n)+22¢¢log(1/log(1/)=n)=n, , DD is is (k,sqrt(n))(k,sqrt(n))-bipartite.-bipartite.
Parameters of Strong DispersersParameters of Strong Dispersers
So can we get So can we get s=O(log(n))s=O(log(n))+log(1/+log(1/))??
no.no.
Can we get Can we get s=ss=snn+log(1/+log(1/)) for some small function for some small function ssnn??
(would imply a (would imply a (k,s(k,snn))-bipartite Ramsey construction)-bipartite Ramsey construction)
no.no.
So what do we get??So what do we get??
An An almost strongalmost strong disperser… disperser…
Almost-strong dispersersAlmost-strong dispersers
A A (k,(k,)) disperser disperser DD is strong in is strong in tt bits if bits if
(s=t+u bits)
(t-bits)
D
x|X|¸ 2k
r
r[t] (m-bits)
Only interesting if Only interesting if m>um>u..
Almost-strong dispersersAlmost-strong dispersers
A A (k,(k,)) disperser disperser DD is strong in t bits if is strong in t bits if
Our construction:Our construction:
t= O(log n+loglog(1/t= O(log n+loglog(1/)) + log(1/)) + log(1/))
u=O(loglog n +loglog(1/u=O(loglog n +loglog(1/)))), ,
m=2m=2¢¢uu
D
x| X| ¸ 2k
r
r[t](m-bits)
D
x| X| ¸ 2k
r
r[t](m-bits)
The constructionThe construction
SE
m=100(log k+loglog(1/))
s’=O(m+log n)
DTUZ
t=10s’+log(1/)
x|X|¸ 2k u=O(log t)
(t,1/2)-disperser(t,1/2)-disperser(t,1/2)-disperser(t,1/2)-disperser
Combinatorial interpretationCombinatorial interpretation
We built a bipartite graph We built a bipartite graph GG on on (V,W)(V,W), , ||VV||==||WW||=2=2nn
Each edge is associated with a list of Each edge is associated with a list of loglog55nn colors, out colors, out
of a rainbow of size of a rainbow of size loglog1010nn..
If If XXµµVV and and YYµµWW have size have size ||XX||==||YY||=n=n2020, then , then E(X,Y)E(X,Y)
contains a complete rainbow. contains a complete rainbow.
Open questionsOpen questions
Show a strong Show a strong (k,(k,)) disperser disperser DD ::{{0,10,1}}nnxx{{0,10,1}}ss!!{{0,10,1}}
with with
Preferrably Preferrably ssnn=log n + O(1)=log n + O(1)..
The End The End