on the error parameter in dispersers

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)



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



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,


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


D

x|X|¸ 2k

(s-bit string)

(s-bits)

r

0/1



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

on the error parameter in dispersers

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