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 Guy Kindler Microsoft Research

this talk: this talk: Goal: better explicit bipartite Ramsey constructions We have: some seeded dispersers and extractors. Observe: bipartite Ramsey µ strong seeded dispersers. Draw a path from where we are to where we want to go. Make some steps on that path.

Entropy (not really…)  Define: The entropy of a set X by H(X)=log 2 ( | X | ) X (n-bit strings)

Ram 0/1 Bipartite Ramsey Graphs  A function Ram: { 0,1 } n x { 0,1 } n ! { 0,1 } is ( k, k ) bipartite Ramsey if 8 X,Y µ { 0,1 } n, H(X),H(Y)>k, Ram(X,Y)= { 0,1 }. (n-bit strings) |X| ¸ 2 k |Y| ¸ 2 k x y

Bipartite Ramsey Graphs  A function Ram: { 0,1 } n x { 0,1 } n ! { 0,1 } is ( k, k ) bipartite Ramsey if 8 X,Y µ { 0,1 } n, H(X),H(Y)>k, Ram(X,Y)= { 0,1 }.  Known to exists for k=O(log n). [GV 88] k=n/2 [?] (O(log n),n/2 )- bipartite Ramsey graph. [BKSSW 05] k=  n [BRSSW 06] k=n 

Seeded Dispersers  D : { 0,1 } n x { 0,1 } s ! { 0,1 } m is a (k,  ) disperser if for H(X) > k, (m-bit strings) D x |X| ¸ 2 k (s-bit string) r  If s > m, take D(x,r)=r [m]  Interesting only when m > s !

Strong Dispersers  D : { 0,1 } n x { 0,1 } s ! { 0,1 } m is a (k,  ) strong disperser if for H(X) > k, D x |X| ¸ 2 k (s-bit string) r (m-bits)(s-bits)

Strong Dispersers  D : { 0,1 } n x { 0,1 } s ! { 0,1 } m is a (k,  ) strong disperser if for H(X) > k, D x |X| ¸ 2 k (s-bit string) r (m-bits)(s-bits) .  For all but  fraction of r ’s,

0/1 Strong Dispersers  D : { 0,1 } n x { 0,1 } s ! { 0,1 } m is a (k,  ) strong disperser if for H(X) > k, D x |X| ¸ 2 k (s-bit string) r (s-bits) .  For all but  fraction of r’s,

 D : { 0,1 } n x { 0,1 } s ! { 0,1 } m is a (k,  ) strong disperser if for H(X) > k,  For all but  fraction of r ’s,  | Y |>  ¢ 2 s, ! 9 r 2 Y s.t. 0/1 Strong Dispersers D x |X| ¸ 2 k (s-bit string) (s-bits) r

0/1 Strong Dispersers  D : { 0,1 } n x { 0,1 } s ! { 0,1 } m is a (k,  ) strong disperser if for H(X) > k, D x |X| ¸ 2 k (s-bit string) r  For all but  fraction of r ’s,  | Y |>  ¢ 2 s, ! 9 r 2 Y s.t.  D is (k,s-log(1/  )) Ramsey!

 D : { 0,1 } n x { 0,1 } s ! { 0,1 } is a (k,  ) strong disperser.  D is (k,s-log(1/  )) Ramsey. k ¸  (log n) s=O(log n)+log(1/  )  In that case D is (k,O(log n)) -Ramsey!  For extractors: s ¸ O(log n)+ 2 ¢ log(1/  )  If s=log(n)+ 2 ¢ log(1/  )=n, D is (k,sqrt(n)) -bipartite. Parameters of Strong Dispersers

So can we get s=O(log(n))+log(1/  ) ?  no.  Can we get s=s n +log(1/  ) for some small function s n ? (would imply a (k,s n ) -bipartite Ramsey construction)  no.  So what do we get??  An almost strong disperser…

Almost-strong dispersers  A (k,  ) disperser D is strong in t bits if (s=t+u bits) (t-bits) D x |X| ¸ 2 k r r [t] (m-bits)  Only interesting if m>u.

Almost-strong dispersers  A (k,  ) disperser D is strong in t bits if  Our construction: t= O(log n+loglog(1/  )) + log(1/  ) u=O(loglog n +loglog(1/  )), m=2 ¢ u

The construction SE m=100(log k+loglog(1/  )) s’=O(m+log n) D TUZ t=10s’+log(1/  ) x |X| ¸ 2 k u=O(log t) (t,1/2)-disperser(t,1/2)-disperser

Combinatorial interpretation  We built a bipartite graph G on (V,W), | V | = | W | =2 n  Each edge is associated with a list of log 5 n colors, out of a rainbow of size log 10 n.  If X µ V and Y µ W have size | X | = | Y | =n 20, then E(X,Y) contains a complete rainbow.

Open questions  Show a strong (k,  ) disperser D : { 0,1 } n x { 0,1 } s ! { 0,1 } with  Preferrably s n =log n + O(1). The End