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On the Error Parameter in Dispersers

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:

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On the Error Parameter in Dispersers

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  1. Guy Kindler Microsoft Research On the Error Parameter in Dispersers Ronen Gradwohl, Omer Reingold, and Amnon Ta-Shma. Joint work with

  2. Guy Kindler Microsoft Research On the Error Parameter in Dispersers Ronen Gradwohl, Omer Reingold, and Amnon Ta-Shma. Joint work with

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

  4. X (n-bit strings) Entropy (not really…) • Define: The entropy of a set X by H(X)=log2(|X|)

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

  6. Bipartite Ramsey Graphs • A function Ram:{0,1}nx{0,1}n!{0,1} is (k,k) bipartite Ramsey if 8X,Yµ{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

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

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

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

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

  11. Strong Dispersers • D:{0,1}nx{0,1}s!{0,1}m is a (k,) strong disperser if for H(X)>k, • For all but  fraction of r’s, • |Y|>¢2s, !9r2Y s.t. x |X|¸ 2k r D (s-bit string) (s-bits) 0/1

  12. Strong Dispersers • D:{0,1}nx{0,1}s!{0,1}m is a (k,) strong disperser if for H(X)>k, • For all but  fraction of r’s, • |Y|>¢2s, !9r2Y s.t. • D is (k,s-log(1/)) Ramsey! x |X|¸ 2k r D (s-bit string) 0/1

  13. Parameters of Strong Dispersers • D:{0,1}nx{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.

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

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

  16. 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

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

  18. Combinatorial interpretation • We built a bipartite graph G on (V,W), |V|=|W|=2n • Each edge is associated with a list of log5n colors, out of a rainbow of size log10n. • If XµV and YµW have size |X|=|Y|=n20, then E(X,Y) contains a complete rainbow.

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

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