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Property Testing and Communication Complexity

Explore property testing algorithms and communication complexity in randomized settings, including direct sums and lower bounds. Learn about linearity testing, disjointness, and direct products. Discover optimal bounds for various properties. Dive into advanced topics like small decision trees, Lipschitz properties, and coding theory.

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Property Testing and Communication Complexity

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  1. Property Testing and Communication Complexity GrigoryYaroslavtsev http://grigory.us

  2. Property Testing [Goldreich, Goldwasser, Ron, Rubinfeld, Sudan] Randomized algorithm Property tester YES YES Accept with probability Accept with probability Don’t care -far No No Reject with probability Reject with probability -far : fraction has to be changed to become YES

  3. Property Testing [Goldreich, Goldwasser, Ron, Rubinfeld, Sudan] Property = set of YES instances Query complexity of testing • = Adaptive queries • = Non-adaptive (all queries at once) • = Queries in rounds () For error :

  4. Communication Complexity [Yao’79] Shared randomness Bob: Alice: … • = min. communication (error ) • min. -round communication (error )

  5. /2-disjointness -linearity [Blais, Brody,Matulef’11] • -linear function: where • -Disjointness: , , iff. Alice: Bob: 0?

  6. /2-disjointness -linearity [Blais, Brody,Matulef’11] • is -linear • is -linear, ½-far from -linear • Test for -linearity using shared randomness • To evaluate exchange and (2 bits)

  7. -Disjointness • [Razborov, Hastad-Wigderson] [Folklore + Dasgupta, Kumar, Sivakumar; Buhrman’12, Garcia-Soriano, Matsliah, De Wolf’12] where [Saglam, Tardos’13] • [Braverman, Garg, Pankratov, Weinstein’13] • (-Intersection) = [Brody, Chakrabarti, Kondapally, Woodruff, Y.] { = times

  8. Communication Direct Sums “Solving m copies of a communication problem requires m times more communication”: • For arbitrary [… Braverman, Rao 10; Barak Braverman, Chen, Rao 11, ….] • In general, can’t go beyond iff, where

  9. Specialized Communication Direct Sums Information cost Communication complexity • [Bar Yossef, Jayram, Kumar,Sivakumar’01] Disjointness • Stronger direct sum for Equality-type problems (a.k.a. “union bound is optimal”) [Molinaro, Woodruff, Y.’13] • Bounds for , (-Set Intersection) via Information Theory [Brody, Chakrabarty, Kondapally, Woodruff, Y.’13]

  10. Direct Sums in Property Testing [Woodruff, Y.] • Testing linearity: is linear if • Equality: decide whether • is linear • is ¼ -far from linear

  11. Direct Sums in Property Testing [Woodruff, Y.] • = = (matching [Blum, Luby, Rubinfeld]) • Strong Direct Sum for Equality[MWY’13]Strong Direct Sum for Testing Linearity

  12. Property Testing Direct Sums [Goldreich’13] • Direct Sum [Woodruff, Y.]: Solve with probability • Direct -Sum[Goldreich’13]: Solve with probability per instance • Direct -Product[Goldreich’13]: All instances are in instance –far from

  13. [Goldreich ‘13] For all properties : • Direct m-Sum (solve all w.p. 2/3 per instance) • Adaptive: • Non-adaptive: • Direct -Product (-far instance?) • Adaptive: • Non-adaptive:

  14. Reduction from Simultaneous Communication [Woodruff] Referee: • min. simultaneous complexity of • GAF: [Babai, Kimmel, Lokam] • GAF if 0 otherwise • , but S(GAF) = Bob: Alice:

  15. Property testing lower bounds via CC • Monotonicity, Juntas, Low Fourier degree, Small Decision Trees [Blais, Brody, Matulef’11] • Small-width OBDD properties [Brody, Matulef, Wu’11] • Lipschitz property [Jha, Raskhodnikova’11] • Codes [Goldreich’13, Gur, Rothblum’13] • Number of relevant variables [Ron, Tsur’13] All functions are over Boolean hypercube

  16. Functions[Blais, Raskhodnikova, Y.] monotone functions over Previous for monotonicity on the line (): • [Ergun, Kannan, Kumar, Rubinfeld, Viswanathan’00] • [Fischer’04]

  17. Functions[Blais, Raskhodnikova, Y.] • Thm. Any non-adaptive tester for monotonicity of has complexity • Proof. • Reduction from Augmented Index • Basis of Walsh functions

  18. Functions[Blais, Raskhodnikova, Y.] • Augmented Index: S, () • [Miltersen, Nisan, Safra, Wigderson, 98]

  19. Functions[Blais, Raskhodnikova, Y.] Walsh functions: For : , where is the -th bit of …

  20. Functions[Blais, Raskhodnikova, Y.] Step functions: For :

  21. Functions[Blais, Raskhodnikova, Y.] • Augmented Index Monotonicity Testing • is monotone • is ¼ -far from monotone • Thus,

  22. Functions[Blais, Raskhodnikova, Y.] • monotone functions over • -Lipschitz functions over • separately convex functions over • convex functions over Thm. [BRY] For all these properties These bounds are optimal for and [Chakrabarty, Seshadhri, ‘13]

  23. Thank you!

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