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##### Property Testing and Communication Complexity

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**Property Testing and Communication Complexity**GrigoryYaroslavtsev http://grigory.us**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**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 :**Communication Complexity [Yao’79]**Shared randomness Bob: Alice: … • = min. communication (error ) • min. -round communication (error )**/2-disjointness -linearity [Blais, Brody,Matulef’11]**• -linear function: where • -Disjointness: , , iff. Alice: Bob: 0?**/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)**-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**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**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]**Direct Sums in Property Testing [Woodruff, Y.]**• Testing linearity: is linear if • Equality: decide whether • is linear • is ¼ -far from linear**Direct Sums in Property Testing [Woodruff, Y.]**• = = (matching [Blum, Luby, Rubinfeld]) • Strong Direct Sum for Equality[MWY’13]Strong Direct Sum for Testing Linearity**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**[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:**Reduction from Simultaneous Communication [Woodruff]**Referee: • min. simultaneous complexity of • GAF: [Babai, Kimmel, Lokam] • GAF if 0 otherwise • , but S(GAF) = Bob: Alice:**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**Functions[Blais, Raskhodnikova, Y.]**monotone functions over Previous for monotonicity on the line (): • [Ergun, Kannan, Kumar, Rubinfeld, Viswanathan’00] • [Fischer’04]**Functions[Blais, Raskhodnikova, Y.]**• Thm. Any non-adaptive tester for monotonicity of has complexity • Proof. • Reduction from Augmented Index • Basis of Walsh functions**Functions[Blais, Raskhodnikova, Y.]**• Augmented Index: S, () • [Miltersen, Nisan, Safra, Wigderson, 98]**Functions[Blais, Raskhodnikova, Y.]**Walsh functions: For : , where is the -th bit of …**Functions[Blais, Raskhodnikova, Y.]**Step functions: For :**Functions[Blais, Raskhodnikova, Y.]**• Augmented Index Monotonicity Testing • is monotone • is ¼ -far from monotone • Thus,**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]