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Property Testing and Communication Complexity
Grigory Yaroslavtsevhttp://grigory.us
Property Testing [Goldreich, Goldwasser, Ron, Rubinfeld, Sudan]
No
YES
Randomized algorithm
Accept with probability
Reject with probability
⇒
⇒⇒
YES
No
Property tester
-far
Accept with probability
Reject with probability
⇒
⇒Don’t care
-far : fraction has to be changed to become YES
Property = set of YES instances
Query complexity of testing • = Adaptive queries• = Non-adaptive (all queries at once)• = Queries in rounds ()
Property Testing [Goldreich, Goldwasser, Ron, Rubinfeld, Sudan]
For error :
Communication Complexity [Yao’79]
Alice: Bob:
𝒇 (𝒙 ,𝒚 )=?
Shared randomness
…
𝒇 (𝒙 ,𝒚 )• = min. communication (error ) • min. -round communication (error )
• -linear function: where • -Disjointness: ,
, iff .
Alice: Bob:
0?
/2-disjointness -linearity [Blais, Brody,Matulef’11]
/2-disjointness -linearity [Blais, Brody,Matulef’11]
• is -linear• is -linear, ½-far from -linear
𝑺⊆ [𝒏 ] ,|𝑺|=𝒌 /𝟐 𝐓⊆ [𝒏 ] ,|𝑻|=𝒌/𝟐𝝌𝑺=⊕𝑖 ∈𝑺 𝑥𝑖 𝝌𝑻=⊕𝑖 ∈𝑻 𝑥 𝑖
𝝌=𝜒 𝑺⊕ 𝜒𝑻
• 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
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]
Specialized Communication Direct Sums
Direct Sums in Property Testing [Woodruff, Y.]
• Testing linearity: is linear if • Equality: decide whether
• is linear• is ¼ -far from linear
𝑺⊆ [𝒏 ] 𝐓⊆ [𝒏 ]𝝌𝑺=⊕𝑖 ∈𝑺(𝑥2 𝑖−1∧𝑥2 𝑖)𝝌𝑻=⊕𝑖 ∈𝑻 (𝑥2 𝑖−1∧ 𝑥2 𝑖)
𝝌=𝜒 𝑺⊕ 𝜒𝑻
• = = (matching [Blum, Luby, Rubinfeld])
• Strong Direct Sum for Equality [MWY’13]Strong Direct Sum for Testing Linearity
Direct Sums in Property Testing [Woodruff, Y.]
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]
Alice: Bob: Referee:
• min. simultaneous complexity of • • GAF: [Babai, Kimmel, Lokam]GAF if 0 otherwise, but S(GAF) =
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]
𝑺⊆ [𝒎 ] 𝒊∈ [𝒎 ] ,𝑺∩ [𝒊−1]
Walsh functions: For :,
where is the -th bit of
Functions [Blais, Raskhodnikova, Y.]
𝒙𝒘 {𝟒 }=¿
𝒙𝒘 {𝟏}=¿
𝒙𝒘 {𝟐}=¿
…
Step functions: For :
Functions [Blais, Raskhodnikova, Y.]
𝒙𝑠𝑡𝑒𝑝2=¿
• Augmented Index Monotonicity Testing
• is monotone• is ¼ -far from monotone• Thus,
𝑺⊆ [𝒎 ]
𝒊∈ [𝒎 ] ,𝑺∩ [𝒊−1]
Functions [Blais, Raskhodnikova, Y.]
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]
Thank you!