property testing and communication complexity grigory yaroslavtsev

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Property Testing and Communication Complexity Grigory Yaroslavtsev http://grigory.us

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Page 1: Property Testing and Communication Complexity Grigory Yaroslavtsev

Property Testing and Communication Complexity

Grigory Yaroslavtsevhttp://grigory.us

Page 2: Property Testing and Communication Complexity Grigory Yaroslavtsev

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

Page 3: Property Testing and Communication Complexity Grigory Yaroslavtsev

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 :

Page 4: Property Testing and Communication Complexity Grigory Yaroslavtsev

Communication Complexity [Yao’79]

Alice: Bob:

𝒇 (𝒙 ,𝒚 )=?

Shared randomness

𝒇 (𝒙 ,𝒚 )• = min. communication (error ) • min. -round communication (error )

Page 5: Property Testing and Communication Complexity Grigory Yaroslavtsev

• -linear function: where • -Disjointness: ,

, iff .

Alice: Bob:

0?

/2-disjointness -linearity [Blais, Brody,Matulef’11]

Page 6: Property Testing and Communication Complexity Grigory Yaroslavtsev

/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)

Page 7: Property Testing and Communication Complexity Grigory Yaroslavtsev

-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

=

Page 8: Property Testing and Communication Complexity Grigory Yaroslavtsev

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

Page 9: Property Testing and Communication Complexity Grigory Yaroslavtsev

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

Page 10: Property Testing and Communication Complexity Grigory Yaroslavtsev

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

Page 11: Property Testing and Communication Complexity Grigory Yaroslavtsev

• = = (matching [Blum, Luby, Rubinfeld])

• Strong Direct Sum for Equality [MWY’13]Strong Direct Sum for Testing Linearity

Direct Sums in Property Testing [Woodruff, Y.]

Page 12: Property Testing and Communication Complexity Grigory Yaroslavtsev

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

Page 13: Property Testing and Communication Complexity Grigory Yaroslavtsev

[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:

Page 14: Property Testing and Communication Complexity Grigory Yaroslavtsev

Reduction from Simultaneous Communication [Woodruff]

Alice: Bob: Referee:

• min. simultaneous complexity of • • GAF: [Babai, Kimmel, Lokam]GAF if 0 otherwise, but S(GAF) =

Page 15: Property Testing and Communication Complexity Grigory Yaroslavtsev

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

Page 16: Property Testing and Communication Complexity Grigory Yaroslavtsev

Functions [Blais, Raskhodnikova, Y.]

monotone functions over

Previous for monotonicity on the line ():• [Ergun, Kannan, Kumar, Rubinfeld, Viswanathan’00]• [Fischer’04]

Page 17: Property Testing and Communication Complexity Grigory Yaroslavtsev

Functions [Blais, Raskhodnikova, Y.]

• Thm. Any non-adaptive tester for monotonicity of has complexity

• Proof. – Reduction from Augmented Index– Basis of Walsh functions

Page 18: Property Testing and Communication Complexity Grigory Yaroslavtsev

Functions [Blais, Raskhodnikova, Y.]

• Augmented Index: S, ()

• [Miltersen, Nisan, Safra, Wigderson, 98]

𝑺⊆ [𝒎 ] 𝒊∈ [𝒎 ] ,𝑺∩ [𝒊−1]

Page 19: Property Testing and Communication Complexity Grigory Yaroslavtsev

Walsh functions: For :,

where is the -th bit of

Functions [Blais, Raskhodnikova, Y.]

𝒙𝒘 {𝟒 }=¿

𝒙𝒘 {𝟏}=¿

𝒙𝒘 {𝟐}=¿

Page 20: Property Testing and Communication Complexity Grigory Yaroslavtsev

Step functions: For :

Functions [Blais, Raskhodnikova, Y.]

𝒙𝑠𝑡𝑒𝑝2=¿

Page 21: Property Testing and Communication Complexity Grigory Yaroslavtsev

• Augmented Index Monotonicity Testing

• is monotone• is ¼ -far from monotone• Thus,

𝑺⊆ [𝒎 ]

𝒊∈ [𝒎 ] ,𝑺∩ [𝒊−1]

Functions [Blais, Raskhodnikova, Y.]

Page 22: Property Testing and Communication Complexity Grigory Yaroslavtsev

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]

Page 23: Property Testing and Communication Complexity Grigory Yaroslavtsev

Thank you!