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Property Testing and Communication Complexity. Grigory Yaroslavtsev 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.

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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

• = 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

[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)

• Direct -Product (-far instance?)

### 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]

• Codes [Goldreich’13, Gur, Rothblum’13]

• Number of relevant variables [Ron, Tsur’13]

All functions are over Boolean hypercube

monotone functions over

Previous for monotonicity on the line ():

• [Ergun, Kannan, Kumar, Rubinfeld, Viswanathan’00]

• [Fischer’04]

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

• Proof.

• Reduction from Augmented Index

• Basis of Walsh functions

• Augmented Index: S, ()

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

Walsh functions: For :

,

where is the -th bit of

Step functions: For :

• Augmented Index Monotonicity Testing

• is monotone

• is ¼ -far from monotone

• Thus,

• 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]