Property testing and communication complexity
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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

Property Testing and Communication Complexity

GrigoryYaroslavtsev

http://grigory.us


Property testing goldreich goldwasser ron rubinfeld sudan

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 sudan1

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

Communication Complexity [Yao’79]

Shared randomness

Bob:

Alice:

  • = min. communication (error )

  • min. -round communication (error )


Property testing and communication complexity

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

  • -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

  • Test for -linearity using shared randomness

  • To evaluate exchange and (2 bits)


D isjointness

-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

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

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

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 y1

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

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

[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

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

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

Functions[Blais, Raskhodnikova, Y.]

monotone functions over

Previous for monotonicity on the line ():

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

  • [Fischer’04]


Functions blais raskhodnikova y1

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 y2

Functions[Blais, Raskhodnikova, Y.]

  • Augmented Index: S, ()

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


Functions blais raskhodnikova y3

Functions[Blais, Raskhodnikova, Y.]

Walsh functions: For :

,

where is the -th bit of


Functions blais raskhodnikova y4

Functions[Blais, Raskhodnikova, Y.]

Step functions: For :


Functions blais raskhodnikova y5

Functions[Blais, Raskhodnikova, Y.]

  • Augmented Index Monotonicity Testing

  • is monotone

  • is ¼ -far from monotone

  • Thus,


Functions blais raskhodnikova y6

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

Thank you!


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