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Property Testing and Communication Complexity

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Property Testing and Communication Complexity

GrigoryYaroslavtsev

http://grigory.us

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 = set of YES instances

Query complexity of testing

- = Adaptive queries
- = Non-adaptive (all queries at once)
- = Queries in rounds ()

For error :

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?

- is -linear
- is -linear, ½-far from -linear

- Test for -linearity using shared randomness
- To evaluate exchange and (2 bits)

- [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

“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]

- Testing linearity: is linear if
- Equality: decide whether

- is linear
- is ¼ -far from linear

- = = (matching [Blum, Luby, Rubinfeld])
- Strong Direct Sum for Equality[MWY’13]Strong Direct Sum for Testing Linearity

- 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

For all properties :

- Direct m-Sum (solve all w.p. 2/3 per instance)
- Adaptive:
- Non-adaptive:

- Direct -Product (-far instance?)
- Adaptive:
- Non-adaptive:

Referee:

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

Bob:

Alice:

- 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

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]