Languages with Bounded Multiparty Communication Complexity Arkadev Chattopadhyay (McGill) Joint work...
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Transcript of Languages with Bounded Multiparty Communication Complexity Arkadev Chattopadhyay (McGill) Joint work...
Languages with Bounded Multiparty Communication Complexity
Arkadev Chattopadhyay (McGill)
Joint work with:
Andreas Krebs (Tubingen)Michal Koucky (Czech Acad. Sciences)
Mario Szegedy (Rutgers)Pascal Tesson (Laval)Denis Therien (McGill)
‘Number on Forehead’ Model
010001110
Player 1
110111001
Player 2111000001
Player 3
110101111101
Cost of protocol is worst case cost. Dk(f) is the cost of best protocol for f
for the worst partition.
A Theorem for k=2Question. What functions can be computed in
constant communication for the worst partition by two players (denoted by CCC2)?
Remark: A priori there is no reason to believe that CCC2 should have any relationship to space-time complexity classes!
Theorem (Szegedy93). Every function in CCC2 can be computed by linear sized ACC0 circuits.
Three Players
Question: What can we say about CCCk, for k ¸ 3?
We show,Theorem 1. CCC3 contains functions
that have arbitrarily large circuit complexity.
We use a coding trick to show this.
The Coding IdeaLet C : {0,1}* ! {0,1}* be an encoding function.
Definition. For a L, define C(L) as follows: y 2 C(L) if there exists x 2 L s.t. C(x)=y.
Observation. If C has efficient encoding and decoding algorithms, then L and C(L) have comparable complexity.
Fact. If the relative distance of C is more than 2/3, then C(L) 2 CCC3.
Proof of Fact
0100011101001
Player X
110111001
Player Y
1
1
0
111000001
Player Z
Find w² x
Find w ² x
w² y ?
w² z ?
w 2 C(L)?
Such Codes ExistFact: Reed-Solomon codes
concatenated with unary codes can be used to carry out this idea!
Remark: Picking a hard L (that is guaranteed to exist by a counting argument), proves our Theorem.
Question. What makes 3 players so powerful?
¯¯¯¯Pr
x
£X
i
= 1nxi ´ q bjP (x) = a¤
¡ 1=q¯¯¯¯·
°n=cd
Pr[P (x) = a]¯¯¯¯Pr
x
£X
i
= 1nxi ´ q bjP (x) = a¤
¡ 1=q¯¯¯¯·
°n=cd
Pr[P (x) = a]¯¯¯¯Pr
x
£X
i
= 1nxi ´ q bjP (x) = a¤
¡ 1=q¯¯¯¯·
°n=cd
Pr[P (x) = a]
¯¯¯¯Pr
x
£X
i
= 1nxi ´ q bjP (x) = a¤
¡ 1=q
¯¯¯¯·
°n=cd
Pr[P (x) = a]
Key Features of Three Players Every pair of input bits is looked at by some
player.
At least a third of the input bits overlap the view of two players.
Each player knows the precise position in the input word of every input bit that he sees.
Question. How useful is the third feature?
Answer. We obtain insight into this question by considering two simple classes of functions.
Neutral Letter and SymmetricityDefinition. Boolean function f : * ! {0,1} has neutral
letter e if for every x,y 2 *, f(xey) = f(xy).
Theorem 2. Every language with a neutral letter that is in CCCk for some fixed k, is regular. We can also give a decidable algebraic characterization of such languages.
Definition. f over is symmetric if for any permutation and any input string x, f(x)=f((x)).
Theorem 3. A symmetric function f is in CCCk for some fixed k iff it is in CCC2.
Promised Partition Let A be a 0-1 matrix of dimension k£ n.Let there be a promise that each column of A has
at most one 1.Definition. PPartn
k(A) is 1 if each column of A has a 1.
Theorem. PPartnk cannot be computed by k
players using c bits of communication for the row-wise partition of inputs if n ¸ HJ(k,2c).
The above Theorem will give us a handle on languages with a neutral letter in CCCk
Partition to Neutral Letter Let f, g be two functions that have alphabet
= {a,b,e}, where e is neutral.
Let w 2 {a,b}* be the minimal word s.t. f(w) g(w), with |w|=m.
Let D3(f) = D3(g) = c.
Claim: PPartm3 can be computed by three players
using 2c bits of communication for the row-wise partition of input bits.
Proof of the Claim
0100
Player X
1000
Player Y
0001
Player Z
Let w = abab, m=|w|=4
x=ebee
z=eeeb
y=aeee
u = x § y § z = eae bee eee eeb •Compute f(u) =f(abb)•Compute g(u) = g(abb)•Output 1 iff f(u) g(u)
ConsequenceCorollary: If f and g are any two functions over a given
alphabet with a neutral letter, can be computed by k players using c bits of communication and they agree on all inputs of length at most HJ(k, 22c), then they must be identical.
Remark: There are only a finite number of such functions over a given alphabet that can be computed by k players communicating c bits, for each fixed k and c.
Fact: This observation can be used to show that languages in CCCk having a neutral letter, are regular.
Conclusion
We omit the characterization for symmetric functions.
If your are interested, please check out the full version on ECCC at
http://eccc.hpi-web.de/eccc-reports/2006/TR06-117/Paper.pdf
THANK YOU!