Arkadev Chattopadhyay TIFR Mumbai Joint with Michael Saks Rutgers A Conjecture f01 n 01 0 1 1 1 1 1 1 1 1 0 ID: 465046
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Slide1
The Power of Super-Log Number of Players
Arkadev
Chattopadhyay
(TIFR, Mumbai)
Joint with:
Michael Saks
(
Rutgers
)Slide2
A Conjecture
f:{0,1}
n
! {0,1}
0
11111111011010111100..............1111110
X1
X2
X3
X
k
(f±g) (X1,X2,,Xk) =
) MAJ ± MAJ ACC0
1
1
0
1
0
0
0
f(g(C1),g(C2),…,g(Cn))
n
Question:
Complexity of (MAJ
± MAJ)?
Observation:
a
la Beigel-Tarui’91
) MAJ ACC0
Proposed by
Babai-Kimmel-Lokam’95
g
:{0,1}
k
!
{0,1} .Slide3
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A
AAAAA
A
Some Upper BoundsSYM ± AND {GIP, Disj,…} Popular NamesSYM ± g {GIP, MAJ ± MAJ, Disj…}
DeterministicO (n/2k
+ k¢ log n ). O(k.(log n)2), k ¸ log n + 2.Grolmusz’91, PudlakBabai-Gal-Kimmel-Lokam’02k ¸ 3Ada-C-Fawzi-Nguyen’12g: compressible and symmetric SYM ± ANY SimultaneousSimultaneousAlmost- Simultaneous O(k.(log n)2), k ¸ log n + 4.Slide4
Block Composition
f:{0,1}
n
! {0,1}
0
11111110111011110............111110X1X2 X3 Xk(fn±gr) (X1,X2,,Xk) =) MAJ ACC0f(g(A1),g(A2),…,g(An))n = 2r = 3Conjecture: Fact: Babai-Gal-Kimmel-Lokam’02g:{0,1}k£r ! {0,1}.
Still Open!
A
1A2
Even for interactive protocolsSlide5
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A
AAAAA
A
Our ResultTheorem: SYMn ± ANYr has a 2-round k-party deterministic protocol of cost when,
Remark 1:
First protocol for r > 1. Remark 2: Corollary: MAJ ± MAJr has efficient protocol when r is poly-log and k is a sufficiently large poly-log. r = O(log log n)Slide6
Main Ingredients
Computing k-1 degree polynomials is easy for k-players.
(Goldman-Hastad’90’s)Degree reduction by basis change. (New Idea)Slide7
Low degree Polynomials
x
3
x
5
x7 Alice Bobx6 x10 x11x2 x8 x9Charlie Alexx1 x4 x12Bob, CharlieAliceAlexAlice, CharlieBob, Alexdeg(P) = 3k = 4 > deg(P) Simultaneous k-partydeterministic protocolCost = O(k¢ log|F|)Slide8
A Polynomial Fantasy
f:{0,1}
n
! {0,1}
(SYM
± g) (C1,C2,,Cn) =Fantasy: Phigh(Ci) = 0 for all i !!g:{0,1}k ! {0,1} µ Fp .Prime p > ng(X) ´ P(X1,,Xk)deg(P) · kP ´ Phigh(X) + Plow(X)deg < kdeg = keasy k-player protocolof cost = k.log(p) Bad Slide9
Shifted Basis
0
1
1
0
001101111000Example:Fact: Bu is a basis for every u 2 {0,1}ku = 0k gives standard basis u-shiftedADef: u is good for A if for all column C of A, u and C agree onsome co-ordinate.1000good0101badno agreementSlide10
Good is Really Good
(SYM
±
g) (C1,C2,,Cn) =Fact: Phigh(C) = 0 for all C if u is good for A.
easy k-player protocol
of cost log(p) Bad uApply u -shift uuZeroed out!Slide11
Good Shifts Are Aplenty
Observation:
If
k À log n + 1, Player k spots many good shifts. Protocol: Player k announces a good shift u.All players compute their portions using u.
Simultaneous!
Cost = k - 1Cost = k¢ log(p) = O(
k¢ log n)
Extends to r = O(log log n).Slide12
Future Direction
Can we go to r = O(log n)?
Is ?
Thank You!