The Power of Super-Log Number of Players

Arkadev Chattopadhyay (TIFR, Mumbai)

Joint with:

Michael Saks (Rutgers)

A Conjecturef:{0,1}n ! {0,1}

¡(n2)

¢

0 1 1 1 1 1 11 1 0 1 1 0 10 1 1 1 1 0 0. . . . . . .. . . . . . .1 1 1 1 1 1 0

X1

X2

X3

Xk

(f±g) (X1,X2,,Xk) =

) MAJ ± MAJ ACC0

1 1 01 000

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} .

Some Upper Bounds

SYM ± AND {GIP, Disj,…}

Popular Names

SYM ± g {GIP, MAJ ± MAJ, Disj…}

Deterministic

O (n/2k + k¢ log n ).

O(k.(log n)2), k ¸ log n + 2.

Grolmusz’91, Pudlak

Babai-Gal-Kimmel-Lokam’02

k ¸ 3

Ada-C-Fawzi-Nguyen’12

g: compressible and symmetric

SYM ± ANY

Simultaneous

Simultaneous

Almost- Simultaneous

O(k.(log n)2), k ¸ log n + 4.

Block Compositionf:{0,1}n ! {0,1}

¡(n2)

¢

0 1 1 1 1 11 1 0 1 1 10 1 1 1 1 0. . . . . .. . . . . .1 1 1 1 1 0

X1

X2

X3

Xk

(fn±gr) (X1,X2,,Xk) =

) MAJ ACC0

f(g(A1),g(A2),…,g(An))n = 2r = 3Conjecture:

Fact:

Babai-Gal-Kimmel-Lokam’02

g:{0,1}k£r ! {0,1}.

Still Open!

A1 A2

Even for interactive protocols

Our Result

Theorem: 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)

Main Ingredients

• Computing k-1 degree polynomials is easy for k-players. (Goldman-Hastad’90’s)

• Degree reduction by basis change. (New Idea)

Low degree Polynomials

x3 x5 x7

Alice Bob

x6 x10 x11

x2 x8 x9

Charlie Alex

x1 x4 x12

Bob, Charlie Alice Alex Alice, CharlieBob, Alex

deg(P) = 3

k = 4 > deg(P)

Simultaneous k-partydeterministic protocol

Cost = O(k¢ log|F|)

A Polynomial Fantasyf:{0,1}n ! {0,1}

¡(n2)

¢

(SYM ± g) (C1,C2,,Cn) =

Fantasy: Phigh(Ci) = 0 for all i !!

g:{0,1}k ! {0,1} µ Fp . Prime p > n

g(X) ´ P(X1,,Xk) deg(P) · k

P ´ Phigh(X) + Plow(X)

deg < kdeg = k

easy k-player protocolof cost = k.log(p) Bad

Shifted Basis

¡(n2)

¢

0 1 1 00 0 1 10 1 1 11 0 0 0

Example:

Fact: Bu is a basis for every u 2 {0,1}k

u = 0k gives standard basis

u-shifted

A

Def: u is good for A if for all column C of A, u and C agree onsome co-ordinate.

1

0

0

0good

0

1

0

1

bad

no agreement

Good is Really Good

¡(n2)

¢

(SYM ± g) (C1,C2,,Cn) =

Fact: Phigh(C) = 0 for all C if u is good for A.

easy k-player protocolof cost log(p) Bad

u

Apply u -shift

u

u

Zeroed out!

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 - 1

Cost = k¢ log(p)

= O(k¢ log n)Extends to r = O(log log n).

Future Direction

• Can we go to r = O(log n)?• Is ?

Thank You!