Simultaneous Multiparty Communication Protocols for Composed Functions

Yassine Hamoudi IRIF, Université Paris Diderot, CNRS

MFCS 2018

Number-On-Forehead model [Chandra, Furst, Lipton’83] 2

F(x1, x2, x3, x4) = ?

x2, x3, x4 x1, x3, x4

x1, x2, x3x1, x2, x4

Player 1 Player 2

Player 4Player 3

F : X1 × ⋯ × Xk → {0,1}

Number-On-Forehead model [Chandra, Furst, Lipton’83] 2

• Player i doesn’t know (⇔ Number-On-Forehead)

• Communicate by broadcasting bits

• Players have unlimited computational power

xi

F(x1, x2, x3, x4) = ?

x2, x3, x4 x1, x3, x4

x1, x2, x3x1, x2, x4

Player 1 Player 2

Player 4Player 3

F : X1 × ⋯ × Xk → {0,1}

No randomness in this talk

Examples 3

• Player 1 sends x2

• Player 2 sends F(x1,…, xk)

An always-O(n) protocol: F is easy / protocol is efficient ⇔ communication cost logO(1)(n)

x1, …, xn ∈ {0, 1}n

Examples 3

Equality: x1 = … = xk?

Two players: Ω(n)

k ≥ 3 players: O(1)

• Player 1 indicates if x2 = … = xk

• Player 2 indicates if x1 = x3

• Player 1 sends x2

• Player 2 sends F(x1,…, xk)

An always-O(n) protocol: F is easy / protocol is efficient ⇔ communication cost logO(1)(n)

x1, …, xn ∈ {0, 1}n

Applications of the NOF model 4

• Branching programs, Ramsey theory [Chandra, Furst, Lipton'83]

• Quasi-random graphs [Chung, Tetali'93]

• Proof complexity [Beame, Pitassi, Segerlind’07]

• Circuit complexity [Håstad, Goldmann’91] [Razborov, Wigderson'93] [Beigel, Tarui’94]

• Data-structures for dynamic problems [Patrascu’10]

Applications of the NOF model 4

• Branching programs, Ramsey theory [Chandra, Furst, Lipton'83]

• Quasi-random graphs [Chung, Tetali'93]

• Proof complexity [Beame, Pitassi, Segerlind’07]

• Circuit complexity [Håstad, Goldmann’91] [Razborov, Wigderson'93] [Beigel, Tarui’94]

• Data-structures for dynamic problems [Patrascu’10]

polysize constant-depth circuits with AND, OR, NOT, MODm gates

F is hard to compute for k ≥ log(n) players ⇒ F is not in ACC0

[Håstad, Goldmann’91][Babai, Gál, Kimmel, Lokam’04]

Conjecture: MAJORITY ∉ ACC0

Conjecture: NP ⊈ ACC0

(log n)ω(1)communication cost:

Best lower bounds so far: Ω̃ ( n2k )

Applications of the NOF model 4

• Branching programs, Ramsey theory [Chandra, Furst, Lipton'83]

• Quasi-random graphs [Chung, Tetali'93]

• Proof complexity [Beame, Pitassi, Segerlind’07]

• Circuit complexity [Håstad, Goldmann’91] [Razborov, Wigderson'93] [Beigel, Tarui’94]

• Data-structures for dynamic problems [Patrascu’10]

polysize constant-depth circuits with AND, OR, NOT, MODm gates

F is hard to compute for k ≥ log(n) players ⇒ F is not in ACC0

[Håstad, Goldmann’91][Babai, Gál, Kimmel, Lokam’04]

Conjecture: MAJORITY ∉ ACC0

Conjecture: NP ⊈ ACC0

Log(n) barrier problem

Find a function that is hard to compute for log(n) or more players in the Number-On-Forehead model.

(log n)ω(1)communication cost:

Best lower bounds so far: Ω̃ ( n2k )

Applications of the NOF model 4

• Branching programs, Ramsey theory [Chandra, Furst, Lipton'83]

• Quasi-random graphs [Chung, Tetali'93]

• Proof complexity [Beame, Pitassi, Segerlind’07]

• Circuit complexity [Håstad, Goldmann’91] [Razborov, Wigderson'93] [Beigel, Tarui’94]

• Data-structures for dynamic problems [Patrascu’10]

polysize constant-depth circuits with AND, OR, NOT, MODm gates

F is hard to compute for k ≥ log(n) players ⇒ F is not in ACC0

[Håstad, Goldmann’91][Babai, Gál, Kimmel, Lokam’04]

Conjecture: MAJORITY ∉ ACC0

Conjecture: NP ⊈ ACC0

Log(n) barrier problem

Find a function that is hard to compute for log(n) or more players in the Number-On-Forehead model.

(log n)ω(1)

simultaneous

communication cost:

even in the simultaneous

NOF model

Best lower bounds so far: Ω̃ ( n2k )

Simultaneous Number-On-Forehead model 5

F(x1, x2, x3, x4) = ?

x2, x3, x4 x1, x3, x4 x1, x2, x3x1, x2, x4

Player 1 Player 2 Player 4Player 3

One-way communication to a referee, no interactions

Referee

g1 g2 g3 gn

Candidates to break the log(n) barrier: the composed functions 6

x1 0 1 0 1

x2 1 1 0 0

⋮ …

xk 1 0 0 1

n bits

k players

• Input: x1, …, xk ∈ {0,1}n • Player i doesn’t see row i

Composed function: f ∘ (g1, …, gn) where f : {0, 1}n → {0, 1}

and gj : {0, 1}k → {0, 1}

f

g1 g2 g3 gn

Candidates to break the log(n) barrier: the composed functions 6

x1 0 1 0 1

x2 1 1 0 0

⋮ …

xk 1 0 0 1

n bits

k players

• Input: x1, …, xk ∈ {0,1}n • Player i doesn’t see row i

Composed function: f ∘ (g1, …, gn) where f : {0, 1}n → {0, 1}

and gj : {0, 1}k → {0, 1}

f

Examples: • Generalized Inner Product: MOD2 ○ (AND,…,AND)• Disjointness: OR ○ (AND,…,AND)• Majority of Majority: MAJ ○ (MAJ,…,MAJ)

g1 g2 g3 gn

Candidates to break the log(n) barrier: the composed functions 6

x1 0 1 0 1

x2 1 1 0 0

⋮ …

xk 1 0 0 1

n bits

k players

• Input: x1, …, xk ∈ {0,1}n • Player i doesn’t see row i

Composed function: f ∘ (g1, …, gn) where f : {0, 1}n → {0, 1}

and gj : {0, 1}k → {0, 1}

f

[Grolmusz’94][Babai, Gál, Kimmel, Lokam’04][Ada, Chattopadhyay, Fawzi, Nguyen’15]

Examples: • Generalized Inner Product: MOD2 ○ (AND,…,AND)• Disjointness: OR ○ (AND,…,AND)• Majority of Majority: MAJ ○ (MAJ,…,MAJ)

There is an efficient simultaneous protocol for f ○ (g1,…,gn) when f is symmetric and k ≥ Ω(log n).

Block-composed functions 7

x1 1 0 1

x2 1 1 0

⋮ … …

xk 1 1 0

tn bitst bits

g1 gn f

k players

Block-composed functions 7

x1 1 0 1

x2 1 1 0

⋮ … …

xk 1 1 0

tn bitst bits

g1 gn f

The simultaneous communication cost of MAJ ◦ (MAJ,…,MAJ)is (log n)ω(1) for t ≥ √n and k ≥ Ω(log n).Unknown even

for t = 2

Conjecture [Babai et. al.’04]

k players

Our result 8

0 1 11 0 1

… …

1 1 0

t bits

g1 f

If t is constant, there is an efficient simultaneous protocol for f ○ (g1,…,gn) when f, g1, …, gn are symmetric and k ≥ Ω(log n).

MAJ ◦ (MAJ,…,MAJ) cannot break the log n barrier for constant t

Theorem:

gn

Our result 8

0 1 11 0 1

… …

1 1 0

t bits

g1 f

If t is constant, there is an efficient simultaneous protocol for f ○ (g1,…,gn) when f, g1, …, gn are symmetric and k ≥ Ω(log n).

MAJ ◦ (MAJ,…,MAJ) cannot break the log n barrier for constant t

Theorem:

gn

Block-width t Model Conditions[Ada, Chattopadhyay, Fawzi, Nguyen’15] 1 simultaneous f symmetric

[Chattopadhyay, Saks’14] log log n non-simultaneous f symmetric [Chattopadhyay, Saks’14] log n non-simultaneous f, g1, …, gn symmetric

Our result constant simultaneous f, g1, …, gn symmetric

Our result 9

0 1 11 0 1

… …

1 1 0

t bits

g1 f

If t is constant, there is an efficient simultaneous protocol for f ○ (g1,…,gn) when f, g1, …, gn are symmetric and k ≥ Ω(log n).

Roadmap (when k = Θ(log n)):

1. Reduce to the case of equal inner functions g = g1 = … = gn

2. Simultaneous protocol for f◦(g,…,g) with a generalization of [Babai et. al.’04]

MAJ ◦ (MAJ,…,MAJ) cannot break the log n barrier for constant t

Theorem:

gn

Step 1: equal inner functions 10

x1 1 0 0

x2 1 1 1

⋮ … …

xk 0 1 1

fg1 gn

B1 Bn

1. Each is decomposed in a basis of symmetric functions:

Step 1: equal inner functions 10

x1 1 0 0

x2 1 1 1

⋮ … …

xk 0 1 1

fg1 gn

B1 Bn

gj gj(x) = ∑aca(gj) ⋅ ma(x)

1. Each is decomposed in a basis of symmetric functions:

2. For each basis element ma, define the matrix Ma where each Bj is repeated times.

Step 1: equal inner functions 10

x1 1 0 0

x2 1 1 1

⋮ … …

xk 0 1 1

fg1 gn

B1 Bn

gj gj(x) = ∑aca(gj) ⋅ ma(x)

ca(gj)

1. Each is decomposed in a basis of symmetric functions:

2. For each basis element ma, define the matrix Ma where each Bj is repeated times.

3. For each ma, compute SUM○(ma,…,ma) on Ma.

Step 1: equal inner functions 10

x1 1 0 0

x2 1 1 1

⋮ … …

xk 0 1 1

fg1 gn

B1 Bn

gj gj(x) = ∑aca(gj) ⋅ ma(x)

ca(gj)

1. Each is decomposed in a basis of symmetric functions:

2. For each basis element ma, define the matrix Ma where each Bj is repeated times.

3. For each ma, compute SUM○(ma,…,ma) on Ma.

Step 1: equal inner functions 10

x1 1 0 0

x2 1 1 1

⋮ … …

xk 0 1 1

fg1 gn

B1 Bn

gj gj(x) = ∑aca(gj) ⋅ ma(x)

ca(gj)

∑aSUM ∘ (ma, …, ma)(Ma) = ∑a ∑j

ca(gj) ⋅ ma(Bj) = ∑jgj(Bj)

1. Each is decomposed in a basis of symmetric functions:

2. For each basis element ma, define the matrix Ma where each Bj is repeated times.

3. For each ma, compute SUM○(ma,…,ma) on Ma.

Step 1: equal inner functions 10

x1 1 0 0

x2 1 1 1

⋮ … …

xk 0 1 1

fg1 gn

B1 Bn

gj gj(x) = ∑aca(gj) ⋅ ma(x)

ca(gj)

∑aSUM ∘ (ma, …, ma)(Ma) = ∑a ∑j

ca(gj) ⋅ ma(Bj) = ∑jgj(Bj)

Enough to compute since f is symmetric. f ∘ (g1, …, gn)

1. Each is decomposed in a basis of symmetric functions:

2. For each basis element ma, define the matrix Ma where each Bj is repeated times.

3. For each ma, compute SUM○(ma,…,ma) on Ma.

Step 1: equal inner functions 10

x1 1 0 0

x2 1 1 1

⋮ … …

xk 0 1 1

fg1 gn

B1 Bn

gj gj(x) = ∑aca(gj) ⋅ ma(x)

ca(gj)

∑aSUM ∘ (ma, …, ma)(Ma) = ∑a ∑j

ca(gj) ⋅ ma(Bj) = ∑jgj(Bj)

Enough to compute since f is symmetric. f ∘ (g1, …, gn)

mod p

Size ≤ k×n2

for prime p ∈ (n,2n)

• For a k×n matrix M, define y(M) = (y0,…,yk) where yi = #columns with exactly i 1's.

→ Knowing y(M) is enough to compute f○(g,…,g) when f and g are symmetric.

11Step 3: equation solving protocol

[Babai, Gál, Kimmel, Lokam’04] Protocol for t = 1:

x1 0 1 1 1 1 1 1

x2 1 1 1 0 0 1 0

x3 1 0 1 1 0 0 0

x4 0 1 1 0 1 0 0

x5 1 1 1 0 1 1 0

y0 = 0y1 = 1y2 = 1y3 = 3y4 = 1y5 = 1

• For a k×n matrix M, define y(M) = (y0,…,yk) where yi = #columns with exactly i 1's.

→ Knowing y(M) is enough to compute f○(g,…,g) when f and g are symmetric.

11Step 3: equation solving protocol

[Babai, Gál, Kimmel, Lokam’04] Protocol for t = 1:

x1 0 1 1 1 1 1 1

x2 1 1 1 0 0 1 0

x3 1 0 1 1 0 0 0

x4 0 1 1 0 1 0 0

x5 1 1 1 0 1 1 0

• For each 1 ≤ i ≤ k, Player i sends y(Mi) where Mi is the submatrix seen by Player i.

→ This will be the only communication part of our protocol.

y0 = 0y1 = 1y2 = 1y3 = 3y4 = 1y5 = 1

M1

→ The referee computes y(M) and then f○(g,…,g).

• For a k×n matrix M, define y(M) = (y0,…,yk) where yi = #columns with exactly i 1's.

→ Knowing y(M) is enough to compute f○(g,…,g) when f and g are symmetric.

11Step 3: equation solving protocol

[Babai, Gál, Kimmel, Lokam’04] Protocol for t = 1:

x1 0 1 1 1 1 1 1

x2 1 1 1 0 0 1 0

x3 1 0 1 1 0 0 0

x4 0 1 1 0 1 0 0

x5 1 1 1 0 1 1 0

• For each 1 ≤ i ≤ k, Player i sends y(Mi) where Mi is the submatrix seen by Player i.

→ This will be the only communication part of our protocol.

• Using y(M1), …, y(Mk), one can define an equation whose only integral solution is y(M).

y0 = 0y1 = 1y2 = 1y3 = 3y4 = 1y5 = 1

M1

→ The referee computes y(M) and then f○(g,…,g).

• For a k×n matrix M, define y(M) = (y0,…,yk) where yi = #columns with exactly i 1's.

→ Knowing y(M) is enough to compute f○(g,…,g) when f and g are symmetric.

11Step 3: equation solving protocol

[Babai, Gál, Kimmel, Lokam’04] Protocol for t = 1:

x1 0 1 1 1 1 1 1

x2 1 1 1 0 0 1 0

x3 1 0 1 1 0 0 0

x4 0 1 1 0 1 0 0

x5 1 1 1 0 1 1 0

• For each 1 ≤ i ≤ k, Player i sends y(Mi) where Mi is the submatrix seen by Player i.

→ This will be the only communication part of our protocol.

• Using y(M1), …, y(Mk), one can define an equation whose only integral solution is y(M).

y0 = 0y1 = 1y2 = 1y3 = 3y4 = 1y5 = 1

M1

We generalize this protocol to t > 1, and show that the corresponding equation admits exactly one integral solution when .k ≥ Ω(log n)

Conclusion 12

• Efficient simultaneous protocol for non-constant t and/or non-symmetric g1,…,gn

• Strong lower bound for k ≥ log n players

→ only general method known: discrepancy method and its variants → [Podolskii, Sherstov’17]: first ω(1) lower bound when k ≥ log n for explicit function

Our result: MAJ ◦ (MAJ,…,MAJ) cannot break the log n barrier for any constant t(in fact, any symmetric f○(g1,…,gn))

Future directions:

arXiv: 1710.01969

Step 0: decreasing the number of players to k = Θ(log n) 13

If g : Y1,…,Yk → Y is symmetric then

is symmetric.

g′ (y1, …, yk′ ) = g(y1, …, yk′ , yk′ +1, …, yk)

Lemma:

any fixed valuesvariables

Step 0: decreasing the number of players to k = Θ(log n) 13

If g : Y1,…,Yk → Y is symmetric then

is symmetric.

g′ (y1, …, yk′ ) = g(y1, …, yk′ , yk′ +1, …, yk)

Lemma:

x1 1 0 0x2 1 1 1

⋮ … …xk’ 0 1 1⋮

xk 1 0 0

f

any fixed valuesvariables

g1 gn

Step 0: decreasing the number of players to k = Θ(log n) 13

If g : Y1,…,Yk → Y is symmetric then

is symmetric.

g′ (y1, …, yk′ ) = g(y1, …, yk′ , yk′ +1, …, yk)

Lemma:

x1 1 0 0x2 1 1 1

⋮ … …xk’ 0 1 1⋮

xk 1 0 0

f

any fixed valuesvariables

g1 gn

x1 1 0 0x2 1 1 1

⋮ … …xk’ 0 1 1⋮

xk 1 0 0

fg′ 1 g′ n