Simultaneous Multiparty Communication Protocols for Composed - PowerPoint PPT Presentation

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 : X 1 × ⋯ × X k → {0,1} Player 1 Player 2 x 2 , x 3 , x 4 x 1 , x 3 , x 4 Player 3 Player 4 F ( x 1 , x 2 , x 3 , x 4 ) = ? x 1 , x 2 , x 4 x 1 , x 2 , x 3

Number-On-Forehead model [Chandra, Furst, Lipton’83] 2 F : X 1 × ⋯ × X k → {0,1} Player 1 Player 2 x 2 , x 3 , x 4 x 1 , x 3 , x 4 Player 3 Player 4 F ( x 1 , x 2 , x 3 , x 4 ) = ? x 1 , x 2 , x 4 x 1 , x 2 , x 3 • Player i doesn’t know ( ⇔ Number-On-Forehead) N x i o r a n d o • Communicate by broadcasting bits m i n n e t h s i s s t a • Players have unlimited computational power l k

Examples 3 x 1 , …, x n ∈ { 0 , 1 } n An always-O(n) protocol: F is easy / protocol is e ffi cient • Player 1 sends x 2 ⇔ communication cost log O (1) ( n ) • Player 2 sends F(x 1 ,…, x k )

Examples 3 x 1 , …, x n ∈ { 0 , 1 } n An always-O(n) protocol: F is easy / protocol is e ffi cient • Player 1 sends x 2 ⇔ communication cost log O (1) ( n ) • Player 2 sends F(x 1 ,…, x k ) Equality: x 1 = … = x k ? Two players: Ω (n) k ≥ 3 players: O(1) • Player 1 indicates if x 2 = … = x k • Player 2 indicates if x 1 = x 3

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] [Håstad, Goldmann’91] [Babai, Gál, Kimmel, Lokam’04] F is hard to compute for ⇒ F is not in ACC 0 k ≥ log(n) players polysize constant-depth circuits (log n ) ω (1) communication cost: with AND, OR, NOT, MOD m gates Ω ( 2 k ) n Best lower bounds so far: ˜ Conjecture: MAJORITY ∉ ACC 0 Conjecture: NP ⊈ ACC 0

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] [Håstad, Goldmann’91] [Babai, Gál, Kimmel, Lokam’04] F is hard to compute for ⇒ F is not in ACC 0 k ≥ log(n) players polysize constant-depth circuits (log n ) ω (1) communication cost: with AND, OR, NOT, MOD m gates Ω ( 2 k ) n Best lower bounds so far: ˜ Conjecture: MAJORITY ∉ ACC 0 Conjecture: NP ⊈ ACC 0 Log(n) barrier problem Find a function that is hard to compute for log(n) or more players in the Number-On-Forehead model.

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] [Håstad, Goldmann’91] [Babai, Gál, Kimmel, Lokam’04] F is hard to compute for ⇒ F is not in ACC 0 k ≥ log(n) players even in the simultaneous polysize constant-depth circuits NOF model (log n ) ω (1) communication cost: with AND, OR, NOT, MOD m gates Ω ( 2 k ) n Best lower bounds so far: ˜ Conjecture: MAJORITY ∉ ACC 0 Conjecture: NP ⊈ ACC 0 Log(n) barrier problem Find a function that is hard to compute for log(n) or more players simultaneous in the Number-On-Forehead model.

Simultaneous Number-On-Forehead model 5 Player 1 Player 2 Player 3 Player 4 x 2 , x 3 , x 4 x 1 , x 3 , x 4 x 1 , x 2 , x 4 x 1 , x 2 , x 3 Referee F ( x 1 , x 2 , x 3 , x 4 ) = ? One-way communication to a referee, no interactions

Candidates to break the log(n) barrier: the composed functions 6 • Input: x 1 , …, x k ∈ {0,1} n • Player i doesn’t see row i n bits x 1 0 1 0 1 x 2 1 1 0 0 k players ⋮ … x k 1 0 0 1 g 1 g n g 2 g 3 f Composed function: f ∘ ( g 1 , …, g n ) where f : { 0 , 1 } n → { 0 , 1 } and g j : { 0 , 1 } k → { 0 , 1 }

Candidates to break the log(n) barrier: the composed functions 6 • Input: x 1 , …, x k ∈ {0,1} n • Player i doesn’t see row i n bits x 1 0 1 0 1 x 2 1 1 0 0 k players ⋮ … x k 1 0 0 1 g 1 g n g 2 g 3 f Composed function: Examples: • Generalized Inner Product: MOD 2 ○ (AND,…,AND) f ∘ ( g 1 , …, g n ) where f : { 0 , 1 } n → { 0 , 1 } • Disjointness: OR ○ (AND,…,AND) and g j : { 0 , 1 } k → { 0 , 1 } • Majority of Majority: MAJ ○ (MAJ,…,MAJ)

Candidates to break the log(n) barrier: the composed functions 6 • Input: x 1 , …, x k ∈ {0,1} n • Player i doesn’t see row i n bits x 1 0 1 0 1 x 2 1 1 0 0 k players ⋮ … x k 1 0 0 1 g 1 g n g 2 g 3 f Composed function: Examples: • Generalized Inner Product: MOD 2 ○ (AND,…,AND) f ∘ ( g 1 , …, g n ) where f : { 0 , 1 } n → { 0 , 1 } • Disjointness: OR ○ (AND,…,AND) and g j : { 0 , 1 } k → { 0 , 1 } • Majority of Majority: MAJ ○ (MAJ,…,MAJ) [Grolmusz’94] There is an e ffi cient simultaneous protocol for [Babai, Gál, Kimmel, Lokam’04] f ○ (g 1 ,…,g n ) when f is symmetric and k ≥ Ω (log n). [Ada, Chattopadhyay, Fawzi, Nguyen’15]

Block-composed functions 7 tn bits t bits x 1 1 0 1 x 2 1 1 0 k players … … ⋮ x k 1 1 0 g 1 f g n

Block-composed functions 7 tn bits t bits x 1 1 0 1 x 2 1 1 0 k players … … ⋮ x k 1 1 0 g 1 f g n Conjecture [Babai et. al.’04] 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

Our result 8 t bits 0 1 1 1 0 1 … … 1 1 0 f g 1 g n Theorem: If t is constant , there is an e ffi cient simultaneous protocol for f ○ (g 1 ,…,g n ) when f, g 1 , …, g n are symmetric and k ≥ Ω (log n). MAJ ◦ (MAJ,…,MAJ) cannot break the log n barrier for constant t

Our result 8 t bits 0 1 1 1 0 1 … … 1 1 0 f g 1 g n Theorem: If t is constant , there is an e ffi cient simultaneous protocol for f ○ (g 1 ,…,g n ) when f, g 1 , …, g n are symmetric and k ≥ Ω (log n). MAJ ◦ (MAJ,…,MAJ) cannot break the log n barrier for constant t Block-width t Model Conditions 1 simultaneous f symmetric [Ada, Chattopadhyay, Fawzi, Nguyen’15] log log n non-simultaneous f symmetric [Chattopadhyay, Saks’14] log n non-simultaneous f, g 1 , …, g n symmetric [Chattopadhyay, Saks’14] Our result constant simultaneous f, g 1 , …, g n symmetric

Our result 9 t bits 0 1 1 1 0 1 … … 1 1 0 f g 1 g n Theorem: If t is constant , there is an e ffi cient simultaneous protocol for f ○ (g 1 ,…,g n ) when f, g 1 , …, g n are symmetric and k ≥ Ω (log n). MAJ ◦ (MAJ,…,MAJ) cannot break the log n barrier for constant t Roadmap (when k = Θ (log n)) : 1. Reduce to the case of equal inner functions g = g 1 = … = g n 2. Simultaneous protocol for f ◦ (g,…,g) with a generalization of [Babai et. al.’04]

Step 1: equal inner functions 10 x 1 1 0 0 x 2 1 1 1 B 1 B n ⋮ … … x k 0 1 1 f g 1 g n

Step 1: equal inner functions 10 x 1 1 0 0 x 2 1 1 1 B 1 B n ⋮ … … x k 0 1 1 f g 1 g n g j ( x ) = ∑ a c a ( g j ) ⋅ m a ( x ) 1. Each is decomposed in a basis of symmetric functions: g j

Step 1: equal inner functions 10 x 1 1 0 0 x 2 1 1 1 B 1 B n ⋮ … … x k 0 1 1 f g 1 g n g j ( x ) = ∑ a c a ( g j ) ⋅ m a ( x ) 1. Each is decomposed in a basis of symmetric functions: g j 2. For each basis element m a , define the matrix M a where each B j is repeated times. c a ( g j )

Step 1: equal inner functions 10 x 1 1 0 0 x 2 1 1 1 B 1 B n ⋮ … … x k 0 1 1 f g 1 g n g j ( x ) = ∑ a c a ( g j ) ⋅ m a ( x ) 1. Each is decomposed in a basis of symmetric functions: g j 2. For each basis element m a , define the matrix M a where each B j is repeated times. c a ( g j ) 3. For each m a , compute SUM ○ (m a ,…,m a ) on M a .

Step 1: equal inner functions 10 x 1 1 0 0 x 2 1 1 1 B 1 B n ⋮ … … x k 0 1 1 f g 1 g n g j ( x ) = ∑ a c a ( g j ) ⋅ m a ( x ) 1. Each is decomposed in a basis of symmetric functions: g j 2. For each basis element m a , define the matrix M a where each B j is repeated times. c a ( g j ) 3. For each m a , compute SUM ○ (m a ,…,m a ) on M a . ∑ a SUM ∘ ( m a , …, m a )( M a ) = ∑ a ∑ j c a ( g j ) ⋅ m a ( B j ) = ∑ j g j ( B j )