Theorem 5 2 t log n then it admits exactly one integral solution. If k the referee recovers the y i 1 i 2 i 3 ’s and computes the output 11 The referee computes: It verifies: Proof sketch b i 1 , i 2 , i 3 = a 1 i 1 , i 2 , i 3 + · · · + a 5 i 1 , i 2 , i 3 y i 1 , i 2 , i 3 ≥ 0 ∑ y i 1 , i 2 , i 3 = n ( k − ( i 1 + i 2 + i 3 )) y i 1 , i 2 , i 3 + ( i 1 + 1 ) y i 1 + 1 , i 2 , i 3 +( i 2 + 1 ) y i 1 , i 2 + 1 , i 3 + ( i 3 + 1 ) y i 1 , i 2 , i 3 + 1 = b i 1 , i 2 , i 3
11 It verifies: The referee computes: Proof sketch b i 1 , i 2 , i 3 = a 1 i 1 , i 2 , i 3 + · · · + a 5 i 1 , i 2 , i 3 y i 1 , i 2 , i 3 ≥ 0 ∑ y i 1 , i 2 , i 3 = n ( k − ( i 1 + i 2 + i 3 )) y i 1 , i 2 , i 3 + ( i 1 + 1 ) y i 1 + 1 , i 2 , i 3 +( i 2 + 1 ) y i 1 , i 2 + 1 , i 3 + ( i 3 + 1 ) y i 1 , i 2 , i 3 + 1 = b i 1 , i 2 , i 3 Theorem If k ≥ 5 2 t log n then it admits exactly one integral solution. → the referee recovers the y i 1 , i 2 , i 3 ’s and computes the output
Decision tree complexity and log-rank conjecture
Conjecture log c rank M F D 2 F For some absolute constant c: log rank M F 12 Log-rank conjecture F : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } Proposition ([MS82]) Let M F ∈ { 0 , 1 } n × n be the communication matrix: M F ( x , y ) = F ( x , y ) . log rank M F ≤ D 2 ( F )
12 For some absolute constant c: Log-rank conjecture F : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } Proposition ([MS82]) Let M F ∈ { 0 , 1 } n × n be the communication matrix: M F ( x , y ) = F ( x , y ) . log rank M F ≤ D 2 ( F ) Conjecture log rank M F ≤ D 2 ( F ) ≤ log c rank M F
0 1 n 0 1 n 0 1 is an AND function if: Examples: Equality x y y , Hamming d x y GAP d x MOD 2 x Interests: • Connections with Decision Tree complexity y , InnerProduct x y mon f [BdW01] • For AND functions: rank M F [BC99] mon f • For XOR functions: rank M F y , etc. 13 NOR x Disjointness x y y , NOR x y f x F x y • A function F XOR and AND functions • A function F : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } is an XOR function if: F ( x , y ) = f ( x ⊕ y ) for some f : { 0 , 1 } n → { 0 , 1 }
0 1 n Examples: Equality x y y , Hamming d x y GAP d x MOD 2 x Interests: • Connections with Decision Tree complexity 13 mon f [BdW01] • For AND functions: rank M F [BC99] mon f • For XOR functions: rank M F y , etc. Disjointness x y y , InnerProduct x y NOR x y , NOR x 0 1 for some f XOR and AND functions • A function F : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } is an XOR function if: F ( x , y ) = f ( x ⊕ y ) • A function F : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } is an AND function if: F ( x , y ) = f ( x ∧ y )
0 1 n Interests: • Connections with Decision Tree complexity 13 • For XOR functions: rank M F for some f 0 1 mon f [BdW01] • For AND functions: rank M F [BC99] mon f XOR and AND functions • A function F : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } is an XOR function if: F ( x , y ) = f ( x ⊕ y ) • A function F : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } is an AND function if: F ( x , y ) = f ( x ∧ y ) Examples: Equality ( x , y ) = NOR ( x ⊕ y ) , Hamming d ( x , y ) = GAP d ( x ⊕ y ) , Disjointness ( x , y ) = NOR ( x ∧ y ) , InnerProduct ( x , y ) = MOD 2 ( x ∧ y ) , etc.
0 1 n for some f 0 1 [BC99] 13 XOR and AND functions • A function F : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } is an XOR function if: F ( x , y ) = f ( x ⊕ y ) • A function F : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } is an AND function if: F ( x , y ) = f ( x ∧ y ) Examples: Equality ( x , y ) = NOR ( x ⊕ y ) , Hamming d ( x , y ) = GAP d ( x ⊕ y ) , Disjointness ( x , y ) = NOR ( x ∧ y ) , InnerProduct ( x , y ) = MOD 2 ( x ∧ y ) , etc. Interests: • For XOR functions: rank M F = mon f • For AND functions: rank M F = mon ⋆ f [BdW01] • Connections with Decision Tree complexity
A decision tree is an ordered tree where each internal node is labeled with a query, and each leaf is labeled with 0 or 1. x 3 x 2 x 1 x 1 x 2 0 1 0 1 0 1 14 Decision tree complexity
14 A decision tree is an ordered tree where each internal node is labeled with a 1 0 1 0 1 0 x 2 x 1 x 1 x 2 x 3 query, and each leaf is labeled with 0 or 1. Decision tree complexity Input: x 1 x 2 x 3 = 011 on a regular decision tree DT ( f ) , RDT ( f ) and QDT ( f )
14 A decision tree is an ordered tree where each internal node is labeled with a 1 0 1 0 1 0 x 2 x 2 query, and each leaf is labeled with 0 or 1. Decision tree complexity x 1 ⊕ x 2 ⊕ x 3 x 1 ⊕ x 2 x 2 ⊕ x 3 Input: x 1 x 2 x 3 = 011 on a parity decision tree DT ⊕ ( f ) , RDT ⊕ ( f ) and QDT ⊕ ( f )
14 A decision tree is an ordered tree where each internal node is labeled with a 1 0 1 0 1 0 x 2 x 1 query, and each leaf is labeled with 0 or 1. Decision tree complexity x 2 ∧ x 3 x 1 ∧ x 3 x 1 ∧ x 3 Input: x 1 x 2 x 3 = 011 on a conjunctive decision tree DT ∧ ( f ) , RDT ∧ ( f ) and QDT ∧ ( f )
log c mon f D 2 F 2 DT log c mon D 2 F Conjecture f f 2 DT f • AND function: log mon 15 f • XOR function: log mon f • Log-rank conjecture for decision trees: • Communication and Decision Tree complexities are polynomially related Connections Proposition ([ZS10]) For any XOR function F ( x , y ) = f ( x ⊕ y ) : D 2 ( F ) ≤ 2 · DT ⊕ ( f ) For any AND function F ( x , y ) = f ( x ∧ y ) : D 2 ( F ) ≤ 2 · DT ∧ ( f )
log c mon f D 2 F log c mon D 2 F 15 f f f 2 DT f • AND function: log mon 2 DT • XOR function: log mon f • Log-rank conjecture for decision trees: • Communication and Decision Tree complexities are polynomially related Connections Proposition ([ZS10]) For any XOR function F ( x , y ) = f ( x ⊕ y ) : D 2 ( F ) ≤ 2 · DT ⊕ ( f ) For any AND function F ( x , y ) = f ( x ∧ y ) : D 2 ( F ) ≤ 2 · DT ∧ ( f ) Conjecture
• Log-rank conjecture for decision trees: • Communication and Decision Tree complexities are polynomially related 15 Connections Proposition ([ZS10]) For any XOR function F ( x , y ) = f ( x ⊕ y ) : D 2 ( F ) ≤ 2 · DT ⊕ ( f ) For any AND function F ( x , y ) = f ( x ∧ y ) : D 2 ( F ) ≤ 2 · DT ∧ ( f ) Conjecture • XOR function: log mon ( f ) ≤ D 2 ( F ) ≤ 2 · DT ⊕ ( f ) ≤ log c mon ( f ) • AND function: log mon ⋆ ( f ) ≤ D 2 ( F ) ≤ 2 · DT ∧ ( f ) ≤ log c mon ⋆ ( f )
16 n 1 [ZS09, BdW01, Raz03] Quantum n Randomized symmetric f : XOR functions Deterministic AND functions Symmetric XOR and AND functions Communication complexity 1 of (nontrivial) XOR and AND functions, for ( ( )) Θ ( n ) Θ ( n − t ( f )) 1 + log n − t ( f ) Θ † ( ( )) Θ ( r ( f )) ( n − t ( f )) 1 + log n − t ( f ) Θ ⋆ (√ ) Θ ( r ( f )) n · ℓ 0 ( f ) + ℓ 1 ( f )
Result: Communication and Decision Tree complexities are polynomially 17 n Regular Parity Conjunctive Deterministic related for symmetric functions. Quantum n Randomized Symmetric functions Decision tree complexities 2 of (nontrivial) symmetric functions: ( ( )) Θ ( n ) Θ ( n ) Θ ( n − t ( f )) 1 + log n − t ( f ) Θ † ( ( )) Θ ( n ) Θ ( r ( f )) ( n − t ( f )) 1 + log n − t ( f ) (√ ) Θ ⋆ (√ ) Θ n · ℓ ( f ) Θ ( r ( f )) n · ℓ 0 ( f ) + ℓ 1 ( f ) 2 [ZS09, BdW01, Raz03, BBC + 01]
17 n Regular Parity Conjunctive Deterministic related for symmetric functions. Quantum n Randomized Symmetric functions Decision tree complexities 2 of (nontrivial) symmetric functions: ( ( )) Θ ( n ) Θ ( n ) Θ ( n − t ( f )) 1 + log n − t ( f ) Θ † ( ( )) Θ ( n ) Θ ( r ( f )) ( n − t ( f )) 1 + log n − t ( f ) (√ ) Θ ⋆ (√ ) Θ n · ℓ ( f ) Θ ( r ( f )) n · ℓ 0 ( f ) + ℓ 1 ( f ) Result: Communication and Decision Tree complexities are polynomially 2 [ZS09, BdW01, Raz03, BBC + 01]
Conclusion
• breaking the log n barrier Our contributions: functions p Future work: • other protocols for larger families of composed functions • log-rank conjecture for XOR and AND functions (using decision tree complexity?) 18 • first efficient simultaneous protocol for Sym ◦ Sym t • full characterization of the decision tree complexities of symmetric • efficient construction for Ramsey numbers over F n
Our contributions: functions p Future work: • other protocols for larger families of composed functions • log-rank conjecture for XOR and AND functions (using decision tree complexity?) 18 • first efficient simultaneous protocol for Sym ◦ Sym t • full characterization of the decision tree complexities of symmetric • efficient construction for Ramsey numbers over F n • breaking the log n barrier
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• log-rank method 26 Equality function Equality ( x 1 , . . . , x k ) = 1 ⇔ x 1 = · · · = x k D 2 ( Equality ) = Ω( n ) R || 2 ( Equality ) = O ( 1 ) • Alice and Bob test x · r = y · r mod 2 for two random r ∈ { 0 , 1 } n D || k ( Equality ) = O ( 1 ) when k > 2 • Player 1 checks x 2 = · · · = x k • Player 2 checks x 1 = x 3 = · · · = x k
and each bottom gate is an AND gate of fan-in k SYM AND AND s k k 27 ACC 0 and Sym + Sym + ( s , k ) = depth-2 circuits whose top gate is a symmetric gate of fan-in s , · · · • ACC 0 ⊂ SYM + ( 2 polylog n , polylog n ) [Yao90, BT94] • f is computed by a SYM + ( s , k − 1 ) circuit ⇒ for any partition of the input between k players, there is a protocol of cost O ( k log s ) computing f
0 f 1 f • r f 28 • f min p f i f i 1 for i p n p n 2 n 1 min p f i f i 2 for i p n p 2 p f i f n 2 for i f p f x depends only on x . Hence f 0 n 0 1 • t f min p f p 1 • min p n 2 f i f n 2 for i p n 2 • min p n 2 Symmetric XOR and AND functions F ( x , y ) = f ( x ⊕ y ) is symmetric iff f is symmetric
0 f 1 f • r f p f min p f i f i 1 for i p n 28 1 p min p f i f i 2 for i p n p 2 • f n 2 for i n 2 n min p • t f min p f p 1 f p • n 2 f i f n 2 for i p n 2 • min p n 2 f i Symmetric XOR and AND functions F ( x , y ) = f ( x ⊕ y ) is symmetric iff f is symmetric → f ( x ) depends only on | x | . Hence f : { 0 , . . . , n } → { 0 , 1 }
28 Symmetric XOR and AND functions F ( x , y ) = f ( x ⊕ y ) is symmetric iff f is symmetric → f ( x ) depends only on | x | . Hence f : { 0 , . . . , n } → { 0 , 1 } • t ( f ) = min { p : f ( p − 1 ) ̸ = f ( p ) } • ℓ 0 ( f ) = min { p ≤ n / 2 : f ( i ) = f ( n / 2 ) for i ∈ [ p , n / 2 ] } • ℓ 1 ( f ) = min { p ≤ n / 2 : f ( i ) = f ( n / 2 ) for i ∈ [ n / 2 , n − p ] } • ℓ ( f ) = min { p : f ( i ) = f ( i + 1 ) for i ∈ [ p , n − p − 1 ] } • r ( f ) = min { p : f ( i ) = f ( i + 2 ) for i ∈ [ p , n − p − 2 ] }
Ramsey numbers and Eval G
• R k Eval G • D k Eval G connections to Ramsey theory Communication complexity: 1 since x 1 x k 0 x 1 x 2 x k 29 Eval G For any Abelian group G and x 1 , . . . , x k ∈ G : Eval G ( x 1 , . . . , x k ) = 1 ⇔ x 1 + · · · + x k = 0
29 Communication complexity: Eval G For any Abelian group G and x 1 , . . . , x k ∈ G : Eval G ( x 1 , . . . , x k ) = 1 ⇔ x 1 + · · · + x k = 0 • R || k ( Eval G ) = O ( 1 ) since x 1 + · · · + x k = 0 ⇔ x 1 = − ( x 2 · · · + x k ) • D k ( Eval G ) → connections to Ramsey theory
• c k G = min # of colors to avoid monochromatic k -dim corner in G k • r k G = size of largest subset of G k without any k -dim corner log c k G 1 Eval G log c k G k -dimensional corner in G k : Ramsey numbers: Chandra, Furst and Lipton [CFL83]: D k k 30 Ramsey numbers ( x 1 , x 2 , . . . , x k ) , ( x 1 + λ, x 2 , . . . , x k ) , ( x 1 , x 2 + λ, . . . , x k ) , . . . , ( x 1 , x 2 , . . . , x k + λ )
log c k G 1 Eval G log c k G k -dimensional corner in G k : Ramsey numbers: Chandra, Furst and Lipton [CFL83]: D k k 30 Ramsey numbers ( x 1 , x 2 , . . . , x k ) , ( x 1 + λ, x 2 , . . . , x k ) , ( x 1 , x 2 + λ, . . . , x k ) , . . . , ( x 1 , x 2 , . . . , x k + λ ) • c ∠ k ( G ) = min # of colors to avoid monochromatic k -dim corner in G k • r ∠ k ( G ) = size of largest subset of G k without any k -dim corner
k -dimensional corner in G k : Ramsey numbers: Chandra, Furst and Lipton [CFL83]: 30 Ramsey numbers ( x 1 , x 2 , . . . , x k ) , ( x 1 + λ, x 2 , . . . , x k ) , ( x 1 , x 2 + λ, . . . , x k ) , . . . , ( x 1 , x 2 , . . . , x k + λ ) • c ∠ k ( G ) = min # of colors to avoid monochromatic k -dim corner in G k • r ∠ k ( G ) = size of largest subset of G k without any k -dim corner log ( c ∠ k ( G )) ≤ D k + 1 ( Eval G ) ≤ k + log ( c ∠ k ( G ))
Chandra, Furst and Lipton [CFL83]: 31 Connections log ( c ∠ k ( G )) ≤ D k + 1 ( Eval G ) ≤ k + log ( c ∠ k ( G ))
• D 3 Eval n • an explicit large corner free set over 2 [ACFN15] p log 2 n p p log n when k • the first explicit large corner-free set over C k 2 p k • c k n p 2 k 2 1 n Our result: n p , of size p nk p log 3 n [CS14] 32 p 1 [LM07] p : • the proofs are easier and cleaner • they can be adapted to any other group [Gre05] Prior work: [ACFN15] p • c k n 2 2 n 2 k 2 n k 1 Ramsey numbers and Eval F n Motivations for G = F n p ∈ Sym ◦ Sym p • Eval F n
• the first explicit large corner-free set over C k 2 p k 32 p k 2 p : • the proofs are easier and cleaner • they can be adapted to any other group [Gre05] p nk Prior work: p , of size n Our result: [ACFN15] Ramsey numbers and Eval F n Motivations for G = F n p ∈ Sym ◦ Sym p • Eval F n • D 3 ( Eval F n p ) = ω ( 1 ) [LM07] ( 2 n / 2 k − 2 n k + 1 ) • c ∠ k ( F n 2 ) ≤ O • an explicit large corner free set over F n 2 [ACFN15] p ) ≤ 2 O ( p log 2 n ) p O ( p log n ) when k > 1 + p log ( 3 n ) [CS14] • c ∠ k ( F n
32 p p nk p : • the proofs are easier and cleaner • they can be adapted to any other group [Gre05] p , of size Our result: Prior work: [ACFN15] Ramsey numbers and Eval F n Motivations for G = F n p ∈ Sym ◦ Sym p • Eval F n • D 3 ( Eval F n p ) = ω ( 1 ) [LM07] ( 2 n / 2 k − 2 n k + 1 ) • c ∠ k ( F n 2 ) ≤ O • an explicit large corner free set over F n 2 [ACFN15] p ) ≤ 2 O ( p log 2 n ) p O ( p log n ) when k > 1 + p log ( 3 n ) [CS14] • c ∠ k ( F n • the first explicit large corner-free set over F n C k 2 p k + k 2
k is seen as a k • n i c M 0 N i n i c M is a corner-free set. 1 i then S k C k 2 p k 33 1 0 k N i If k log n log 1 k p 1 and N i k i p p k n c p nk k 2 i n p For any c p Definitions: • M n p n matrix over p • d c c j Hamming distance between columns c and c j number of columns at distance i to c in M k p , N k 0 and N 0 N k 1 0 such that k i n : S k c M Results Our contribution: the first explicit large corner-free set in F n
0 N i n i c M is a corner-free set. 1 i then S k C k 2 p k 33 p N i If k log n log 1 1 k 1 and N i 0 i p p k n c p nk k 2 k i N k 0 such that For any c k p , N k 0 and N 0 k 1 k Definitions: i n : S k c M n p p Results Our contribution: the first explicit large corner-free set in F n • M ∈ ( F n p ) k is seen as a k × n matrix over F p • d ( c , c j ) = Hamming distance between columns c and c j • n i , c ( M ) = number of columns at distance i to c in M
1 i then S k C k 2 p k 33 1 k 2 p nk c n p k p i k and N i 1 p log 1 S k If k p Definitions: log n Results Our contribution: the first explicit large corner-free set in F n • M ∈ ( F n p ) k is seen as a k × n matrix over F p • d ( c , c j ) = Hamming distance between columns c and c j • n i , c ( M ) = number of columns at distance i to c in M For any c ∈ F k p , N k = 0 and N 0 , . . . , N k − 1 ≥ 0 such that ∑ k i = 0 N i = n : p ) k : ∀ i ∈ { 0 , . . . , k } , n i , c ( M ) = N i } c = { M ∈ ( F n is a corner-free set.
33 S k p nk n p k i 1 log n log p Definitions: Results Our contribution: the first explicit large corner-free set in F n • M ∈ ( F n p ) k is seen as a k × n matrix over F p • d ( c , c j ) = Hamming distance between columns c and c j • n i , c ( M ) = number of columns at distance i to c in M For any c ∈ F k p , N k = 0 and N 0 , . . . , N k − 1 ≥ 0 such that ∑ k i = 0 N i = n : p ) k : ∀ i ∈ { 0 , . . . , k } , n i , c ( M ) = N i } c = { M ∈ ( F n is a corner-free set. ⌈ ⌉ ) ( p − 1 ) i ⌊( k ⌋ If k ≥ and N i = then | S k c | ≥ ( ) C k 2 p k + k 2 1 + p − 1
The log n barrier and composed functions
34 Player 1 ( x 1 ) f g n g 3 g 2 g 1 k . . . n Player k ( x k ) Player 2 ( x 2 ) Composed functions Given f : { 0 , 1 } n → { 0 , 1 } and − → g = ( g 1 , . . . , g n ) where g i : { 0 , 1 } k → { 0 , 1 } : f ◦ − → g ( x 1 , . . . , x k ) = f ( . . . , g i ( x 1 , i , . . . , x k , i ) , . . . ) x 1 , 1 x 1 , 2 x 1 , 3 x 1 , n x 2 , 1 x 2 , 2 x 2 , 3 x 2 , n · · · x k , 1 x k , 2 x k , 3 x k , n
• most of the important functions: GIP 35 AND • major open problems still unknown for composed functions Sym Sym AND NOR Sym, DISJ Sym MAJ MAJ Sym, Sym MOD 2 Definitions: • very simple structure Motivations: Composed functions • f ◦ g if g 1 = · · · = g n • Symmetric = invariant under any permutation of the input • Any ◦ − − → Any, Any ◦ Any, Sym ◦ − − → Any, Sym ◦ Sym...
35 Definitions: • major open problems still unknown for composed functions • very simple structure Motivations: Composed functions • f ◦ g if g 1 = · · · = g n • Symmetric = invariant under any permutation of the input • Any ◦ − − → Any, Any ◦ Any, Sym ◦ − − → Any, Sym ◦ Sym... • most of the important functions: GIP = MOD 2 ◦ AND ∈ Sym ◦ Sym, MAJ ◦ MAJ ∈ Sym ◦ Sym, DISJ = NOR ◦ AND ∈ Sym ◦ Sym
log 2 n • D k f log 3 n • D k f log 3 n • D k f none of the functions in Sym 36 Comp [BGKL04] Any can break the log n barrier Any [ACFN15] Sym g for f g for f Sym g g AND [Gro94] Sym g for f g log n : When k Prior work Conjecture ([BKL95]): MAJ ◦ MAJ breaks the log n barrier
none of the functions in Sym 36 Any [ACFN15] Any can break the log n barrier Prior work Conjecture ([BKL95]): MAJ ◦ MAJ breaks the log n barrier When k = Ω( log n ) : ( ) • D k ( f ◦ g ) = O for f ◦ g ∈ Sym ◦ AND [Gro94] log 2 n ( ) • D || k ( f ◦ g ) = O for f ◦ g ∈ Sym ◦ Comp [BGKL04] log 3 n g ∈ Sym ◦ − − → k ( f ◦ − → for f ◦ − → ( ) • D || g ) = O log 3 n
36 Any [ACFN15] Any can break the log n barrier Prior work Conjecture ([BKL95]): MAJ ◦ MAJ breaks the log n barrier When k = Ω( log n ) : ( ) • D k ( f ◦ g ) = O for f ◦ g ∈ Sym ◦ AND [Gro94] log 2 n ( ) • D || k ( f ◦ g ) = O for f ◦ g ∈ Sym ◦ Comp [BGKL04] log 3 n g ∈ Sym ◦ − − → k ( f ◦ − → for f ◦ − → ( ) • D || g ) = O log 3 n → none of the functions in Sym ◦ − − →
0 1 k t n breaks the barrier 37 Player 2 ( x 2 ) MAJ • Conjecture : MAJ 0 1 • MAJ t f g n g 1 k Player k ( x k ) Player 1 ( x 1 ) . . . Composed functions of block-width t t · n x 1 , 1 x 1 , t x 1 , tn x 2 , 1 x 2 , t x 2 , tn · · · · · · x k , 1 x k , t x k , tn
37 Player 1 ( x 1 ) f . . . g n g 1 k Player k ( x k ) Player 2 ( x 2 ) Composed functions of block-width t t · n x 1 , 1 x 1 , t x 1 , tn x 2 , 1 x 2 , t x 2 , tn · · · · · · x k , 1 x k , t x k , tn • MAJ t : { 0 , 1 } k · t → { 0 , 1 } • Conjecture : MAJ ◦ MAJ √ n breaks the barrier
0 1 n g n where g i g i x 1 i Any p Any Any p Sym 38 p Given f 0 1 and g g 1 k g x 1 0 1 : f g n x k f x k i Any Any p f g 1 . . . Player k ( x k ) Player 2 ( x 2 ) Player 1 ( x 1 ) k Composed functions of block-width t t · n x 1 , 1 x 1 , t x 1 , tn x 2 , 1 x 2 , t x 2 , tn · · · · · · x k , 1 x k , t x k , tn { 0 , 1 } t ∼ F 2 t
0 1 n g n where g i g i x 1 i Any p Any Any p Sym 38 k g n f Given f 0 1 and g g 1 f p 0 1 : g 2 g x 1 x k f x k i Any Any p g 3 g 1 . . n Player k ( x k ) Player 2 ( x 2 ) Player 1 ( x 1 ) k . Composed functions of block-width t x 1 , 1 x 1 , 2 x 1 , 3 x 1 , n x 2 , 1 x 2 , 2 x 2 , 3 x 2 , n · · · x k , 1 x k , 2 x k , 3 x k , n F 2 t
38 Player 1 ( x 1 ) f g n g 3 g 2 g 1 k n Player 2 ( x 2 ) Player k ( x k ) . . . Composed functions of block-width t x 1 , 1 x 1 , 2 x 1 , 3 x 1 , n x 2 , 1 x 2 , 2 x 2 , 3 x 2 , n · · · x k , 1 x k , 2 x k , 3 x k , n F 2 t Given f : { 0 , 1 } n → { 0 , 1 } and − → g = ( g 1 , . . . , g n ) where g i : F k p → { 0 , 1 } : f ◦ − → g ( x 1 , . . . , x k ) = f ( . . . , g i ( x 1 , i , . . . , x k , i ) , . . . ) → Any ◦ − − → Any p , Any ◦ Any p , Sym ◦ Any p , . . .
log 3 n • D k f Any 2 [ACFN15] • D k f Any p and p New results for constant p : • D k f Sym p ( k • D k f Comp p ( k • MAJ MAJ t cannot break the barrier for constant t Sym polylog n ) 39 g g g Sym polylog n ) polylogn for f polylog n [CS14] polylogn for f g Sym g polylogn for f g Sym g for f g polylog n : When k Prior work Conjecture: MAJ ◦ MAJ √ log n breaks the log n barrier
• D k f Sym p ( k • D k f Comp p ( k • MAJ MAJ t cannot break the barrier for constant t g polylogn for f g Sym 39 polylog n ) New results for constant p : polylogn for f g Sym polylog n ) g Prior work Conjecture: MAJ ◦ MAJ √ log n breaks the log n barrier When k = Ω( polylog n ) : g ∈ Sym ◦ − − → k ( f ◦ − → for f ◦ − → ( ) • D || g ) = O log 3 n Any 2 [ACFN15] • D k ( f ◦ g ) = O ( polylogn ) for f ◦ g ∈ Sym ◦ − − → Any p and p ≤ polylog n [CS14]
39 New results for constant p : Prior work Conjecture: MAJ ◦ MAJ √ log n breaks the log n barrier When k = Ω( polylog n ) : g ∈ Sym ◦ − − → k ( f ◦ − → for f ◦ − → ( ) • D || g ) = O log 3 n Any 2 [ACFN15] • D k ( f ◦ g ) = O ( polylogn ) for f ◦ g ∈ Sym ◦ − − → Any p and p ≤ polylog n [CS14] • D || k ( f ◦ g ) = O ( polylogn ) for f ◦ g ∈ Sym ◦ Sym p ( k = polylog n ) • D || k ( f ◦ g ) = O ( polylogn ) for f ◦ g ∈ Sym ◦ Comp p ( k ≥ polylog n ) • MAJ ◦ MAJ t cannot break the barrier for constant t
40 1 1 2 1 0 0 0 2 1 0 1 2 2 0 0 3 0 0 3 1 0 1 0 2 0 Recovering the y i j ’s is enough since f and g are symmetric 0 2 3 2 g g g f 0 0 1 2 0 2 3 1 1 0 0 0 0 2 1 1 0 1 3 0 1 Proof sketch Symmetric f and g over F 3 : · · · y i , j = # columns with i one’s and j two’s
40 1 1 2 1 0 0 0 2 1 0 1 2 2 0 0 3 0 0 3 1 0 1 0 2 0 Recovering the y i j ’s is enough since f and g are symmetric 0 2 3 1 g g g f 0 0 1 2 0 2 3 2 1 1 0 0 0 2 1 0 1 0 1 3 0 Proof sketch Symmetric f and g over F 3 : · · · y i , j = # columns with i one’s and j two’s → y 0 , 0 = 1
40 1 1 2 1 0 0 0 2 1 0 1 2 2 0 0 3 0 0 3 1 0 1 0 2 0 Recovering the y i j ’s is enough since f and g are symmetric 0 2 3 1 g g g f 0 0 1 2 0 2 3 2 1 1 0 0 0 2 1 0 1 0 1 3 0 Proof sketch Symmetric f and g over F 3 : · · · y i , j = # columns with i one’s and j two’s → y 0 , 0 = 1 , y 1 , 0 = 2
40 1 1 2 1 0 0 0 2 1 0 1 2 2 0 0 3 0 0 3 1 0 1 0 2 0 Recovering the y i j ’s is enough since f and g are symmetric 0 2 3 1 g g g f 0 0 1 2 0 2 3 2 1 1 0 0 0 2 1 0 1 0 1 3 0 Proof sketch Symmetric f and g over F 3 : · · · y i , j = # columns with i one’s and j two’s → y 0 , 0 = 1 , y 1 , 0 = 2 , y 1 , 1 = 1
40 2 0 1 2 1 0 0 0 2 1 0 1 1 3 2 0 3 0 0 3 1 0 1 0 2 0 0 2 0 1 g g g f 0 0 1 2 0 2 2 3 1 0 0 0 3 1 0 1 1 1 0 2 Proof sketch Symmetric f and g over F 3 : · · · y i , j = # columns with i one’s and j two’s → y 0 , 0 = 1 , y 1 , 0 = 2 , y 1 , 1 = 1 , . . . Recovering the y i , j ’s is enough since f and g are symmetric
41 3 0 0 0 2 1 0 1 2 1 2 0 0 2 0 3 1 0 1 0 2 0 • Player 1 sends to the referee: a 1 • Players 2 to 5 do the same 1 0 1 3 0 1 2 0 2 3 2 1 1 0 0 1 1 0 0 2 0 0 3 0 2 0 1 1 Proof sketch i , j = # columns she sees with i one’s and j two’s → a 1 0 , 0 = 2 , a 1 1 , 0 = 1 , a 1 1 , 1 = 3 , . . .
41 3 0 0 0 2 1 0 1 2 1 2 0 0 2 0 3 1 0 1 0 2 0 • Player 1 sends to the referee: a 1 • Players 2 to 5 do the same 1 0 1 3 0 1 2 0 2 3 2 1 1 0 0 1 1 0 0 2 0 0 3 0 2 0 1 1 Proof sketch i , j = # columns she sees with i one’s and j two’s → a 1 0 , 0 = 2 , a 1 1 , 0 = 1 , a 1 1 , 1 = 3 , . . .
42 0 0 1 2 1 2 0 3 0 0 3 1 0 1 2 0 0 The referee computes: Note that: b i j k i j y i j i 1 y i 1 j j 1 y i j 1 1 2 0 0 0 1 2 0 2 3 2 1 1 1 0 3 1 0 1 3 0 1 2 1 0 0 2 0 1 0 0 2 0 Proof sketch b i , j = a 1 i , j + · · · + a 5 i , j
42 0 0 1 2 1 2 0 3 0 0 3 1 0 1 2 2 0 The referee computes: Note that: b i j k i j y i j i 1 y i 1 j j 1 y i j 1 0 1 0 0 0 1 2 0 2 3 2 1 1 1 0 3 0 1 1 3 0 1 2 1 0 0 2 0 1 0 0 2 0 Proof sketch b i , j = a 1 i , j + · · · + a 5 i , j • b 0 , 0 =
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