Depth Lower Bound against Circuits with Sparse Orientation Sajin - PowerPoint PPT Presentation

Depth Lower Bound against Circuits with Sparse Orientation Sajin Koroth 1 Jayalal Sarma 1 1 Algorithms and Complexity Theory Lab Department of Computer Science IIT Madras Chennai Theory Day, 2013 1 / 17

Circuit Model A circuit family C computing a Boolean function f is such that for each n ∈ N , ∀ x ∈ { 0 , 1 } n f ( x ) = C n ( x ) , C n ∈ C C 3 C 4 x 4 x 1 x 2 x 3 x 1 x 2 x 3 2 / 17

Circuit Model A circuit family C computing a Boolean function f is such that for each n ∈ N , ∀ x ∈ { 0 , 1 } n f ( x ) = C n ( x ) , C n ∈ C C 3 C 4 A computation model which can compute any Boolean function . x 4 x 1 x 2 x 3 x 1 x 2 x 3 2 / 17

Circuit Model A circuit family C computing a Boolean function f is such that for each n ∈ N , ∀ x ∈ { 0 , 1 } n f ( x ) = C n ( x ) , C n ∈ C C 3 C 4 A computation model which can compute any Boolean function . A circuit is a Direct Acyclic Graph x 4 x 1 x 2 x 3 x 1 x 2 x 3 2 / 17

Circuit Model A circuit family C computing a Boolean function f is such that for each n ∈ N , ∀ x ∈ { 0 , 1 } n f ( x ) = C n ( x ) , C n ∈ C C 3 C 4 A computation model which can compute any Boolean function . A circuit is a Direct Acyclic Graph Important Parameters : Size : Number of gates(internal nodes), Depth : The longest path from root to any leaf . x 4 x 1 x 2 x 3 x 1 x 2 x 3 2 / 17

Circuit Model A circuit family C computing a Boolean function f is such that for each n ∈ N , ∀ x ∈ { 0 , 1 } n f ( x ) = C n ( x ) , C n ∈ C C 3 C 4 A computation model which can compute any Boolean function . A circuit is a Direct Acyclic Graph Important Parameters : Size : Number of gates(internal nodes), Depth : The longest path from root to any leaf . x 4 x 1 x 2 x 3 x 1 x 2 x 3 Assumption : Internal gates are {∧ , ∨ , ¬} with fan-in { 2 , 2 , 1 } respectively. (Bounded fan-in) 2 / 17

Circuit Lower bounds - motivation Nice combinatorial structure (graphs with additional information) 3 / 17

Circuit Lower bounds - motivation Nice combinatorial structure (graphs with additional information) Strong reasons to believe NP �⊂ PolySize ∗ whereas P ⊂ PolySize . Hence NP �⊂ PolySize = ⇒ P � = NP ∗ If NP has polynomial size circuits then PH = Σ 2 3 / 17

Circuit Lower bounds - motivation Nice combinatorial structure (graphs with additional information) Strong reasons to believe NP �⊂ PolySize ∗ whereas P ⊂ PolySize . Hence NP �⊂ PolySize = ⇒ P � = NP Depth captures “efficient” parallel computation. 3 / 17

Circuit Lower bounds - motivation Nice combinatorial structure (graphs with additional information) Strong reasons to believe NP �⊂ PolySize ∗ whereas P ⊂ PolySize . Hence NP �⊂ PolySize = ⇒ P � = NP Depth captures “efficient” parallel computation. Depth ( Clique ) ? = ω ((log n ) k ) ( NP vs NC k † ) - does NP have efficient parallel algorithms ? † NC k - poly-size, poly-log((log n ) k ) depth 3 / 17

Restrictions on the model Known circuit lower bounds against general circuits are very weak (5 n size and 3 log n depth) 4 / 17

Restrictions on the model Known circuit lower bounds against general circuits are very weak (5 n size and 3 log n depth) Strong lower bound against monotone circuits 4 / 17

Restrictions on the model Known circuit lower bounds against general circuits are very weak (5 n size and 3 log n depth) Strong lower bound against monotone circuits A circuit is monotone iff it has no NOT( ¬ ) gates. 4 / 17

Restrictions on the model Known circuit lower bounds against general circuits are very weak (5 n size and 3 log n depth) Strong lower bound against monotone circuits A circuit is monotone iff it has no NOT( ¬ ) gates. Monotone circuits computes monotone functions (and every monotone function is computed by a monotone circuit) 4 / 17

Restrictions on the model Known circuit lower bounds against general circuits are very weak (5 n size and 3 log n depth) Strong lower bound against monotone circuits A circuit is monotone iff it has no NOT( ¬ ) gates. Monotone circuits computes monotone functions (and every monotone function is computed by a monotone circuit) A function is monotone iff ∀ x ≤ y , f ( x ) ≤ f ( y ) 4 / 17

Restrictions on the model Known circuit lower bounds against general circuits are very weak (5 n size and 3 log n depth) Strong lower bound against monotone circuits A circuit is monotone iff it has no NOT( ¬ ) gates. Monotone circuits computes monotone functions (and every monotone function is computed by a monotone circuit) A function is monotone iff ∀ x ≤ y , f ( x ) ≤ f ( y ) A function is monotone iff changing bits from 0 to 1 in the input cannot decrease the function value. 4 / 17

Monotone Depth lower bounds Clique { n , n / 2 } ( x ) ‡ where x is the undirected adjacency list of a graph is a monotone function, as adding edges cannot remove a clique. It is an NP -complete problem. ‡ Clique { n , n / 2 } ( x ) = 1 iff G x is an n -vertex graph containing a clique of size n / 2 § Ran Raz and Avi Wigderson. “Monotone circuits for matching require linear depth”. In: J. ACM 39.3 (July 1992), pp. 736–744. 5 / 17

Monotone Depth lower bounds Clique { n , n / 2 } ( x ) ‡ where x is the undirected adjacency list of a graph is a monotone function, as adding edges cannot remove a clique. It is an NP -complete problem. Raz and Wigderson : Perfect matching and Clique § requires Ω( n ) depth ‡ Clique { n , n / 2 } ( x ) = 1 iff G x is an n -vertex graph containing a clique of size n / 2 § Ran Raz and Avi Wigderson. “Monotone circuits for matching require linear depth”. In: J. ACM 39.3 (July 1992), pp. 736–744. 5 / 17

Karchmer Wigderson Game 2 bit Parity Circuit Aim Alice is given x ∈ f − (1) and Bob is given y ∈ f − (0), goal is to find i ∈ [ n ] such that x i � = y i 2 bit Parity KW protocol 1. Send x 1 2. If x 1 != y 1 output 1 and stop, else send y 2 3. Output 2 0 1 1 0 1 1 0 0 Alice x=01 Bob y=11 Alice x=01 Bob y=11 6 / 17

Karchmer Wigderson Game 2 bit Parity Circuit Aim Alice is given x ∈ f − (1) and Bob is given y ∈ f − (0), goal is to find i ∈ [ n ] such that x i � = y i 2 bit Parity KW protocol 0 1. Send x 1 2. If x 1 != y 1 output 1 and stop, else send y 2 3. Output 2 Alice x=01 Bob y=11 0 1 1 0 1 1 0 0 A x 1 =0 Alice x=01 Bob y=11 6 / 17

Karchmer Wigderson Game 2 bit Parity Circuit Aim Alice is given x ∈ f − (1) and Bob is given y ∈ f − (0), goal is to find i ∈ [ n ] such that x i � = y i 2 bit Parity KW protocol 1. Send x 1 2. If x 1 != y 1 output 1 and stop, else send y 2 0 3. Output 2 0 Alice x=01 Bob y=11 A 0 1 1 0 1 1 0 0 x 1 =0 B y=11 Alice x=01 Bob 1 6 / 17

Karchmer Wigderson Game 2 bit Parity Circuit Aim Alice is given x ∈ f − (1) and Bob is given y ∈ f − (0), goal is to find i ∈ [ n ] such that x i � = y i 2 bit Parity KW protocol 1. Send x 1 2. If x 1 != y 1 output 1 and stop, else send y 2 0 3. Output 2 0 Alice x=10 Bob y=11 A 0 1 1 0 1 1 0 0 x 1 =0 x 1 =1 B B Alice x=01 Bob y=11 1 2 6 / 17

Karchmer Wigderson Game 2 bit Parity Circuit Aim Alice is given x ∈ f − (1) and Bob is given y ∈ f − (0), goal is to find i ∈ [ n ] such that x i � = y i 2 bit Parity KW protocol 1. Send x 1 2. If x 1 != y 1 output 1 and stop, else send y 2 0 3. Output 2 0 Alice x=10 Bob y=00 A 0 1 1 0 1 1 0 0 x 1 =0 x 1 =1 B B Alice x=01 Bob y=11 1 2 6 / 17

Karchmer Wigderson Game 2 bit Parity Circuit Aim Alice is given x ∈ f − (1) and Bob is given y ∈ f − (0), goal is to find i ∈ [ n ] such that x i � = y i 2 bit Parity KW protocol 1. Send x 1 2. If x 1 != y 1 output 1 and stop, else send y 2 0 3. Output 2 0 Alice x=10 Bob y=00 A 0 1 1 0 1 1 0 0 x 1 =0 x 1 =1 B B Alice x=01 Bob y=11 1 1 2 6 / 17

Karchmer Wigderson Game 2 bit Parity Circuit Aim Alice is given x ∈ f − (1) and Bob is given y ∈ f − (0), goal is to find i ∈ [ n ] such that x i � = y i 2 bit Parity KW protocol 1. Send x 1 2. If x 1 != y 1 output 1 and stop, else send y 2 0 3. Output 2 0 Alice x=01 Bob y=00 A 0 1 1 0 1 1 0 0 x 1 =0 x 1 =1 B B Alice x=01 Bob y=11 1 1 2 6 / 17

Karchmer Wigderson Game 2 bit Parity Circuit Aim Alice is given x ∈ f − (1) and Bob is given y ∈ f − (0), goal is to find i ∈ [ n ] such that x i � = y i 2 bit Parity KW protocol 1. Send x 1 2. If x 1 != y 1 output 1 and stop, else send y 2 0 3. Output 2 0 Alice x=01 Bob y=00 A 0 1 1 0 1 1 0 0 x 1 =0 x 1 =1 B B Alice x=01 Bob y=11 1 2 1 2 6 / 17

Monotone KW-game ( KW + ( f )) KW + ( f ) Alice is given x ∈ f − 1 (1) and Bob is given y ∈ f − 1 (0) for a monotone function f . Goal : Find i ∈ [ n ] such that x i = 1 and y i = 0. 7 / 17

Monotone KW-game ( KW + ( f )) KW + ( f ) Alice is given x ∈ f − 1 (1) and Bob is given y ∈ f − 1 (0) for a monotone function f . Goal : Find i ∈ [ n ] such that x i = 1 and y i = 0. KW + ( f ) = MDepth ( f ) For every d -length protocol solving KW + ( f ) there is a corresponding d -depth monotone circuit computing f and vice versa. 7 / 17

Generalizing Monotone Lower bounds Raz and Wigderson : KW + ( f ) for Perfect matching and Clique ¶ is Ω( n ). ¶ Ran Raz and Avi Wigderson. “Monotone circuits for matching require linear depth”. In: J. ACM 39.3 (July 1992), pp. 736–744. 8 / 17