branching program size lower bounds via projective
play

Branching Program size lower bounds via Projective Dimension Sajin - PowerPoint PPT Presentation

Sep 24, 2023 •164 likes •1.2k views

Branching Program size lower bounds via Projective Dimension Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) Indian Institute of Technology, Madras Theory Lunch, Technion Sajin Koroth (joint work with Krishnamoorthy

  1. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  2. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  3. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  4. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  5. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  6. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  7. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  8. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  9. Projective Dimension and BP size lower bounds Projective Dimension Measure on bipartite graphs introduced by Pudlak and Rodl Graph G ( U , V , E ) . Assign subspaces from F d to vertices so that ( x , y ) ∈ E ⇐ ⇒ φ ( x ) ∩ φ ( y ) � = { 0 } Smallest such d : pd F ( G ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 7 / 23

  10. Projective Dimension and BP size lower bounds Projective Dimension Measure on bipartite graphs introduced by Pudlak and Rodl Graph G ( U , V , E ) . Assign subspaces from F d to vertices so that ( x , y ) ∈ E ⇐ ⇒ φ ( x ) ∩ φ ( y ) � = { 0 } Smallest such d : pd F ( G ) . x 1 x 2 x 3 x 4 00 00 01 01 10 10 11 11 G Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 7 / 23

  11. Projective Dimension and BP size lower bounds Projective Dimension Measure on bipartite graphs introduced by Pudlak and Rodl Graph G ( U , V , E ) . Assign subspaces from F d to vertices so that ( x , y ) ∈ E ⇐ ⇒ φ ( x ) ∩ φ ( y ) � = { 0 } Smallest such d : pd F ( G ) . x 1 x 2 x 3 x 4 span { e 1 } span { e 2 } 00 00 span { e 2 } span { e 1 } 01 01 span { e 2 } span { e 1 } 10 10 span { e 1 } span { e 2 } 11 11 pd F ( G ) = 2 G Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 7 / 23

  12. Projective Dimension and BP size lower bounds bpsize ( f ) ≥ pd ( f ) Theorem, (Pudlak and Rodl (1992)) Over any F , bpsize ( f ) ≥ pd F ( G f ) . To define the bipartite graph G f associated with a function f on 2 n variables, take some natural partition of the variable set into two equal parts Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 8 / 23

  13. Projective Dimension and BP size lower bounds bpsize ( f ) ≥ pd ( f ) Theorem, (Pudlak and Rodl (1992)) Over any F , bpsize ( f ) ≥ pd F ( G f ) . To define the bipartite graph G f associated with a function f on 2 n variables, take some natural partition of the variable set into two equal parts Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 8 / 23

  14. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  15. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  16. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  17. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  18. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  19. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  20. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  21. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  22. Projective Dimension and BP size lower bounds Proof contd. Let ( x , y ) be an input. And H x be the edge-subgraph of the branching program whose edges query variables in x . Similarly define H y . After the transformation f ( x , y ) = 1 iff H x ∪ H y contains a cycle Make sure that for any ( x , y ) s.t. f ( x , y ) = 1 this unique cycle has edges from both H x and H y . Any linear dependence in span { φ ( x ) , φ ( y ) } corresponds to a cycle in H x ∪ H y Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 10 / 23

  23. Projective Dimension and BP size lower bounds Proof contd. Let ( x , y ) be an input. And H x be the edge-subgraph of the branching program whose edges query variables in x . Similarly define H y . After the transformation f ( x , y ) = 1 iff H x ∪ H y contains a cycle Make sure that for any ( x , y ) s.t. f ( x , y ) = 1 this unique cycle has edges from both H x and H y . Any linear dependence in span { φ ( x ) , φ ( y ) } corresponds to a cycle in H x ∪ H y Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 10 / 23

  24. Projective Dimension and BP size lower bounds Proof contd. Let ( x , y ) be an input. And H x be the edge-subgraph of the branching program whose edges query variables in x . Similarly define H y . After the transformation f ( x , y ) = 1 iff H x ∪ H y contains a cycle Make sure that for any ( x , y ) s.t. f ( x , y ) = 1 this unique cycle has edges from both H x and H y . Any linear dependence in span { φ ( x ) , φ ( y ) } corresponds to a cycle in H x ∪ H y Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 10 / 23

  25. Projective Dimension and BP size lower bounds Proof contd. Let ( x , y ) be an input. And H x be the edge-subgraph of the branching program whose edges query variables in x . Similarly define H y . After the transformation f ( x , y ) = 1 iff H x ∪ H y contains a cycle Make sure that for any ( x , y ) s.t. f ( x , y ) = 1 this unique cycle has edges from both H x and H y . Any linear dependence in span { φ ( x ) , φ ( y ) } corresponds to a cycle in H x ∪ H y Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 10 / 23

  26. Projective Dimension and BP size lower bounds Known Bounds on pd F (Existential) N vertex bipartite G such that pd F ( G ) Field Result �� � N Infinite Babai et.al, 2002 Ω log N � √ � Ω N Finite Pudlak and Rodl, 1992 (Explicit) G = Complement of N perfect matchings. pd R ( G ) = Ω( log N ) � � N (Upper bounds) Bipartite G , pd R ( G ) = O log N To summarize, we only know linear lower bounds for projective dimension of explicit functions. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 11 / 23

  27. Projective Dimension and BP size lower bounds Known Bounds on pd F (Existential) N vertex bipartite G such that pd F ( G ) Field Result �� � N Infinite Babai et.al, 2002 Ω log N � √ � Ω N Finite Pudlak and Rodl, 1992 (Explicit) G = Complement of N perfect matchings. pd R ( G ) = Ω( log N ) � � N (Upper bounds) Bipartite G , pd R ( G ) = O log N To summarize, we only know linear lower bounds for projective dimension of explicit functions. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 11 / 23

  28. Projective Dimension and BP size lower bounds Known Bounds on pd F (Existential) N vertex bipartite G such that pd F ( G ) Field Result �� � N Infinite Babai et.al, 2002 Ω log N � √ � Ω N Finite Pudlak and Rodl, 1992 (Explicit) G = Complement of N perfect matchings. pd R ( G ) = Ω( log N ) � � N (Upper bounds) Bipartite G , pd R ( G ) = O log N To summarize, we only know linear lower bounds for projective dimension of explicit functions. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 11 / 23

  29. Projective Dimension and BP size lower bounds Known Bounds on pd F (Existential) N vertex bipartite G such that pd F ( G ) Field Result �� � N Infinite Babai et.al, 2002 Ω log N � √ � Ω N Finite Pudlak and Rodl, 1992 (Explicit) G = Complement of N perfect matchings. pd R ( G ) = Ω( log N ) � � N (Upper bounds) Bipartite G , pd R ( G ) = O log N To summarize, we only know linear lower bounds for projective dimension of explicit functions. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 11 / 23

  30. Gap Between Projective Dimension and BP Size An exponential gap! Our Result, a similar result known for Formulas and Graph Complexity by Jukna There exists (non-explicit) function f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } such that pd ( f ) = O ( n ) , but bpsize ( f ) = Ω( 2 n / n ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 12 / 23

  31. Gap Between Projective Dimension and BP Size An exponential gap! Our Result, a similar result known for Formulas and Graph Complexity by Jukna There exists (non-explicit) function f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } such that pd ( f ) = O ( n ) , but bpsize ( f ) = Ω( 2 n / n ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 12 / 23

  32. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  33. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  34. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  35. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  36. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  37. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  38. Bridging the Gap : Bitwise Projective Dimension Bridging the gap The gap example gave an assignment which is of low projective dimension, but it may not be easy (read poly in n ) to describe The assignment constructed from branching program by Pudlak and Rodl is easy to describe. There are 4 n subspaces, 2 for each of the 2 n bits whose various spans create all the subspaces assigned to the 2 n + 2 n vertices of the bipartite graph For each i ∈ [ 2 n ] and b ∈ { 0 , 1 } , look at the edges querying x i = b . The span of the vectors assigned to these edges constitute these building block sub-spaces. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 14 / 23

  39. Bridging the Gap : Bitwise Projective Dimension Bridging the gap The gap example gave an assignment which is of low projective dimension, but it may not be easy (read poly in n ) to describe The assignment constructed from branching program by Pudlak and Rodl is easy to describe. There are 4 n subspaces, 2 for each of the 2 n bits whose various spans create all the subspaces assigned to the 2 n + 2 n vertices of the bipartite graph For each i ∈ [ 2 n ] and b ∈ { 0 , 1 } , look at the edges querying x i = b . The span of the vectors assigned to these edges constitute these building block sub-spaces. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 14 / 23

  40. Bridging the Gap : Bitwise Projective Dimension Bridging the gap The gap example gave an assignment which is of low projective dimension, but it may not be easy (read poly in n ) to describe The assignment constructed from branching program by Pudlak and Rodl is easy to describe. There are 4 n subspaces, 2 for each of the 2 n bits whose various spans create all the subspaces assigned to the 2 n + 2 n vertices of the bipartite graph For each i ∈ [ 2 n ] and b ∈ { 0 , 1 } , look at the edges querying x i = b . The span of the vectors assigned to these edges constitute these building block sub-spaces. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 14 / 23

  41. Bridging the Gap : Bitwise Projective Dimension Bridging the gap The gap example gave an assignment which is of low projective dimension, but it may not be easy (read poly in n ) to describe The assignment constructed from branching program by Pudlak and Rodl is easy to describe. There are 4 n subspaces, 2 for each of the 2 n bits whose various spans create all the subspaces assigned to the 2 n + 2 n vertices of the bipartite graph For each i ∈ [ 2 n ] and b ∈ { 0 , 1 } , look at the edges querying x i = b . The span of the vectors assigned to these edges constitute these building block sub-spaces. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 14 / 23

  42. Bridging the Gap : Bitwise Projective Dimension Bitwise Decomposable Projective Dimension Definition For f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } , bpdim ( f ) ≤ d if there exists C = { U a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , D = { V a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , such that U x i � � φ ( x ) = span i ∈ [ n ] , i Each U a i is a span of difference of standard basis vectors. Similarly each V i a If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m . Similar condition for D . C , D subspaces from F d 2 Main Result bitpdim ( f ) = Ω( bpsize ( f ) 1 / 6 ) Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 15 / 23

  43. Bridging the Gap : Bitwise Projective Dimension Bitwise Decomposable Projective Dimension Definition For f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } , bpdim ( f ) ≤ d if there exists C = { U a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , D = { V a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , such that U x i � � φ ( x ) = span i ∈ [ n ] , i Each U a i is a span of difference of standard basis vectors. Similarly each V i a If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m . Similar condition for D . C , D subspaces from F d 2 Main Result bitpdim ( f ) = Ω( bpsize ( f ) 1 / 6 ) Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 15 / 23

  44. Bridging the Gap : Bitwise Projective Dimension Bitwise Decomposable Projective Dimension Definition For f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } , bpdim ( f ) ≤ d if there exists C = { U a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , D = { V a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , such that U x i � � φ ( x ) = span i ∈ [ n ] , i Each U a i is a span of difference of standard basis vectors. Similarly each V i a If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m . Similar condition for D . C , D subspaces from F d 2 Main Result bitpdim ( f ) = Ω( bpsize ( f ) 1 / 6 ) Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 15 / 23

  45. Bridging the Gap : Bitwise Projective Dimension Bitwise Decomposable Projective Dimension Definition For f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } , bpdim ( f ) ≤ d if there exists C = { U a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , D = { V a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , such that U x i � � φ ( x ) = span i ∈ [ n ] , i Each U a i is a span of difference of standard basis vectors. Similarly each V i a If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m . Similar condition for D . C , D subspaces from F d 2 Main Result bitpdim ( f ) = Ω( bpsize ( f ) 1 / 6 ) Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 15 / 23

  46. Bridging the Gap : Bitwise Projective Dimension Bitwise Decomposable Projective Dimension Definition For f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } , bpdim ( f ) ≤ d if there exists C = { U a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , D = { V a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , such that U x i � � φ ( x ) = span i ∈ [ n ] , i Each U a i is a span of difference of standard basis vectors. Similarly each V i a If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m . Similar condition for D . C , D subspaces from F d 2 Main Result bitpdim ( f ) = Ω( bpsize ( f ) 1 / 6 ) Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 15 / 23

  47. Bridging the Gap : Bitwise Projective Dimension bitpdim assignment from Branching Programs Excpet for, If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m , all the other conditions are satisfied by Pudlak Rodl Construction Modify the branching program so that no two edges which share an end vertex query variables from the same partition This can be done by blowing up the size of the given branching program by a factor of at most 4. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 16 / 23

  48. Bridging the Gap : Bitwise Projective Dimension bitpdim assignment from Branching Programs Excpet for, If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m , all the other conditions are satisfied by Pudlak Rodl Construction Modify the branching program so that no two edges which share an end vertex query variables from the same partition This can be done by blowing up the size of the given branching program by a factor of at most 4. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 16 / 23

  49. Bridging the Gap : Bitwise Projective Dimension bitpdim assignment from Branching Programs Excpet for, If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m , all the other conditions are satisfied by Pudlak Rodl Construction Modify the branching program so that no two edges which share an end vertex query variables from the same partition This can be done by blowing up the size of the given branching program by a factor of at most 4. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 16 / 23

  50. Bridging the Gap : Bitwise Projective Dimension bitpdim assignment from Branching Programs Excpet for, If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m , all the other conditions are satisfied by Pudlak Rodl Construction Modify the branching program so that no two edges which share an end vertex query variables from the same partition This can be done by blowing up the size of the given branching program by a factor of at most 4. y i =1 x i x i y i y i =0 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 16 / 23

  51. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) We describe a space bounded algorithm which given the bitpdim assignment as an advice, and two inputs ( x , y ) computes whether f ( x , y ) = 1. implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i Argue that any linear dependence in span { φ ( x ) ∪ φ ( y ) } is a cycle in G ∗ . Coordinate-wise disjointedness of the basis vectors constituting U x i i and U x j ensure that there is no cycle involving just edges from H x j Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 17 / 23

  52. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) We describe a space bounded algorithm which given the bitpdim assignment as an advice, and two inputs ( x , y ) computes whether f ( x , y ) = 1. implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i Argue that any linear dependence in span { φ ( x ) ∪ φ ( y ) } is a cycle in G ∗ . Coordinate-wise disjointedness of the basis vectors constituting U x i i and U x j ensure that there is no cycle involving just edges from H x j Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 17 / 23

  53. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) We describe a space bounded algorithm which given the bitpdim assignment as an advice, and two inputs ( x , y ) computes whether f ( x , y ) = 1. implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i Argue that any linear dependence in span { φ ( x ) ∪ φ ( y ) } is a cycle in G ∗ . Coordinate-wise disjointedness of the basis vectors constituting U x i i and U x j ensure that there is no cycle involving just edges from H x j Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 17 / 23

  54. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) We describe a space bounded algorithm which given the bitpdim assignment as an advice, and two inputs ( x , y ) computes whether f ( x , y ) = 1. implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i Argue that any linear dependence in span { φ ( x ) ∪ φ ( y ) } is a cycle in G ∗ . Coordinate-wise disjointedness of the basis vectors constituting U x i i and U x j ensure that there is no cycle involving just edges from H x j Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 17 / 23

  55. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) Given ( x , y ) implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i f ( x , y ) = 1 iff there is a cycle in G ∗ check for a cycle in G ∗ . Can be done in space 5log | G ∗ | | G ∗ | = bitpdim ( f ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 18 / 23

  56. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) Given ( x , y ) implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i f ( x , y ) = 1 iff there is a cycle in G ∗ check for a cycle in G ∗ . Can be done in space 5log | G ∗ | | G ∗ | = bitpdim ( f ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 18 / 23

  57. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) Given ( x , y ) implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i f ( x , y ) = 1 iff there is a cycle in G ∗ check for a cycle in G ∗ . Can be done in space 5log | G ∗ | | G ∗ | = bitpdim ( f ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 18 / 23

  58. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) Given ( x , y ) implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i f ( x , y ) = 1 iff there is a cycle in G ∗ check for a cycle in G ∗ . Can be done in space 5log | G ∗ | | G ∗ | = bitpdim ( f ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 18 / 23

  59. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds The best bitpdim lower bound we get from the best known branching programs lower bounds is only sub-linear The best known pd lower bound is linear Can we get a super-linear lower bound ? Yes, but the proof we could come up with relies on using Nechiporuk’s method Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 19 / 23

  60. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds The best bitpdim lower bound we get from the best known branching programs lower bounds is only sub-linear The best known pd lower bound is linear Can we get a super-linear lower bound ? Yes, but the proof we could come up with relies on using Nechiporuk’s method Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 19 / 23

  61. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds The best bitpdim lower bound we get from the best known branching programs lower bounds is only sub-linear The best known pd lower bound is linear Can we get a super-linear lower bound ? Yes, but the proof we could come up with relies on using Nechiporuk’s method Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 19 / 23

  62. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds The best bitpdim lower bound we get from the best known branching programs lower bounds is only sub-linear The best known pd lower bound is linear Can we get a super-linear lower bound ? Yes, but the proof we could come up with relies on using Nechiporuk’s method Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 19 / 23

  63. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  64. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  65. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  66. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  67. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  68. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  69. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  70. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  71. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  72. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  73. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  74. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  75. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  76. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  77. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

  78. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

  79. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

  80. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

  81. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

  82. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013,

Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013,

Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013, Warsaw Andrew Drucker Kernel-Size Lower Bounds Part 2/3 Andrew Drucker Kernel-Size Lower Bounds Note These slides are a slightly revised version

803 views • 51 slides
Kernel-size lower bounds: accompanying exercises Andrew Drucker April 19, 2013 These

Kernel-size lower bounds: accompanying exercises Andrew Drucker April 19, 2013 These

Kernel-size lower bounds: accompanying exercises Andrew Drucker April 19, 2013 These exercises are intended to aimed at building background to understand recent work on complexity-theoretic evidence for kernel-size lower bounds [BDFH09, FS11,

587 views • 12 slides
Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013,

Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013,

Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013, Warsaw Andrew Drucker Kernel-Size Lower Bounds Part 1/3 Andrew Drucker Kernel-Size Lower Bounds Note These slides are taken (with minor

1.3k views • 95 slides
Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013,

Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013,

Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013, Warsaw Andrew Drucker Kernel-Size Lower Bounds Part 3/3 Andrew Drucker Kernel-Size Lower Bounds Note These slides are taken (with minor

1.04k views • 88 slides
172 Plan 1. Space Complexity in Resolution 2. Space lower bounds for Random Formulas 3.

172 Plan 1. Space Complexity in Resolution 2. Space lower bounds for Random Formulas 3.

172 Plan 1. Space Complexity in Resolution 2. Space lower bounds for Random Formulas 3. Combinatorial Characterization of width 4. Width vs Space 5. Feasible Interpolation for Resolution and size lower bounds 6. Proof Search,

1.52k views • 83 slides
Conspiracies between Learning Algorithms, Lower Bounds, and Pseudorandomness Igor Carboni

Conspiracies between Learning Algorithms, Lower Bounds, and Pseudorandomness Igor Carboni

Conspiracies between Learning Algorithms, Lower Bounds, and Pseudorandomness Igor Carboni Oliveira University of Oxford Joint work with Rahul Santhanam (Oxford) Context Minor algorithmic improvements imply lower bounds (Williams, 2010). NEXP

696 views • 30 slides
Lower bounds for parameterized problems Dniel Marx 1 1 Institute for Computer Science and

Lower bounds for parameterized problems Dniel Marx 1 1 Institute for Computer Science and

Lower bounds for parameterized problems Dniel Marx 1 1 Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI) Budapest, Hungary ICERM Providence, RI April 26, 2014 1 Lower bounds So far we have seen positive

1.14k views • 94 slides
Lower Bounds on the Probability of a Finite Union of Events Jun Yang (joint work with Fady

Lower Bounds on the Probability of a Finite Union of Events Jun Yang (joint work with Fady

Lower Bounds on the Probability of a Finite Union of Events Jun Yang (joint work with Fady Alajaji and Glen Takahara) Department of Mathematics and Statistics, Queens University, Kingston, Canada ISIT 2014 Lower Bounds on P ( N Jun

228 views • 19 slides
Circuit Lower-bounds Lecture 24 Weak circuits are indeed weak 1 Circuit Lower-bounds 2

Circuit Lower-bounds Lecture 24 Weak circuits are indeed weak 1 Circuit Lower-bounds 2

Circuit Lower-bounds Lecture 24 Weak circuits are indeed weak 1 Circuit Lower-bounds 2 Circuit Lower-bounds Today: 2 Circuit Lower-bounds Today: PARITY AC 0 2 Circuit Lower-bounds Today: PARITY AC 0 Two different proofs! (Latter

1.59k views • 137 slides
Stronger Connections Between Circuit Analysis and Circuit Lower Bounds, via PCPs of Proximity

Stronger Connections Between Circuit Analysis and Circuit Lower Bounds, via PCPs of Proximity

Stronger Connections Between Circuit Analysis and Circuit Lower Bounds, via PCPs of Proximity Lijie Chen Ryan Williams Context: The Algorithmic Method for Proving Circuit Lower Bounds Proving limitations on non-uniform circuits is extremely

796 views • 31 slides
Techniques to lower bound extension complexity Thomas Rothvoss UW Seattle Known lower bounds on

Techniques to lower bound extension complexity Thomas Rothvoss UW Seattle Known lower bounds on

Techniques to lower bound extension complexity Thomas Rothvoss UW Seattle Known lower bounds on extended formulation Kaibel, Razborovs Inform. SA Weltge symmetry arg. theory + Fourier COR/ yes yes yes ? TSP [KW13]

1.25k views • 108 slides
Exponential Lower Bounds for Polytopes in Combinatorial Optimization Ronald de Wolf Joint with

Exponential Lower Bounds for Polytopes in Combinatorial Optimization Ronald de Wolf Joint with

Exponential Lower Bounds for Polytopes in Combinatorial Optimization Ronald de Wolf Joint with Samuel Fiorini (ULB), Serge Massar (ULB), Sebastian Pokutta (Erlangen), Hans Raj Tiwary (ULB) Exponential Lower Bounds for Polytopes in

1.19k views • 94 slides
Lower Bounds for Geometric Diameter Problems Herv e Fournier University of Versailles

Lower Bounds for Geometric Diameter Problems Herv e Fournier University of Versailles

Lower Bounds for Geometric Diameter Problems Herv e Fournier University of Versailles St-Quentin en Yvelines Antoine Vigneron INRA Jouy-en-Josas Lower Bounds for Geometric Diameter Problems p.1/40 Outline Review of previous work on

563 views • 40 slides
Lecture 2. Upper and lower bounds for subgaussian matrices The -net method refined 1 Random

Lecture 2. Upper and lower bounds for subgaussian matrices The -net method refined 1 Random

Lecture 2. Upper and lower bounds for subgaussian matrices The -net method refined 1 Random processes. Multiscale -net method: Dudleys inequality 2 Upper and lower bounds Our goal: upper and lower bounds on random matrices. In Lecture

535 views • 17 slides
COMP 3170 - Analysis of Algorithms & Data Structures Shahin Kamali Lower Bounds CLRS 8.1

COMP 3170 - Analysis of Algorithms & Data Structures Shahin Kamali Lower Bounds CLRS 8.1

COMP 3170 - Analysis of Algorithms & Data Structures Shahin Kamali Lower Bounds CLRS 8.1 University of Manitoba Lower Bounds Introductions Assume you design an algorithm that solves a given problem P in ( n 2 ). Further exploration

1.58k views • 61 slides
Better Time-Space Lower Bounds for SAT and Related Problems Ryan Williams Carnegie Mellon

Better Time-Space Lower Bounds for SAT and Related Problems Ryan Williams Carnegie Mellon

Better Time-Space Lower Bounds for SAT and Related Problems Ryan Williams Carnegie Mellon University June 12, 2005 0-0 Introduction Few super-linear time lower bounds known for natural problems in NP (Existing ones generally use a restricted

1.28k views • 106 slides
A Measure of Space for Computing over the Reals Paulin Jacob e de Naurois LIPN - Universit

A Measure of Space for Computing over the Reals Paulin Jacob e de Naurois LIPN - Universit

A Measure of Space for Computing over the Reals Paulin Jacob e de Naurois LIPN - Universit e Paris XIII A Measure of Space for Computing over the Reals p. 1/24 Plan of the Talk The BSS Model of Computation over the Reals Michauxs

1.11k views • 70 slides
Computing Over Generalisations of the Reals Transfinite Computability Surreal Numbers Lorenzo

Computing Over Generalisations of the Reals Transfinite Computability Surreal Numbers Lorenzo

Computing Over Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire Spaces Bristol, 4 February 2020 Computing Over Generalisations of the Reals Transfinite Computability Surreal Numbers Lorenzo Galeotti

715 views • 70 slides
MA/CSSE 474 Theory of Computation Computational Complexity Announcements Don't forget the

MA/CSSE 474 Theory of Computation Computational Complexity Announcements Don't forget the

2/16/2012 MA/CSSE 474 Theory of Computation Computational Complexity Announcements Don't forget the course evaluations on Banner Web. If a 90%+ response rate for either section, everyone in that section gets a 5% bonus on the final

446 views • 29 slides
Fundamentele Informatica 3 voorjaar 2014 A slide from lecture 6

Fundamentele Informatica 3 voorjaar 2014 A slide from lecture 6

Fundamentele Informatica 3 voorjaar 2014 A slide from lecture 6 http://www.liacs.nl/home/rvvliet/fi3/ Definition 8.10. Unrestricted grammars Rudy van Vliet An unrestricted grammar is a 4-tuple G = ( V, , S, P ), where V kamer 124 Snellius,

389 views • 3 slides
Grammars A grammar is a 4-tuple ( N, , S, P ) where Towards more complex grammar systems Some

Grammars A grammar is a 4-tuple ( N, , S, P ) where Towards more complex grammar systems Some

Grammars A grammar is a 4-tuple ( N, , S, P ) where Towards more complex grammar systems Some basic formal language theory N is a finite set of non-terminals is a finite set of terminal symbols , with N = Detmar Meurers:

437 views • 6 slides
INF2080 Context-Sensitive Langugaes Daniel Lupp Universitetet i Oslo 15th February 2017

INF2080 Context-Sensitive Langugaes Daniel Lupp Universitetet i Oslo 15th February 2017

INF2080 Context-Sensitive Langugaes Daniel Lupp Universitetet i Oslo 15th February 2017 Department of University of Informatics Oslo INF2080 Lecture :: 15th February 1 / 20 Context-Free Grammar Definition (Context-Free Grammar) A

825 views • 66 slides
randomized computation Sometimes randomness helps in computation. randomized computation Augment

randomized computation Sometimes randomness helps in computation. randomized computation Augment

randomized computation Sometimes randomness helps in computation. randomized computation Augment our usual Turing machines with a read-only tape of random bits . start 0 2 accept 1 # # 0 1 1 0

736 views • 24 slides
Advanced Topics in Theoretical Computer Science Part 5: Complexity (Part 2) 5.07.2016 Viorica

Advanced Topics in Theoretical Computer Science Part 5: Complexity (Part 2) 5.07.2016 Viorica

Advanced Topics in Theoretical Computer Science Part 5: Complexity (Part 2) 5.07.2016 Viorica Sofronie-Stokkermans Universit at Koblenz-Landau e-mail: sofronie@uni-koblenz.de 1 Contents Recall: Turing machines and Turing computability

525 views • 35 slides

More recommend