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The KRW conjecture Results and Open problems Or Meir Introduction 1 Known results 2 Proof strategy 3 Future directions 4 Depth complexity Let f : { 0 , 1 } n { 0 , 1 } . The depth complexity D ( f ) is the depth of the shallowest

  1. The KRW conjecture Results and Open problems Or Meir

  2. Introduction 1 Known results 2 Proof strategy 3 Future directions 4

  3. Depth complexity Let f : { 0 , 1 } n → { 0 , 1 } . The depth complexity D ( f ) is the depth of the shallowest circuit for f . Captures the complexity of parallel computation.

  4. Depth complexity Let f : { 0 , 1 } n → { 0 , 1 } . The depth complexity D ( f ) is the depth of the shallowest circuit for f . Captures the complexity of parallel computation. We only consider circuits with fan-in 2 .

  5. Depth complexity Let f : { 0 , 1 } n → { 0 , 1 } . The depth complexity D ( f ) is the depth of the shallowest circuit for f . Captures the complexity of parallel computation. We only consider circuits with fan-in 2 . Major frontier: Explicit f with D ( f ) = ω (log n ) . a.k.a. P � = NC 1 .

  6. Composition [Karchmer-Raz-Wigderson-91]: We need to understand composition.

  7. Composition [Karchmer-Raz-Wigderson-91]: We need to understand composition. Let f : { 0 , 1 } m → { 0 , 1 } , g : { 0 , 1 } n → { 0 , 1 } . The composition f ⋄ g : { 0 , 1 } m × n → { 0 , 1 } is n g f . m a ( f ⋄ g )( X ) . X .

  8. Composition [Karchmer-Raz-Wigderson-91]: We need to understand composition. Let f : { 0 , 1 } m → { 0 , 1 } , g : { 0 , 1 } n → { 0 , 1 } . The composition f ⋄ g : { 0 , 1 } m × n → { 0 , 1 } is n g f . m a ( f ⋄ g )( X ) . X .

  9. Composition [Karchmer-Raz-Wigderson-91]: We need to understand composition. Let f : { 0 , 1 } m → { 0 , 1 } , g : { 0 , 1 } n → { 0 , 1 } . The composition f ⋄ g : { 0 , 1 } m × n → { 0 , 1 } is n g f . m a ( f ⋄ g )( X ) . X .

  10. Composition [Karchmer-Raz-Wigderson-91]: We need to understand composition. Let f : { 0 , 1 } m → { 0 , 1 } , g : { 0 , 1 } n → { 0 , 1 } . The composition f ⋄ g : { 0 , 1 } m × n → { 0 , 1 } is n g f . m a ( f ⋄ g )( X ) . X .

  11. The KRW conjecture g D ( f ) f f . a ( f ⋄ g )( X ) . X . . . . D ( g ) g g . . . . . . X 1 X m Clearly, D ( f ⋄ g ) ≤ D ( f ) + D ( g ) .

  12. The KRW conjecture g D ( f ) f f . a ( f ⋄ g )( X ) . X . . . . D ( g ) g g . . . . . . X 1 X m Clearly, D ( f ⋄ g ) ≤ D ( f ) + D ( g ) . KRW conjecture: ∀ f, g : D ( f ⋄ g ) ≈ D ( f ) + D ( g ) .

  13. The KRW conjecture g D ( f ) f f . a ( f ⋄ g )( X ) . X . . . . D ( g ) g g . . . . . . X 1 X m Clearly, D ( f ⋄ g ) ≤ D ( f ) + D ( g ) . KRW conjecture: ∀ f, g : D ( f ⋄ g ) ≈ D ( f ) + D ( g ) . Theorem [KRW91]: the conjecture implies that P � = NC 1 .

  14. Outline Introduction 1 Known results 2 Proof strategy 3 Future directions 4

  15. Karchmer-Wigderson relations Relate D ( f ) to complexity of a communication problem KW f .

  16. Karchmer-Wigderson relations Relate D ( f ) to complexity of a communication problem KW f . The KW relation KW f is defined as follows: Alice gets x ∈ f − 1 (0) . Bob gets y ∈ f − 1 (1) . Clearly, x � = y , so ∃ i s.t. x i � = y i . Want to find such i .

  17. Karchmer-Wigderson relations Relate D ( f ) to complexity of a communication problem KW f . The KW relation KW f is defined as follows: Alice gets x ∈ f − 1 (0) . Bob gets y ∈ f − 1 (1) . Clearly, x � = y , so ∃ i s.t. x i � = y i . Want to find such i . Theorem [KW88]: D ( f ) = C ( KW f ) .

  18. Karchmer-Wigderson relations Relate D ( f ) to complexity of a communication problem KW f . The KW relation KW f is defined as follows: Alice gets x ∈ f − 1 (0) . Bob gets y ∈ f − 1 (1) . Clearly, x � = y , so ∃ i s.t. x i � = y i . Want to find such i . Theorem [KW88]: D ( f ) = C ( KW f ) . Only deterministic protocols!

  19. Karchmer-Wigderson relations Relate D ( f ) to complexity of a communication problem KW f . The KW relation KW f is defined as follows: Alice gets x ∈ f − 1 (0) . Bob gets y ∈ f − 1 (1) . Clearly, x � = y , so ∃ i s.t. x i � = y i . Want to find such i . Theorem [KW88]: D ( f ) = C ( KW f ) . Only deterministic protocols! KRW conjecture: C ( KW f ⋄ g ) ≈ C ( KW f ) + C ( KW g )

  20. KRW and KW Can we use KW games to attack the KRW conjecture? What does KW f ⋄ g look like? Recall: f ⋄ g maps { 0 , 1 } m × n to { 0 , 1 } . Alice Bob g g f f . . a . 0 1 . X b Y . .

  21. KRW and KW Can we use KW games to attack the KRW conjecture? What does KW f ⋄ g look like? Recall: f ⋄ g maps { 0 , 1 } m × n to { 0 , 1 } . Alice Bob g g f f . . a . 0 1 . X b Y . .

  22. KRW and KW Can we use KW games to attack the KRW conjecture? What does KW f ⋄ g look like? Recall: f ⋄ g maps { 0 , 1 } m × n to { 0 , 1 } . Alice Bob g g f f . . a . 0 1 . X b Y . .

  23. KRW and KW Can we use KW games to attack the KRW conjecture? What does KW f ⋄ g look like? Recall: f ⋄ g maps { 0 , 1 } m × n to { 0 , 1 } . Alice Bob g g f f . . a . 0 1 . X b Y . .

  24. KRW and KW Can we use KW games to attack the KRW conjecture? What does KW f ⋄ g look like? Recall: f ⋄ g maps { 0 , 1 } m × n to { 0 , 1 } . Alice Bob g g f f . . a . 0 1 . X b Y . .

  25. KRW and KW Can we use KW games to attack the KRW conjecture? What does KW f ⋄ g look like? Recall: f ⋄ g maps { 0 , 1 } m × n to { 0 , 1 } . Alice Bob g g f f . . a . 0 1 . X b Y . .

  26. KRW and KW Can we use KW games to attack the KRW conjecture? What does KW f ⋄ g look like? Recall: f ⋄ g maps { 0 , 1 } m × n to { 0 , 1 } . Alice Bob g g f f . . a . 0 1 . X b Y . . Hence, C ( KW f ⋄ g ) ≤ C ( KW f ) + C ( KW g ) . KRW conjecture: the obvious protocol is essentially optimal.

  27. The universal relation The KRW conjecture is hard. [KRW91] suggested a starting point.

  28. The universal relation The KRW conjecture is hard. [KRW91] suggested a starting point. The universal relation U n is: Alice gets x ∈ { 0 , 1 } n . Bob gets y ∈ { 0 , 1 } n . x � = y . Wish to find i s.t. x i � = y i .

  29. The universal relation The KRW conjecture is hard. [KRW91] suggested a starting point. The universal relation U n is: Alice gets x ∈ { 0 , 1 } n . Bob gets y ∈ { 0 , 1 } n . x � = y . Wish to find i s.t. x i � = y i . Easy to prove: C (U n ) ≥ n .

  30. The universal relation The KRW conjecture is hard. [KRW91] suggested a starting point. The universal relation U n is: Alice gets x ∈ { 0 , 1 } n . Bob gets y ∈ { 0 , 1 } n . x � = y . Wish to find i s.t. x i � = y i . Easy to prove: C (U n ) ≥ n . [KRW91] suggested to study U m ⋄ U n .

  31. The composition of the universal relation [KRW91] suggested to study the composition U m ⋄ U n . Alice Bob a X b Y a � = b . If a i � = b i then X i � = Y i .

  32. The composition of the universal relation [KRW91] suggested to study the composition U m ⋄ U n . Alice Bob a X b Y a � = b . If a i � = b i then X i � = Y i .

  33. The composition of the universal relation [KRW91] suggested to study the composition U m ⋄ U n . Alice Bob a X b Y a � = b . If a i � = b i then X i � = Y i .

  34. The composition of the universal relation [KRW91] suggested to study the composition U m ⋄ U n . Alice Bob a X b Y a � = b . If a i � = b i then X i � = Y i .

  35. The composition of the universal relation Goal: C (U m ⋄ U n ) = C (U m ) + C (U n ) ≥ m + n . Alice Bob a X b Y

  36. The composition of the universal relation Goal: C (U m ⋄ U n ) = C (U m ) + C (U n ) ≥ m + n . Challenge was met by [Edmonds-Impagliazzo-Rudich-S’gall-91]. Alice Bob a X b Y

  37. The composition of the universal relation Goal: C (U m ⋄ U n ) = C (U m ) + C (U n ) ≥ m + n . Challenge was met by [Edmonds-Impagliazzo-Rudich-S’gall-91]. Alternative proof obtained by [H˚ astad-Wigderson-93]. Alice Bob a X b Y

  38. Composing a function and the universal relation An analog of KRW conjecture for KW f ⋄ U n for any f . [Gavinsky-M-Weinstein-Wigderson-14] Alice Bob f f a 0 1 X b Y If a i � = b i then X j � = Y j . The obvious protocol works.

  39. Composing a function and the universal relation An analog of KRW conjecture for KW f ⋄ U n for any f . [Gavinsky-M-Weinstein-Wigderson-14] Quantative improvement by [Koroth-M-18]. Alice Bob f f a 0 1 X b Y If a i � = b i then X j � = Y j . The obvious protocol works.

  40. Composing any function and parity [Dinur-M-16]: (Re-)proved KRW conjecture for f ⋄ � n

  41. Composing any function and parity [Dinur-M-16]: (Re-)proved KRW conjecture for f ⋄ � n Actually, this case was already implicit in [H˚ astad 98].

  42. Composing any function and parity [Dinur-M-16]: (Re-)proved KRW conjecture for f ⋄ � n Actually, this case was already implicit in [H˚ astad 98]. However, our proof was very different, and more in line with the other works on the KRW conjecture.

  43. Outline Introduction 1 Known results 2 Proof strategy 3 Future directions 4

  44. Why should the obvious protocol be optimal? Alice Bob g g f f . . a . 0 1 . X b Y . .

  45. Why should the obvious protocol be optimal? Alice Bob g g f f . . a . 0 1 . X b Y . . The players must solve KW g on some row.

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