the krw conjecture
play

The KRW conjecture Results and Open problems Or Meir Introduction - PowerPoint PPT Presentation

Nov 06, 2022 •361 likes •1.34k views

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.

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

From the Birch and Swinnerton Dyer Conjecture to the GL 2 Main Conjecture for elliptic curves by

From the Birch and Swinnerton Dyer Conjecture to the GL 2 Main Conjecture for elliptic curves by

X International Workshop on Differential Equations, Number Theory, Data Analysis Methods and Geometry University of Havana, February 19-23, 2007 From the Birch and Swinnerton Dyer Conjecture to the GL 2 Main Conjecture for elliptic curves by

549 views • 32 slides
Holonomic D -Modules, the Dixmie Conjecture and the Jacobian Conjecture Vladimir Bavula

Holonomic D -Modules, the Dixmie Conjecture and the Jacobian Conjecture Vladimir Bavula

Holonomic D -Modules, the Dixmie Conjecture and the Jacobian Conjecture Vladimir Bavula Talks/DOKBONN 1 Plan 1. Holonomic D -Modules and the Inequality of Bernstein. 2. The Jacobian Conjecture, the Dixmier Prob- lem and a Question

457 views • 13 slides
THE CACCETTA-H AGGKVIST CONJECTURE Adrian Bondy What is a beautiful conjecture? The

THE CACCETTA-H AGGKVIST CONJECTURE Adrian Bondy What is a beautiful conjecture? The

THE CACCETTA-H AGGKVIST CONJECTURE Adrian Bondy What is a beautiful conjecture? The mathematicians patterns, like the painters or the poets must be beautiful; the ideas, like the colors or the words must fit together in a

796 views • 43 slides
Orbital profile and orbit algebra of oligomorphic permutation groups Conjecture of Macpherson

Orbital profile and orbit algebra of oligomorphic permutation groups Conjecture of Macpherson

Age and profile Conjecture of Cameron Conjecture of Macpherson Tools and intermediary results Slow growths Bonus slides Orbital profile and orbit algebra of oligomorphic permutation groups Conjecture of Macpherson Sminaire Lotharingien de

1.39k views • 101 slides
Entanglement and Spacetime Jennifer Lin + a conjecture about what the RT area is counting in the

Entanglement and Spacetime Jennifer Lin + a conjecture about what the RT area is counting in the

Entanglement and Spacetime Jennifer Lin + a conjecture about what the RT area is counting in the bulk: 1704.07763 + work in progress October 7, 2017 1 / 33 Plan 1. (15 min) Review of RT formula and applications. 2. (25 min) Conjecture: RT

499 views • 33 slides
The Kadison-Singer Problem in Mathematics and Engineering Lecture 2: The Paving Conjecture, the R

The Kadison-Singer Problem in Mathematics and Engineering Lecture 2: The Paving Conjecture, the R

The Kadison-Singer Problem in Mathematics and Engineering Lecture 2: The Paving Conjecture, the R -Conjecture, the Bourgain-Tzafriri Conjecture Master Course on the Kadison-Singer Problem University of Copenhagen Pete Casazza The Frame

1.21k views • 96 slides
Semidefinite method and Caccetta-H aggvist conjecture Jan Volec ETH Z urich joint work

Semidefinite method and Caccetta-H aggvist conjecture Jan Volec ETH Z urich joint work

Semidefinite method and Caccetta-H aggvist conjecture Jan Volec ETH Z urich joint work with Jean-S ebastien Sereni and R emi De Joannis De Verclos Caccetta-H aggvist conjecture Conjecture (Caccetta-H aggvist, 1978) Every

798 views • 58 slides
Exact solution of the Erd os-S os conjecture Mikl os Ajtai J anos Koml os, Mikl

Exact solution of the Erd os-S os conjecture Mikl os Ajtai J anos Koml os, Mikl

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Exact solution of the Erd os-S os conjecture Mikl os Ajtai J anos Koml os, Mikl os Simonovits, Endre Szemer edi Alfr ed

936 views • 70 slides
On the state of Wans Conjecture Newton Polygon of f HP () Ordinarity Phong Le

On the state of Wans Conjecture Newton Polygon of f HP () Ordinarity Phong Le

Wans Conjecture Phong Le L On the state of Wans Conjecture Newton Polygon of f HP () Ordinarity Phong Le Decomposition Theorems Department of Mathematics Niagara University April 2012 Upstate Number Theory Conference

699 views • 32 slides
An introduction to the Jacobian conjecture Damiano Fulghesu Minnesota State University Moorhead

An introduction to the Jacobian conjecture Damiano Fulghesu Minnesota State University Moorhead

An introduction to the Jacobian conjecture Damiano Fulghesu Minnesota State University Moorhead October 12, 2010 Damiano Fulghesu An introduction to the Jacobian conjecture Origin of the conjecture. First formulated by O. Keller in 1939.

944 views • 82 slides
On the CRON Conjecture Tom J. Ameloot Jan Van den Bussche Hasselt University, Belgium 1 On the

On the CRON Conjecture Tom J. Ameloot Jan Van den Bussche Hasselt University, Belgium 1 On the

On the CRON Conjecture Tom J. Ameloot Jan Van den Bussche Hasselt University, Belgium 1 On the CRON Conjecture Introduction Declarative networking Dedalus Hellerstein: Causality Required Only for Non-monotonicity (CRON) Program

1k views • 37 slides
Wieners Conjecture About Transformations . . . Transformation Groups Examples of . . .

Wieners Conjecture About Transformations . . . Transformation Groups Examples of . . .

Formulation of the . . . Wieners Conjecture: . . . How Wieners . . . Different Assignment . . . Wieners Conjecture About Transformations . . . Transformation Groups Examples of . . . Definition and Main . . . Helps Predict Which

204 views • 16 slides
Searching and Sorting The Millennium Challenge Problems Searching and Sorting Birch and

Searching and Sorting The Millennium Challenge Problems Searching and Sorting Birch and

Eric Roberts Handout #62 CS 106A March 5, 2010 Searching and Sorting The Millennium Challenge Problems Searching and Sorting Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equations P vs NP Poincar Conjecture

275 views • 4 slides
Progress on the Real -Conjecture Pascal Koiran LIP, Ecole Normale Sup erieure de Lyon

Progress on the Real -Conjecture Pascal Koiran LIP, Ecole Normale Sup erieure de Lyon

Progress on the Real -Conjecture Pascal Koiran LIP, Ecole Normale Sup erieure de Lyon Fields Institute, May 2012 The -Conjecture [Shub-Smale95] ( f ) = length of smallest straight-line program for f Z [ X ]. No constants are

980 views • 58 slides
Colmez Conjecture in Average Shou-Wu Zhang Princeton University May 28, 2015 Shou-Wu Zhang

Colmez Conjecture in Average Shou-Wu Zhang Princeton University May 28, 2015 Shou-Wu Zhang

Colmez Conjecture in Average Shou-Wu Zhang Princeton University May 28, 2015 Shou-Wu Zhang Colmez Conjecture in Average Faltings Heights A / K : abelian variety defined over a number field of dim g . Shou-Wu Zhang Colmez Conjecture in

1.08k views • 89 slides
The Two Hyperplane Conjecture David Jerison (MIT) In honor of Steve Hofmann, ICMAT, May 2018

The Two Hyperplane Conjecture David Jerison (MIT) In honor of Steve Hofmann, ICMAT, May 2018

The Two Hyperplane Conjecture David Jerison (MIT) In honor of Steve Hofmann, ICMAT, May 2018 David Jerison The Two Hyperplane Conjecture Outline The two hyperplane conjecture Whence it came (level sets) Where it may lead

659 views • 35 slides
Product-Term Based Synthesizable Embedded Programmable Logic Cores Andy Yan, Dr. Steven Wilton

Product-Term Based Synthesizable Embedded Programmable Logic Cores Andy Yan, Dr. Steven Wilton

Product-Term Based Synthesizable Embedded Programmable Logic Cores Andy Yan, Dr. Steven Wilton SoC Research Lab, University of British Columbia Vancouver, BC Canada Overview of Contributions Propose new architectural family for

355 views • 16 slides
Systems I Logic Design I Topics Topics Digital logic Logic gates Simple

Systems I Logic Design I Topics Topics Digital logic Logic gates Simple

Systems I Logic Design I Topics Topics Digital logic Logic gates Simple combinational logic circuits A Simple C statement.. C = A + B; A + B; C = What pieces of hardware do you think you might need? pieces of hardware do

226 views • 19 slides
Pre-Proposal Conference Houston Independent School District PURCHASING SERVICES Date: January 7,

Pre-Proposal Conference Houston Independent School District PURCHASING SERVICES Date: January 7,

Pre-Proposal Conference Houston Independent School District PURCHASING SERVICES Date: January 7, 2020 Presenter: Teresa Y. Browder Category Specialist Introduction: RFP #20-12-01 Capital Program and Project Management Software Houston ISD

339 views • 32 slides
Your Guide to Financial Aid Types of Financial Aid. Financial Aid Timeline MAC College Money

Your Guide to Financial Aid Types of Financial Aid. Financial Aid Timeline MAC College Money

Your Guide to Financial Aid Types of Financial Aid. Financial Aid Timeline MAC College Money Program Types of Financial Aid $ GRANTS $ SCHOLARSHIPS $ WORK STUDY $ LOANS Types of Financial Aid GRANTS are free money awarded based on a

348 views • 33 slides
Motivation Classical Technique: For AND-OR-NOT circuit for function on bit strings Build AND-NOT

Motivation Classical Technique: For AND-OR-NOT circuit for function on bit strings Build AND-NOT

Matrix Decompositions and Quantum Circuit Design Stephen S. Bullock (joint with Vivek V.Shende,Igor L.Markov, U.M. EECS) M athematical and C omputational S ciences D ivision Division Seminar National Institute of Standards and Technology

388 views • 36 slides
Development of the ELMB Development of the ELMB Henk Boterenbrood LECC2003, Sep 29 - Oct 03

Development of the ELMB Development of the ELMB Henk Boterenbrood LECC2003, Sep 29 - Oct 03

Development of the ELMB Development of the ELMB Henk Boterenbrood LECC2003, Sep 29 - Oct 03 ELMB: Embedded Local Monitor Board Credit-card sized plug-on board programmable: microcontroller (8-bit, 4MHz) communication: CAN-interface

485 views • 23 slides
Signature Analyzer Presented By : Andrew ONeil-Smith, Amy Zamon, Austin Clifton EEN 315

Signature Analyzer Presented By : Andrew ONeil-Smith, Amy Zamon, Austin Clifton EEN 315

Signature Analyzer Presented By : Andrew ONeil-Smith, Amy Zamon, Austin Clifton EEN 315 Group 2 For : Sayan Maity, TA University of Miami November 18, 2013 Project Goals Understand Shift Registers Reinforce sequential design

630 views • 17 slides
in ZNTU Galyna Tabunshchyk Prof. Software Tools Department Development of Embedded System

in ZNTU Galyna Tabunshchyk Prof. Software Tools Department Development of Embedded System

DesIRE Implementation in ZNTU Galyna Tabunshchyk Prof. Software Tools Department Development of Embedded System Courses with implementation Development of Embedded System Courses with implementation GTU, Georgia of Innovative Virtual

653 views • 53 slides

More recommend