1. Depth-Bounded Communication Complexity for Distributed Computation Student: Jie-Hong Jiang Mentor: Prof. Robert Brayton EE249 Class Project 12/3/2002

2. Motivation In system-on-chip design, computation tasks may be localized in some particular locations (e.g. analog/digital separations) Avoid long distance (delay) communication An embedded system interacts with its environment, which may span over a large area Localize computation tasks Communication costs among different locations may be quite high (due to implementation, noise, delay, etc.) Minimize communication links and depths

3. Problem formulation Given a computation task T(I,O), two physically separated parties A(I 1,O 1 ) and B(I 2,O 2 ) want to fulfill task T using minimum amount of communication within a specified depth Assume X 1  X 2 = X and X 1  X 2 = , where X = {I, O}

4. Prior work Communication complexity has been intensively studied in the community of theoretical computer science since 1979 Yao ’ s formulation is the most well-studied Assume the two parties in communication have unbounded computation power Use protocol tree to represent the communication behavior The height of tree = bits communicated

5. What has been missing ? Communication depths Sharing of communication links

6. Categorization Combinational instances with one-sided outputs, i.e. (O 1 = O, O 2 =  ) Combinational instances with two-sided outputs, i.e. (O 1  , O 2   ) Sequential instances (finite-state machines)

7. Combinational instance: One-sided output (O 1 = O, O 2 =  ) Reduce functional matrix representation Merge identical rows and columns Equivalent communication complexity analysis Use multi-valued representation for multi- output functions

8. Combinational instance: One-sided output (O 1 = O, O 2 =  ) Communication depths should be captured in embedded system design Assume the two parties in communication have unbounded computation power. This is fine even for combinational implementations.

9. Combinational instance: One-sided output (O 1 = O, O 2 =  ) Slicing functional matrices vs. building protocol trees Limited alternating communication Lower bounds Depth-1,  lg (#column)  Depth-k, min all protocol {  i=1,…,k  lg (max #branch at level i)  }

10. Combinational instance: Two-sided output (O 1  , O 2   ) Reduce functional matrix representation Use multi-valued representation for multi- output functions column merging row merging

11. Combinational instance: Two-sided output (O 1  , O 2   ) Sharing communication links may result in combinational cycles Sometimes is essential to achieve minimum communication Might possibly have ambiguous causalities (cause bi- stable, oscillation behaviors)

12. Sequential instance Degenerate state equivalence relation between two parties in communication Take advantage of this partial information to reduce interaction Compute partial information by Galois connection Approximate by combinational techniques

13. Conclusions and future work We give a formulation for the analysis of the depth-bounded communication complexity problem Effective techniques need to be explored Language equation formulation ? Game-theoretic formulation ? Nash equilibrium may not even be local optimal for selfish row and column players Cooperative games