DepthBounded Communication Complexity for Distributed Computation Student JieHong

Depth-Bounded Communication Complexity for Distributed Computation Student: Jie-Hong Jiang Mentor: Prof. Robert Brayton EE 249 Class Project 12/3/2002

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

Problem formulation n 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 n Assume X 1 X 2 = X and X 1 X 2 = , where X = {I, O}

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

What has been missing ? n n Communication depths Sharing of communication links

Categorization n 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)

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

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

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

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

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

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

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