Bit Complexity of Breaking and Achieving Symmetry in Chains and Rings.

Introduction: Yao’s Problem (1979) What is the number of bits that two processors, A and B, have to communicate to each other in order to compute a function f(x,y) of their respective private inputs: x and y.

Yao’s Problem in the Distributed Computing Model Two identical, unnamed processor, each one provided with a value – its identity; the two processors have to find the larger identity.

The Distributed Computing Model Network of processors. Asynchronous communication. Failure free. Links are FIFO. Number of links denoted by.

The Distributed Computing Model (Continued) Processors have distinct ids from the set Distributed algorithm. Scheduler.

The Tasks Ahead Consensus : all processors must output the same bit. Leader : one processor outputs 1, the rest output 0. MaxF : the same as Leader, except that the processor to output 1 must be the one with the maximal id.

Bit and Message Complexity Message Complexity : the number of messages needed to send, in order to solve a task. Denoted MsgC(task). Bit Complexity : the number of bits needed to send, in order to solve a task. Denoted BitC(task).

Achieving symmetry with two processors Theorem In chain of two processors with,. Proof Each processor sends the parity of its id to the other processor, and both decide on the (say) OR of these bit.

Achieving symmetry with two processors (continued) Proof Lets partition the set of ids into 3 groups: according to the first bit: 0, 1 or nothing. Assume that We get that every decision is 0, which is a contradiction to our assumption.

Breaking symmetry with two processors Theorem In chain of two processors, Proof The two processors exchange their identities without the last bit; the larger abridged identity wins, and if they are equal, the identity with the unsent bit 1 wins.

Breaking symmetry with two processors (continued) Proof (lower bound)

Chains n + 1 processors are linked with n links. Easy case: Every processor in the chain has an id.

Consensus in a Chain A 2n algorithm is easy. Theorem (no proof) For a chain with n links the message complexity of Consensus is at least 2n.

Leader in a chain: General Case

Leader in a chain: Even Length Chain, a better result The idea: some of the messages can be discarded. The result:

Leader in a Chain: Hard Case In the easy case, every processor had an id. Here, only the two processors in the edge have ids. The idea: the edges sends their id in the network. The result: