Streaming Communication Protocols

Abstract

We define the Streaming Communication model that combines the main aspects of communication complexity and streaming. We consider two agents that want to compute some function that depends on inputs that are distributed to each agent. The inputs arrive as data streams and each agent has a bounded memory. Agents are allowed to communicate with each other and also update their memory based on the input bit they read and the previous message they received.We provide tight tradeoffs between the necessary resources, i.e. communication and memory, for some of the canonical problems from communication complexity by proving a strong general lower bound technique. Second, we analyze the Approximate Matching problem and show that the complexity of this problem (i.e. the achievable approximation ratio) in the one-way variant of our model is strictly different both from the streaming complexity and the one-way communication complexity thereof. * This work has been partially supported by the ERC project QCC and the French ANR Blanc project RDAM † Π +1 errs. First, when the protocol g +1 x +1 , y +1 = a +1 and the protocol exits (and hence outputs 1 − a +1 ); second, when the protocol completes the simulation of Π and Π errs. We can upper bound the error probability as(2)We know that Pr[Π errs] ≤ ε and hence it remains to show that Pr[Π +1 exits when g +1 (x +1 , y +1 ) = a +1 ] ≤ δ. First, recall that for all possible inputs (x +1 , y +1 ) with g +1 (x +1 , y +1 ) = a +1 , the expected number of communication rounds on block + 1 in Π is bounded by R +1 . Using the results in section 2.4, the protocol Π on block +1 can be simulated by Alice and Bob in the usual communication model with a multiplicative overhead S+2 log t +1 +1. Hence, the expected communication cost of step 2 is R +1 (S + 2 log t +1 + 1).The communication cost at step 3 is equal to S, and so, the total expected cost of Π +1 , when g +1 (x +1 , y +1 ) = a +1 , is smaller than R +1 (S + 2 log t +1 + 1) + S = δA +1 . Thus the probability the protocol exits because the transcript reached A +1 in this case is smaller than δ by Markov inequality. Hence we established that Π +1 is a protocol for g +1 with error ε + δ using ≤ A +1 bits in the worst case over inputs. This concludes the proof.