The Proof That N

is similar to the proof of Theorem 2.4 ((rst direction), together with the second part of Lemma 4.3 that guarantees that D 1 (e) = D 1 (e f) D 1 (e g). Using the last two theorems, we get Let us now brieey discuss the case of computing general relations and not necessarily functions. The equality in Lemma 4.3 part (1), does not hold anymore. However, by FKN91] the two sides cannot be too far. As a result, Theorem 4.4 is changed as well and it claims: Let R and S be two relations. Then Simple lower bound for monotone clique using a communication game", Information Processing Letters, 41, pp. 221-226 (1992). K71] V. Khrapchenko, \A method of determining lower bounds for the complexity of-schemes", Now, we can come back to the proof of Theorem 4.1. To analyze the two-round deterministic communication complexity of a relation R, we deene the following hypergraph H 2 R. The vertices are again all the pairs in f0; 1g n f0; 1g n. The hyperedges are all the rectangles of the form A f0; 1g n , where A f0; 1g n. Now, for each such hyperedge e we deene its weight to be D 1 (e), the one-way deterministic communication complexity of computing R on the sub-domain e. in the opposite direction, let us concentrate for a while on the case of computing functions. We need the following lemma.