1983 Reports

Lower Bounds on Communication Complexity

Duris, Pavol; Galil, Zvi; Schnitger, Georg

We prove the following four results on communication complexity: 1) For every k ≥ 2, the language Lk of encodings of directed graphs of out degree one that contain a path of length k+1 from the first vertex to the last vertex and can be recognized by exchanging O(k log n) bits using a simple k-round protocol requires exchanging Ω(n1/2/k4log3n) bits if any (k-1)- round protocol is used. 2) For every k ≥ 1 and for infinitely many n ≥ 1, there exists a collection of sets Lnk ⊆ {0,1}2n that can be recognized by exchanging O(k log n) bits using a k-round protocol, and any (k-1)-round protocol recognizing Lnk requires exchanging Ω(n/k) bits. 3) Given a set L ⊆ {0,1}2n, there is a set L ⊆{0,1}8n such that any (k-round) protocol recognizing L can be transformed to a (k-round) fixed partition protocol recognizing L with the same communication complexity, and vice versa. 4) For every integer function f, 1 ≤ f(n) ≤ n, there are languages recognized by a one round deterministic protocol exchanging f(n) bits, but not by any nondeterministic protocol exchanging f(n)-1 bits. The first two results show in an incomparable way an exponential gap between (k-1)-round and k-round protocols, settling a conjecture by Papadimitriou and Sipser. The third result shows that as long as we are interested in existence proofs, a fixed partition of the input is not a restriction. The fourth result extends a result by Papadimitriou and Sipser who showed that for every integer function f, 1 ≤ f(n) ≤ n, there is a language accepted by a deterministic protocol exchanging f(n) bits but not by any deterministic protocol exchanging f(n) - 1 bits.