Lower bounds on communication complexity

Abstract

We prove the following four results on communication complexity: (1) For every k ≥ 2, the language 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 can be recognized by exchanging O ( k log n ) 1 bits using a simple k -round protocol and requires the exchange of Ω( n 1 2 (k 4 log 3 n) ) bits by any ( k − 1 )-round protocol. (2) For every k ≥ 1 and for infinitely many n ≥ 1, there exists a collection of sets L k n ⊆ {0, 1} 2 n that can be recognized by exchanging O ( k log n ) bits using a k -round protocol, and any ( k − 1 )-round protocol recognizing L k n requires the exchange of Ω( n k ) bits. (3) Given a set L ⊆ {0, 1} 2 n , 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 recognizable 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.