The linear-array conjecture in communication complexity is false

A linear array network consists of k + 1 processors P₀, P₁,..., P_k with links only between Pi and P_i+i (0 ≤ i < k). It is required to compute some boolean function f(x, y) in this network, where x is initially stored at P₀ and y at P_k. Let D_k(f) be the (total) number of bits that must be exchanged to compute/in worst case. Clearly, D_k(f) ≤ k · D(f), where D(f) is the standard two-party communication complexity of/. Tiwari proved that for almost all functions D_k(f) ≥ k(D(f) - 0(1)) and conjectured that this is true for all functions. In this paper we disprove Tiwari's conjecture, by exhibiting an infinite family of functions for which D_k(f) is essentially at most 3/4k · D(f). This construction also leads to progress on another major problem in this area: It is easy to bound the two-party communication complexity of any function, given the least number of monochromatic rectangles in any partition of the input space. How tight are such bounds? We exhibit certain functions, for which the (two-party) communication complexity is twice larger than the best lower bound obtainable this way.