The Linear-Array Conjecture in Communication Complexity Is False

network consists of k+1 processors \(\) with links only between \(\) and \(\) (0≤i<k). It is required to compute some boolean function f(x,y) in this network, where initially x is stored at \(\) and y is stored at \(\). Let \(\) be the (total) number of bits that must be exchanged to compute f in worst case. Clearly, \(\), where D(f) is the standard two-party communication complexity of f. Tiwari proved that for almost all functions \(\) and conjectured that this is true for all functions.